1 | ////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id$"; |
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3 | category="General purpose"; |
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4 | info=" |
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5 | LIBRARY: grobcov.lib Groebner Cover for parametric ideals. |
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6 | PURPOSE: Comprehensive Groebner Systems, Groebner Cover, Canonical Forms. |
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7 | The library contains Montes's algorithms to compute the |
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8 | canonical Groebner cover of a parametric ideal as described in |
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9 | the paper: |
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10 | |
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11 | Montes A., Wibmer M., |
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12 | Groebner Bases for Polynomial Systems with parameters. |
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13 | Journal of Symbolic Computation 45 (2010) 1391-1425. |
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14 | |
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15 | The central routine is grobcov. Given a parametric |
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16 | ideal, grobcov outputs its canonical Groebner cover, consisting |
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17 | of a set of pairs of (basis, segment). The basis (after |
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18 | normalization) is the reduced Groebner basis for each point |
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19 | of the segment. The segments are disjoint, locally closed |
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20 | and correspond to constant lpp (leading power product) |
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21 | of the basis, and are represented in canonical prime |
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22 | representation. The segments are disjoint and cover the |
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23 | whole parameter space. The output is canonical, it only |
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24 | depends on the given parametric ideal and the monomial order. |
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25 | This is much more than a simple comprehensive Groebner system. |
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26 | The algorithm grobcov allows options to solve partially the |
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27 | problem when the whole automatic algorithm does not finish |
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28 | in reasonable time. |
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29 | |
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30 | grobcov uses a first algorithm cgsdr that outputs a disjoint |
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31 | reduced comprehensive Groebner system with constant lpp. |
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32 | cgsdr can be called directly if only a disjoint reduced |
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33 | comprehensive Groebner system is required. |
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34 | |
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35 | Two other routines: gencase1 and multigrobcov can be used |
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36 | in problems with basis of the generic case equal to 1 |
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37 | (for example in automatic geometric theorem discovering) |
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38 | that allow to obtain partial results even when grobcov does |
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39 | not finish in reasonable time. |
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40 | |
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41 | For completeness, the library also contains the algorithms |
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42 | with similar purposes contained in the old library redcgs.lib. |
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43 | These algorithms are, in general, less efficient and do not |
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44 | ensure a canonical results, even if they are similar to the |
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45 | results obtained with grobcov. |
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46 | The old routines are no more recommended and remain in |
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47 | this library for didactic purposes. These are |
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48 | cgsdrold, grobcovold, buildtreetoMaple, cantreetoMaple. |
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49 | |
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50 | AUTHORS: Antonio Montes , Hans Schoenemann. |
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51 | OVERVIEW: see \"Groebner Bases for Polynomial Systems with parameters\" |
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52 | Montes A., Wibmer M., |
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53 | Journal of Symbolic Computation 45 (2010) 1391-1425. |
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54 | (http://www-ma2.upc.edu/~montes/). |
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55 | |
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56 | NOTATIONS: All given and determined polynomials and ideals are in the |
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57 | @* basering Q[a][x]; (a=parameters, x=variables) |
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58 | @* After defining the ring, the main routines |
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59 | @* grobcov, cgsdr, gencase1, multigrobcov |
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60 | @* generate the global rings |
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61 | @* @R (Q[a][x]), |
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62 | @* @P (Q[a]), |
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63 | @* @RP (Q[x,a]) |
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64 | @* that are used inside and killed before the output. |
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65 | @* If you want to use some internal routine you must |
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66 | @* create before the above rings by calling setglobalrings(); |
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67 | @* because most of the internal routines use these rings. |
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68 | @* The call to the basic routines grobcov, cgsdr, gencase1, multigrobcov |
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69 | @* or even the older grobcovold, cgsdrold will kill these rings. |
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70 | |
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71 | PROCEDURES: |
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72 | |
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73 | grobcov(F); Is the basic routine giving the canonical |
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74 | Groebner cover of the parametric ideal F. |
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75 | This routine accepts many options, that |
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76 | allow to obtain results even when the canonical |
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77 | computation does not finish in reasonable time. |
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78 | |
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79 | cgsdr(F); Is the procedure for obtaining a first disjoint, |
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80 | reduced comprehensive Groebner system that |
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81 | is used in grobcov, but that can be used |
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82 | independently if only the CGS is required. |
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83 | It is a more efficient version of buildtree |
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84 | that does not output the complete discussion tree |
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85 | but only the terminal vertices giving the |
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86 | disjoint reduced comprehensive Groebner system. |
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87 | |
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88 | gencase1(F); Returns the segment of the generic case when his |
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89 | basis is 1. This is useful for automatic discovering |
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90 | of geometrical theorems, as it gives the components |
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91 | where a solution exists and is much more efficient |
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92 | than the complete computation of grobcov. |
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93 | |
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94 | multigrobcov(F); In problems like automatic discovery of theorems, |
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95 | when grobcov does not give the answer in reasonable |
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96 | time, and the generic case is expected to |
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97 | have basis 1, one can try with multigrobcov procedure |
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98 | to obtain an answer over the different irreducible |
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99 | components: the generic case with basis 1, and the |
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100 | components not corresponding to the generic case. To |
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101 | deduce from its result the true Groebner cover one |
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102 | must discuss theoretically in which segment |
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103 | must be located the intersecting parts in the |
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104 | different irreducible components. |
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105 | |
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106 | setglobalrings(); Generates the global rings @R, @P and @PR that are |
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107 | respectively the rings Q[a][x], Q[a], Q[x,a]. |
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108 | It is called inside each of the fundamental routines of the |
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109 | library: grobcov, cgsdr, gencase1, multigrobcov, as well as |
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110 | by the old routines cgsdrold, grobcovold and killed |
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111 | before the output. |
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112 | If the user want to use some other internal routine, |
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113 | then setglobalrings() is to be called before, as |
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114 | the rings @R, @P and @RP are needed in most of them. |
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115 | globally, and more internal routines can be used, but |
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116 | These rings are destroyed by the call to any of the basic |
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117 | routines. |
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118 | |
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119 | pdivi(f,F); Performs a pseudodivision of a parametric polynomial |
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120 | by a parametric ideal. |
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121 | |
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122 | pnormalform(f,N,W); Reduces a parametric polynomial f by a reduced-representation |
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123 | (N,W) of null and non-null conditions over the parameters. |
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124 | Before using it setglobalrings() must be called. |
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125 | |
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126 | Also included from the old library redcgs.lib the following routines |
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127 | |
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128 | cgsdrold(F); Similar to cgsdr using the algorithm buildtree |
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129 | of the old library. |
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130 | grobcovold(F); Similar to grobcov with the algorithms of the old |
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131 | library. |
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132 | buildtreetoMaple(T); Writes into a file the output of cgsdrold called |
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133 | with option ('old',0) into a text file that is Maple |
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134 | readable and can be plotted in Maple using |
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135 | the tplot routine of the library dpgb. |
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136 | cantreetoMaple(M); Writes into a text file the output of grobcovold called |
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137 | with option ('out',1), that is readable |
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138 | in Maple and can be plotted using the routine |
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139 | plotcantree of the Maple library dpgb. |
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140 | |
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141 | SEE ALSO: compregb_lib |
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142 | "; |
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143 | |
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144 | LIB "primdec.lib"; |
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145 | LIB "qhmoduli.lib"; |
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146 | |
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147 | // ************ Begin of the grobcov library ********************* |
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148 | |
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149 | // Library grobcov.lib |
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150 | // (Groebner cover): |
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151 | // Initial data: 6-9-2009 |
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152 | // Release 1: |
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153 | // Final data: 30-12-2010 |
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154 | // Contains also the old redcgs.lib library that was created |
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155 | // Initial data: 21-1-2008 |
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156 | // Release 1: |
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157 | // Final data: 3-7-2008 |
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158 | // Given and determined polynomials and ideals are in the |
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159 | // basering Q[a][x]; |
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160 | |
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161 | // ************ Begin of buildtree ****************************** |
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162 | |
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163 | proc setglobalrings() |
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164 | "USAGE: setglobalrings(); |
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165 | No arguments |
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166 | RETURN: After its call the rings @R=Q[a][x], @P=Q[a], @RP=Q[x,a] are |
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167 | defined as global variables. |
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168 | NOTE: It is called internally by the fundamental routines of the |
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169 | library grobcov, cgsdr, gencase1, muligrobcov as well as by the |
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170 | old ones grobcovold,cgsdrold, and killed before the output. |
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171 | The user does not need to call it, except when it is interested |
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172 | in using some internal routine of the library that |
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173 | uses these rings. |
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174 | The basering R, must be of the form Q[a][x], a=parameters, |
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175 | x=variables, and should be defined previously. |
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176 | KEYWORDS: ring, rings |
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177 | EXAMPLE: setglobalrings; shows an example" |
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178 | { |
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179 | if (defined(@P)==1) |
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180 | { |
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181 | kill @P; kill @R; kill @RP; |
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182 | } |
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183 | def RR=basering; |
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184 | def @R=basering; // must be of the form K[a][x], a=parameters, x=variables |
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185 | def Rx=ringlist(RR); |
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186 | def @P=ring(Rx[1]); |
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187 | list Lx; |
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188 | Lx[1]=0; |
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189 | Lx[2]=Rx[2]+Rx[1][2]; |
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190 | Lx[3]=Rx[1][3]; |
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191 | Lx[4]=Rx[1][4]; |
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192 | Rx[1]=0; |
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193 | def D=ring(Rx); |
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194 | def @RP=D+@P; |
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195 | exportto(Top,@R); // global ring K[a][x] |
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196 | exportto(Top,@P); // global ring K[a] |
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197 | exportto(Top,@RP); // global ring K[x,a] with product order |
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198 | setring(RR); |
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199 | } |
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200 | example |
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201 | { "EXAMPLE:"; echo = 2; |
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202 | ring R=(0,a,b),(x,y,z),dp; |
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203 | setglobalrings(); |
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204 | @R; |
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205 | @P; |
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206 | @RP; |
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207 | } |
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208 | |
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209 | //*************Auxilliary routines************** |
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210 | |
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211 | // cld : clears denominators of an ideal and normalizes to content 1 |
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212 | // can be used in @R or @P or @RP |
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213 | // input: |
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214 | // ideal J (J can be also poly), but the output is an ideal; |
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215 | // output: |
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216 | // ideal Jc (the new form of ideal J without denominators and |
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217 | // normalized to content 1) |
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218 | static proc cld(ideal J) |
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219 | { |
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220 | if (size(J)==0){return(ideal(0));} |
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221 | def RR=basering; |
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222 | setring(@RP); |
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223 | def Ja=imap(RR,J); |
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224 | ideal Jb; |
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225 | if (size(Ja)==0){return(ideal(0));} |
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226 | int i; |
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227 | def j=0; |
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228 | for (i=1;i<=ncols(Ja);i++){if (size(Ja[i])!=0){j++; Jb[j]=cleardenom(Ja[i]);}} |
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229 | setring(RR); |
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230 | def Jc=imap(@RP,Jb); |
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231 | return(Jc); |
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232 | } |
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233 | |
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234 | static proc memberpos(f,J) |
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235 | //"USAGE: memberpos(f,J); |
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236 | // (f,J) expected (polynomial,ideal) |
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237 | // or (int,list(int)) |
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238 | // or (int,intvec) |
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239 | // or (intvec,list(intvec)) |
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240 | // or (list(int),list(list(int))) |
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241 | // or (ideal,list(ideal)) |
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242 | // or (list(intvec), list(list(intvec))). |
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243 | // The ring can be @R or @P or @RP or any other. |
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244 | //RETURN: The list (t,pos) t int; pos int; |
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245 | // t is 1 if f belongs to J and 0 if not. |
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246 | // pos gives the position in J (or 0 if f does not belong). |
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247 | //EXAMPLE: memberpos; shows an example" |
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248 | { |
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249 | int pos=0; |
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250 | int i=1; |
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251 | int j; |
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252 | int t=0; |
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253 | int nt; |
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254 | if (typeof(J)=="ideal"){nt=ncols(J);} |
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255 | else{nt=size(J);} |
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256 | if ((typeof(f)=="poly") or (typeof(f)=="int")) |
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257 | { // (poly,ideal) or |
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258 | // (poly,list(poly)) |
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259 | // (int,list(int)) or |
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260 | // (int,intvec) |
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261 | i=1; |
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262 | while(i<=nt) |
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263 | { |
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264 | if (f==J[i]){return(list(1,i));} |
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265 | i++; |
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266 | } |
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267 | return(list(0,0)); |
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268 | } |
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269 | else |
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270 | { |
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271 | if ((typeof(f)=="intvec") or ((typeof(f)=="list") and (typeof(f[1])=="int"))) |
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272 | { // (intvec,list(intvec)) or |
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273 | // (list(int),list(list(int))) |
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274 | i=1; |
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275 | t=0; |
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276 | pos=0; |
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277 | while((i<=nt) and (t==0)) |
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278 | { |
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279 | t=1; |
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280 | j=1; |
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281 | if (size(f)!=size(J[i])){t=0;} |
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282 | else |
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283 | { |
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284 | while ((j<=size(f)) and t) |
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285 | { |
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286 | if (f[j]!=J[i][j]){t=0;} |
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287 | j++; |
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288 | } |
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289 | } |
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290 | if (t){pos=i;} |
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291 | i++; |
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292 | } |
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293 | if (t){return(list(1,pos));} |
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294 | else{return(list(0,0));} |
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295 | } |
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296 | else |
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297 | { |
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298 | if (typeof(f)=="ideal") |
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299 | { // (ideal,list(ideal)) |
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300 | i=1; |
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301 | t=0; |
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302 | pos=0; |
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303 | while((i<=nt) and (t==0)) |
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304 | { |
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305 | t=1; |
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306 | j=1; |
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307 | if (ncols(f)!=ncols(J[i])){t=0;} |
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308 | else |
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309 | { |
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310 | while ((j<=ncols(f)) and t) |
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311 | { |
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312 | if (f[j]!=J[i][j]){t=0;} |
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313 | j++; |
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314 | } |
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315 | } |
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316 | if (t){pos=i;} |
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317 | i++; |
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318 | } |
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319 | if (t){return(list(1,pos));} |
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320 | else{return(list(0,0));} |
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321 | } |
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322 | else |
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323 | { |
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324 | if ((typeof(f)=="list") and (typeof(f[1])=="intvec")) |
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325 | { // (list(intvec),list(list(intvec))) |
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326 | i=1; |
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327 | t=0; |
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328 | pos=0; |
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329 | while((i<=nt) and (t==0)) |
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330 | { |
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331 | t=1; |
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332 | j=1; |
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333 | if (size(f)!=size(J[i])){t=0;} |
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334 | else |
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335 | { |
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336 | while ((j<=size(f)) and t) |
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337 | { |
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338 | if (f[j]!=J[i][j]){t=0;} |
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339 | j++; |
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340 | } |
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341 | } |
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342 | if (t){pos=i;} |
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343 | i++; |
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344 | } |
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345 | if (t){return(list(1,pos));} |
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346 | else{return(list(0,0));} |
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347 | } |
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348 | } |
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349 | } |
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350 | } |
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351 | } |
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352 | //example |
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353 | //{ "EXAMPLE:"; echo = 2; |
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354 | // list L=(7,4,5,1,1,4,9); |
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355 | // memberpos(1,L); |
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356 | // > |
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357 | //} |
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358 | |
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359 | |
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360 | static proc subset(J,K) |
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361 | //"USAGE: subset(J,K); |
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362 | // (J,K) expected (ideal,ideal) |
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363 | // or (list, list) |
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364 | //RETURN: 1 if all the elements of J are in K, 0 if not. |
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365 | //EXAMPLE: subset; shows an example;" |
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366 | { |
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367 | int i=1; |
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368 | int nt; |
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369 | if (typeof(J)=="ideal"){nt=ncols(J);} |
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370 | else{nt=size(J);} |
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371 | if (size(J)==0){return(1);} |
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372 | while(i<=nt) |
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373 | { |
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374 | if (memberpos(J[i],K)[1]){i++;} |
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375 | else {return(0);} |
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376 | } |
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377 | return(1); |
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378 | } |
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379 | //example |
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380 | //{ "EXAMPLE:"; echo = 2; |
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381 | // list J=list(7,3,2); |
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382 | // list K=list(1,2,3,5,7,8); |
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383 | // subset(J,K); |
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384 | //} |
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385 | |
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386 | // elimintfromideal: elimine the constant numbers from an ideal |
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387 | // (designed for W, nonnull conditions) |
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388 | // input: ideal J |
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389 | // output:ideal K with the elements of J that are non constants, in the ring @P |
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390 | static proc elimintfromideal(ideal J) |
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391 | { |
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392 | int i; |
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393 | int j=0; |
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394 | ideal K; |
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395 | if (size(J)==0){return(ideal(0));} |
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396 | for (i=1;i<=ncols(J);i++){if (size(variables(J[i])) !=0){j++; K[j]=J[i];}} |
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397 | return(K); |
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398 | } |
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399 | |
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400 | // simpqcoeffs : simplifies a quotient of two polynomials |
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401 | // input: two coeficients (or terms), that are considered as a quotient |
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402 | // output: the two coeficients reduced without common factors |
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403 | static proc simpqcoeffs(poly n,poly m) |
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404 | { |
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405 | def nc=content(n); |
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406 | def mc=content(m); |
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407 | def gc=gcd(nc,mc); |
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408 | ideal s=n/gc,m/gc; |
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409 | return (s); |
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410 | } |
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411 | |
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412 | // pdivi : pseudodivision of a poly f by an ideal F in a parametric ideal |
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413 | // Q[a][x] |
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414 | // input: |
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415 | // poly f0 (in the parametric ring @R) |
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416 | // ideal F0 (in the parametric ring @R) |
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417 | // output: |
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418 | // list (poly r, ideal q, poly mu) |
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419 | proc pdivi(poly f,ideal F) |
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420 | "USAGE: pdivi(f,F); |
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421 | poly f: the polynomial to be divided |
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422 | ideal F: the divisor ideal |
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423 | RETURN: A list (poly r, ideal q, poly m). r is the remainder of the |
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424 | pseudodivision, q is the set of quotients, and m is the |
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425 | factor by which f is to be multiplied. |
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426 | NOTE: pseudodivision of a poly f by an ideal F in @R. Returns a |
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427 | list (r,q,m) such that m*f=r+sum(q.G), and no lpp of a divisor |
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428 | divides a pp of r. |
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429 | KEYWORDS: division, reduce |
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430 | EXAMPLE: pdivi; shows an example" |
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431 | { |
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432 | int i; |
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433 | int j; |
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434 | poly r=0; |
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435 | poly mu=1; |
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436 | def p=f; |
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437 | ideal q; |
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438 | for (i=1; i<=size(F); i++){q[i]=0;} |
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439 | ideal lpf; |
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440 | ideal lcf; |
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441 | for (i=1;i<=size(F);i++){lpf[i]=leadmonom(F[i]);} |
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442 | for (i=1;i<=size(F);i++){lcf[i]=leadcoef(F[i]);} |
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443 | poly lpp; |
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444 | poly lcp; |
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445 | poly qlm; |
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446 | poly nu; |
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447 | poly rho; |
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448 | int divoc=0; |
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449 | ideal qlc; |
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450 | while (p!=0) |
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451 | { |
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452 | i=1; |
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453 | divoc=0; |
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454 | lpp=leadmonom(p); |
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455 | lcp=leadcoef(p); |
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456 | while (divoc==0 and i<=size(F)) |
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457 | { |
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458 | qlm=lpp/lpf[i]; |
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459 | if (qlm!=0) |
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460 | { |
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461 | qlc=simpqcoeffs(lcp,lcf[i]); |
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462 | nu=qlc[2]; |
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463 | mu=mu*nu; |
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464 | rho=qlc[1]*qlm; |
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465 | p=nu*p-rho*F[i]; |
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466 | r=nu*r; |
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467 | for (j=1;j<=size(F);j++){q[j]=nu*q[j];} |
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468 | q[i]=q[i]+rho; |
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469 | divoc=1; |
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470 | } |
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471 | else {i++;} |
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472 | } |
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473 | if (divoc==0) |
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474 | { |
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475 | r=r+lcp*lpp; |
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476 | p=p-lcp*lpp; |
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477 | } |
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478 | } |
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479 | list res=r,q,mu; |
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480 | return(res); |
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481 | } |
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482 | example |
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483 | { "EXAMPLE:"; echo = 2; |
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484 | ring R=(0,a,b,c),(x,y),dp; |
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485 | "Divisor="; |
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486 | poly f=(ab-ac)*xy+(ab)*x+(5c); |
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487 | "Dividends="; |
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488 | ideal F=ax+b,cy+a; |
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489 | "(Remainder, quotients, factor)="; |
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490 | def r=pdivi(f,F); |
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491 | r; |
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492 | "Verifying the division: r[3]*f-(r[2][1]*F[1]+r[2][2]*F[2])-r[1] ="; |
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493 | r[3]*f-(r[2][1]*F[1]+r[2][2]*F[2])-r[1]; |
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494 | } |
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495 | |
---|
496 | // pspol : S-poly of two polynomials in @R |
---|
497 | // @R |
---|
498 | // input: |
---|
499 | // poly f (given in the ring @R) |
---|
500 | // poly g (given in the ring @R) |
---|
501 | // output: |
---|
502 | // list (S, red): S is the S-poly(f,g) and red is a Boolean variable |
---|
503 | // if red==1 then S reduces by Buchberger 1st criterion (not used) |
---|
504 | static proc pspol(poly f,poly g) |
---|
505 | { |
---|
506 | def lcf=leadcoef(f); |
---|
507 | def lcg=leadcoef(g); |
---|
508 | def lpf=leadmonom(f); |
---|
509 | def lpg=leadmonom(g); |
---|
510 | def v=gcd(lpf,lpg); |
---|
511 | def s=simpqcoeffs(lcf,lcg); |
---|
512 | def vf=lpf/v; |
---|
513 | def vg=lpg/v; |
---|
514 | poly S=s[2]*vg*f-s[1]*vf*g; |
---|
515 | return(S); |
---|
516 | } |
---|
517 | |
---|
518 | // facvar: Returns all the free-square factors of the elements |
---|
519 | // of ideal J (non repeated). Integer factors are ignored, |
---|
520 | // even 0 is ignored. It can be called from ideal @R, but |
---|
521 | // the given ideal J must only contain poynomials in the |
---|
522 | // parameters. |
---|
523 | // Operates in the ring @P, but can be called from ring @R, |
---|
524 | // and the ideal @P must be defined calling first setglobalrings(); |
---|
525 | // input: ideal J |
---|
526 | // output: ideal Jc: Returns all the free-square factors of the elements |
---|
527 | // of ideal J (non repeated). Integer factors are ignored, |
---|
528 | // even 0 is ignored. It can be called from ideal @R, but |
---|
529 | // the given ideal J must only contain poynomials in the |
---|
530 | // parameters. |
---|
531 | static proc facvar(ideal J) |
---|
532 | //"USAGE: facvar(J); |
---|
533 | // J: an ideal in the parameters |
---|
534 | //RETURN: all the free-square factors of the elements |
---|
535 | // of ideal J (non repeated). Integer factors are ignored, |
---|
536 | // even 0 is ignored. It can be called from ideal @R, but |
---|
537 | // the given ideal J must only contain poynomials in the |
---|
538 | // parameters. |
---|
539 | //NOTE: Operates in the ring @P, and the ideal J must contain only |
---|
540 | // polynomials in the parameters, but can be called from ring @R. |
---|
541 | //KEYWORDS: factor |
---|
542 | //EXAMPLE: facvar; shows an example" |
---|
543 | { |
---|
544 | int i; |
---|
545 | def RR=basering; |
---|
546 | setring(@P); |
---|
547 | def Ja=imap(RR,J); |
---|
548 | if(size(Ja)==0){return(ideal(0));} |
---|
549 | Ja=elimintfromideal(Ja); // also in ideal @P |
---|
550 | ideal Jb; |
---|
551 | if (size(Ja)==0){Jb=ideal(0);} |
---|
552 | else |
---|
553 | { |
---|
554 | for (i=1;i<=ncols(Ja);i++){if(size(Ja[i])!=0){Jb=Jb,factorize(Ja[i],1);}} |
---|
555 | Jb=simplify(Jb,2+4+8); |
---|
556 | Jb=cld(Jb); |
---|
557 | Jb=elimintfromideal(Jb); // also in ideal @P |
---|
558 | } |
---|
559 | setring(RR); |
---|
560 | def Jc=imap(@P,Jb); |
---|
561 | return(Jc); |
---|
562 | } |
---|
563 | //example |
---|
564 | //{ "EXAMPLE:"; echo = 2; |
---|
565 | // ring R=(0,a,b,c),(x,y,z),dp; |
---|
566 | // setglobalrings(); |
---|
567 | // ideal J=a2-b2,a2-2ab+b2,abc-bc; |
---|
568 | // facvar(J); |
---|
569 | //} |
---|
570 | |
---|
571 | // Wred: eliminate the factors in the polynom f that are in W |
---|
572 | // in ring @RP |
---|
573 | // input: |
---|
574 | // poly f: |
---|
575 | // ideal W of non-null conditions (already supposed that it is facvar) |
---|
576 | // output: |
---|
577 | // poly f2 where the non-null conditions in W have been dropped from f |
---|
578 | static proc Wred(poly f, ideal W) |
---|
579 | { |
---|
580 | if (f==0){return(f);} |
---|
581 | def RR=basering; |
---|
582 | setring(@RP); |
---|
583 | def ff=imap(RR,f); |
---|
584 | def RPW=imap(RR,W); |
---|
585 | def l=factorize(ff,2); |
---|
586 | int i; |
---|
587 | poly f1=1; |
---|
588 | for(i=1;i<=size(l[1]);i++) |
---|
589 | { |
---|
590 | if ((memberpos(l[1][i],RPW)[1]) or (memberpos(-l[1][i],RPW)[1])){;} |
---|
591 | else{f1=f1*((l[1][i])^(l[2][i]));} |
---|
592 | } |
---|
593 | setring(RR); |
---|
594 | def f2=imap(@RP,f1); |
---|
595 | return(f2); |
---|
596 | } |
---|
597 | |
---|
598 | // pnormalform: reduces a polynomial wrt a red-spec dividing by N and eliminating factors in W. |
---|
599 | // called in the ring @R |
---|
600 | // operates in the ring @RP |
---|
601 | // both ideals must be defined calling first setglobalrings(); |
---|
602 | // input: |
---|
603 | // poly f |
---|
604 | // ideal N (depends only on the parameters) |
---|
605 | // ideal W (depends only on the parameters) |
---|
606 | // (N,W) must be a red-spec (depends only on the parameters) |
---|
607 | // output: poly f2 reduced wrt to the red-spec (N,W) |
---|
608 | // note: for security a lot of work is done. If (N,W) is already a red-spec |
---|
609 | // it should be simplified |
---|
610 | proc pnormalform(poly f, ideal N, ideal W) |
---|
611 | "USAGE: pnormalform(f,N,W); |
---|
612 | f: the polynomial to be reduced modulo (N,W) a reduced representation |
---|
613 | of a segment in the parameters. |
---|
614 | N: the null conditions ideal |
---|
615 | W: the non-null conditions (set of irreducible polynomials) |
---|
616 | RETURN: a reduced polynomial g of f, whose coefficients are reduced |
---|
617 | modulo N and having no factor in W. |
---|
618 | NOTE: Should be called from ring Q[a][x], and the global rings @R, @P |
---|
619 | and @RP must be defined. These rings can be created by calling |
---|
620 | previously setglobalrings(); |
---|
621 | Ideals N and W must be given by polynomials |
---|
622 | in the parameters forming a reduced-representation (see |
---|
623 | definition in the paper). |
---|
624 | KEYWORDS: division, pdivi, reduce |
---|
625 | EXAMPLE: pnormalform; shows an example" |
---|
626 | { |
---|
627 | def RR=basering; |
---|
628 | setglobalrings(); |
---|
629 | setring(@RP); |
---|
630 | def fa=imap(RR,f); |
---|
631 | def Na=imap(RR,N); |
---|
632 | def Wa=imap(RR,W); |
---|
633 | option(redSB); |
---|
634 | Na=std(Na); |
---|
635 | def r=cld(reduce(fa,Na)); |
---|
636 | def f1=Wred(r[1],Wa); |
---|
637 | setring(RR); |
---|
638 | def f2=imap(@RP,f1); |
---|
639 | return(f2) |
---|
640 | } |
---|
641 | example |
---|
642 | { "EXAMPLE:"; echo = 2; |
---|
643 | ring R=(0,a,b,c),(x,y),dp; |
---|
644 | setglobalrings(); |
---|
645 | poly f=(b^2-1)*x^3*y+(c^2-1)*x*y^2+(c^2*b-b)*x+(a-bc)*y; |
---|
646 | ideal N=(ab-c)*(a-b),(a-bc)*(a-b); |
---|
647 | ideal W=a^2-b^2,bc; |
---|
648 | def r=redspec(N,W); |
---|
649 | pnormalform(f,r[1],r[2]); |
---|
650 | } |
---|
651 | |
---|
652 | // idint: ideal intersection |
---|
653 | // in the ring @P. |
---|
654 | // it works in an extended ring |
---|
655 | // input: two ideals in the ring @P |
---|
656 | // output the intersection of both (is not a GB) |
---|
657 | static proc idint(ideal I, ideal J) |
---|
658 | { |
---|
659 | def RR=basering; |
---|
660 | ring T=0,t,lp; |
---|
661 | def K=T+RR; |
---|
662 | setring(K); |
---|
663 | def Ia=imap(RR,I); |
---|
664 | def Ja=imap(RR,J); |
---|
665 | ideal IJ; |
---|
666 | int i; |
---|
667 | for(i=1;i<=size(Ia);i++){IJ[i]=t*Ia[i];} |
---|
668 | for(i=1;i<=size(Ja);i++){IJ[size(Ia)+i]=(1-t)*Ja[i];} |
---|
669 | ideal eIJ=eliminate(IJ,t); |
---|
670 | setring(RR); |
---|
671 | return(imap(K,eIJ)); |
---|
672 | } |
---|
673 | |
---|
674 | // redspec: generates a red-representation |
---|
675 | // called in any ring |
---|
676 | // it changes to the ring @P |
---|
677 | // So the globalrings @P, @RP, @R, must be created before |
---|
678 | // using it by calling setglobalrings(); |
---|
679 | // input: |
---|
680 | // ideal N : the ideal of null-conditions |
---|
681 | // ideal W : set of non-null polynomials: |
---|
682 | // if W corresponds to no non null conditions then W=ideal(0) |
---|
683 | // otherwise it should be given as an ideal. |
---|
684 | // returns: list (Na,Wa,DGN) |
---|
685 | // the completely reduced representation: |
---|
686 | // Na = ideal reduced and radical of the red-spec |
---|
687 | // facvar(Wa) = ideal the reduced non-null set of polynomials of the red-spec. |
---|
688 | // if it corresponds to no non null conditions then it is ideal(0) |
---|
689 | // otherwise the ideal is returned. |
---|
690 | // DGN = the list of prime ideals associated to Na (uses primASSGTZ in "primdec.lib") |
---|
691 | // none of the polynomials in facvar(Wa) are contained in none of the ideals in DGN |
---|
692 | // If the given conditions are not compatible, then N=ideal(1) and DGN=list(ideal(1)) |
---|
693 | proc redspec(ideal Ni, ideal Wi) |
---|
694 | //"USAGE: redspec(N,W); |
---|
695 | // N: null conditions ideal |
---|
696 | // W: set of non-null polynomials (ideal) |
---|
697 | //RETURN: a list (N1,W1,L1) containing a red-representation of the segment (N,W). |
---|
698 | // N1 is the radical reduced ideal characterizing the segment. |
---|
699 | // V(N1) is the Zariski closure of the segment (N,W). |
---|
700 | // The segment S=V(N1) \ V(h), where h=prod(w in W1) |
---|
701 | // N1 is uniquely determined and no prime component of N1 contains none of |
---|
702 | // the polynomials in W1. The polynomials in W1 are prime and reduced |
---|
703 | // wrt N1, and are considered non-null on the segment. |
---|
704 | // L1 contains the list of prime components of N1. |
---|
705 | //NOTE: Called from ring @R it works in ring @P, that must be defined |
---|
706 | // by the call to setglobalrings(); |
---|
707 | // Used in the old library redcgs.lib. |
---|
708 | //KEYWORDS: representation |
---|
709 | //EXAMPLE: redspec; shows an example" |
---|
710 | { |
---|
711 | ideal Nc; |
---|
712 | ideal Wc; |
---|
713 | def RR=basering; |
---|
714 | setring(@P); |
---|
715 | def N=imap(RR,Ni); |
---|
716 | def W=imap(RR,Wi); |
---|
717 | ideal Wa; |
---|
718 | ideal Wb; |
---|
719 | if(size(W)==0){Wa=ideal(0);} |
---|
720 | //when there are no non-null conditions then W=ideal(0) |
---|
721 | else |
---|
722 | { |
---|
723 | Wa=facvar(W); |
---|
724 | } |
---|
725 | if (size(N)==0) |
---|
726 | { |
---|
727 | setring(RR); |
---|
728 | Wc=imap(@P,Wa); |
---|
729 | return(list(ideal(0), Wc, list(ideal(0)))); |
---|
730 | } |
---|
731 | int i; |
---|
732 | list LNb; |
---|
733 | list LNa; |
---|
734 | def LN=minGTZ(N); |
---|
735 | for (i=1;i<=size(LN);i++) |
---|
736 | { |
---|
737 | option(redSB); |
---|
738 | LNa[i]=std(LN[i]); |
---|
739 | } |
---|
740 | poly h=1; |
---|
741 | if (size(Wa)!=0) |
---|
742 | { |
---|
743 | for(i=1;i<=size(Wa);i++){h=h*Wa[i];} |
---|
744 | } |
---|
745 | ideal Na; |
---|
746 | intvec save_opt=option(get); |
---|
747 | if (size(N)!=0 and (size(LNa)>0)) |
---|
748 | { |
---|
749 | option(returnSB); |
---|
750 | Na=intersect(LNa[1..size(LNa)]); |
---|
751 | option(redSB); |
---|
752 | Na=std(Na); |
---|
753 | option(set,save_opt); |
---|
754 | } |
---|
755 | attrib(Na,"isSB",1); |
---|
756 | if (reduce(h,Na,1)==0) |
---|
757 | { |
---|
758 | setring(RR); |
---|
759 | Wc=imap(@P,Wa); |
---|
760 | return(list (ideal(1),Wc,list(ideal(1)))); |
---|
761 | } |
---|
762 | i=1; |
---|
763 | while(i<=size(LNa)) |
---|
764 | { |
---|
765 | if (reduce(h,LNa[i],1)==0){LNa=delete(LNa,i);} |
---|
766 | else{ i++;} |
---|
767 | } |
---|
768 | if (size(LNa)==0) |
---|
769 | { |
---|
770 | setring(RR); |
---|
771 | return(list(ideal(1),ideal(0),list(ideal(1)))); |
---|
772 | } |
---|
773 | option(returnSB); |
---|
774 | ideal Nb=intersect(LNa[1..size(LNa)]); |
---|
775 | option(redSB); |
---|
776 | Nb=std(Nb); |
---|
777 | option(set,save_opt); |
---|
778 | if (size(Wa)==0) |
---|
779 | { |
---|
780 | setring(RR); |
---|
781 | Nc=imap(@P,Nb); |
---|
782 | Wc=imap(@P,Wa); |
---|
783 | LNb=imap(@P,LNa); |
---|
784 | return(list(Nc,Wc,LNb)); |
---|
785 | } |
---|
786 | Wb=ideal(0); |
---|
787 | attrib(Nb,"isSB",1); |
---|
788 | for (i=1;i<=size(Wa);i++){Wb[i]=reduce(Wa[i],Nb);} |
---|
789 | Wb=facvar(Wb); |
---|
790 | if (size(LNa)!=0) |
---|
791 | { |
---|
792 | setring(RR); |
---|
793 | Nc=imap(@P,Nb); |
---|
794 | Wc=imap(@P,Wb); |
---|
795 | LNb=imap(@P,LNa); |
---|
796 | return(list(Nc,Wc,LNb)) |
---|
797 | } |
---|
798 | else |
---|
799 | { |
---|
800 | setring(RR); |
---|
801 | Nd=imap(@P,Nb); |
---|
802 | Wc=imap(@P,Wb); |
---|
803 | kill LNb; |
---|
804 | list LNb; |
---|
805 | return(list(Nd,Wc,LNb)) |
---|
806 | } |
---|
807 | } |
---|
808 | //example |
---|
809 | //{ "EXAMPLE:"; echo = 2; |
---|
810 | // ring r=(0,a,b,c),(x,y),dp; |
---|
811 | // setglobalrings(); |
---|
812 | // ideal N=(ab-c)*(a-b),(a-bc)*(a-b); |
---|
813 | // ideal W=a^2-b^2,bc; |
---|
814 | // redspec(N,W); |
---|
815 | //} |
---|
816 | |
---|
817 | // lesspol: compare two polynomials by its leading power products |
---|
818 | // input: two polynomials f,g in the ring @R |
---|
819 | // output: 0 if f<g, 1 if f>=g |
---|
820 | static proc lesspol(poly f, poly g) |
---|
821 | { |
---|
822 | if (leadmonom(f)<leadmonom(g)){return(1);} |
---|
823 | else |
---|
824 | { |
---|
825 | if (leadmonom(g)<leadmonom(f)){return(0);} |
---|
826 | else |
---|
827 | { |
---|
828 | if (leadcoef(f)<leadcoef(g)){return(1);} |
---|
829 | else {return(0);} |
---|
830 | } |
---|
831 | } |
---|
832 | } |
---|
833 | |
---|
834 | // delfromideal: deletes the i-th polynomial from the ideal F |
---|
835 | static proc delfromideal(ideal F, int i) |
---|
836 | { |
---|
837 | int j; |
---|
838 | ideal G; |
---|
839 | if (size(F)<i){ERROR("delfromideal was called incorrect arguments");} |
---|
840 | if (size(F)<=1){return(ideal(0));} |
---|
841 | if (i==0){return(F);} |
---|
842 | for (j=1;j<=size(F);j++) |
---|
843 | { |
---|
844 | if (j!=i){G[size(G)+1]=F[j];} |
---|
845 | } |
---|
846 | return(G); |
---|
847 | } |
---|
848 | |
---|
849 | // delidfromid: deletes the polynomials in J that are in I |
---|
850 | // input: ideals I,J |
---|
851 | // output: the ideal J without the polynomials in I |
---|
852 | static proc delidfromid(ideal I, ideal J) |
---|
853 | { |
---|
854 | int i; list r; |
---|
855 | ideal JJ=J; |
---|
856 | for (i=1;i<=size(I);i++) |
---|
857 | { |
---|
858 | r=memberpos(I[i],JJ); |
---|
859 | if (r[1]) |
---|
860 | { |
---|
861 | JJ=delfromideal(JJ,r[2]); |
---|
862 | } |
---|
863 | } |
---|
864 | return(JJ); |
---|
865 | } |
---|
866 | |
---|
867 | // sortideal: sorts the polynomials in an ideal by lm in ascending order |
---|
868 | static proc sortideal(ideal Fi) |
---|
869 | { |
---|
870 | def RR=basering; |
---|
871 | setring(@RP); |
---|
872 | def F=imap(RR,Fi); |
---|
873 | def H=F; |
---|
874 | ideal G; |
---|
875 | int i; |
---|
876 | int j; |
---|
877 | poly p; |
---|
878 | while (size(H)!=0) |
---|
879 | { |
---|
880 | j=1; |
---|
881 | p=H[1]; |
---|
882 | for (i=1;i<=size(H);i++) |
---|
883 | { |
---|
884 | if(lesspol(H[i],p)){j=i;p=H[j];} |
---|
885 | } |
---|
886 | G[size(G)+1]=p; |
---|
887 | H=delfromideal(H,j); |
---|
888 | } |
---|
889 | setring(RR); |
---|
890 | def GG=imap(@RP,G); |
---|
891 | return(GG); |
---|
892 | } |
---|
893 | |
---|
894 | // mingb: given a basis (gb reducing) it |
---|
895 | // order the polynomials is ascending order and |
---|
896 | // eliminate the polynomials whose lpp is divisible by some |
---|
897 | // smaller one |
---|
898 | static proc mingb(ideal F) |
---|
899 | { |
---|
900 | int t; int i; int j; |
---|
901 | def H=sortideal(F); |
---|
902 | ideal G; |
---|
903 | if (ncols(H)<=1){return(H);} |
---|
904 | G=H[1]; |
---|
905 | for (i=2; i<=ncols(H); i++) |
---|
906 | { |
---|
907 | t=1; |
---|
908 | j=1; |
---|
909 | while (t and (j<i)) |
---|
910 | { |
---|
911 | if((leadmonom(H[i])/leadmonom(H[j]))!=0) {t=0;} |
---|
912 | j++; |
---|
913 | } |
---|
914 | if (t) {G[size(G)+1]=H[i];} |
---|
915 | } |
---|
916 | return(G); |
---|
917 | } |
---|
918 | |
---|
919 | // redgb: given a minimal basis (gb reducing) it |
---|
920 | // reduces each polynomial wrt to the others |
---|
921 | static proc redgb(ideal F, ideal N, ideal W) |
---|
922 | { |
---|
923 | ideal G; |
---|
924 | ideal H; |
---|
925 | int i; |
---|
926 | if (size(F)==0){return(ideal(0));} |
---|
927 | for (i=1;i<=size(F);i++) |
---|
928 | { |
---|
929 | H=delfromideal(F,i); |
---|
930 | G[i]=pnormalform(pdivi(F[i],H)[1],N,W); |
---|
931 | } |
---|
932 | return(G); |
---|
933 | } |
---|
934 | |
---|
935 | //********************Main routines for buildtree****************** |
---|
936 | |
---|
937 | // splitspec: a new leading coefficient f is given to a red-spec |
---|
938 | // then splitspec computes the two new red-spec by |
---|
939 | // considering it null, and non null. |
---|
940 | // in ring @P |
---|
941 | // given f, and the red-spec (N,W) |
---|
942 | // it outputs the null and the non-null red-spec adding f. |
---|
943 | // if some of the output representations has N=1 then |
---|
944 | // there must be no split and buildtree must continue on |
---|
945 | // the compatible red-spec |
---|
946 | // input: poly f coefficient to split if needed |
---|
947 | // list r=(N,W,LN) redspec |
---|
948 | // output: list L = list(ideal N0, ideal W0), list(ideal N1, ideal W1), cond |
---|
949 | static proc splitspec(poly fi, list ri) |
---|
950 | { |
---|
951 | def RR=basering; |
---|
952 | def Ni=ri[1]; |
---|
953 | def Wi=ri[2]; |
---|
954 | setring(@P); |
---|
955 | def f=imap(RR,fi); |
---|
956 | def N=imap(RR,Ni); |
---|
957 | def W=imap(RR,Wi); |
---|
958 | f=Wred(f,W); |
---|
959 | def N0=N; |
---|
960 | def W1=W; |
---|
961 | N0[size(N0)+1]=f; |
---|
962 | def r0=redspec(N0,W); |
---|
963 | W1[size(W1)+1]=f; |
---|
964 | def r1=redspec(N,W1); |
---|
965 | setring(RR); |
---|
966 | def ra0=imap(@P,r0); |
---|
967 | def ra1=imap(@P,r1); |
---|
968 | def cond=imap(@P,f); |
---|
969 | return (list(ra0,ra1,cond)); |
---|
970 | } |
---|
971 | |
---|
972 | // redcoefs |
---|
973 | // 15/09/2010 |
---|
974 | static proc redcoefs(poly f, ideal N) |
---|
975 | { |
---|
976 | def f1=f; int test0=1; poly lc; poly lm; |
---|
977 | poly lc1; |
---|
978 | def RR=basering; |
---|
979 | setring(@P); |
---|
980 | poly lcp; |
---|
981 | def Np=imap(RR,N); |
---|
982 | attrib(Np,"isSB",1); |
---|
983 | setring(RR); |
---|
984 | while((test0==1) and (f1<>0)) |
---|
985 | { |
---|
986 | lc=leadcoef(f1); |
---|
987 | lm=leadmonom(f1); |
---|
988 | setring(@P); |
---|
989 | lcp=imap(RR,lc); |
---|
990 | lcp=reduce(lcp,Np); |
---|
991 | setring(RR); |
---|
992 | lc1=imap(@P,lcp); |
---|
993 | if(lc1<>0){test0=0;} |
---|
994 | f1=f1+(lc1-lc)*lm; |
---|
995 | } |
---|
996 | return(f1); |
---|
997 | } |
---|
998 | |
---|
999 | // discusspolys: given a basis B and a red-spec (N,W), it analyzes the |
---|
1000 | // leadcoef of the polynomials in B until it finds |
---|
1001 | // that one of them can be either null or non null. |
---|
1002 | // If at the end only the non null option is compatible |
---|
1003 | // then the reduced B has all the leadcoef non null. |
---|
1004 | // Else recbuildtree must split. |
---|
1005 | // ring @R |
---|
1006 | // input: ideal B |
---|
1007 | // ideal N |
---|
1008 | // ideal W (a reduced-representation) |
---|
1009 | // output: list of ((N0,W0,LN0),(N1,W1,LN1),Br,cond) |
---|
1010 | // cond is the condition to branch |
---|
1011 | static proc discusspolys(ideal B, list r) |
---|
1012 | { |
---|
1013 | poly f; poly f1; poly f2; |
---|
1014 | poly cond; |
---|
1015 | def N=r[1]; def W=r[2]; def LN=r[3]; |
---|
1016 | def Ba=B; def F=B; |
---|
1017 | ideal N0=1; def W0=W; list LN0=ideal(1); |
---|
1018 | def N1=N; def W1=W; def LN1=LN; |
---|
1019 | list L; |
---|
1020 | list M; list M0; list M1; |
---|
1021 | list rr; |
---|
1022 | if (size(B)==0) |
---|
1023 | { |
---|
1024 | M0=N0,W0,LN0; // incompatible |
---|
1025 | M1=N1,W1,LN1; |
---|
1026 | M=M0,M1,B,poly(1); |
---|
1027 | return(M); |
---|
1028 | } |
---|
1029 | while ((size(F)!=0) and ((N0[1]==1) or (N1[1]==1))) |
---|
1030 | { |
---|
1031 | f=F[1]; |
---|
1032 | F=delfromideal(F,1); |
---|
1033 | f1=pnormalform(f,N,W); |
---|
1034 | rr=memberpos(f,Ba); |
---|
1035 | if (f1!=0) |
---|
1036 | { |
---|
1037 | Ba[rr[2]]=f1; |
---|
1038 | if (pardeg(leadcoef(f1))!=0) |
---|
1039 | { |
---|
1040 | f2=Wred(leadcoef(f1),W); |
---|
1041 | L=splitspec(f2,list(N,W,LN)); |
---|
1042 | N0=L[1][1]; W0=L[1][2]; LN0=L[1][3]; N1=L[2][1]; W1=L[2][2]; LN1=L[2][3]; |
---|
1043 | cond=L[3]; |
---|
1044 | } |
---|
1045 | } |
---|
1046 | else |
---|
1047 | { |
---|
1048 | Ba=delfromideal(Ba,rr[2]); |
---|
1049 | N0=ideal(1); //F=ideal(0); |
---|
1050 | } |
---|
1051 | } |
---|
1052 | M0=N0,W0,LN0; |
---|
1053 | M1=N1,W1,LN1; |
---|
1054 | M=M0,M1,Ba,cond; |
---|
1055 | return(M); |
---|
1056 | } |
---|
1057 | |
---|
1058 | // discussSpolys: given a basis B and a red-spec (N,W), it analyzes the |
---|
1059 | // leadcoef of the polynomials in B until it finds |
---|
1060 | // that one of them can be either null or non null. |
---|
1061 | // If at the end only the non null option is compatible |
---|
1062 | // then the reduced B has all the leadcoef non null. |
---|
1063 | // Else recbuildtree must split. |
---|
1064 | // ring @R |
---|
1065 | // input: ideal B |
---|
1066 | // ideal N |
---|
1067 | // ideal W (a reduced-representation) |
---|
1068 | // list P current set of pairs of polynomials from B to be tested. |
---|
1069 | // output: list of (N0,W0,LN0),(N1,W1,LN1),Br,Pr,cond] |
---|
1070 | // list Pr the not checked list of pairs. |
---|
1071 | static proc discussSpolys(ideal B, list r, list P) |
---|
1072 | { |
---|
1073 | int i; int j; int k; |
---|
1074 | int npols; int nSpols; int tt; |
---|
1075 | poly cond=1; |
---|
1076 | poly lm; poly lpf; poly lpg; |
---|
1077 | def F=B; def Pa=P; list Pa0; |
---|
1078 | def N=r[1]; def W=r[2]; def LN=r[3]; |
---|
1079 | ideal N0=1; def W0=W; list LN0=ideal(1); |
---|
1080 | def N1=N; def W1=W; def LN1=LN; |
---|
1081 | ideal Bw; |
---|
1082 | poly S; |
---|
1083 | list L; list L0; list L1; |
---|
1084 | list M; list M0; list M1; |
---|
1085 | list pair; |
---|
1086 | list KK; int loc; |
---|
1087 | int crit; |
---|
1088 | poly h; |
---|
1089 | if (size(B)==0) |
---|
1090 | { |
---|
1091 | M0=N0,W0,LN0; |
---|
1092 | M1=N1,W1,LN1; |
---|
1093 | M=M0,M1,ideal(0),Pa,cond; |
---|
1094 | return(M); |
---|
1095 | } |
---|
1096 | tt=1; |
---|
1097 | i=1; |
---|
1098 | while ((tt) and (i<=size(B))) |
---|
1099 | { |
---|
1100 | h=B[i]; |
---|
1101 | for (j=1;j<=npars(@R);j++) |
---|
1102 | { |
---|
1103 | h=subst(h,par(j),0); |
---|
1104 | } |
---|
1105 | if (h!=B[i]){tt=0;} |
---|
1106 | i++; |
---|
1107 | } |
---|
1108 | if (tt) |
---|
1109 | { |
---|
1110 | //"T_ a non parametric system occurred"; |
---|
1111 | def RR=basering; |
---|
1112 | def RL=ringlist(RR); |
---|
1113 | RL[1]=0; |
---|
1114 | def LRR=ring(RL); |
---|
1115 | setring(LRR); |
---|
1116 | def BP=imap(RR,B); |
---|
1117 | option(redSB); |
---|
1118 | BP=std(BP); |
---|
1119 | setring(RR); |
---|
1120 | B=imap(LRR,BP); |
---|
1121 | M0=ideal(1),W0,LN0; |
---|
1122 | M1=N1,W1,LN1; |
---|
1123 | M=M0,M1,B,list(),cond; |
---|
1124 | return(M); |
---|
1125 | } |
---|
1126 | if (size(Pa)==0){npols=size(B); Pa=orderingpairs(F); nSpols=size(Pa);} |
---|
1127 | while ((size(Pa)!=0) and (N0[1]==1) or (N1[1]==1)) |
---|
1128 | { |
---|
1129 | pair=Pa[1]; |
---|
1130 | i=pair[1]; |
---|
1131 | j=pair[2]; |
---|
1132 | Pa=delete(Pa,1); |
---|
1133 | // Buchberger 1st criterion (not needed here, it is already eliminated |
---|
1134 | // when creating the list of pairs |
---|
1135 | for (k=1;k<=size(Pa);k++){Pa0[k]=delete(Pa[k],3);} |
---|
1136 | crit=0; |
---|
1137 | if (not(crit)) |
---|
1138 | { |
---|
1139 | S=pspol(F[i],F[j]); |
---|
1140 | KK=pdivi(S,F); |
---|
1141 | S=KK[1]; |
---|
1142 | if (S!=0) |
---|
1143 | { |
---|
1144 | S=pnormalform(S,N,W); |
---|
1145 | if (S!=0) |
---|
1146 | { |
---|
1147 | L=discusspolys(ideal(S),list(N,W,LN)); |
---|
1148 | N0=L[1][1]; |
---|
1149 | W0=L[1][2]; |
---|
1150 | LN0=L[1][3]; |
---|
1151 | N1=L[2][1]; |
---|
1152 | W1=L[2][2]; |
---|
1153 | LN1=L[2][3]; |
---|
1154 | S=L[3][1]; |
---|
1155 | cond=L[4]; |
---|
1156 | if (S==1) |
---|
1157 | { |
---|
1158 | M0=ideal(1),W0,list(ideal(1)); |
---|
1159 | M1=N1,W1,LN1; |
---|
1160 | M=M0,M1,ideal(1),list(),cond; |
---|
1161 | return(M); |
---|
1162 | } |
---|
1163 | if (S!=0) |
---|
1164 | { |
---|
1165 | F[size(F)+1]=S; |
---|
1166 | npols=size(F); |
---|
1167 | for (k=1;k<size(F);k++) |
---|
1168 | { |
---|
1169 | lm=lcmlmonoms(F[k],S); |
---|
1170 | // Buchberger 1st criterion |
---|
1171 | lpf=leadmonom(F[k]); |
---|
1172 | lpg=leadmonom(S); |
---|
1173 | if (lpf*lpg!=lm) |
---|
1174 | { |
---|
1175 | pair=k,size(F),lm; |
---|
1176 | Pa=placepairinlist(pair,Pa); |
---|
1177 | nSpols=size(Pa); |
---|
1178 | } |
---|
1179 | } |
---|
1180 | if (N0[1]==1){N=N1; W=W1; LN=LN1;} |
---|
1181 | } |
---|
1182 | } |
---|
1183 | } |
---|
1184 | } |
---|
1185 | } |
---|
1186 | M0=N0,W0,LN0; |
---|
1187 | M1=N1,W1,LN1; |
---|
1188 | M=M0,M1,F,Pa,cond; |
---|
1189 | return(M); |
---|
1190 | } |
---|
1191 | |
---|
1192 | // lcmlmonoms: computes the lcm of the leading monomials |
---|
1193 | // of the polynomils f and g |
---|
1194 | // ring @R |
---|
1195 | static proc lcmlmonoms(poly f,poly g) |
---|
1196 | { |
---|
1197 | def lf=leadmonom(f); |
---|
1198 | def lg=leadmonom(g); |
---|
1199 | def gls=gcd(lf,lg); |
---|
1200 | return((lf*lg)/gls); |
---|
1201 | } |
---|
1202 | |
---|
1203 | // placepairinlist |
---|
1204 | // 15/09/2010 |
---|
1205 | // input: given a new pair of the form (i,j,lmij) |
---|
1206 | // and a list of pairs of the same form |
---|
1207 | // ring @R |
---|
1208 | // output: it inserts the new pair in ascending order of lmij |
---|
1209 | static proc placepairinlist(list pair,list P) |
---|
1210 | { |
---|
1211 | list Pr; |
---|
1212 | if (size(P)==0){Pr=insert(P,pair); return(Pr);} |
---|
1213 | if (pair[3]<P[1][3]){Pr=insert(P,pair); return(Pr);} |
---|
1214 | if (pair[3]>=P[size(P)][3]){Pr=insert(P,pair,size(P)); return(Pr);} |
---|
1215 | kill Pr; |
---|
1216 | list Pr; |
---|
1217 | int j; |
---|
1218 | int i=1; |
---|
1219 | int loc=0; |
---|
1220 | while((i<=size(P)) and (loc==0)) |
---|
1221 | { |
---|
1222 | if (pair[3]>=P[i][3]){j=i; i++;} |
---|
1223 | else{loc=1; j=i-1;} |
---|
1224 | } |
---|
1225 | Pr=insert(P,pair,j); |
---|
1226 | return(Pr); |
---|
1227 | } |
---|
1228 | |
---|
1229 | // orderingpairs: |
---|
1230 | // input: ideal F |
---|
1231 | // output: list of ordered pairs (i,j,lcmij) of F in ascending order of lcmij |
---|
1232 | // if a pair verifies Buchberger 1st criterion it is not stored |
---|
1233 | // ring @R |
---|
1234 | static proc orderingpairs(ideal F) |
---|
1235 | { |
---|
1236 | int i; |
---|
1237 | int j; |
---|
1238 | poly lm; |
---|
1239 | poly lpf; |
---|
1240 | poly lpg; |
---|
1241 | list P; |
---|
1242 | list pair; |
---|
1243 | if (size(F)<=1){return(P);} |
---|
1244 | for (i=1;i<=size(F)-1;i++) |
---|
1245 | { |
---|
1246 | for (j=i+1;j<=size(F);j++) |
---|
1247 | { |
---|
1248 | lm=lcmlmonoms(F[i],F[j]); |
---|
1249 | // Buchberger 1st criterion |
---|
1250 | lpf=leadmonom(F[i]); |
---|
1251 | lpg=leadmonom(F[j]); |
---|
1252 | if (lpf*lpg!=lm) |
---|
1253 | { |
---|
1254 | pair=(i,j,lm); |
---|
1255 | P=placepairinlist(pair,P); |
---|
1256 | } |
---|
1257 | } |
---|
1258 | } |
---|
1259 | return(P); |
---|
1260 | } |
---|
1261 | |
---|
1262 | // Buchberger 2nd criterion |
---|
1263 | // input: integers i,j |
---|
1264 | // list P of pairs of the form (i,j) not yet verified |
---|
1265 | // ring @R |
---|
1266 | // not used (it increases time) |
---|
1267 | static proc criterion(int i, int j, list P, ideal B) |
---|
1268 | { |
---|
1269 | def lcmij=lcmlmonoms(B[i],B[j]); |
---|
1270 | int crit=0; |
---|
1271 | int k=1; |
---|
1272 | list ik; list jk; |
---|
1273 | while ((k<=size(B)) and (crit==0)) |
---|
1274 | { |
---|
1275 | if ((k!=i) and (k!=j)) |
---|
1276 | { |
---|
1277 | if (i<k){ik=i,k;} else{ik=k,i;} |
---|
1278 | if (j<k){jk=i,k;} else{jk=k,j;} |
---|
1279 | if (not((memberpos(ik,P)[1]) or (memberpos(jk,P)[1]))) |
---|
1280 | { |
---|
1281 | if ((lcmij)/leadmonom(B[k])!=0){crit=1;} |
---|
1282 | } |
---|
1283 | } |
---|
1284 | k++; |
---|
1285 | } |
---|
1286 | return(crit); |
---|
1287 | } |
---|
1288 | |
---|
1289 | // buildtree: Basic routine of the old redcgs.lib generating a |
---|
1290 | // first reduced CGS |
---|
1291 | // it will define the rings @R, @P and @RP as global rings |
---|
1292 | // and the list @T a global list that will be killed at the output |
---|
1293 | // input: ideal F in ring K[a][x]; |
---|
1294 | // output: list T of lists whose list elements are of the form |
---|
1295 | // T[i]=list(list lab, boolean terminal, ideal B, ideal N, ideal W, list of ideals decomp of N, |
---|
1296 | // ideal of monomials lpp); |
---|
1297 | // all the ideals are in the ring K[a][x]; |
---|
1298 | static proc buildtree(ideal F, list #) |
---|
1299 | //"USAGE: buildtree(F); |
---|
1300 | // F: ideal in Q[a][x] (parameters and variables) to be discussed. |
---|
1301 | // It outputs the whole discussion tree to construct the |
---|
1302 | // first disjoint reduced CGS. It is the old version of the new |
---|
1303 | // cgsdr routine. It remains on the library for didactic purposes |
---|
1304 | // and is, in general, less efficient. |
---|
1305 | // Also, for some problems where cgsdr does stack, sometimes |
---|
1306 | // buildtree is able to obtain the result. |
---|
1307 | // The output of buildtree contains the whole information about the discussion |
---|
1308 | // process (the whole tree discussion) and can be reduced to |
---|
1309 | // somewhat similar to the output of cgsdr after calling |
---|
1310 | // setglobalrings(); then applying finalcases and then groupsegments to the |
---|
1311 | // output of buidtree. This is automatically done by the routine |
---|
1312 | // cgsdrold also contained in the library, that outputs only the |
---|
1313 | // CGS like the new cgsdr. |
---|
1314 | // |
---|
1315 | //RETURN: Returns a list T describing the complete discussion tree |
---|
1316 | // for obtaining a reduced and disjoint comprehensive |
---|
1317 | // Groebner system (CGS) of the ideal F of Q[a][x] with |
---|
1318 | // constant leading power products (lpp) of the reduced Groebner |
---|
1319 | // basis. |
---|
1320 | // The first element of the list is the root, and contains |
---|
1321 | // [1] label: intvec(-1) |
---|
1322 | // [2] number of children : int |
---|
1323 | // [3] the ideal F |
---|
1324 | // [4], [5], [6] the red-representation of the segment |
---|
1325 | // (null, non-null conditions, prime components of the null |
---|
1326 | // conditions) given (as option). |
---|
1327 | // ideal (0), ideal (1), list(ideal(0)) is assumed if |
---|
1328 | // no optional conditions are given. |
---|
1329 | // [7] the set of lpp of ideal F |
---|
1330 | // [8] condition that was taken to reach the vertex |
---|
1331 | // (poly 1, for the root). |
---|
1332 | // The remaining elements of the list represent vertices of the tree: |
---|
1333 | // with the same structure: |
---|
1334 | // [1] label: intvec (1,0,0,1,...) gives its position in the tree: |
---|
1335 | // first branch condition is taken non-null, second null,... |
---|
1336 | // [2] number of children (0 if it is a terminal vertex) |
---|
1337 | // [3] the specialized ideal with the previous assumed conditions |
---|
1338 | // to reach the vertex |
---|
1339 | // [4],[5],[6] the red-representation of the segment corresponding |
---|
1340 | // to the previous assumed conditions to reach the vertex |
---|
1341 | // [7] the set of lpp of the specialized ideal at this stage |
---|
1342 | // [8] condition that was taken to reach the vertex from the |
---|
1343 | // father's vertex (that was taken non-null if the last |
---|
1344 | // integer in the label is 1, and null if it is 0) |
---|
1345 | // The terminal vertices form a disjoint partition of the parameter space |
---|
1346 | // whose bases specialize to the reduced Groebner basis of the |
---|
1347 | // specialized ideal on each point of the segment and preserve |
---|
1348 | // the lpp. So they form a disjoint reduced CGS. |
---|
1349 | //NOTE: The basering R, must be of the form Q[a][x], a=parameters, |
---|
1350 | // x=variables, and should be defined previously. The ideal must |
---|
1351 | // be defined on R. |
---|
1352 | // The call of finalcases applied to the output of buildtree |
---|
1353 | // selects the terminal vertices forming the disjoint and reduced |
---|
1354 | // CGS. To obtain the output similar |
---|
1355 | // to that of the new cgsdr procedure one can call instead |
---|
1356 | // cgsdrold. |
---|
1357 | // |
---|
1358 | // The content of buildtree can be written in a file that is readable |
---|
1359 | // by Maple in order to plot its content using buildtreetoMaple; |
---|
1360 | // The file written by buildtreetoMaple when is read in a Maple |
---|
1361 | // worksheet can be plotted using the dbgb routine tplot; |
---|
1362 | // |
---|
1363 | //KEYWORDS: CGS, disjoint, reduced, comprehensive Groebner system |
---|
1364 | //EXAMPLE: buildtree; shows an example" |
---|
1365 | { |
---|
1366 | list @T; |
---|
1367 | exportto(Top,@T); |
---|
1368 | setglobalrings(); |
---|
1369 | int i; |
---|
1370 | int j; |
---|
1371 | poly f; |
---|
1372 | poly cond=1; |
---|
1373 | list LN; |
---|
1374 | LN[1]=ideal(0); |
---|
1375 | def N=ideal(0); |
---|
1376 | def W=ideal(1); |
---|
1377 | int comment=0; |
---|
1378 | list L=#; |
---|
1379 | for(i=1;i<=size(L) div 2;i++) |
---|
1380 | { |
---|
1381 | if(L[2*i-1]=="null"){N=L[2*i];} |
---|
1382 | else |
---|
1383 | { |
---|
1384 | if(L[2*i-1]=="nonnull"){W=L[2*i];} |
---|
1385 | else |
---|
1386 | { |
---|
1387 | if(L[2*i-1]=="comment"){comment=L[2*i];} |
---|
1388 | } |
---|
1389 | } |
---|
1390 | } |
---|
1391 | ideal B; |
---|
1392 | if(equalideals(N,ideal(0))==0) |
---|
1393 | { |
---|
1394 | def LL=redspec(N,W); |
---|
1395 | N=LL[1]; |
---|
1396 | W=LL[2]; |
---|
1397 | LN=LL[3]; |
---|
1398 | for (i=1;i<=size(F);i++) |
---|
1399 | { |
---|
1400 | f=pnormalform(F[i],N,W); |
---|
1401 | if (f!=0){B[size(B)+1]=f;} |
---|
1402 | } |
---|
1403 | } |
---|
1404 | else {B=F;} |
---|
1405 | def lpp=ideal(0); |
---|
1406 | if (size(B)==0){lpp=ideal(0);} |
---|
1407 | else |
---|
1408 | { |
---|
1409 | for (i=1;i<=size(B);i++){lpp[i]=leadmonom(B[i]);} |
---|
1410 | // lpp=ideal of lead power product of the polys in B |
---|
1411 | } |
---|
1412 | intvec lab=-1; |
---|
1413 | int term=0; |
---|
1414 | list root; |
---|
1415 | root[1]=lab; |
---|
1416 | root[2]=term; |
---|
1417 | root[3]=B; |
---|
1418 | root[4]=N; |
---|
1419 | root[5]=W; |
---|
1420 | root[6]=LN; |
---|
1421 | root[7]=lpp; |
---|
1422 | root[8]=cond; |
---|
1423 | @T[1]=root; |
---|
1424 | list P; |
---|
1425 | recbuildtree(root,P); |
---|
1426 | def T=@T; |
---|
1427 | kill @T; |
---|
1428 | kill @P; kill @RP; kill @R; |
---|
1429 | return(T) |
---|
1430 | } |
---|
1431 | //example |
---|
1432 | //{ "EXAMPLE:"; echo = 2; |
---|
1433 | // ring R=(0,a0,a1,a2,a3,a4),(x1,x2,x3),dp; |
---|
1434 | // "Casas conjecture for degree 4"; |
---|
1435 | // ideal F=x1^4+(4*a3)*x1^3+(6*a2)*x1^2+(4*a1)*x1+(a0), |
---|
1436 | // x1^3+(3*a3)*x1^2+(3*a2)*x1+(a1), |
---|
1437 | // x2^4+(4*a3)*x2^3+(6*a2)*x2^2+(4*a1)*x2+(a0), |
---|
1438 | // x2^2+(2*a3)*x2+(a2), |
---|
1439 | // x3^4+(4*a3)*x3^3+(6*a2)*x3^2+(4*a1)*x3+(a0), |
---|
1440 | // x3+(a3); |
---|
1441 | // def T=buildtree(F); "buildtree(F)="; T; |
---|
1442 | // setglobalrings(); |
---|
1443 | // def FC=finalcases(T); |
---|
1444 | // "finalcases(buildtree(F))="; FC; |
---|
1445 | // "groupsegments(finalcases(buildtree(F)))="; |
---|
1446 | // groupsegments(FC); |
---|
1447 | // buildtreetoMaple(T,"Tb","Tb.txt"); " "; |
---|
1448 | // "Compare with cgsdrold"; " "; |
---|
1449 | // def CDR=cgsdrold(F); |
---|
1450 | // "cgsdrold(F)="; CDR; |
---|
1451 | //} |
---|
1452 | |
---|
1453 | // recbuildtree: auxilliary recursive routine called by buildtree |
---|
1454 | static proc recbuildtree(list v, list P) |
---|
1455 | { |
---|
1456 | def vertex=v; |
---|
1457 | int i; |
---|
1458 | int j; |
---|
1459 | int pos; |
---|
1460 | list P0; |
---|
1461 | list P1; |
---|
1462 | poly f; |
---|
1463 | def lab=vertex[1]; |
---|
1464 | if ((size(lab)>1) and (lab[1]==-1)) |
---|
1465 | {lab=lab[2..size(lab)];} |
---|
1466 | def term=vertex[2]; |
---|
1467 | def B=vertex[3]; |
---|
1468 | def N=vertex[4]; |
---|
1469 | def W=vertex[5]; |
---|
1470 | def LN=vertex[6]; |
---|
1471 | def lpp=vertex[7]; |
---|
1472 | def cond=vertex[8]; |
---|
1473 | def lab0=lab; |
---|
1474 | def lab1=lab; |
---|
1475 | if ((size(lab)==1) and (lab[1]==-1)) |
---|
1476 | { |
---|
1477 | lab0=0; |
---|
1478 | lab1=1; |
---|
1479 | } |
---|
1480 | else |
---|
1481 | { |
---|
1482 | lab0[size(lab)+1]=0; |
---|
1483 | lab1[size(lab)+1]=1; |
---|
1484 | } |
---|
1485 | list vertex0; |
---|
1486 | list vertex1; |
---|
1487 | ideal B0; |
---|
1488 | ideal lpp0; |
---|
1489 | ideal lpp1; |
---|
1490 | ideal N0=1; |
---|
1491 | def W0=ideal(0); |
---|
1492 | list LN0=ideal(1); |
---|
1493 | def B1=B; |
---|
1494 | def N1=N; |
---|
1495 | def W1=W; |
---|
1496 | list LN1=LN; |
---|
1497 | list L; |
---|
1498 | if (size(P)==0) |
---|
1499 | { |
---|
1500 | L=discusspolys(B,list(N,W,LN)); |
---|
1501 | N0=L[1][1]; |
---|
1502 | W0=L[1][2]; |
---|
1503 | LN0=L[1][3]; |
---|
1504 | N1=L[2][1]; |
---|
1505 | W1=L[2][2]; |
---|
1506 | LN1=L[2][3]; |
---|
1507 | B1=L[3]; |
---|
1508 | cond=L[4]; |
---|
1509 | } |
---|
1510 | if ((size(B1)!=0) and (N0[1]==1)) |
---|
1511 | { |
---|
1512 | L=discussSpolys(B1,list(N1,W1,LN1),P); |
---|
1513 | N0=L[1][1]; |
---|
1514 | W0=L[1][2]; |
---|
1515 | LN0=L[1][3]; |
---|
1516 | N1=L[2][1]; |
---|
1517 | W1=L[2][2]; |
---|
1518 | LN1=L[2][3]; |
---|
1519 | B1=L[3]; |
---|
1520 | P1=L[4]; |
---|
1521 | cond=L[5]; |
---|
1522 | lpp=ideal(0); |
---|
1523 | for (i=1;i<=size(B1);i++){lpp[i]=leadmonom(B1[i]);} |
---|
1524 | } |
---|
1525 | vertex[3]=B1; |
---|
1526 | vertex[4]=N1; // unnecessary |
---|
1527 | vertex[5]=W1; // unnecessary |
---|
1528 | vertex[6]=LN1;// unnecessary |
---|
1529 | vertex[7]=lpp; |
---|
1530 | vertex[8]=cond; |
---|
1531 | if (size(@T)>0) |
---|
1532 | { |
---|
1533 | pos=size(@T)+1; |
---|
1534 | @T[pos]=vertex; |
---|
1535 | } |
---|
1536 | if ((N0[1]!=1) and (N1[1]!=1)) |
---|
1537 | { |
---|
1538 | vertex1[1]=lab1; |
---|
1539 | vertex1[2]=0; |
---|
1540 | vertex1[3]=B1; |
---|
1541 | vertex1[4]=N1; |
---|
1542 | vertex1[5]=W1; |
---|
1543 | vertex1[6]=LN1; |
---|
1544 | vertex1[7]=lpp1; |
---|
1545 | vertex1[8]=cond; |
---|
1546 | if (size(B1)==0){B0=ideal(0); lpp0=ideal(0);} |
---|
1547 | else |
---|
1548 | { |
---|
1549 | j=1; |
---|
1550 | lpp0=ideal(0); |
---|
1551 | for (i=1;i<=size(B1);i++) |
---|
1552 | { |
---|
1553 | f=pnormalform(B1[i],N0,W0); |
---|
1554 | if (f!=0){B0[j]=f; lpp0[j]=leadmonom(f);j++;} |
---|
1555 | } |
---|
1556 | } |
---|
1557 | vertex0[1]=lab0; |
---|
1558 | vertex0[2]=0; |
---|
1559 | vertex0[3]=B0; |
---|
1560 | vertex0[4]=N0; |
---|
1561 | vertex0[5]=W0; |
---|
1562 | vertex0[6]=LN0; |
---|
1563 | vertex0[7]=lpp0; |
---|
1564 | vertex0[8]=cond; |
---|
1565 | recbuildtree(vertex0,P0); |
---|
1566 | recbuildtree(vertex1,P1); |
---|
1567 | } |
---|
1568 | else |
---|
1569 | { |
---|
1570 | if (equalideals(N1,ideal(1))==0) |
---|
1571 | { |
---|
1572 | vertex[2]=1; |
---|
1573 | B1=mingb(B1); |
---|
1574 | vertex[3]=redgb(B1,N1,W1); |
---|
1575 | vertex[4]=N1; |
---|
1576 | vertex[5]=W1; |
---|
1577 | vertex[6]=LN1; |
---|
1578 | lpp=ideal(0); |
---|
1579 | for (i=1;i<=size(vertex[3]);i++){lpp[i]=leadmonom(vertex[3][i]);} |
---|
1580 | vertex[7]=lpp; |
---|
1581 | vertex[8]=cond; |
---|
1582 | @T[pos]=vertex; |
---|
1583 | //print(vertex); |
---|
1584 | } |
---|
1585 | } |
---|
1586 | } |
---|
1587 | |
---|
1588 | // RtoPrep |
---|
1589 | // Computes the P-representaion of a R-representaion (N,W,L) of a set |
---|
1590 | // input: |
---|
1591 | // ideal N (null conditions, must be radical) |
---|
1592 | // ideal W (non-null conditions ideal) |
---|
1593 | // list L must contain the radical decomposition of N. |
---|
1594 | // output: |
---|
1595 | // the ((p_1,(p_11,..,p_1k_1)),..,(p_r,(p_r1,..,p_rk_r))); |
---|
1596 | // the Prep of V(N) \ V(h), where h=prod(w in W). |
---|
1597 | static proc RtoPrep(ideal N, ideal W, list L) |
---|
1598 | { |
---|
1599 | int i; int j; list L0; |
---|
1600 | if (N[1]==1) |
---|
1601 | { |
---|
1602 | L0[1]=list(ideal(1),list(ideal(1))); |
---|
1603 | return(L0); |
---|
1604 | } |
---|
1605 | def RR=basering; |
---|
1606 | setring(@P); |
---|
1607 | ideal Np=imap(RR,N); |
---|
1608 | ideal Wp=imap(RR,W); |
---|
1609 | list Lp=imap(RR,L); |
---|
1610 | poly h=1; |
---|
1611 | for (i=1;i<=size(Wp);i++){h=h*Wp[i];} |
---|
1612 | list r; list Ti; list LL; |
---|
1613 | for (i=1;i<=size(Lp);i++) |
---|
1614 | { |
---|
1615 | Ti=minGTZ(Lp[i]+h); |
---|
1616 | for(j=1;j<=size(Ti);j++) |
---|
1617 | { |
---|
1618 | option(redSB); |
---|
1619 | Ti[j]=std(Ti[j]); |
---|
1620 | } |
---|
1621 | //list LL[i]; |
---|
1622 | LL[i]=list(Lp[i],Ti); |
---|
1623 | } |
---|
1624 | setring(RR); |
---|
1625 | return(imap(@P,LL)); |
---|
1626 | } |
---|
1627 | |
---|
1628 | // groupRtoPrep |
---|
1629 | // input: L (list) is the output of groupsegments |
---|
1630 | // output: LL (list) the same list but the segments are expressed |
---|
1631 | // in canonical representations: |
---|
1632 | // ( (lpp, (lab BuildTree, basis, |
---|
1633 | // ((P_1),(P_{11},...,P_{1t1})) |
---|
1634 | // ... |
---|
1635 | // ((P_j),(P_{j1},...,P_{jtj})) |
---|
1636 | // ) |
---|
1637 | // ... |
---|
1638 | // (lab BuildTree, basis, |
---|
1639 | // ((P_1),(P_{11},...,P_{1t1})) |
---|
1640 | // ... |
---|
1641 | // ((P_j),(P_{j1},...,P_{jtj})) |
---|
1642 | // ) |
---|
1643 | // ) |
---|
1644 | // ... |
---|
1645 | // (lpp, (lab BuildTree, basis, |
---|
1646 | // ((P_1),(P_{11},...,P_{1t1})) |
---|
1647 | // ... |
---|
1648 | // ((P_j),(P_{j1},...,P_{jtj})) |
---|
1649 | // ) |
---|
1650 | // ... |
---|
1651 | // (lab BuildTree, basis, |
---|
1652 | // ((P_1),(P_{11},...,P_{1t1})) |
---|
1653 | // ... |
---|
1654 | // ((P_j),(P_{j1},...,P_{jtj})) |
---|
1655 | // ) |
---|
1656 | // ) |
---|
1657 | // ) |
---|
1658 | static proc groupRtoPrep(list L) |
---|
1659 | { |
---|
1660 | int i; int j; |
---|
1661 | list LL; list ct; |
---|
1662 | // size(L)=number of lpp-segments |
---|
1663 | for (i=1;i<=size(L);i++) |
---|
1664 | { |
---|
1665 | LL[i]=list(); |
---|
1666 | LL[i][1]=L[i][1]; |
---|
1667 | // L[i][1]=lpp |
---|
1668 | LL[i][2]=list(); |
---|
1669 | for (j=1;j<=size(L[i][2]);j++) |
---|
1670 | { |
---|
1671 | ct=RtoPrep(L[i][2][j][3],L[i][2][j][4],L[i][2][j][5]); |
---|
1672 | LL[i][2][j]=list(); |
---|
1673 | LL[i][2][j][1]=L[i][2][j][1]; |
---|
1674 | // L[i][2][j][1]=label |
---|
1675 | LL[i][2][j][2]=L[i][2][j][2]; |
---|
1676 | // L[i][2][j][2]=basis |
---|
1677 | LL[i][2][j][3]=ct; |
---|
1678 | } |
---|
1679 | } |
---|
1680 | return(LL); |
---|
1681 | } |
---|
1682 | |
---|
1683 | // NEW |
---|
1684 | // input: L (list) is the output of groupsegments |
---|
1685 | // output: LL (list) the same list but the segments are expressed |
---|
1686 | // in canonical representations: |
---|
1687 | // ( (lpp, (lab BuildTree, basis, |
---|
1688 | // ((1,u1),(lab,child,P_1)), |
---|
1689 | // ((1,1,1),(lab,child,P_{11})), |
---|
1690 | // ... |
---|
1691 | // ((1,1,t1),(lab,child,P_{1t1})), |
---|
1692 | // ... |
---|
1693 | // ((1,u1),(lab,child,P_u1)), |
---|
1694 | // ((1,u1,1),(lab,child,P_{u1,1})), |
---|
1695 | // ... |
---|
1696 | // ((1,u1,tu),(lab,child,P_{u1,tu})), |
---|
1697 | // (lab BuildTree, basis, |
---|
1698 | // ((1,u2),(lab,child,P_2)), |
---|
1699 | // ((1,u1+1,1),(lab,child,P_{21})), |
---|
1700 | // ... |
---|
1701 | // ((1,u1+1,t2),(lab,child,P_{2,t2})), |
---|
1702 | // ... |
---|
1703 | // ((1,u1+..+ut),(lab,child,P_ut)), |
---|
1704 | // ((1,u1+..+ut,1),(lab,child,P_{ut,1})), |
---|
1705 | // ... |
---|
1706 | // ((1,u1+..+ut,tu),(lab,child,P_{ut,tu})), |
---|
1707 | // ... |
---|
1708 | static proc groupredtocan(list L) |
---|
1709 | { |
---|
1710 | int i; int j; |
---|
1711 | list LL; list ct; |
---|
1712 | for (i=1;i<=size(L);i++) |
---|
1713 | { |
---|
1714 | LL[i]=list(); |
---|
1715 | LL[i][1]=L[i][1]; |
---|
1716 | LL[i][2]=list(); |
---|
1717 | for (j=1;j<=size(L[i][2]);j++) |
---|
1718 | { |
---|
1719 | ct=redtocanspec(intvec(i),j-1,list(L[i][2][j][3],L[i][2][j][4],L[i][2][j][5])); |
---|
1720 | LL[i][2][j]=list(); |
---|
1721 | LL[i][2][j][1]=L[i][2][j][1]; |
---|
1722 | LL[i][2][j][2]=L[i][2][j][2]; |
---|
1723 | LL[i][2][j][3]=ct; |
---|
1724 | } |
---|
1725 | } |
---|
1726 | return(LL); |
---|
1727 | } |
---|
1728 | |
---|
1729 | //****************End of BuildTree************************************* |
---|
1730 | |
---|
1731 | //****************Begin BuildTree To Maple***************************** |
---|
1732 | |
---|
1733 | // buildtreetoMaple: writes the list provided by buildtree to a file |
---|
1734 | // containing the table representing it in Maple |
---|
1735 | |
---|
1736 | // writes the list L=buildtree(F) to a file "writefile" that |
---|
1737 | // is readable by Maple whith name T |
---|
1738 | // input: |
---|
1739 | // L: the list output by buildtree |
---|
1740 | // T: the name (string) of the output table in Maple |
---|
1741 | // writefile: the name of the datafile where the output is to be stored |
---|
1742 | // output: |
---|
1743 | // the result is written on the datafile "writefile" containig |
---|
1744 | // the assignement to the table with name "T" |
---|
1745 | proc buildtreetoMaple(list L, string T, string writefile) |
---|
1746 | "USAGE: buildtreetoMaple(T, TM, writefile); |
---|
1747 | T: is the list provided by grobcovold called with option "old",0; |
---|
1748 | TM: is the name (string) of the table variable in Maple that will represent |
---|
1749 | the output of cgsdrold; |
---|
1750 | writefile: is the name (string) of the file whereas to write the |
---|
1751 | content. |
---|
1752 | RETURN: writes the list provided by grobcovold called with option "old",0, |
---|
1753 | (old buildtree) to a file containing the table representing it in |
---|
1754 | Maple. |
---|
1755 | KEYWORDS: cgsdrold, buildtree, Maple |
---|
1756 | EXAMPLE: buildtreetoMaple; shows an example" |
---|
1757 | { |
---|
1758 | def R=basering; |
---|
1759 | if(size(T[1])!=8) |
---|
1760 | { |
---|
1761 | " 'Warning!' cgsdrold must be called with option 'old' set to 0 to be operative"; |
---|
1762 | return(); |
---|
1763 | } |
---|
1764 | short=0; |
---|
1765 | poly cond; |
---|
1766 | int i; |
---|
1767 | link LLw=":w "+writefile; |
---|
1768 | string La=string("table(",T,");"); |
---|
1769 | write(LLw, La); |
---|
1770 | close(LLw); |
---|
1771 | link LLa=":a "+writefile; |
---|
1772 | def RL=ringlist(R); |
---|
1773 | list p=RL[1][2]; |
---|
1774 | string param=string(p[1]); |
---|
1775 | if (size(p)>1) |
---|
1776 | { |
---|
1777 | for(i=2;i<=size(p);i++){param=string(param,",",p[i]);} |
---|
1778 | } |
---|
1779 | list v=RL[2]; |
---|
1780 | string vars=string(v[1]); |
---|
1781 | if (size(v)>1) |
---|
1782 | { |
---|
1783 | for(i=2;i<=size(v);i++){vars=string(vars,",",v[i]);} |
---|
1784 | } |
---|
1785 | list xord; |
---|
1786 | list pord; |
---|
1787 | if (RL[1][3][1][1]=="dp"){pord=string("tdeg(",param);} |
---|
1788 | if (RL[1][3][1][1]=="lp"){pord=string("plex(",param);} |
---|
1789 | if (RL[3][1][1]=="dp"){xord=string("tdeg(",vars);} |
---|
1790 | if (RL[3][1][1]=="lp"){xord=string("plex(",vars);} |
---|
1791 | write(LLa,string(T,"[[9]]:=",xord,");")); |
---|
1792 | write(LLa,string(T,"[[10]]:=",pord,");")); |
---|
1793 | write(LLa,string(T,"[[11]]:=true; ")); |
---|
1794 | list S; |
---|
1795 | for (i=1;i<=size(L);i++) |
---|
1796 | { |
---|
1797 | if (L[i][2]==0) |
---|
1798 | { |
---|
1799 | cond=L[i][8]; |
---|
1800 | S=btcond(T,L[i],cond); |
---|
1801 | write(LLa,S[1]); |
---|
1802 | write(LLa,S[2]); |
---|
1803 | } |
---|
1804 | S=btbasis(T,L[i]); |
---|
1805 | write(LLa,S); |
---|
1806 | S=btN(T,L[i]); |
---|
1807 | write(LLa,S); |
---|
1808 | S=btW(T,L[i]); |
---|
1809 | write(LLa,S); |
---|
1810 | if (L[i][2]==1) {S=btterminal(T,L[i]); write(LLa,S);} |
---|
1811 | S=btlpp(T,L[i]); |
---|
1812 | write(LLa,S); |
---|
1813 | } |
---|
1814 | close(LLa); |
---|
1815 | } |
---|
1816 | example |
---|
1817 | { "EXAMPLE:"; echo = 2; |
---|
1818 | ring R=(0,a1,a2,a3,a4),(x1,x2,x3,x4),dp; |
---|
1819 | ideal F=x4-a4+a2, |
---|
1820 | x1+x2+x3+x4-a1-a3-a4, |
---|
1821 | x1*x3*x4-a1*a3*a4, |
---|
1822 | x1*x3+x1*x4+x2*x3+x3*x4-a1*a4-a1*a3-a3*a4; |
---|
1823 | def T=cgsdrold(F,"old",0); "T="; T; |
---|
1824 | buildtreetoMaple(T,"Tb","Tb.txt"); |
---|
1825 | } |
---|
1826 | |
---|
1827 | // auxiliary routine called by buildtreetoMaple |
---|
1828 | // input: |
---|
1829 | // list L: element i of the list of buildtree(F) |
---|
1830 | // output: |
---|
1831 | // the string of T[[lab,1]]:=label; in Maple |
---|
1832 | static proc btterminal(string T, list L) |
---|
1833 | { |
---|
1834 | int i; |
---|
1835 | string Li; |
---|
1836 | string term; |
---|
1837 | string coma=","; |
---|
1838 | if (L[2]==0){term="false";} else {term="true";} |
---|
1839 | def lab=L[1]; |
---|
1840 | string slab; |
---|
1841 | if ((size(lab)==1) and lab[1]==-1) |
---|
1842 | {slab="";coma="";} //if (size(lab)==0) |
---|
1843 | else |
---|
1844 | { |
---|
1845 | slab=string(lab[1]); |
---|
1846 | if (size(lab)>=1) |
---|
1847 | { |
---|
1848 | for (i=2;i<=size(lab);i++){slab=string(slab,",",lab[i]);} |
---|
1849 | } |
---|
1850 | } |
---|
1851 | Li=string(T,"[[",slab,coma,"6]]:=",term,"; "); |
---|
1852 | return(Li); |
---|
1853 | } |
---|
1854 | |
---|
1855 | // auxiliary routine called by buildtreetoMaple |
---|
1856 | // input: |
---|
1857 | // list L: element i of the list of buildtree(F) |
---|
1858 | // output: |
---|
1859 | // the string of T[[lab,3]] (basis); in Maple |
---|
1860 | static proc btbasis(string T, list L) |
---|
1861 | { |
---|
1862 | int i; |
---|
1863 | string Li; |
---|
1864 | string coma=","; |
---|
1865 | def lab=L[1]; |
---|
1866 | string slab; |
---|
1867 | if ((size(lab)==1) and lab[1]==-1) |
---|
1868 | {slab="";coma="";} //if (size(lab)==0) |
---|
1869 | else |
---|
1870 | { |
---|
1871 | slab=string(lab[1]); |
---|
1872 | if (size(lab)>=1) |
---|
1873 | { |
---|
1874 | for (i=2;i<=size(lab);i++){slab=string(slab,",",lab[i]);} |
---|
1875 | } |
---|
1876 | } |
---|
1877 | Li=string(T,"[[",slab,coma,"3]]:=[",L[3],"]; "); |
---|
1878 | return(Li); |
---|
1879 | } |
---|
1880 | |
---|
1881 | // auxiliary routine called by buildtreetoMaple |
---|
1882 | // input: |
---|
1883 | // list L: element i of the list of buildtree(F) |
---|
1884 | // output: |
---|
1885 | // the string of T[[lab,4]] (null conditions ideal); in Maple |
---|
1886 | static proc btN(string T, list L) |
---|
1887 | { |
---|
1888 | int i; |
---|
1889 | string Li; |
---|
1890 | string coma=","; |
---|
1891 | def lab=L[1]; |
---|
1892 | string slab; |
---|
1893 | if ((size(lab)==1) and lab[1]==-1) |
---|
1894 | {slab=""; coma="";} |
---|
1895 | else |
---|
1896 | { |
---|
1897 | slab=string(lab[1]); |
---|
1898 | if (size(lab)>=1) |
---|
1899 | { |
---|
1900 | for (i=2;i<=size(lab);i++){slab=string(slab,",",lab[i]);} |
---|
1901 | } |
---|
1902 | } |
---|
1903 | if ((size(lab)==1) and lab[1]==-1) |
---|
1904 | {Li=string(T,"[[",slab,coma,"4]]:=[ ]; ");} |
---|
1905 | else |
---|
1906 | {Li=string(T,"[[",slab,coma,"4]]:=[",L[4],"]; ");} |
---|
1907 | return(Li); |
---|
1908 | } |
---|
1909 | |
---|
1910 | // auxiliary routine called by buildtreetoMaple |
---|
1911 | // input: |
---|
1912 | // list L: element i of the list of buildtree(F) |
---|
1913 | // output: |
---|
1914 | // the string of T[[lab,5]] (null conditions ideal); in Maple |
---|
1915 | static proc btW(string T, list L) |
---|
1916 | { |
---|
1917 | int i; |
---|
1918 | string Li; |
---|
1919 | string coma=","; |
---|
1920 | def lab=L[1]; |
---|
1921 | string slab; |
---|
1922 | if ((size(lab)==1) and lab[1]==-1) |
---|
1923 | {slab=""; coma="";} |
---|
1924 | else |
---|
1925 | { |
---|
1926 | slab=string(lab[1]); |
---|
1927 | if (size(lab)>=1) |
---|
1928 | { |
---|
1929 | for (i=2;i<=size(lab);i++){slab=string(slab,",",lab[i]);} |
---|
1930 | } |
---|
1931 | } |
---|
1932 | if (size(L[5])==0) |
---|
1933 | {Li=string(T,"[[",slab,coma,"5]]:={ }; ");} |
---|
1934 | else |
---|
1935 | {Li=string(T,"[[",slab,coma,"5]]:={",L[5],"}; ");} |
---|
1936 | return(Li); |
---|
1937 | } |
---|
1938 | |
---|
1939 | // auxiliary routine called by buildtreetoMaple |
---|
1940 | // input: |
---|
1941 | // list L: element i of the list of buildtree(F) |
---|
1942 | // output: |
---|
1943 | // the string of T[[lab,12]] (lpp); in Maple |
---|
1944 | static proc btlpp(string T, list L) |
---|
1945 | { |
---|
1946 | int i; |
---|
1947 | string Li; |
---|
1948 | string coma=",";; |
---|
1949 | def lab=L[1]; |
---|
1950 | string slab; |
---|
1951 | if ((size(lab)==1) and lab[1]==-1) |
---|
1952 | {slab=""; coma="";} |
---|
1953 | else |
---|
1954 | { |
---|
1955 | slab=string(lab[1]); |
---|
1956 | if (size(lab)>=1) |
---|
1957 | { |
---|
1958 | for (i=2;i<=size(lab);i++){slab=string(slab,",",lab[i]);} |
---|
1959 | } |
---|
1960 | } |
---|
1961 | if (size(L[7])==0) |
---|
1962 | { |
---|
1963 | Li=string(T,"[[",slab,coma,"12]]:=[ ]; "); |
---|
1964 | } |
---|
1965 | else |
---|
1966 | { |
---|
1967 | Li=string(T,"[[",slab,coma,"12]]:=[",L[7],"]; "); |
---|
1968 | } |
---|
1969 | return(Li); |
---|
1970 | } |
---|
1971 | |
---|
1972 | // auxiliary routine called by buildtreetoMaple |
---|
1973 | // input: |
---|
1974 | // list L: element i of the list of buildtree(F) |
---|
1975 | // output: |
---|
1976 | // the list of strings of (T[[lab,0]]=0,T[[lab,1]]<>0); in Maple |
---|
1977 | static proc btcond(string T, list L, poly cond) |
---|
1978 | { |
---|
1979 | int i; |
---|
1980 | string Li1; |
---|
1981 | string Li2; |
---|
1982 | def lab=L[1]; |
---|
1983 | string slab; |
---|
1984 | string coma=",";; |
---|
1985 | if ((size(lab)==1) and lab[1]==-1) |
---|
1986 | {slab=""; coma="";} |
---|
1987 | else |
---|
1988 | { |
---|
1989 | slab=string(lab[1]); |
---|
1990 | if (size(lab)>=1) |
---|
1991 | { |
---|
1992 | for (i=2;i<=size(lab);i++){slab=string(slab,",",lab[i]);} |
---|
1993 | } |
---|
1994 | } |
---|
1995 | Li1=string(T,"[[",slab+coma,"0]]:=",L[8],"=0; "); |
---|
1996 | Li2=string(T,"[[",slab+coma,"1]]:=",L[8],"<>0; "); |
---|
1997 | return(list(Li1,Li2)); |
---|
1998 | } |
---|
1999 | |
---|
2000 | //*****************End of BuildtreetoMaple********************* |
---|
2001 | |
---|
2002 | //*****************Begin of Selectcases************************ |
---|
2003 | |
---|
2004 | // given an intvec with sum=n |
---|
2005 | // it returns the list of intvect with the sum=n+1 |
---|
2006 | static proc comp1(intvec l) |
---|
2007 | { |
---|
2008 | list L; |
---|
2009 | int p=size(l); |
---|
2010 | int i; |
---|
2011 | if (p==0){return(l);} |
---|
2012 | if (p==1){return(list(intvec(l[1]+1)));} |
---|
2013 | L[1]=intvec((l[1]+1),l[2..p]); |
---|
2014 | L[p]=intvec(l[1..p-1],(l[p]+1)); |
---|
2015 | for (i=2;i<p;i++) |
---|
2016 | { |
---|
2017 | L[i]=intvec(l[1..(i-1)],(l[i]+1),l[(i+1)..p]); |
---|
2018 | } |
---|
2019 | return(L); |
---|
2020 | } |
---|
2021 | |
---|
2022 | // comp: p-compositions of n |
---|
2023 | // input |
---|
2024 | // int n; |
---|
2025 | // int p; |
---|
2026 | // return |
---|
2027 | // the list of all intvec (p-composition of n) |
---|
2028 | static proc comp(int n,int p) |
---|
2029 | { |
---|
2030 | if (n<0){ERROR("comp was called with negative argument");} |
---|
2031 | if (n==0){return(list(0:p));} |
---|
2032 | int i; |
---|
2033 | int k; |
---|
2034 | list L1=comp(n-1,p); |
---|
2035 | list L=comp1(L1[1]); |
---|
2036 | list l; |
---|
2037 | list la; |
---|
2038 | for (i=2; i<=size(L1);i++) |
---|
2039 | { |
---|
2040 | l=comp1(L1[i]); |
---|
2041 | for (k=1;k<=size(l);k++) |
---|
2042 | { |
---|
2043 | if(not(memberpos(l[k],L)[1])) |
---|
2044 | {L[size(L)+1]=l[k];} |
---|
2045 | } |
---|
2046 | } |
---|
2047 | return(L); |
---|
2048 | } |
---|
2049 | |
---|
2050 | // given the matrices of coefficients and monomials m amd m1 of |
---|
2051 | // two polynomials (the first one contains all the terms of f |
---|
2052 | // and the second only those of f |
---|
2053 | // it returns the list with the comon monomials and the list of coefficients |
---|
2054 | // of the polynomial f with zeroes if necessary. |
---|
2055 | static proc adaptcoef(matrix m, matrix m1) |
---|
2056 | { |
---|
2057 | int i; |
---|
2058 | int j; |
---|
2059 | int ncm=ncols(m); |
---|
2060 | int ncm1=ncols(m1); |
---|
2061 | ideal T; |
---|
2062 | for (i=1;i<=ncm;i++){T[i]=m[1,i];} |
---|
2063 | ideal C; |
---|
2064 | for (i=1;i<=ncm;i++){C[i]=0;} |
---|
2065 | for (i=1;i<=ncm;i++) |
---|
2066 | { |
---|
2067 | j=1; |
---|
2068 | while((j<ncm1) and (m1[1,j]>m[1,i])){j++;} |
---|
2069 | if (m1[1,j]==m[1,i]){C[i]=m1[2,j];} |
---|
2070 | } |
---|
2071 | return(list(T,C)); |
---|
2072 | } |
---|
2073 | |
---|
2074 | // given teh ideal of non-null conditions and an intvec lambda |
---|
2075 | // with the exponents of each w in W |
---|
2076 | // it returns the polynomial prod (w_i)^(lambda_i). |
---|
2077 | static proc WW(ideal W, intvec lambda) |
---|
2078 | { |
---|
2079 | if (size(W)==0){return(poly(1));} |
---|
2080 | poly w=1; |
---|
2081 | int i; |
---|
2082 | for (i=1;i<=ncols(W);i++) |
---|
2083 | { |
---|
2084 | w=w*(W[i])^(lambda[i]); |
---|
2085 | } |
---|
2086 | return(w); |
---|
2087 | } |
---|
2088 | |
---|
2089 | // given a polynomial f and the non-null conditions W |
---|
2090 | // WPred eliminates the factors in f that are in W |
---|
2091 | // ring @PAB |
---|
2092 | // input: |
---|
2093 | // poly f: |
---|
2094 | // ideal W of non-null conditions (already supposed that it is facvar) |
---|
2095 | // output: |
---|
2096 | // poly f2 where the non-null conditions in W have been dropped from f |
---|
2097 | static proc WPred(poly f, ideal W) |
---|
2098 | { |
---|
2099 | if (f==0){return(f);} |
---|
2100 | def l=factorize(f,2); |
---|
2101 | int i; |
---|
2102 | poly f1=1; |
---|
2103 | for(i=1;i<=size(l[1]);i++) |
---|
2104 | { |
---|
2105 | if (memberpos(l[1][i],W)[1]){;} |
---|
2106 | else{f1=f1*((l[1][i])^(l[2][i]));} |
---|
2107 | } |
---|
2108 | return(f1); |
---|
2109 | } |
---|
2110 | |
---|
2111 | //genimage |
---|
2112 | // ring @R |
---|
2113 | //input: |
---|
2114 | // poly f1, idel N1,ideal W1,poly f2, ideal N2, ideal W2 |
---|
2115 | // corresponding to two polynomials having the same lpp |
---|
2116 | // f1 in the redspec given by N1,W1, f2 in the redspec given by N2,W2 |
---|
2117 | //output: |
---|
2118 | // the list of (ideal GG, list(list r1, list r2)) |
---|
2119 | // where g an ideal whose elements have the same lpp as f1 and f2 |
---|
2120 | // that specialize well to f1 in N1,W1 and to f2 in N2,W2. |
---|
2121 | // If it doesn't exist a genimage, then g=ideal(0). |
---|
2122 | static proc genimage(poly f1, ideal N1, ideal W1, poly f2, ideal N2, ideal W2) |
---|
2123 | { |
---|
2124 | int i; ideal W12; poly ff1; poly g1=0; ideal GG; |
---|
2125 | int tt=1; |
---|
2126 | // detect weather f1 reduces to 0 on segment 2 |
---|
2127 | ff1=pnormalform(f1,N2,W2); |
---|
2128 | if (ff1==0) |
---|
2129 | { |
---|
2130 | // detect weather N1 is included in N2 |
---|
2131 | def RR=basering; |
---|
2132 | setring @P; |
---|
2133 | def NP1=imap(RR,N1); |
---|
2134 | def NP2=imap(RR,N2); |
---|
2135 | attrib(NP2,"isSB",1); |
---|
2136 | poly nr; |
---|
2137 | i=1; |
---|
2138 | while ((tt) and (i<=size(NP1))) |
---|
2139 | { |
---|
2140 | nr=reduce(NP1[i],NP2); |
---|
2141 | if (nr!=0){tt=0;} |
---|
2142 | i++; |
---|
2143 | } |
---|
2144 | setring(RR); |
---|
2145 | } |
---|
2146 | else{tt=0;} |
---|
2147 | if (tt==1) |
---|
2148 | { |
---|
2149 | // detect weather W1 intersect W2 is non-empty |
---|
2150 | for (i=1;i<=size(W1);i++) |
---|
2151 | { |
---|
2152 | if (memberpos(W1[i],W2)[1]) |
---|
2153 | { |
---|
2154 | W12[size(W12)+1]=W1[i]; |
---|
2155 | } |
---|
2156 | else |
---|
2157 | { |
---|
2158 | if (nonnull(W1[i],N2,W2)) |
---|
2159 | { |
---|
2160 | W12[size(W12)+1]=W1[i]; |
---|
2161 | } |
---|
2162 | } |
---|
2163 | } |
---|
2164 | for (i=1;i<=size(W2);i++) |
---|
2165 | { |
---|
2166 | if (not(memberpos(W2[i],W12)[1])) |
---|
2167 | { |
---|
2168 | W12[size(W12)+1]=W2[i]; |
---|
2169 | } |
---|
2170 | } |
---|
2171 | } |
---|
2172 | if (tt==1){g1=extendpoly(f1,N1,W12);} |
---|
2173 | if (g1!=0) |
---|
2174 | { |
---|
2175 | if (pnormalform(g1,N1,W1)==0) |
---|
2176 | { |
---|
2177 | GG=f1,g1; |
---|
2178 | } |
---|
2179 | else |
---|
2180 | { |
---|
2181 | GG=g1; |
---|
2182 | } |
---|
2183 | return(GG); |
---|
2184 | } |
---|
2185 | |
---|
2186 | // begins the second step; |
---|
2187 | int bound=6; |
---|
2188 | // in ring @R |
---|
2189 | int j; int g=0; int alpha; int r1; int s1=1; int s2=1; |
---|
2190 | poly G; |
---|
2191 | matrix qT; |
---|
2192 | matrix T; |
---|
2193 | ideal N10; |
---|
2194 | poly GT; |
---|
2195 | ideal N12=N1,N2; |
---|
2196 | def varx=maxideal(1); |
---|
2197 | int nx=size(varx); |
---|
2198 | poly pvarx=1; |
---|
2199 | for (i=1;i<=nx;i++){pvarx=pvarx*varx[i];} |
---|
2200 | def m=coef(43*f1+157*f2,pvarx); |
---|
2201 | def m1=coef(f1,pvarx); |
---|
2202 | def m2=coef(f2,pvarx); |
---|
2203 | list L1=adaptcoef(m,m1); |
---|
2204 | list L2=adaptcoef(m,m2); |
---|
2205 | ideal Tm=L1[1]; |
---|
2206 | ideal c1=L1[2]; |
---|
2207 | ideal c2=L2[2]; |
---|
2208 | poly ww1; |
---|
2209 | poly ww2; |
---|
2210 | poly cA1; |
---|
2211 | poly cB1; |
---|
2212 | matrix TT; |
---|
2213 | poly H; |
---|
2214 | list r; |
---|
2215 | ideal q; |
---|
2216 | poly mu; |
---|
2217 | ideal N; |
---|
2218 | |
---|
2219 | // in ring @PAB |
---|
2220 | list Px=ringlist(@P); |
---|
2221 | list v="@A","@B"; |
---|
2222 | Px[2]=Px[2]+v; |
---|
2223 | def npx=size(Px[3][1][2]); |
---|
2224 | Px[3][1][2]=1:(npx+size(v)); |
---|
2225 | def @PAB=ring(Px); |
---|
2226 | setring(@PAB); |
---|
2227 | |
---|
2228 | poly PH; |
---|
2229 | ideal NP; |
---|
2230 | list rP; |
---|
2231 | def PN1=imap(@R,N1); |
---|
2232 | def PW1=imap(@R,W1); |
---|
2233 | def PN2=imap(@R,N2); |
---|
2234 | def PW2=imap(@R,W2); |
---|
2235 | def a1=imap(@R,c1); |
---|
2236 | def a2=imap(@R,c2); |
---|
2237 | matrix PT; |
---|
2238 | ideal PN; |
---|
2239 | ideal PN12=PN1,PN2; |
---|
2240 | PN=liftstd(PN12,PT); |
---|
2241 | list compos1; |
---|
2242 | list compos2; |
---|
2243 | list compos0; |
---|
2244 | intvec comp0; |
---|
2245 | poly w1=0; |
---|
2246 | poly w2=0; |
---|
2247 | poly h; |
---|
2248 | poly cA=0; |
---|
2249 | poly cB=0; |
---|
2250 | int t=0; |
---|
2251 | list l; |
---|
2252 | poly h1; |
---|
2253 | g=0; |
---|
2254 | while ((g<=bound) and not(t)) |
---|
2255 | { |
---|
2256 | compos0=comp(g,2); |
---|
2257 | r1=1; |
---|
2258 | while ((r1<=size(compos0)) and not(t)) |
---|
2259 | { |
---|
2260 | comp0=compos0[r1]; |
---|
2261 | if (comp0[1]<=bound div 2) |
---|
2262 | { |
---|
2263 | compos1=comp(comp0[1],ncols(PW1)); |
---|
2264 | s1=1; |
---|
2265 | while ((s1<=size(compos1)) and not(t)) |
---|
2266 | { |
---|
2267 | if (comp0[2]<=bound div 2) |
---|
2268 | { |
---|
2269 | compos2=comp(comp0[2],ncols(PW2)); |
---|
2270 | s2=1; |
---|
2271 | while ((s2<=size(compos2)) and not(t)) |
---|
2272 | { |
---|
2273 | w1=WW(PW1,compos1[s1]); |
---|
2274 | w2=WW(PW2,compos2[s2]); |
---|
2275 | h=@A*w1*a1[1]-@B*w2*a2[1]; |
---|
2276 | h=reduce(h,PN); |
---|
2277 | if (h==0){cA=1;cB=-1;} |
---|
2278 | else |
---|
2279 | { |
---|
2280 | l=factorize(h,2); |
---|
2281 | h1=1; |
---|
2282 | for(i=1;i<=size(l[1]);i++) |
---|
2283 | { |
---|
2284 | if ((memberpos(@A,variables(l[1][i]))[1]) or (memberpos(@B,variables(l[1][i]))[1])) |
---|
2285 | {h1=h1*l[1][i];} |
---|
2286 | } |
---|
2287 | cA=diff(h1,@B); |
---|
2288 | cB=diff(h1,@A); |
---|
2289 | } |
---|
2290 | if ((cA!=0) and (cB!=0) and (jet(cA,0)==cA) and (jet(cB,0)==cB)) |
---|
2291 | { |
---|
2292 | t=1; |
---|
2293 | alpha=1; |
---|
2294 | while((t) and (alpha<=ncols(a1))) |
---|
2295 | { |
---|
2296 | h=cA*w1*a1[alpha]+cB*w2*a2[alpha]; |
---|
2297 | if (not(reduce(h,PN,1)==0)){t=0;} |
---|
2298 | alpha++; |
---|
2299 | } |
---|
2300 | } |
---|
2301 | else{t=0;} |
---|
2302 | s2++; |
---|
2303 | } |
---|
2304 | } |
---|
2305 | s1++; |
---|
2306 | } |
---|
2307 | } |
---|
2308 | r1++; |
---|
2309 | } |
---|
2310 | g++; |
---|
2311 | } |
---|
2312 | setring(@R); |
---|
2313 | ww1=imap(@PAB,w1); |
---|
2314 | ww2=imap(@PAB,w2); |
---|
2315 | T=imap(@PAB,PT); |
---|
2316 | N=imap(@PAB,PN); |
---|
2317 | cA1=imap(@PAB,cA); |
---|
2318 | cB1=imap(@PAB,cB); |
---|
2319 | if (t) |
---|
2320 | { |
---|
2321 | G=0; |
---|
2322 | for (alpha=1;alpha<=ncols(Tm);alpha++) |
---|
2323 | { |
---|
2324 | H=cA1*ww1*c1[alpha]+cB1*ww2*c2[alpha]; |
---|
2325 | setring(@PAB); |
---|
2326 | PH=imap(@R,H); |
---|
2327 | PN=imap(@R,N); |
---|
2328 | rP=division(PH,PN); |
---|
2329 | setring(@R); |
---|
2330 | r=imap(@PAB,rP); |
---|
2331 | if (r[2][1]!=0){ERROR("the division is not null and it should be");} |
---|
2332 | q=r[1]; |
---|
2333 | qT=transpose(matrix(q)); |
---|
2334 | N10=N12; |
---|
2335 | for (i=size(N1)+1;i<=size(N1)+size(N2);i++){N10[i]=0;} |
---|
2336 | G=G+(cA1*ww1*c1[alpha]-(matrix(N10)*T*qT)[1,1])*Tm[alpha]; |
---|
2337 | } |
---|
2338 | GG=ideal(G); |
---|
2339 | } |
---|
2340 | else{GG=ideal(0);} |
---|
2341 | return(GG); |
---|
2342 | } |
---|
2343 | |
---|
2344 | // purpose: given a polynomial f (in the reduced basis) |
---|
2345 | // the null-conditions ideal N in the segment |
---|
2346 | // end the set of non-null polynomials common to the segment and |
---|
2347 | // a new segment, |
---|
2348 | // to obtain an equivalent polynomial with a leading coefficient |
---|
2349 | // that is non-null in the second segment. |
---|
2350 | // input: |
---|
2351 | // poly f: a polynomials of the reduced basis in the segment (N,W) |
---|
2352 | // ideal N: the null-conditions ideal in the segment |
---|
2353 | // ideal W12: the set of non-null polynomials common to the segment and |
---|
2354 | // a second segment |
---|
2355 | static proc extendpoly(poly f, ideal N, ideal W12) |
---|
2356 | { |
---|
2357 | int bound=4; |
---|
2358 | ideal cfs; |
---|
2359 | ideal cfsn; |
---|
2360 | ideal ppfs; |
---|
2361 | poly p=f; |
---|
2362 | poly fn; |
---|
2363 | poly lm; poly lc; |
---|
2364 | int tt=0; |
---|
2365 | int i; |
---|
2366 | while (p!=0) |
---|
2367 | { |
---|
2368 | lm=leadmonom(p); |
---|
2369 | lc=leadcoef(p); |
---|
2370 | cfs[size(cfs)+1]=lc; |
---|
2371 | ppfs[size(ppfs)+1]=lm; |
---|
2372 | p=p-lc*lm; |
---|
2373 | } |
---|
2374 | def lcf=cfs[1]; |
---|
2375 | int r1=0; int s1; |
---|
2376 | def RR=basering; |
---|
2377 | setring @P; |
---|
2378 | list compos1; |
---|
2379 | poly w1; |
---|
2380 | ideal q; |
---|
2381 | def lcfp=imap(RR,lcf); |
---|
2382 | def W=imap(RR,W12); |
---|
2383 | def Np=imap(RR,N); |
---|
2384 | def cfsp=imap(RR,cfs); |
---|
2385 | ideal cfspn; |
---|
2386 | matrix T; |
---|
2387 | ideal H=lcfp,Np; |
---|
2388 | def G=liftstd(H,T); |
---|
2389 | list r; |
---|
2390 | while ((r1<=bound) and not(tt)) |
---|
2391 | { |
---|
2392 | compos1=comp(r1,ncols(W)); |
---|
2393 | s1=1; |
---|
2394 | while ((s1<=size(compos1)) and not(tt)) |
---|
2395 | { |
---|
2396 | w1=WW(W,compos1[s1]); |
---|
2397 | cfspn=ideal(0); |
---|
2398 | cfspn[1]=w1; |
---|
2399 | tt=1; |
---|
2400 | i=2; |
---|
2401 | while ((i<=size(cfsp)) and (tt)) |
---|
2402 | { |
---|
2403 | r=division(w1*cfsp[i],G); |
---|
2404 | if (r[2][1]!=0){tt=0;} |
---|
2405 | else |
---|
2406 | { |
---|
2407 | q=r[1]; |
---|
2408 | cfspn[i]=(T*transpose(matrix(q)))[1,1]; |
---|
2409 | } |
---|
2410 | i++; |
---|
2411 | } |
---|
2412 | s1++; |
---|
2413 | } |
---|
2414 | r1++; |
---|
2415 | } |
---|
2416 | setring RR; |
---|
2417 | if (tt) |
---|
2418 | { |
---|
2419 | cfsn=imap(@P,cfspn); |
---|
2420 | fn=0; |
---|
2421 | for (i=1;i<=size(ppfs);i++) |
---|
2422 | { |
---|
2423 | fn=fn+cfsn[i]*ppfs[i]; |
---|
2424 | } |
---|
2425 | } |
---|
2426 | else{fn=0;} |
---|
2427 | return(fn); |
---|
2428 | } |
---|
2429 | |
---|
2430 | // nonnull |
---|
2431 | // ring @P (or @R) |
---|
2432 | // input: |
---|
2433 | // poly f |
---|
2434 | // ideal N |
---|
2435 | // ideal W |
---|
2436 | // output: |
---|
2437 | // 1 if f is nonnull in the segment (N,W) |
---|
2438 | // 0 if it can be zero |
---|
2439 | static proc nonnull(poly f, ideal N, ideal W) |
---|
2440 | { |
---|
2441 | int tt; |
---|
2442 | ideal N0=N; |
---|
2443 | N0[size(N0)+1]=f; |
---|
2444 | poly h=1; |
---|
2445 | int i; |
---|
2446 | for (i=1;i<=size(W);i++){h=h*W[i];} |
---|
2447 | def RR=basering; |
---|
2448 | setring(@P); |
---|
2449 | list Px=ringlist(@P); |
---|
2450 | list v="@C"; |
---|
2451 | Px[2]=Px[2]+v; |
---|
2452 | def npx=size(Px[3][1][2]); |
---|
2453 | Px[3][1][1]="dp"; |
---|
2454 | Px[3][1][2]=1:(npx+size(v)); |
---|
2455 | def @PC=ring(Px); |
---|
2456 | setring(@PC); |
---|
2457 | def N1=imap(RR,N0); |
---|
2458 | def h1=imap(RR,h); |
---|
2459 | ideal G=1-@C*h1; |
---|
2460 | G=G+N1; |
---|
2461 | option(redSB); |
---|
2462 | ideal G1=std(G); |
---|
2463 | if (G1[1]==1){tt=1;} else{tt=0;} |
---|
2464 | setring(RR); |
---|
2465 | return(tt); |
---|
2466 | } |
---|
2467 | |
---|
2468 | // decide |
---|
2469 | // input: |
---|
2470 | // given two corresponding polynomials g1 and g2 with the same lpp |
---|
2471 | // g1 belonging to the basis in the segment N1,W1 |
---|
2472 | // g2 belonging to the basis in the segment N2,W2 |
---|
2473 | // output: |
---|
2474 | // an ideal (with a single polynomial or more if a sheaf is needed) |
---|
2475 | // that specializes well on both segments to g1 and g2 respectivelly. |
---|
2476 | // If ideal(0) is output, then no such polynomial nor sheaf exists. |
---|
2477 | static proc decide(poly g1, ideal N1, ideal W1, poly g2, ideal N2, ideal W2) |
---|
2478 | { |
---|
2479 | poly S; |
---|
2480 | poly S1; |
---|
2481 | poly S2; |
---|
2482 | S=leadcoef(g2)*g1-leadcoef(g1)*g2; |
---|
2483 | def RR=basering; |
---|
2484 | setring(@RP); |
---|
2485 | def SR=imap(RR,S); |
---|
2486 | def N1R=imap(RR,N1); |
---|
2487 | def N2R=imap(RR,N2); |
---|
2488 | attrib(N1R,"isSB",1); |
---|
2489 | attrib(N2R,"isSB",1); |
---|
2490 | poly S1R=reduce(SR,N1R); |
---|
2491 | poly S2R=reduce(SR,N2R); |
---|
2492 | setring(RR); |
---|
2493 | S1=imap(@RP,S1R); |
---|
2494 | S2=imap(@RP,S2R); |
---|
2495 | if ((S2==0) and (nonnull(leadcoef(g1),N2,W2))){return(ideal(g1));} |
---|
2496 | if ((S1==0) and (nonnull(leadcoef(g2),N1,W1))){return(ideal(g2));} |
---|
2497 | if ((S1==0) and (S2==0)) |
---|
2498 | { |
---|
2499 | return(ideal(g1,g2)); |
---|
2500 | } |
---|
2501 | return(ideal(genimage(g1,N1,W1,g2,N2,W2))); |
---|
2502 | } |
---|
2503 | |
---|
2504 | // input: the tree (list) from buildtree output |
---|
2505 | // output: the list of terminal vertices. |
---|
2506 | static proc finalcases(list T) |
---|
2507 | //"USAGE: finalcases(T); |
---|
2508 | // T is the list provided by buildtree |
---|
2509 | //RETURN: A list with the CGS determined by buildtree. |
---|
2510 | // Each element of the list represents one segment |
---|
2511 | // of the terminal vertices of buildtree givieng the CGS. |
---|
2512 | // The list elements have the following structure: |
---|
2513 | // [1]: label (an intvec(1,0,..)) that indicates the position |
---|
2514 | // in the buildtree but that is irrelevant for the CGS |
---|
2515 | // [2]: 1 (integer) it is also irrelevant and indicates |
---|
2516 | // that this was a terminal vertex in buildtree. |
---|
2517 | // [3]: the reduced basis of the segment. |
---|
2518 | // [4], [5], [6]: the red-representation of the segment |
---|
2519 | // [4] are the null-conditions radical ideal N, |
---|
2520 | // [5] are the non-null polynomials set (ideal) W, |
---|
2521 | // [6] is the set of prime components (ideals) of N. |
---|
2522 | // [7]: is the set of lpp |
---|
2523 | // [8]: poly 1 (irrelevant) is the condition to branch (but no |
---|
2524 | // more branch is necessary in the discussion, so 1 is the result. |
---|
2525 | //NOTE: It can be called having as argument the list output by buildtree |
---|
2526 | //KEYWORDS: buildtree, buildtreetoMaple, CGS |
---|
2527 | //EXAMPLE: finalcases; shows an example" |
---|
2528 | { |
---|
2529 | int i; |
---|
2530 | list L; |
---|
2531 | for (i=1;i<=size(T);i++) |
---|
2532 | { |
---|
2533 | if (T[i][2]) |
---|
2534 | {L[size(L)+1]=T[i];} |
---|
2535 | } |
---|
2536 | return(L); |
---|
2537 | } |
---|
2538 | //example |
---|
2539 | //{ "EXAMPLE:"; echo = 2; |
---|
2540 | // ring R=(0,a1,a2,a3,a4),(x1,x2,x3,x4),dp; |
---|
2541 | // ideal F=x4-a4+a2, x1+x2+x3+x4-a1-a3-a4, x1*x3*x4-a1*a3*a4, x1*x3+x1*x4+x2*x3+x3*x4-a1*a4-a1*a3-a3*a4; |
---|
2542 | // def T=buildtree(F); |
---|
2543 | // setglobalrings(); |
---|
2544 | // finalcases(T); |
---|
2545 | //} |
---|
2546 | |
---|
2547 | // input: the list of terminal vertices of buildtree (output of finalcases) |
---|
2548 | // output: the same terminal vertices grouped by lpp |
---|
2549 | static proc groupsegments(list T) |
---|
2550 | { |
---|
2551 | int i; |
---|
2552 | list L; |
---|
2553 | list lpp; |
---|
2554 | list lp; |
---|
2555 | list ls; |
---|
2556 | int n=size(T); |
---|
2557 | lpp[1]=T[n][7]; |
---|
2558 | L[1]=list(lpp[1],list(list(T[n][1],T[n][3],T[n][4],T[n][5],T[n][6]))); |
---|
2559 | if (n>1) |
---|
2560 | { |
---|
2561 | for (i=1;i<=size(T)-1;i++) |
---|
2562 | { |
---|
2563 | lp=memberpos(T[n-i][7],lpp); |
---|
2564 | if(lp[1]==1) |
---|
2565 | { |
---|
2566 | ls=L[lp[2]][2]; |
---|
2567 | ls[size(ls)+1]=list(T[n-i][1],T[n-i][3],T[n-i][4],T[n-i][5],T[n-i][6]); |
---|
2568 | L[lp[2]][2]=ls; |
---|
2569 | } |
---|
2570 | else |
---|
2571 | { |
---|
2572 | lpp[size(lpp)+1]=T[n-i][7]; |
---|
2573 | L[size(L)+1]=list(T[n-i][7],list(list(T[n-i][1],T[n-i][3],T[n-i][4],T[n-i][5],T[n-i][6]))); |
---|
2574 | } |
---|
2575 | } |
---|
2576 | } |
---|
2577 | //"L in groupsegments="; L; |
---|
2578 | return(L); |
---|
2579 | } |
---|
2580 | |
---|
2581 | // eliminates repeated elements form an ideal |
---|
2582 | static proc elimrepeated(ideal F) |
---|
2583 | { |
---|
2584 | int i; |
---|
2585 | int j; |
---|
2586 | ideal FF; |
---|
2587 | FF[1]=F[1]; |
---|
2588 | for (i=2;i<=ncols(F);i++;) |
---|
2589 | { |
---|
2590 | if (not(memberpos(F[i],FF)[1])) |
---|
2591 | { |
---|
2592 | FF[size(FF)+1]=F[i]; |
---|
2593 | } |
---|
2594 | } |
---|
2595 | return(FF); |
---|
2596 | } |
---|
2597 | |
---|
2598 | // decide F is the same as decide but allows as first element a sheaf F |
---|
2599 | static proc decideF(ideal F,ideal N,ideal W, poly f2, ideal N2, ideal W2) |
---|
2600 | { |
---|
2601 | int i; |
---|
2602 | ideal G=F; |
---|
2603 | ideal g; |
---|
2604 | if (ncols(F)==1) {return(decide(F[1],N,W,f2,N2,W2));} |
---|
2605 | for (i=1;i<=ncols(F);i++) |
---|
2606 | { |
---|
2607 | G=G+decide(F[i],N,W,f2,N2,W2); |
---|
2608 | } |
---|
2609 | return(elimrepeated(G)); |
---|
2610 | } |
---|
2611 | |
---|
2612 | // newredspec |
---|
2613 | // input: two redspec in the form of N,W and Nj,Wj |
---|
2614 | // output: a redspec representing the minimal redspec segment that contains |
---|
2615 | // both input segments. |
---|
2616 | static proc newredspec(ideal N,ideal W, ideal Nj, ideal Wj) |
---|
2617 | { |
---|
2618 | ideal nN; |
---|
2619 | ideal nW; |
---|
2620 | int u; |
---|
2621 | def RR=basering; |
---|
2622 | setring(@P); |
---|
2623 | list r; |
---|
2624 | def Np=imap(RR,N); |
---|
2625 | def Wp=imap(RR,W); |
---|
2626 | def Njp=imap(RR,Nj); |
---|
2627 | def Wjp=imap(RR,Wj); |
---|
2628 | Np=intersect(Np,Njp); |
---|
2629 | ideal WR; |
---|
2630 | for(u=1;u<=size(Wjp);u++) |
---|
2631 | { |
---|
2632 | if(nonnull(Wjp[u],Np,Wp)){WR[size(WR)+1]=Wjp[u];} |
---|
2633 | } |
---|
2634 | for(u=1;u<=size(Wp);u++) |
---|
2635 | { |
---|
2636 | if((not(memberpos(Wp[u],WR)[1])) and (nonnull(Wp[u],Njp,Wjp))) |
---|
2637 | { |
---|
2638 | WR[size(WR)+1]=Wp[u]; |
---|
2639 | } |
---|
2640 | } |
---|
2641 | r=redspec(Np,WR); |
---|
2642 | option(redSB); |
---|
2643 | Np=std(r[1]); |
---|
2644 | Wp=r[2]; |
---|
2645 | setring(RR); |
---|
2646 | nN=imap(@P,Np); |
---|
2647 | nW=imap(@P,Wp); |
---|
2648 | return(list(nN,nW)); |
---|
2649 | } |
---|
2650 | |
---|
2651 | // selectcases |
---|
2652 | // input: |
---|
2653 | // list bT: the list output by buildtree. |
---|
2654 | // output: |
---|
2655 | // list L it contins the list of segments allowing a common |
---|
2656 | // reduced basis. The elements of L are of the form |
---|
2657 | // list (lpp,B,list(list(N,W,L),..list(N,W,L)) ) |
---|
2658 | static proc selectcases(list bT) |
---|
2659 | { |
---|
2660 | list T=groupsegments(finalcases(bT)); |
---|
2661 | //NEW |
---|
2662 | //groupredtocan(T); |
---|
2663 | list T0=bT[1]; |
---|
2664 | // first element of the list of buildtree |
---|
2665 | list TT0; |
---|
2666 | TT0[1]=list(T0[7],T0[3],list(list(T0[4],T0[5],T0[6]))); |
---|
2667 | // first element of the output of selectcases |
---|
2668 | list T1=T; // the initial list; it is only actualized (split) |
---|
2669 | // when a segment is completly revised (all split are |
---|
2670 | // already be considered); |
---|
2671 | // ( (lpp, ((lab,B,N,W,L),.. ()) ), .. (..) ) |
---|
2672 | list TT; // the output list ( (lpp,B,((N,W,L),..()) ),.. (..) ) |
---|
2673 | // case i |
---|
2674 | list S1; // the segments in case i T1[i][2]; ( (lab,B,N,W,L),..() ) |
---|
2675 | list S2; // the segments in case i that are being summarized in |
---|
2676 | // actual segment ( (N,W,L),..() ) |
---|
2677 | list S3; // the segments in case i that cannot be summarized in |
---|
2678 | // the actual case. When the case is finished a new case |
---|
2679 | // is created with them ( (lab,B,N,W,L),..() ) |
---|
2680 | list s3; // list of integers s whose segment cannot be summarized |
---|
2681 | // in the actual case |
---|
2682 | ideal lpp; // the summarized lpp (can contain repetitions) |
---|
2683 | ideal lppi;// in process of sumarizing lpp (can contain repetitions) |
---|
2684 | ideal B; // the summarized B (can contain polynomials with |
---|
2685 | // the same lpp (sheaves)) |
---|
2686 | ideal Bi; // in process of summarizing B (can contain polynomials with |
---|
2687 | // the same lpp (sheaves)) |
---|
2688 | ideal N; // the summarized N |
---|
2689 | ideal W; // the summarized W |
---|
2690 | ideal F; // the summarized poly j (can contain a sheaf instead of |
---|
2691 | // a single poly) |
---|
2692 | ideal FF; // the same as F but it can be ideal(0) |
---|
2693 | poly lpj; |
---|
2694 | poly fj; |
---|
2695 | ideal Nj; |
---|
2696 | ideal Wj; |
---|
2697 | ideal G; |
---|
2698 | int i; // the index of the case i in T1; |
---|
2699 | int j; // the index of the polynomial j of the basis |
---|
2700 | int s; // the index of the segment s in S1; |
---|
2701 | int u; |
---|
2702 | int tests; // true if al the polynomial in segment s have been generalized; |
---|
2703 | list r; |
---|
2704 | // initializing the new list |
---|
2705 | i=1; |
---|
2706 | while(i<=size(T1)) |
---|
2707 | { |
---|
2708 | S1=T1[i][2]; // ((lab,B,N,W,L)..) of the segments in case i |
---|
2709 | if (size(S1)==1) |
---|
2710 | { |
---|
2711 | TT[i]=list(T1[i][1],S1[1][2],list(list(S1[1][3],S1[1][4],S1[1][5]))); |
---|
2712 | } |
---|
2713 | else |
---|
2714 | { |
---|
2715 | S2=list(); |
---|
2716 | S3=list(); // ((lab,B,N,W,L)..) of the segments in case i to |
---|
2717 | // create another segment i+1 |
---|
2718 | s3=list(); |
---|
2719 | B=S1[1][2]; |
---|
2720 | Bi=ideal(0); |
---|
2721 | lpp=T1[i][1]; |
---|
2722 | j=1; |
---|
2723 | tests=1; |
---|
2724 | while (j<=size(S1[1][2])) |
---|
2725 | { // j desings the new j-th polynomial |
---|
2726 | N=S1[1][3]; |
---|
2727 | W=S1[1][4]; |
---|
2728 | F=ideal(S1[1][2][j]); |
---|
2729 | s=2; |
---|
2730 | while (s<=size(S1) and not(memberpos(s,s3)[1])) |
---|
2731 | { // s desings the new segment s |
---|
2732 | fj=S1[s][2][j]; |
---|
2733 | Nj=S1[s][3]; |
---|
2734 | Wj=S1[s][4]; |
---|
2735 | FF=decideF(F,N,W,fj,Nj,Wj); |
---|
2736 | if (FF[1]==0) |
---|
2737 | { |
---|
2738 | if (@ish) |
---|
2739 | { |
---|
2740 | "Warning: Dealing with an homogeneous ideal"; |
---|
2741 | "mrcgs was not able to summarize all lpp cases into a single segment"; |
---|
2742 | "Please send a mail with your Problem to antonio.montes@upc.edu"; |
---|
2743 | "You found a counterexample of the complete success of the actual mrcgs algorithm"; |
---|
2744 | //NEW |
---|
2745 | "f1:"; F; "N1:"; N; "W1:"; W; "f2:"; fj; "N2:"; Nj; "W2:"; Wj; |
---|
2746 | } |
---|
2747 | S3[size(S3)+1]=S1[s]; |
---|
2748 | s3[size(s3)+1]=s; |
---|
2749 | tests=0; |
---|
2750 | } |
---|
2751 | else |
---|
2752 | { |
---|
2753 | F=FF; |
---|
2754 | lpj=leadmonom(fj); |
---|
2755 | r=newredspec(N,W,Nj,Wj); |
---|
2756 | N=r[1]; |
---|
2757 | W=r[2]; |
---|
2758 | } |
---|
2759 | s++; |
---|
2760 | } |
---|
2761 | if (Bi[1]==0){Bi=FF;} |
---|
2762 | else |
---|
2763 | { |
---|
2764 | Bi=Bi+FF; |
---|
2765 | } |
---|
2766 | j++; |
---|
2767 | } |
---|
2768 | if (tests) |
---|
2769 | { |
---|
2770 | B=Bi; |
---|
2771 | lpp=ideal(0); |
---|
2772 | for (u=1;u<=size(B);u++){lpp[u]=leadmonom(B[u]);} |
---|
2773 | } |
---|
2774 | for (s=1;s<=size(T1[i][2]);s++) |
---|
2775 | { |
---|
2776 | if (not(memberpos(s,s3)[1])) |
---|
2777 | { |
---|
2778 | S2[size(S2)+1]=list(S1[s][3],S1[s][4],S1[s][5]); |
---|
2779 | } |
---|
2780 | } |
---|
2781 | TT[i]=list(lpp,B,S2); |
---|
2782 | // for (s=1;s<=size(s3);s++){S1=delete(S1,s);} |
---|
2783 | T1[i][2]=S2; |
---|
2784 | if (size(S3)>0){T1=insert(T1,list(T1[i][1],S3),i);} |
---|
2785 | } |
---|
2786 | i++; |
---|
2787 | } |
---|
2788 | for (i=1;i<=size(TT);i++){TT0[i+1]=TT[i];} |
---|
2789 | return(TT0); |
---|
2790 | } |
---|
2791 | |
---|
2792 | //*****************End of Selectcases************************** |
---|
2793 | |
---|
2794 | //*****************Begin of CanTree**************************** |
---|
2795 | |
---|
2796 | // equalideals |
---|
2797 | // input: 2 ideals F and G; |
---|
2798 | // output: 1 if they are identical (the same polynomials in the same order) |
---|
2799 | // 0 else |
---|
2800 | static proc equalideals(ideal F, ideal G) |
---|
2801 | { |
---|
2802 | int i=1; int t=1; |
---|
2803 | if (size(F)!=size(G)){return(0);} |
---|
2804 | while ((i<=size(F)) and (t)) |
---|
2805 | { |
---|
2806 | if (F[i]!=G[i]){t=0;} |
---|
2807 | i++; |
---|
2808 | } |
---|
2809 | return(t); |
---|
2810 | } |
---|
2811 | |
---|
2812 | // delintvec |
---|
2813 | // input: intvec V |
---|
2814 | // int i |
---|
2815 | // output: |
---|
2816 | // intvec W (equal to V but the coordinate i is deleted |
---|
2817 | static proc delintvec(intvec V, int i) |
---|
2818 | { |
---|
2819 | int j; |
---|
2820 | intvec W; |
---|
2821 | for (j=1;j<i;j++){W[j]=V[j];} |
---|
2822 | for (j=i+1;j<=size(V);j++){W[j-1]=V[j];} |
---|
2823 | return(W); |
---|
2824 | } |
---|
2825 | |
---|
2826 | // redtocanspec |
---|
2827 | // Computes the canonical representation of a redspec (N,W,L). |
---|
2828 | // input: |
---|
2829 | // ideal N (null conditions, must be radical) |
---|
2830 | // ideal W (non-null conditions ideal) |
---|
2831 | // list L must contain the radical decomposition of N. |
---|
2832 | // output: |
---|
2833 | // the list of elements of the (ideal N1,list(ideal M11,..,ideal M1k)) |
---|
2834 | // determining the canonical representation of the difference of |
---|
2835 | // V(N) \ V(h), where h=prod(w in W). |
---|
2836 | static proc redtocanspec(intvec lab, int child, list rs) |
---|
2837 | { |
---|
2838 | ideal N=rs[1]; ideal W=rs[2]; list L=rs[3]; |
---|
2839 | intvec labi; intvec labij; |
---|
2840 | int childi; |
---|
2841 | int i; int j; list L0; |
---|
2842 | L0[1]=list(lab,size(L)); |
---|
2843 | if (W[1]==0) |
---|
2844 | { |
---|
2845 | for (i=1;i<=size(L);i++) |
---|
2846 | { |
---|
2847 | labi=lab,child+i; |
---|
2848 | L0[size(L0)+1]=list(labi,1,L[i]); |
---|
2849 | labij=labi,1; |
---|
2850 | L0[size(L0)+1]=list(labij,0,ideal(1)); |
---|
2851 | } |
---|
2852 | return(L0); |
---|
2853 | } |
---|
2854 | if (N[1]==1) |
---|
2855 | { |
---|
2856 | L0[1]=list(lab,1); |
---|
2857 | labi=lab,child+1; |
---|
2858 | L0[size(L0)+1]=list(labi,1,ideal(1)); |
---|
2859 | labij=labi,1; |
---|
2860 | L0[size(L0)+1]=list(labij,0,ideal(1)); |
---|
2861 | } |
---|
2862 | def RR=basering; |
---|
2863 | setring(@P); |
---|
2864 | ideal Np=imap(RR,N); |
---|
2865 | ideal Wp=imap(RR,W); |
---|
2866 | poly h=1; |
---|
2867 | for (i=1;i<=size(Wp);i++){h=h*Wp[i];} |
---|
2868 | list Lp=imap(RR,L); |
---|
2869 | list r; list Ti; list LL; |
---|
2870 | LL[1]=list(lab,size(Lp)); |
---|
2871 | for (i=1;i<=size(Lp);i++) |
---|
2872 | { |
---|
2873 | Ti=minGTZ(Lp[i]+h); |
---|
2874 | for(j=1;j<=size(Ti);j++) |
---|
2875 | { |
---|
2876 | option(redSB); |
---|
2877 | Ti[j]=std(Ti[j]); |
---|
2878 | } |
---|
2879 | labi=lab,child+i; |
---|
2880 | childi=size(Ti); |
---|
2881 | LL[size(LL)+1]=list(labi,childi,Lp[i]); |
---|
2882 | for (j=1;j<=childi;j++) |
---|
2883 | { |
---|
2884 | labij=labi,j; |
---|
2885 | LL[size(LL)+1]=list(labij,0,Ti[j]); |
---|
2886 | } |
---|
2887 | } |
---|
2888 | LL[1]=list(lab,size(Lp)); |
---|
2889 | setring(RR); |
---|
2890 | return(imap(@P,LL)); |
---|
2891 | } |
---|
2892 | |
---|
2893 | // difftocanspec |
---|
2894 | // Computes the canonical representation of a diffspec V(N) \ V(M) |
---|
2895 | // input: |
---|
2896 | // intvec lab: label where to hang the canspec |
---|
2897 | // list N ideal of null conditions. |
---|
2898 | // ideal M ideal of the variety to be substacted |
---|
2899 | // output: |
---|
2900 | // the list of elements determining the canonical representation of |
---|
2901 | // the difference V(N) \ V(M): |
---|
2902 | // ( (intvec(i),children), ...(lab, children, prime ideal),...) |
---|
2903 | static proc difftocanspec(intvec lab, int child, ideal N, ideal M) |
---|
2904 | { |
---|
2905 | int i; int j; list LLL; |
---|
2906 | def RR=basering; |
---|
2907 | setring(@P); |
---|
2908 | ideal Np=imap(RR,N); |
---|
2909 | ideal Mp=imap(RR,M); |
---|
2910 | def L=minGTZ(Np); |
---|
2911 | for(j=1;j<=size(L);j++) |
---|
2912 | { |
---|
2913 | option(redSB); |
---|
2914 | L[j]=std(L[j]); |
---|
2915 | } |
---|
2916 | intvec labi; intvec labij; |
---|
2917 | int childi; |
---|
2918 | list LL; |
---|
2919 | if ((Mp[1]==0) or ((size(L)==1) and (L[1][1]==1))) |
---|
2920 | { |
---|
2921 | //LL[1]=list(lab,1); |
---|
2922 | //labi=lab,1; |
---|
2923 | //LL[2]=list(labi,1,ideal(1)); |
---|
2924 | //labij=labi,1; |
---|
2925 | //LL[3]=list(labij,0,ideal(1)); |
---|
2926 | setring(RR); |
---|
2927 | return(LLL); |
---|
2928 | } |
---|
2929 | list r; list Ti; |
---|
2930 | def k=0; |
---|
2931 | LL[1]=list(lab,0); |
---|
2932 | for (i=1;i<=size(L);i++) |
---|
2933 | { |
---|
2934 | Ti=minGTZ(L[i]+Mp); |
---|
2935 | for(j=1;j<=size(Ti);j++) |
---|
2936 | { |
---|
2937 | option(redSB); |
---|
2938 | Ti[j]=std(Ti[j]); |
---|
2939 | } |
---|
2940 | if (not((size(Ti)==1) and (equalideals(L[i],Ti[1])))) |
---|
2941 | { |
---|
2942 | k++; |
---|
2943 | labi=lab,child+k; |
---|
2944 | childi=size(Ti); |
---|
2945 | LL[size(LL)+1]=list(labi,childi,L[i]); |
---|
2946 | for (j=1;j<=childi;j++) |
---|
2947 | { |
---|
2948 | labij=labi,j; |
---|
2949 | LL[size(LL)+1]=list(labij,0,Ti[j]); |
---|
2950 | } |
---|
2951 | } |
---|
2952 | else{setring(RR); return(LLL);} |
---|
2953 | } |
---|
2954 | if (size(LL)>0) |
---|
2955 | { |
---|
2956 | LL[1]=list(lab,k); |
---|
2957 | setring(RR); |
---|
2958 | return(imap(@P,LL)); |
---|
2959 | } |
---|
2960 | else {setring(RR); return(LLL);} |
---|
2961 | } |
---|
2962 | |
---|
2963 | // tree |
---|
2964 | // purpose: given a label and the list L of vertices of the tree, |
---|
2965 | // whose content |
---|
2966 | // are of the form list(intvec lab, int children, ideal P) |
---|
2967 | // to obtain the vertex and its position |
---|
2968 | // input: |
---|
2969 | // intvec lab: label of the vertex |
---|
2970 | // list: L the list containing the vertices |
---|
2971 | // output: |
---|
2972 | // list V the vertex list(lab, children, P) |
---|
2973 | static proc tree(intvec lab,list L) |
---|
2974 | { |
---|
2975 | int i=0; int tt=1; list V; intvec labi; |
---|
2976 | while ((i<size(L)) and (tt)) |
---|
2977 | { |
---|
2978 | i++; |
---|
2979 | labi=L[i][1]; |
---|
2980 | if (labi==lab) |
---|
2981 | { |
---|
2982 | V=list(L[i],i); |
---|
2983 | tt=0; |
---|
2984 | } |
---|
2985 | } |
---|
2986 | if (tt==0){return(V);} |
---|
2987 | else{return(list(list(intvec(0)),0));} |
---|
2988 | } |
---|
2989 | |
---|
2990 | // GCR (generalized canonical representation) |
---|
2991 | // new structure of a GCR |
---|
2992 | |
---|
2993 | // L is a list of vertices V of the GCR. |
---|
2994 | // first vertex=list(intvec lab, int children, ideal lpp, ideal B) |
---|
2995 | // other vertices=list(intvec lab, int children, ideal P) |
---|
2996 | // the individual vertices can be accessed with the function tree |
---|
2997 | // by the call V=tree(lab,L), that outputs the vertex if it exists |
---|
2998 | // and its position in L, or nothing if it does not exist. |
---|
2999 | // The first element of the list must be the root of the tree and has |
---|
3000 | // label lab=i, and other information. |
---|
3001 | |
---|
3002 | // example: |
---|
3003 | // the canonical representation |
---|
3004 | // V(a^2-ac-ba+c-abc) \ (union( V(b,a), V(c,a), V(b,a-c), V(c,a-b))) |
---|
3005 | // is represented by the list |
---|
3006 | // L=((intvec(i),children=1,lpp,B),(intvec(i,1),4,ideal(a^2-ac-ba+c-abc)), |
---|
3007 | // (intvec(i,1,1),0,ideal(b,a)), (intvec(i,1,2),0,ideal(c,a)), |
---|
3008 | // (intvec(i,1,3),0,ideal(b,a-c)), (intvec(i,1,4),0,ideal(c,a-b)) |
---|
3009 | // ) |
---|
3010 | // example: |
---|
3011 | // the canonical representation |
---|
3012 | // (V(a)\(union(V(c,a),V(b+c,a),V(b,a)))) union |
---|
3013 | // (V(b)\(union(V(b,a),V(b,a-c)))) union |
---|
3014 | // (V(c)\(union(V(c,a),V(c,a-b)))) |
---|
3015 | // is represented by the list |
---|
3016 | // L=((i,children=3,lpp,B), |
---|
3017 | // (intvec(i,1),3,ideal(a)), |
---|
3018 | // (intvec(i,1,1),0,(c,a)),(intvec(i,1,2),0,(b+c,a)),(intvec(i,1,3),0,(b,a)), |
---|
3019 | // (intvec(i,2),2,ideal(b)), |
---|
3020 | // (intvec(i,2,1),0,(b,a)),(intvec(i,2,2),0,(b,a-c)), |
---|
3021 | // (intvec(i,3),2,ideal(c)), |
---|
3022 | // (intvec(i,3,1),0,(c,a)),(intvec(i,3,2),0,(c,a-b)) |
---|
3023 | // ) |
---|
3024 | // If L is the list in the last example, the call |
---|
3025 | // tree(intvec(i,2,1),L) will output ((intvec(i,2,1),0,(b,a)),7) |
---|
3026 | |
---|
3027 | // GCR |
---|
3028 | // input: list T is supposed to be an element L[i] of selectcases: |
---|
3029 | // T= list( ideal lpp, ideal B, list(N,W,L),.., list(N,W,L)) |
---|
3030 | // output: the list L of vertices being the GCR of the addition of |
---|
3031 | // all the segments in T. |
---|
3032 | // list(list(intvec lab, int children, ideal lpp, ideal B), |
---|
3033 | // list(intvec lab, int children, ideal P),.. |
---|
3034 | // ) |
---|
3035 | static proc GCR(intvec lab, list case) |
---|
3036 | { |
---|
3037 | int i; int ii; int t; |
---|
3038 | list @L; |
---|
3039 | @L[1]=list(lab,0,case[1],case[2]); |
---|
3040 | exportto(Top,@L); |
---|
3041 | int j; |
---|
3042 | list u; intvec labu; int childu; |
---|
3043 | list v; intvec labv; int childv; |
---|
3044 | list T=case[3]; |
---|
3045 | for (j=1;j<=size(T);j++) |
---|
3046 | { |
---|
3047 | t=addcase(lab,T[j]); |
---|
3048 | deletebrotherscontaining(lab); |
---|
3049 | } |
---|
3050 | relabelingindices(lab,lab); |
---|
3051 | list L=@L; |
---|
3052 | kill @L; |
---|
3053 | return(L); |
---|
3054 | } |
---|
3055 | |
---|
3056 | // sorbylab: |
---|
3057 | // pupose: given the list of mrcgs to order is by increasing label |
---|
3058 | static proc sortbylab(list L) |
---|
3059 | { |
---|
3060 | int n=L[1][2]; |
---|
3061 | int i; int j; |
---|
3062 | list H=L; |
---|
3063 | list LL; |
---|
3064 | list L1; |
---|
3065 | //LL[1]=L[1]; |
---|
3066 | //H=delete(H,1); |
---|
3067 | while (size(H)!=0) |
---|
3068 | { |
---|
3069 | j=1; |
---|
3070 | L1=H[1]; |
---|
3071 | for (i=1;i<=size(H);i++) |
---|
3072 | { |
---|
3073 | if(lesslab(H[i],L1)){j=i;L1=H[j];} |
---|
3074 | } |
---|
3075 | LL[size(LL)+1]=L1; |
---|
3076 | H=delete(H,j); |
---|
3077 | } |
---|
3078 | return(LL); |
---|
3079 | } |
---|
3080 | |
---|
3081 | // lesslab |
---|
3082 | // purpose: given two elements of the list of mrcgs it |
---|
3083 | // returns 1 if the label of the first is less than that of the second |
---|
3084 | static proc lesslab(list l1, list l2) |
---|
3085 | { |
---|
3086 | intvec lab1=l1[1]; |
---|
3087 | intvec lab2=l2[1]; |
---|
3088 | int n1=size(lab1); |
---|
3089 | int n2=size(lab2); |
---|
3090 | int n=n1; |
---|
3091 | if (n2<n1){n=n2;} |
---|
3092 | int tt=0; |
---|
3093 | int j=1; |
---|
3094 | while ((lab1[j]==lab2[j]) and (j<n)){j++;} |
---|
3095 | if (lab1[j]<lab2[j]){tt=1;} |
---|
3096 | if ((j==n) and (lab1[j]==lab2[j]) and (n2>n1)){tt=1;} |
---|
3097 | return(tt); |
---|
3098 | } |
---|
3099 | |
---|
3100 | // cantree |
---|
3101 | // input: the list provided by selectcases |
---|
3102 | // output: the list providing the canonicaltree |
---|
3103 | static proc cantree(list S) |
---|
3104 | { |
---|
3105 | string method=" "; |
---|
3106 | list T0=S[1]; |
---|
3107 | // first element of the list of selectcases |
---|
3108 | int i; int j; |
---|
3109 | list L; |
---|
3110 | list T; |
---|
3111 | L[1]=list(intvec(0),size(S)-1,T0[1],T0[2],T0[3][1],method); |
---|
3112 | for (i=2;i<=size(S);i++) |
---|
3113 | { |
---|
3114 | T=GCR(intvec(i-1),S[i]); |
---|
3115 | T=sortbylab(T); |
---|
3116 | for (j=1;j<=size(T);j++) |
---|
3117 | {L[size(L)+1]=T[j];} |
---|
3118 | } |
---|
3119 | return(L); |
---|
3120 | } |
---|
3121 | |
---|
3122 | // addcase |
---|
3123 | // recursive routine that adds to the list @L, (an alredy GCR) |
---|
3124 | // a new redspec rs=(N,W,L); |
---|
3125 | // and returns the test t whose value is |
---|
3126 | // 0 if the new canspec is not to be hung to the fathers vertex, |
---|
3127 | // 1 if yes. |
---|
3128 | static proc addcase(intvec labu, list rs) |
---|
3129 | { |
---|
3130 | int i; int j; int childu; ideal Pu; |
---|
3131 | list T; int nchildu; |
---|
3132 | def N=rs[1]; def W=rs[2]; def PN=rs[3]; |
---|
3133 | ideal NN; ideal MM; |
---|
3134 | int tt=1; |
---|
3135 | poly h=1; for (i=1;i<=size(W);i++){h=h*W[i];} |
---|
3136 | list u=tree(labu,@L); childu=u[1][2]; |
---|
3137 | list v; intvec labv; int childv; list w; intvec labw; |
---|
3138 | if (childu>0) |
---|
3139 | { |
---|
3140 | v=firstchild(u[1][1]); |
---|
3141 | while(v[2][1]!=0) |
---|
3142 | { |
---|
3143 | labv=v[1][1]; |
---|
3144 | w=firstchild(labv); |
---|
3145 | while(w[2][1]!=0) |
---|
3146 | { |
---|
3147 | labw=w[1][1]; |
---|
3148 | if(addcase(labw,rs)==0) |
---|
3149 | {tt=0;} |
---|
3150 | w=nextbrother(labw); |
---|
3151 | } |
---|
3152 | u=tree(labu,@L); |
---|
3153 | childu=u[1][2]; |
---|
3154 | v=nextbrother(v[1][1]); |
---|
3155 | } |
---|
3156 | deletebrotherscontaining(labu); |
---|
3157 | relabelingindices(labu,labu); |
---|
3158 | } |
---|
3159 | if (tt==1) |
---|
3160 | { |
---|
3161 | u=tree(labu,@L); |
---|
3162 | nchildu=lastchildrenindex(labu); |
---|
3163 | if (size(labu)==1) |
---|
3164 | { |
---|
3165 | T=redtocanspec(labu,nchildu,rs); |
---|
3166 | tt=0; |
---|
3167 | } |
---|
3168 | else |
---|
3169 | { |
---|
3170 | NN=N; |
---|
3171 | if (containedP(u[1][3],N)){tt=0;} |
---|
3172 | for (i=1;i<=size(u[1][3]);i++) |
---|
3173 | { |
---|
3174 | NN[size(NN)+1]=u[1][3][i]; |
---|
3175 | } |
---|
3176 | MM=NN; |
---|
3177 | MM[size(MM)+1]=h; |
---|
3178 | T=difftocanspec(labu,nchildu,NN,MM); |
---|
3179 | } |
---|
3180 | if (size(T)>0) |
---|
3181 | { |
---|
3182 | @L[u[2]][2]=@L[u[2]][2]+T[1][2]; |
---|
3183 | for (i=2;i<=size(T);i++){@L[size(@L)+1]=T[i];} |
---|
3184 | if (size(labu)>1) |
---|
3185 | { |
---|
3186 | simplifynewadded(labu); |
---|
3187 | } |
---|
3188 | } |
---|
3189 | else{tt=1;} |
---|
3190 | } |
---|
3191 | return(tt); |
---|
3192 | } |
---|
3193 | |
---|
3194 | // reduceR |
---|
3195 | // reduces the polynomial f wrt N, in the ring @P |
---|
3196 | static proc reduceR(poly f, ideal N) |
---|
3197 | { |
---|
3198 | def RR=basering; |
---|
3199 | setring(@P); |
---|
3200 | poly fP=imap(RR,f); |
---|
3201 | ideal NP=imap(RR,N); |
---|
3202 | attrib(NP,"isSB",1); |
---|
3203 | poly rp=reduce(fP,NP); |
---|
3204 | setring(RR); |
---|
3205 | return(imap(@P,rp)); |
---|
3206 | } |
---|
3207 | |
---|
3208 | // containedP |
---|
3209 | // returns 1 if ideal Pu is contained in ideal Pv |
---|
3210 | // returns 0 if not |
---|
3211 | // in ring @P |
---|
3212 | static proc containedP(ideal Pu,ideal Pv) |
---|
3213 | { |
---|
3214 | int t=1; |
---|
3215 | int n=ncols(Pu); |
---|
3216 | int i=0; |
---|
3217 | poly r=0; |
---|
3218 | while ((t) and (i<n)) |
---|
3219 | { |
---|
3220 | i++; |
---|
3221 | r=reduceR(Pu[i],Pv); |
---|
3222 | if (r!=0){t=0;} |
---|
3223 | } |
---|
3224 | return(t); |
---|
3225 | } |
---|
3226 | |
---|
3227 | // simplifynewadded |
---|
3228 | // auxiliary routine of addcase |
---|
3229 | // when a new redspec is added to a non terminal vertex, |
---|
3230 | // it is applied to simplify the addition. |
---|
3231 | // When Pu==Pv, the children of w are hung from u fathers |
---|
3232 | // and deleted the whole new addition. |
---|
3233 | // Finally, deletebrotherscontaining is applied to u fathers |
---|
3234 | // in order to eliminate branches contained. |
---|
3235 | static proc simplifynewadded(intvec labu) |
---|
3236 | { |
---|
3237 | int t; int ii; int k; int kk; int j; |
---|
3238 | intvec labfu=delintvec(labu,size(labu)); list fu; int childfu; |
---|
3239 | list u=tree(labu,@L); int childu=u[1][2]; ideal Pu=u[1][3]; |
---|
3240 | list v; intvec labv; int childv; ideal Pv; |
---|
3241 | list w; intvec labw; intvec nlab; list ww; |
---|
3242 | if (childu>0) |
---|
3243 | { |
---|
3244 | v=firstchild(u[1][1]); labv=v[1][1]; childv=v[1][2]; Pv=v[1][3]; |
---|
3245 | ii=0; |
---|
3246 | t=0; |
---|
3247 | while ((not(t)) and (ii<childu)) |
---|
3248 | { |
---|
3249 | ii++; |
---|
3250 | if (equalideals(Pu,Pv)) |
---|
3251 | { |
---|
3252 | fu=tree(labfu,@L); |
---|
3253 | childfu=fu[1][2]; |
---|
3254 | j=lastchildrenindex(fu[1][1])+1; |
---|
3255 | k=0; |
---|
3256 | w=firstchild(v[1][1]); |
---|
3257 | childv=v[1][2]; |
---|
3258 | for (kk=1;kk<=childv;kk++) |
---|
3259 | { |
---|
3260 | if (kk<childv){ww=nextbrother(w[1][1]);} |
---|
3261 | nlab=labfu,j; |
---|
3262 | @L[w[2]][1]=nlab; |
---|
3263 | j++; |
---|
3264 | if (kk<childv){w=ww;} |
---|
3265 | } |
---|
3266 | childfu=fu[1][2]+childv-1; |
---|
3267 | @L[fu[2]][2]=childfu; |
---|
3268 | @L[v[2]][2]=0; |
---|
3269 | t=1; |
---|
3270 | deleteverts(labu); |
---|
3271 | } |
---|
3272 | } |
---|
3273 | } |
---|
3274 | deletebrotherscontaining(labfu); |
---|
3275 | } |
---|
3276 | |
---|
3277 | // given the the label labfu of the vertex fu it returns the last |
---|
3278 | // int of the label of the last existing children. |
---|
3279 | // if no child exists, then it ouputs 0. |
---|
3280 | static proc lastchildrenindex(intvec labfu) |
---|
3281 | { |
---|
3282 | int i; |
---|
3283 | int lastlabi; intvec labi; intvec labfi; |
---|
3284 | int lastlab=0; |
---|
3285 | for (i=1;i<=size(@L);i++) |
---|
3286 | { |
---|
3287 | labi=@L[i][1]; |
---|
3288 | if (size(labi)>1) |
---|
3289 | { |
---|
3290 | labfi=delintvec(labi,size(labi)); |
---|
3291 | if (labfu==labfi) |
---|
3292 | { |
---|
3293 | lastlabi=labi[size(labi)]; |
---|
3294 | if (lastlab<lastlabi) |
---|
3295 | { |
---|
3296 | lastlab=lastlabi; |
---|
3297 | } |
---|
3298 | } |
---|
3299 | } |
---|
3300 | } |
---|
3301 | return(lastlab); |
---|
3302 | } |
---|
3303 | |
---|
3304 | // given the the vertex u it provides the next brother of u. |
---|
3305 | // if it does not exist, then it ouputs v=list(list(intvec(0)),0) |
---|
3306 | static proc nextbrother(intvec labu) |
---|
3307 | { |
---|
3308 | list L; int i; int j; list next; |
---|
3309 | int lastlabu=labu[size(labu)]; |
---|
3310 | intvec labfu=delintvec(labu,size(labu)); |
---|
3311 | int lastlabi; intvec labi; intvec labfi; |
---|
3312 | for (i=1;i<=size(@L);i++) |
---|
3313 | { |
---|
3314 | labi=@L[i][1]; |
---|
3315 | if (size(labi)>1) |
---|
3316 | { |
---|
3317 | labfi=delintvec(labi,size(labi)); |
---|
3318 | if (labfu==labfi) |
---|
3319 | { |
---|
3320 | lastlabi=labi[size(labi)]; |
---|
3321 | if (lastlabu<lastlabi) |
---|
3322 | {L[size(L)+1]=list(lastlabi,list(@L[i],i));} |
---|
3323 | } |
---|
3324 | } |
---|
3325 | } |
---|
3326 | if (size(L)==0){return(list(intvec(0),0));} |
---|
3327 | next=L[1]; |
---|
3328 | for (i=2;i<=size(L);i++) |
---|
3329 | { |
---|
3330 | if (L[i][1]<next[1]){next=L[i];} |
---|
3331 | } |
---|
3332 | return(next[2]); |
---|
3333 | } |
---|
3334 | |
---|
3335 | // gives the first child of vertex fu |
---|
3336 | static proc firstchild(labfu) |
---|
3337 | { |
---|
3338 | intvec labfu0=labfu; |
---|
3339 | labfu0[size(labfu0)+1]=0; |
---|
3340 | return(nextbrother(labfu0)); |
---|
3341 | } |
---|
3342 | |
---|
3343 | // purpose: eliminate the children vertices of fu and all its descendents |
---|
3344 | // whose prime ideal Pu contains a prime ideal Pv of some brother vertex w. |
---|
3345 | static proc deletebrotherscontaining(intvec labfu) |
---|
3346 | { |
---|
3347 | int i; int t; |
---|
3348 | list fu=tree(labfu,@L); |
---|
3349 | int childfu=fu[1][2]; |
---|
3350 | list u; intvec labu; ideal Pu; |
---|
3351 | list v; intvec labv; ideal Pv; |
---|
3352 | u=firstchild(labfu); |
---|
3353 | for (i=1;i<=childfu;i++) |
---|
3354 | { |
---|
3355 | labu=u[1][1]; |
---|
3356 | Pu=u[1][3]; |
---|
3357 | v=firstchild(fu[1][1]); |
---|
3358 | t=1; |
---|
3359 | while ((t) and (v[2]!=0)) |
---|
3360 | { |
---|
3361 | labv=v[1][1]; |
---|
3362 | Pv=v[1][3]; |
---|
3363 | if (labu!=labv) |
---|
3364 | { |
---|
3365 | if (containedP(Pv,Pu)) |
---|
3366 | { |
---|
3367 | deleteverts(labu); |
---|
3368 | fu=tree(labfu,@L); |
---|
3369 | @L[fu[2]][2]=fu[1][2]-1; |
---|
3370 | t=0; |
---|
3371 | } |
---|
3372 | } |
---|
3373 | if (t!=0) |
---|
3374 | { |
---|
3375 | v=nextbrother(v[1][1]); |
---|
3376 | } |
---|
3377 | } |
---|
3378 | if (i<childfu) |
---|
3379 | { |
---|
3380 | u=nextbrother(u[1][1]); |
---|
3381 | } |
---|
3382 | } |
---|
3383 | } |
---|
3384 | |
---|
3385 | // purpose: delete all descendent vertices from u included u |
---|
3386 | // from the list @L. |
---|
3387 | // It must be noted that after the operation, the number of children |
---|
3388 | // in fathers vertex must be decreased in 1 unitity. This operation is not |
---|
3389 | // performed inside this recursive routine. |
---|
3390 | static proc deleteverts(intvec labu) |
---|
3391 | { |
---|
3392 | int i; int ii; list v; intvec labv; |
---|
3393 | list u=tree(labu,@L); |
---|
3394 | int childu=u[1][2]; |
---|
3395 | @L=delete(@L,u[2]); |
---|
3396 | if (childu>0) |
---|
3397 | { |
---|
3398 | v=firstchild(labu); |
---|
3399 | labv=v[1][1]; |
---|
3400 | for (ii=1;ii<=childu;ii++) |
---|
3401 | { |
---|
3402 | deleteverts(labv); |
---|
3403 | if (ii<childu) |
---|
3404 | { |
---|
3405 | v=nextbrother(v[1][1]); |
---|
3406 | labv=v[1][1]; |
---|
3407 | } |
---|
3408 | } |
---|
3409 | } |
---|
3410 | } |
---|
3411 | |
---|
3412 | // purpose: starting from vertex olab (initially nlab=olab) |
---|
3413 | // relabels the vertices of @L to be consecutive |
---|
3414 | static proc relabelingindices(intvec olab, intvec nlab) |
---|
3415 | { |
---|
3416 | int i; |
---|
3417 | intvec nlabi; intvec labv; |
---|
3418 | list u=tree(olab,@L); |
---|
3419 | int childu=u[1][2]; |
---|
3420 | list v; |
---|
3421 | if (childu==0){@L[u[2]][1]=nlab;} |
---|
3422 | else |
---|
3423 | { |
---|
3424 | v=firstchild(u[1][1]); |
---|
3425 | @L[u[2]][1]=nlab; |
---|
3426 | i=1; |
---|
3427 | while(v[2]!=0) |
---|
3428 | { |
---|
3429 | labv=v[1][1]; |
---|
3430 | nlabi=nlab,i; |
---|
3431 | relabelingindices(labv,nlabi); |
---|
3432 | v=nextbrother(labv); |
---|
3433 | i++; |
---|
3434 | } |
---|
3435 | } |
---|
3436 | } |
---|
3437 | |
---|
3438 | // mrcgs |
---|
3439 | // input: F = ideal in ring R=Q[a][x] |
---|
3440 | // output: a list L representing the tree of the mrcgs. |
---|
3441 | static proc mrcgs(ideal F, list #) |
---|
3442 | //"USAGE: mrcgs(F); |
---|
3443 | // F is the ideal from which to obtain the Minimal Reduced CGS. |
---|
3444 | // From the old library redcgs.lib. |
---|
3445 | // Alternatively, as option: |
---|
3446 | // mrcgs(F,L); |
---|
3447 | // Options: We can give a list of options in the list L |
---|
3448 | // of the form |
---|
3449 | // ("null",ideal N,"nonnull",ideal W,"comment",0-1). |
---|
3450 | // One can give none till 3 of these options by giving the |
---|
3451 | // name of the option and the content. |
---|
3452 | // When options "null" and/or "nonnull" are given, then the |
---|
3453 | // parameter space is restricted to V(N)\V(h), where h is the product of |
---|
3454 | // the non null polynomials in W. If the option ("comment",1) is set, |
---|
3455 | // then information about the total number of segments of the |
---|
3456 | // output is printed. |
---|
3457 | // By default N=ideal(0), W=ideal(1), ("comment",0). |
---|
3458 | // mrcgs is the fundamental routine of the old library redcgs.lib, |
---|
3459 | // computing the minimal reduced comprehensive Groebner system. |
---|
3460 | //RETURN: The list T representing the Minimal Reduced CGS. |
---|
3461 | // The description given here is identical for rcgs and crcgs. |
---|
3462 | // The elements of the list T computed by mrcgs are lists representing |
---|
3463 | // a rooted tree. |
---|
3464 | // Each element of the list T has the two first entries with the following content: |
---|
3465 | // [1]: The label (intvec) representing the position in the rooted |
---|
3466 | // tree: 0 for the root (and this is a special element) |
---|
3467 | // i for the root of the segment i |
---|
3468 | // (i,...) for the children of the segment i |
---|
3469 | // [2]: the number of children (int) of the vertex. |
---|
3470 | // There thus three kind of vertices: |
---|
3471 | // (1) the root (first element labelled 0), |
---|
3472 | // (2) the vertices labelled with a single integer i, |
---|
3473 | // (3) the rest of vertices labelled with more indices. |
---|
3474 | // Description of the root. Vertex type (1) |
---|
3475 | // There is a special vertex (the first one) whose content is |
---|
3476 | // the following: |
---|
3477 | // [3] lpp of the given ideal |
---|
3478 | // [4] the given ideal |
---|
3479 | // [5] the red-representation of the (optional) given null and non-null |
---|
3480 | // conditions (see redspec for the description). |
---|
3481 | // [6] MRCGS (to remember which algorithm has been used). If the |
---|
3482 | // algorithm used is rcgs of crcgs then this will be stated |
---|
3483 | // at this vertex (RCGS or CRCGS). |
---|
3484 | // Description of vertices type (2). These are the vertices that |
---|
3485 | // initiate a segment, and are labelled with a single integer. |
---|
3486 | // [3] lpp (ideal) of the reduced basis. If they are repeated lpp's this |
---|
3487 | // will correspond to a sheaf. |
---|
3488 | // [4] the reduced basis (ideal) of the segment. |
---|
3489 | // Description of vertices type (3). These vertices have as first |
---|
3490 | // label i and descend form vertex i in the position of the label |
---|
3491 | // (i,...). They contain moreover a unique prime ideal in the parameters |
---|
3492 | // and form ascending chains of ideals. |
---|
3493 | // How is to be read the mrcgs tree? The vertices with an even number of |
---|
3494 | // integers in the label are to be considered as additive and those |
---|
3495 | // with an odd number of integers in the label are to be considered as |
---|
3496 | // substraction. As an example consider the following vertices: |
---|
3497 | // v1=((i),2,lpp,B), |
---|
3498 | // v2=((i,1),2,P_(i,1)), |
---|
3499 | // v3=((i,1,1),2,P_(i,1,1)), |
---|
3500 | // v4=((i,1,1,1),1,P_(i,1,1,1)), |
---|
3501 | // v5=((i,1,1,1,1),0,P_(i,1,1,1,1)), |
---|
3502 | // v6=((i,1,1,2),1,P_(i,1,1,2)), |
---|
3503 | // v7=((i,1,1,2,1),0,P_(i,1,1,2,1)), |
---|
3504 | // v8=((i,1,2),0,P_(i,1,2)), |
---|
3505 | // v9=((i,2),1,P_(i,2)), |
---|
3506 | // v10=((i,2,1),0,P_(i,2,1)), |
---|
3507 | // They represent the segment: |
---|
3508 | // (V(i,1)\(((V(i,1,1) \ ((V(i,1,1,1) \ V(i,1,1,1,1)) u (V(i,1,1,2) \ V(i,1,1,2,1))))) |
---|
3509 | // u V(i,1,2))) u (V(i,2) \ V(i,2,1)) |
---|
3510 | // and can also be represented by |
---|
3511 | // (V(i,1) \ (V(i,1,1) u V(i,1,2))) u |
---|
3512 | // (V(i,1,1,1) \ V(i,1,1,1)) u |
---|
3513 | // (V(i,1,1,2) \ V(i,1,1,2,1)) u |
---|
3514 | // (V(i,2) \ V(i,2,1)) |
---|
3515 | // where V(i,j,..) = V(P_(i,j,..)) |
---|
3516 | //NOTE: There are three fundamental routines in the old library redcgs.lib: |
---|
3517 | // mrcgs, rcgs and crcgs. |
---|
3518 | // mrcgs (Minimal Reduced CGS) is an algorithm that packs so much as it |
---|
3519 | // is able to do (using algorithms adhoc) the segments with the same lpp, |
---|
3520 | // obtaining the minimal number of segments. The hypothesis is that this |
---|
3521 | // is very close to be canonical, but there is no proof of the uniqueness |
---|
3522 | // of this minimal packing. Moreover, the segments obtained are not |
---|
3523 | // locally closed, i.e. there are not always the difference of two varieties, |
---|
3524 | // but are a union of differences of varieties. |
---|
3525 | // The output can be visualized using cantreetoMaple, that will |
---|
3526 | // write a file with the content of mrcgs that can be read in Maple |
---|
3527 | // and plotted using the Maple plotcantree routine of the Monte's dpgb library |
---|
3528 | //KEYWORDS: rcgs, crcgs, buildtree, cantreetoMaple, |
---|
3529 | //EXAMPLE: mrcgs; shows an example" |
---|
3530 | { |
---|
3531 | int i=1; |
---|
3532 | int @ish=1; |
---|
3533 | exportto(Top,@ish); |
---|
3534 | while((@ish) and (i<=size(F))) |
---|
3535 | { |
---|
3536 | @ish=ishomog(F[i]); |
---|
3537 | i++; |
---|
3538 | } |
---|
3539 | int comment=0; |
---|
3540 | def N=ideal(0); |
---|
3541 | def W=ideal(1); |
---|
3542 | list L=#; |
---|
3543 | for(i=1;i<=size(L) div 2;i++) |
---|
3544 | { |
---|
3545 | if(L[2*i-1]=="null"){N=L[2*i];} |
---|
3546 | else |
---|
3547 | { |
---|
3548 | if(L[2*i-1]=="nonnull"){W=L[2*i];} |
---|
3549 | else |
---|
3550 | { |
---|
3551 | if(L[2*i-1]=="comment"){comment=L[2*i];} |
---|
3552 | } |
---|
3553 | } |
---|
3554 | } |
---|
3555 | def RR=basering; |
---|
3556 | list LL=buildtree(F, #); |
---|
3557 | setglobalrings(); |
---|
3558 | list S=selectcases(LL); |
---|
3559 | list T=cantree(S); |
---|
3560 | if(equalideals(N,ideal(0))==0) |
---|
3561 | { |
---|
3562 | T=reduceconds(T,N,W); |
---|
3563 | } |
---|
3564 | T[1][6]="MRCGS"; |
---|
3565 | T[1][4]=F; |
---|
3566 | for (i=1;i<=size(F);i++) |
---|
3567 | { |
---|
3568 | T[1][3][i]=leadmonom(F[i]); |
---|
3569 | } |
---|
3570 | kill @ish; |
---|
3571 | kill @P; kill @RP; kill @R; |
---|
3572 | return(T); |
---|
3573 | } |
---|
3574 | //example |
---|
3575 | //{ "EXAMPLE:"; echo = 2; |
---|
3576 | // ring R=(0,a1,a2,a3,a4),(x1,x2,x3,x4),dp; |
---|
3577 | // ideal F=x4-a4+a2, x1+x2+x3+x4-a1-a3-a4, x1*x3*x4-a1*a3*a4, x1*x3+x1*x4+x2*x3+x3*x4-a1*a4-a1*a3-a3*a4; |
---|
3578 | // "System="; F; |
---|
3579 | // def T=mrcgs(F); |
---|
3580 | // setglobalrings(); |
---|
3581 | // "mrcgs(F)="; T; |
---|
3582 | // cantreetoMaple(T,"Tm","Tm.txt"); |
---|
3583 | // "cantodiffcgs(T)="; cantodiffcgs(T); |
---|
3584 | // kill R; |
---|
3585 | // ring R=(0,b,c,d,e,f),(x,y),dp; |
---|
3586 | // ideal F1=x^2+b*y^2+2*c*x*y+2*d*x+2*e*y+f, 2*x+2*c*y+2*d, 2*b*y+2*c*x+2*e; |
---|
3587 | // "System="; F1; |
---|
3588 | // def T1=mrcgs(F1); |
---|
3589 | // setglobalrings(); |
---|
3590 | // "mrcgs(F1)="; T1; |
---|
3591 | // cantreetoMaple(T1,"T1m","T1m.txt"); |
---|
3592 | //} |
---|
3593 | |
---|
3594 | // reduceconds: when null and nonnull conditions are specified it |
---|
3595 | // takes the output of cantree and reduces the tree |
---|
3596 | // assuming the null and nonnull conditions |
---|
3597 | // input: list T (the output of cantree computed with null and nonull conditions |
---|
3598 | // ideal N: null conditions |
---|
3599 | // ideal W: non-null conditions |
---|
3600 | // output: the list T assuming the null and non-null conditions |
---|
3601 | static proc reduceconds(list T,ideal N,ideal W) |
---|
3602 | { |
---|
3603 | int i; intvec lab; intvec labfu; list fu; int j; int t; |
---|
3604 | list @L=T; |
---|
3605 | exportto(Top,@L); |
---|
3606 | int n=size(W); |
---|
3607 | for (i=2;i<=size(@L);i++) |
---|
3608 | { |
---|
3609 | t=0; j=0; |
---|
3610 | while ((not(t)) and (j<n)) |
---|
3611 | { |
---|
3612 | j++; |
---|
3613 | if (size(@L[i][1])>1) |
---|
3614 | { |
---|
3615 | if (memberpos(W[j],@L[i][3])[1]) |
---|
3616 | { |
---|
3617 | t=1; |
---|
3618 | @L[i][3]=ideal(1); |
---|
3619 | } |
---|
3620 | } |
---|
3621 | } |
---|
3622 | } |
---|
3623 | for (i=2;i<=size(@L);i++) |
---|
3624 | { |
---|
3625 | if (size(@L[i][1])>1) |
---|
3626 | { |
---|
3627 | @L[i][3]=delidfromid(N,@L[i][3]); |
---|
3628 | } |
---|
3629 | } |
---|
3630 | for (i=2;i<=size(@L);i++) |
---|
3631 | { |
---|
3632 | if ((size(@L[i][1])>1) and (size(@L[i][1]) mod 2==1) and (equalideals(@L[i][3],ideal(0)))) |
---|
3633 | { |
---|
3634 | lab=@L[i][1]; |
---|
3635 | labfu=delintvec(lab,size(lab)); |
---|
3636 | fu=tree(labfu,@L); |
---|
3637 | @L[fu[2]][2]=@L[fu[2]][2]-1; |
---|
3638 | deleteverts(lab); |
---|
3639 | } |
---|
3640 | } |
---|
3641 | for (j=2; j<=size(@L); j++) |
---|
3642 | { |
---|
3643 | if (@L[j][2]>0) |
---|
3644 | { |
---|
3645 | deletebrotherscontaining(@L[j][1]); |
---|
3646 | } |
---|
3647 | } |
---|
3648 | for (i=1;i<=@L[1][2];i++) |
---|
3649 | { |
---|
3650 | relabelingindices(intvec(i),intvec(i)); |
---|
3651 | } |
---|
3652 | list TT=@L; |
---|
3653 | kill @L; |
---|
3654 | return(TT); |
---|
3655 | } |
---|
3656 | |
---|
3657 | //**************End of cantree****************************** |
---|
3658 | |
---|
3659 | //**************Begin of CanTreeTo Maple******************** |
---|
3660 | |
---|
3661 | // cantreetoMaple |
---|
3662 | // input: list L: the output of cantree |
---|
3663 | // string T: the name of the table of Maple that represents L |
---|
3664 | // in Maple |
---|
3665 | // string writefile: the name of the file where the table T |
---|
3666 | // is written |
---|
3667 | proc cantreetoMaple(list L, string T, string writefile) |
---|
3668 | "USAGE: cantreetoMaple(T, TM, writefile); |
---|
3669 | T: is the list provided by grobcovold with option ("out",1), |
---|
3670 | TM: is the name (string) of the table variable in Maple that will |
---|
3671 | represent the output of the fundamental routines, |
---|
3672 | writefile: is the name (string) of the file where to write the content. |
---|
3673 | RETURN: writes the list provided by grobcovold to a file |
---|
3674 | containing the table representing it in Maple. |
---|
3675 | NOTE: It can be called from the output of grobcovold with option ("out",1) |
---|
3676 | KEYWORDS: grobcovold, Maple |
---|
3677 | EXAMPLE: cantreetoMaple; shows an example" |
---|
3678 | { |
---|
3679 | short=0; |
---|
3680 | if(size(L[1])!=6) |
---|
3681 | { |
---|
3682 | " 'Warning!' grobcovold must be called with option 'out' set to 1 to be operative"; |
---|
3683 | return(); |
---|
3684 | } |
---|
3685 | int i; |
---|
3686 | def R=basering; |
---|
3687 | list L0=L[1]; |
---|
3688 | int numcases=L0[2]; |
---|
3689 | link LLw=":w "+writefile; |
---|
3690 | string La=string("table(",T,");"); |
---|
3691 | write(LLw, La); |
---|
3692 | close(LLw); |
---|
3693 | link LLa=":a "+writefile; |
---|
3694 | def RL=ringlist(R); |
---|
3695 | list p=RL[1][2]; |
---|
3696 | string param=string(p[1]); |
---|
3697 | if (size(p)>1) |
---|
3698 | { |
---|
3699 | for(i=2;i<=size(p);i++){param=string(param,",",p[i]);} |
---|
3700 | } |
---|
3701 | list v=RL[2]; |
---|
3702 | string vars=string(v[1]); |
---|
3703 | if (size(v)>1) |
---|
3704 | { |
---|
3705 | for(i=2;i<=size(v);i++){vars=string(vars,",",v[i]);} |
---|
3706 | } |
---|
3707 | list xord; |
---|
3708 | list pord; |
---|
3709 | if (RL[1][3][1][1]=="dp"){pord=string("tdeg(",param);} |
---|
3710 | else |
---|
3711 | { |
---|
3712 | if (RL[1][3][1][1]=="lp"){pord=string("plex(",param);} |
---|
3713 | } |
---|
3714 | if (RL[3][1][1]=="dp"){xord=string("tdeg(",vars);} |
---|
3715 | else |
---|
3716 | { |
---|
3717 | if (RL[3][1][1]=="lp"){xord=string("plex(",vars);} |
---|
3718 | } |
---|
3719 | write(LLa,string(T,"[[___xord]]:=",xord,");")); |
---|
3720 | write(LLa,string(T,"[[___pord]]:=",pord,");")); |
---|
3721 | //write(LLa,string(T,"[[11]]:=true; ")); |
---|
3722 | list S; |
---|
3723 | S=string(T,"[[0]]:=",numcases,";"); |
---|
3724 | write(LLa,S); |
---|
3725 | S=string(T,"[[___method]]:=",L[1][6],";"); |
---|
3726 | // Method L[1][6]; |
---|
3727 | write(LLa,S); |
---|
3728 | S=string(T,"[[___basis]]:=[",L0[4],"];"); |
---|
3729 | write(LLa,S); |
---|
3730 | S=string(T,"[[___nullcond]]:=[",L0[5][1],"];"); |
---|
3731 | write(LLa,S); |
---|
3732 | S=string(T,"[[___notnullcond]]:={",L0[5][2],"};"); |
---|
3733 | write(LLa,S); |
---|
3734 | for (i=1;i<=numcases;i++) |
---|
3735 | { |
---|
3736 | S=ctlppbasis(T,L,intvec(i)); |
---|
3737 | write(LLa,S[1]); |
---|
3738 | write(LLa,S[2]); |
---|
3739 | write(LLa,S[3]); |
---|
3740 | //write(LLa,S[4]); |
---|
3741 | ctrecwrite(LLa, L, T, intvec(i),S[4]); |
---|
3742 | } |
---|
3743 | close(LLa); |
---|
3744 | } |
---|
3745 | example |
---|
3746 | { "EXAMPLE:"; echo = 2; |
---|
3747 | ring R=(0,b,c,d,e,f),(x,y),dp; |
---|
3748 | ideal F=x^2+b*y^2+2*c*x*y+2*d*x+2*e*y+f, 2*x+2*c*y+2*d, 2*b*y+2*c*x+2*e; |
---|
3749 | def T=grobcovold(F,"out",1); |
---|
3750 | T; |
---|
3751 | cantreetoMaple(T,"Tm","Tm.txt"); |
---|
3752 | } |
---|
3753 | |
---|
3754 | // ctlppbasis: auxiliary cantreetoMaple routine |
---|
3755 | // input: |
---|
3756 | // string T: the name of the table in Maple |
---|
3757 | // intvec lab: the label of the case |
---|
3758 | // ideal B: the basis of the case |
---|
3759 | // output: |
---|
3760 | // the string of T[[lab]] (basis); in Maple |
---|
3761 | static proc ctlppbasis(string T, list L, intvec lab) |
---|
3762 | { |
---|
3763 | list u; |
---|
3764 | intvec lab0=lab,0; |
---|
3765 | u=tree(lab,L); |
---|
3766 | list Li; |
---|
3767 | Li[1]=string(T,"[[",lab,",___lpp]]:=[",u[1][3],"]; "); |
---|
3768 | Li[2]=string(T,"[[",lab,"]]:=[",u[1][4],"]; "); |
---|
3769 | Li[3]=string(T,"[[",lab0,"]]:=",u[1][2],"; "); |
---|
3770 | Li[4]=u[1][2]; |
---|
3771 | return(Li); |
---|
3772 | } |
---|
3773 | |
---|
3774 | // ctlppbasis: auxiliary cantreetoMaple routine |
---|
3775 | // recursive routine to write all elements |
---|
3776 | static proc ctrecwrite(LLa, list L, string T, intvec lab, int n) |
---|
3777 | { |
---|
3778 | int i; |
---|
3779 | intvec labi; intvec labi0; |
---|
3780 | string S; |
---|
3781 | list u; |
---|
3782 | for (i=1;i<=n;i++) |
---|
3783 | { |
---|
3784 | labi=lab,i; |
---|
3785 | u=tree(labi,L); |
---|
3786 | S=string(T,"[[",labi,"]]:=[",u[1][3],"];"); |
---|
3787 | write(LLa,S); |
---|
3788 | labi0=labi,0; |
---|
3789 | S=string(T,"[[",labi0,"]]:=",u[1][2],";"); |
---|
3790 | write(LLa,S); |
---|
3791 | ctrecwrite(LLa, L, T, labi, u[1][2]); |
---|
3792 | } |
---|
3793 | } |
---|
3794 | |
---|
3795 | //**************End of CanTreeTo Maple******************** |
---|
3796 | |
---|
3797 | //**************Begin homogenizing************************ |
---|
3798 | |
---|
3799 | // ishomog: |
---|
3800 | // Purpose: test if a polynomial is homogeneous in the variables or not |
---|
3801 | // input: poly f |
---|
3802 | // output 1 if f is homogeneous, 0 if not |
---|
3803 | static proc ishomog(f) |
---|
3804 | { |
---|
3805 | int i; poly r; int d; int dr; |
---|
3806 | if (f==0){return(1);} |
---|
3807 | d=deg(f); dr=d; r=f; |
---|
3808 | while ((d==dr) and (r!=0)) |
---|
3809 | { |
---|
3810 | r=r-lead(r); |
---|
3811 | dr=deg(r); |
---|
3812 | } |
---|
3813 | if (r==0){return(1);} |
---|
3814 | else{return(0);} |
---|
3815 | } |
---|
3816 | |
---|
3817 | static proc rcgs(ideal F, list #) |
---|
3818 | //"USAGE: rcgs(F); |
---|
3819 | // F is the ideal from which to obtain the Reduced CGS. |
---|
3820 | // From the old library redcgs.lib. |
---|
3821 | // Alternatively, as option: |
---|
3822 | // rcgs(F,L); |
---|
3823 | // Options: We can give a list of options in the list L |
---|
3824 | // of the form |
---|
3825 | // ("null",ideal N,"nonnull",ideal W,"comment",int comment). |
---|
3826 | // One can give none till 3 of these options by giving the |
---|
3827 | // name of the option and the content. |
---|
3828 | // When options "null" and/or "nonnull" are given, then the |
---|
3829 | // parameter space is restricted to V(N)\V(h), where h is the product of |
---|
3830 | // the non null polynomials in W. If the option "comment" is set to 1, |
---|
3831 | // then information about the total number of segments of the |
---|
3832 | // output is printed. |
---|
3833 | // By default N=ideal(0) and W=ideal(1). |
---|
3834 | // rcgs is the a routine whose output segments are always |
---|
3835 | // locally closed and correspond to homogenizing the basis |
---|
3836 | // compute its mrcgs and then reduce and de-homogenizing the result. |
---|
3837 | // The result is a Reduced Comprehensive Groebner System. |
---|
3838 | //RETURN: The list T representing the Reduced CGS. |
---|
3839 | // The description given here is identical for mrcgs and crcgs. |
---|
3840 | // The elements of the list T computed by rcgs are lists representing |
---|
3841 | // a rooted tree. |
---|
3842 | // Each element of the list T has the two first entries with the following content: |
---|
3843 | // [1]: The label (intvec) representing the position in the rooted |
---|
3844 | // tree: 0 for the root (and this is a special element) |
---|
3845 | // i for the root of the segment i |
---|
3846 | // (i,...) for the children of the segment i |
---|
3847 | // [2]: the number of children (int) of the vertex. |
---|
3848 | // There thus three kind of vertices: |
---|
3849 | // (1) the root (first element labelled 0), |
---|
3850 | // (2) the vertices labelled with a single integer i, |
---|
3851 | // (3) the rest of vertices labelled with more indices. |
---|
3852 | // Description of the root. Vertex type (1) |
---|
3853 | // There is a special vertex (the first one) whose content is |
---|
3854 | // the following: |
---|
3855 | // [3] lpp of the given ideal |
---|
3856 | // [4] the given ideal |
---|
3857 | // [5] the red-representation of the (optional) given null and non-null conditions |
---|
3858 | // (see redspec for the description) |
---|
3859 | // [6] RCGS (to remember which algorithm has been used). If the |
---|
3860 | // algorithm used is mrcgs of crcgs then this will be stated |
---|
3861 | // at this vertex (MRCGS or CRCGS). |
---|
3862 | // Description of vertices type (2). These are the vertices that |
---|
3863 | // initiate a segment, and are labelled with a single integer. |
---|
3864 | // [3] lpp (ideal) of the reduced basis. If they are repeated lpp's this |
---|
3865 | // will correspond to a sheaf. |
---|
3866 | // [4] the reduced basis (ideal) of the segment. |
---|
3867 | // Description of vertices type (3). These vertices have as first |
---|
3868 | // label i and descend form vertex i in the position of the label |
---|
3869 | // (i,...). They contain moreover a unique prime ideal in the parameters |
---|
3870 | // and form ascending chains of ideals. |
---|
3871 | // How is to be read the rcgs tree? The vertices with an even number of |
---|
3872 | // integers in the label are to be considered as additive and those |
---|
3873 | // with an odd number of integers in the label are to be considered as |
---|
3874 | // substraction. As an example consider the following vertices: |
---|
3875 | // v1=((i),2,lpp,B), |
---|
3876 | // v2=((i,1),2,P_(i,1)), |
---|
3877 | // v3=((i,1,1),2,P_(i,1,1)), |
---|
3878 | // v4=((i,1,1,1),1,P_(i,1,1,1)), |
---|
3879 | // v5=((i,1,1,1,1),0,P_(i,1,1,1,1)), |
---|
3880 | // v6=((i,1,1,2),1,P_(i,1,1,2)), |
---|
3881 | // v7=((i,1,1,2,1),0,P_(i,1,1,2,1)), |
---|
3882 | // v8=((i,1,2),0,P_(i,1,2)), |
---|
3883 | // v9=((i,2),1,P_(i,2)), |
---|
3884 | // v10=((i,2,1),0,P_(i,2,1)), |
---|
3885 | // They represent the segment: |
---|
3886 | // (V(i,1)\(((V(i,1,1) \ ((V(i,1,1,1) \ V(i,1,1,1,1)) u (V(i,1,1,2) \ V(i,1,1,2,1))))) |
---|
3887 | // u V(i,1,2))) u (V(i,2) \ V(i,2,1)) |
---|
3888 | // and can also be represented by |
---|
3889 | // (V(i,1) \ (V(i,1,1) u V(i,1,2))) u |
---|
3890 | // (V(i,1,1,1) \ V(i,1,1,1)) u |
---|
3891 | // (V(i,1,1,2) \ V(i,1,1,2,1)) u |
---|
3892 | // (V(i,2) \ V(i,2,1)) |
---|
3893 | // where V(i,j,..) = V(P_(i,j,..)) |
---|
3894 | //NOTE: There are three fundamental routines in the old library redcgs.lib: |
---|
3895 | // mrcgs, rcgs and crcgs. |
---|
3896 | // The output can be visualized using cantreetoMaple, that will |
---|
3897 | // write a file with the content of rcgs that can be read in Maple |
---|
3898 | // and plotted using the Maple plotcantree routine of the Monte's dpgb library |
---|
3899 | //KEYWORDS: mrcgs, crcgs, buildtree, cantreetoMaple, |
---|
3900 | //EXAMPLE: rcgs; shows an example" |
---|
3901 | { |
---|
3902 | int j; int i; |
---|
3903 | poly f; |
---|
3904 | int comment=0; |
---|
3905 | def N=ideal(0); |
---|
3906 | def W=ideal(1); |
---|
3907 | list L=#; |
---|
3908 | for(i=1;i<=size(L) div 2;i++) |
---|
3909 | { |
---|
3910 | if(L[2*i-1]=="null"){N=L[2*i];} |
---|
3911 | else |
---|
3912 | { |
---|
3913 | if(L[2*i-1]=="nonnull"){W=L[2*i];} |
---|
3914 | else |
---|
3915 | { |
---|
3916 | if(L[2*i-1]=="comment"){comment=L[2*i];} |
---|
3917 | } |
---|
3918 | } |
---|
3919 | } |
---|
3920 | i=1; int postred=0; |
---|
3921 | int ish=1; |
---|
3922 | while ((ish) and (i<=size(F))) |
---|
3923 | { |
---|
3924 | ish=ishomog(F[i]); |
---|
3925 | i++; |
---|
3926 | } |
---|
3927 | if (ish){return(mrcgs(F, #));} |
---|
3928 | def RR=basering; |
---|
3929 | list RRL=ringlist(RR); |
---|
3930 | //if (RRL[3][1][1]!="dp"){ERROR("the order must be dp");} |
---|
3931 | poly @t; |
---|
3932 | ring H=0,@t,dp; |
---|
3933 | def RH=RR+H; |
---|
3934 | setring(RH); |
---|
3935 | def FH=imap(RR,F); |
---|
3936 | list u; ideal B; ideal lpp; intvec lab; |
---|
3937 | FH=homog(FH,@t); |
---|
3938 | def Nh=imap(RR,N); |
---|
3939 | def Wh=imap(RR,W); |
---|
3940 | list LL; |
---|
3941 | if ((size(Nh)>0) or (size(Wh)>0)) |
---|
3942 | { |
---|
3943 | LL=mrcgs(FH,list("null",Nh,"nonnull",Wh)); |
---|
3944 | } |
---|
3945 | else |
---|
3946 | { |
---|
3947 | LL=mrcgs(FH); |
---|
3948 | } |
---|
3949 | setglobalrings(); |
---|
3950 | LL[1][3]=subst(LL[1][3],@t,1); |
---|
3951 | LL[1][4]=subst(LL[1][4],@t,1); |
---|
3952 | for (i=1; i<=LL[1][2]; i++) |
---|
3953 | { |
---|
3954 | lab=intvec(i); |
---|
3955 | u=tree(lab,LL); |
---|
3956 | postred=difflpp(u[1][3]); |
---|
3957 | B=sortideal(subst(LL[u[2]][4],@t,1)); |
---|
3958 | lpp=sortideal(subst(LL[u[2]][3],@t,1)); |
---|
3959 | if (memberpos(1,B)[1]){B=ideal(1); lpp=ideal(1);} |
---|
3960 | if (postred) |
---|
3961 | { |
---|
3962 | lpp=ideal(0); |
---|
3963 | B=postredgb(mingb(B)); |
---|
3964 | for (j=1;j<=size(B);j++){lpp[j]=leadmonom(B[j]);} |
---|
3965 | } |
---|
3966 | else{"Sheaves present, not reduced bases in the case lpp = ";lpp;} |
---|
3967 | LL[u[2]][4]=B; |
---|
3968 | LL[u[2]][3]=lpp; |
---|
3969 | } |
---|
3970 | setring(RR); |
---|
3971 | list LLL=imap(RH,LL); |
---|
3972 | kill @P; kill @R; kill @RP; |
---|
3973 | LLL[1][6]="RCGS"; |
---|
3974 | return(LLL); |
---|
3975 | } |
---|
3976 | //example |
---|
3977 | //{ "EXAMPLE:"; echo = 2; |
---|
3978 | // ring R=(0,b,c,d,e,f),(x,y),dp; |
---|
3979 | // ideal F=x^2+b*y^2+2*c*x*y+2*d*x+2*e*y+f, 2*x+2*c*y+2*d, 2*b*y+2*c*x+2*e; |
---|
3980 | // def T=rcgs(F); |
---|
3981 | // T; |
---|
3982 | // cantreetoMaple(T,"Tr","Tr.txt"); |
---|
3983 | // cantodiffcgs(T); |
---|
3984 | //} |
---|
3985 | |
---|
3986 | static proc difflpp(ideal lpp) |
---|
3987 | { |
---|
3988 | int t=1; int i; |
---|
3989 | poly lp1=lpp[1]; |
---|
3990 | poly lp; |
---|
3991 | i=2; |
---|
3992 | while ((i<=size(lpp)) and (t)) |
---|
3993 | { |
---|
3994 | lp=lpp[i]; |
---|
3995 | if (lp==lp1){t=0;} |
---|
3996 | lp1=lp; |
---|
3997 | i++; |
---|
3998 | } |
---|
3999 | return(t); |
---|
4000 | } |
---|
4001 | |
---|
4002 | // redgb: given a minimal bases (gb reducing) it |
---|
4003 | // reduces each polynomial wrt to the others |
---|
4004 | static proc postredgb(ideal F) |
---|
4005 | { |
---|
4006 | ideal G; |
---|
4007 | ideal H; |
---|
4008 | int i; |
---|
4009 | if (size(F)==0){return(ideal(0));} |
---|
4010 | for (i=1;i<=size(F);i++) |
---|
4011 | { |
---|
4012 | H=delfromideal(F,i); |
---|
4013 | G[i]=pdivi(F[i],H)[1]; |
---|
4014 | } |
---|
4015 | return(G); |
---|
4016 | } |
---|
4017 | |
---|
4018 | static proc crcgs(ideal F, list #) |
---|
4019 | //"USAGE: crcgs(F); |
---|
4020 | // F is the ideal from which to obtain the Canonical Reduced CGS. |
---|
4021 | // From the old library redcgs.lib. |
---|
4022 | // Alternatively, as option: |
---|
4023 | // crcgs(F,L); |
---|
4024 | // Options: We can give a list of options in the list L |
---|
4025 | // of the form |
---|
4026 | // ("null",ideal N,"nonnull",ideal W,"comment",int comment). |
---|
4027 | // One can give none till 3 of these options by giving the |
---|
4028 | // name of the option and the content. |
---|
4029 | // When options "null" and/or "nonnull" are given, then the |
---|
4030 | // parameter space is restricted to V(N)\V(h), where h is the product of |
---|
4031 | // the non null polynomials in W. If the option "comment" is set to 1, |
---|
4032 | // then information about the total number of segments of the |
---|
4033 | // output is printed. |
---|
4034 | // By default N=ideal(0) and W=ideal(1). |
---|
4035 | // crcgs is a routine whose output segments are always |
---|
4036 | // locally closed and correspond to homogenizing the ideal |
---|
4037 | // compute its mrcgs and then reduce and de-homogenizing the result. |
---|
4038 | // The result is in principle the Canonical Comprehensive Groebner System, |
---|
4039 | // similar to the result obtained by the fundamental routine grobcov, |
---|
4040 | // but the output is less friendly and not certified to be always |
---|
4041 | // the canonical Groebner cover. |
---|
4042 | //RETURN: The list T representing the canonical Reduced CGS. |
---|
4043 | // The description given here is identical for mrcgs and rcgs. |
---|
4044 | // The elements of the list T computed by crcgs are lists representing |
---|
4045 | // a rooted tree. |
---|
4046 | // Each element of the list T has the two first entries with the following content: |
---|
4047 | // [1]: The label (intvec) representing the position in the rooted |
---|
4048 | // tree: 0 for the root (and this is a special element) |
---|
4049 | // i for the root of the segment i |
---|
4050 | // (i,...) for the children of the segment i |
---|
4051 | // [2]: the number of children (int) of the vertex. |
---|
4052 | // There thus three kind of vertices: |
---|
4053 | // (1) the root (first element labelled 0), |
---|
4054 | // (2) the vertices labelled with a single integer i, |
---|
4055 | // (3) the rest of vertices labelled with more indices. |
---|
4056 | // Description of the root. Vertex type (1) |
---|
4057 | // There is a special vertex (the first one) whose content is |
---|
4058 | // the following: |
---|
4059 | // [3] lpp of the given ideal |
---|
4060 | // [4] the given ideal |
---|
4061 | // [5] the red-representation of the (optional) given null and non-null conditions |
---|
4062 | // (see redspec for the description) |
---|
4063 | // [6] CRCGS (to remember which algorithm has been used). If the |
---|
4064 | // algorithm used is mrcgs of rcgs then this will be stated |
---|
4065 | // at this vertex (MRCGS or RCGS). |
---|
4066 | // Description of vertices type (2). These are the vertices that |
---|
4067 | // initiate a segment, and are labelled with a single integer. |
---|
4068 | // [3] lpp (ideal) of the reduced basis. If they are repeated lpp's this |
---|
4069 | // will correspond to a sheaf. |
---|
4070 | // [4] the reduced basis (ideal) of the segment. |
---|
4071 | // Description of vertices type (3). These vertices have as first |
---|
4072 | // label i and descend form vertex i in the position of the label |
---|
4073 | // (i,...). They contain moreover a unique prime ideal in the parameters |
---|
4074 | // and form ascending chains of ideals. |
---|
4075 | // How is to be read the crcgs tree? The vertices with an even number of |
---|
4076 | // integers in the label are to be considered as additive and those |
---|
4077 | // with an odd number of integers in the label are to be considered as |
---|
4078 | // substraction. As an example consider the following vertices: |
---|
4079 | // v1=((i),2,lpp,B), |
---|
4080 | // v2=((i,1),2,P_(i,1)), |
---|
4081 | // v3=((i,1,1),2,P_(i,1,1)), |
---|
4082 | // v4=((i,1,1,1),1,P_(i,1,1,1)), |
---|
4083 | // v5=((i,1,1,1,1),0,P_(i,1,1,1,1)), |
---|
4084 | // v6=((i,1,1,2),1,P_(i,1,1,2)), |
---|
4085 | // v7=((i,1,1,2,1),0,P_(i,1,1,2,1)), |
---|
4086 | // v8=((i,1,2),0,P_(i,1,2)), |
---|
4087 | // v9=((i,2),1,P_(i,2)), |
---|
4088 | // v10=((i,2,1),0,P_(i,2,1)), |
---|
4089 | // They represent the segment: |
---|
4090 | // (V(i,1)\(((V(i,1,1) \ ((V(i,1,1,1) \ V(i,1,1,1,1)) u (V(i,1,1,2) \ V(i,1,1,2,1))))) |
---|
4091 | // u V(i,1,2))) u (V(i,2) \ V(i,2,1)) |
---|
4092 | // and can also be represented by |
---|
4093 | // (V(i,1) \ (V(i,1,1) u V(i,1,2))) u |
---|
4094 | // (V(i,1,1,1) \ V(i,1,1,1)) u |
---|
4095 | // (V(i,1,1,2) \ V(i,1,1,2,1)) u |
---|
4096 | // (V(i,2) \ V(i,2,1)) |
---|
4097 | // where V(i,j,..) = V(P_(i,j,..)) |
---|
4098 | //NOTE: There are three fundamental routines in the old library redcgs.lib: |
---|
4099 | // mrcgs, rcgs and crcgs. |
---|
4100 | // The output can be visualized using cantreetoMaple, that will |
---|
4101 | // write a file with the content of rcgs that can be read in Maple |
---|
4102 | // and plotted using the Maple plotcantree routine of the Monte's dpgb library |
---|
4103 | //KEYWORDS: mrcgs, crcgs, buildtree, cantreetoMaple, |
---|
4104 | //EXAMPLE: rcgs; shows an example" |
---|
4105 | { |
---|
4106 | int ish=1; int i=1; |
---|
4107 | while ((ish) and (i<=size(F))) |
---|
4108 | { |
---|
4109 | ish=ishomog(F[i]); |
---|
4110 | i++; |
---|
4111 | } |
---|
4112 | if (ish){return(mrcgs(F, #));} |
---|
4113 | def RR=basering; |
---|
4114 | // int comment=0; |
---|
4115 | // def N=ideal(0); |
---|
4116 | // def W=ideal(1); |
---|
4117 | // list L=#; |
---|
4118 | // for(i=1;i<=size(L) div 2;i++) |
---|
4119 | // { |
---|
4120 | // if(L[2*i-1]=="null"){N=L[2*i];} |
---|
4121 | // else |
---|
4122 | // { |
---|
4123 | // if(L[2*i-1]=="nonnull"){W=L[2*i];} |
---|
4124 | // else |
---|
4125 | // { |
---|
4126 | // if(L[2*i-1]=="comment"){comment=L[2*i];} |
---|
4127 | // } |
---|
4128 | // } |
---|
4129 | // } |
---|
4130 | setglobalrings(); |
---|
4131 | setring(@RP); |
---|
4132 | ideal FP=imap(RR,F); |
---|
4133 | option(redSB); |
---|
4134 | def G=std(FP); |
---|
4135 | setring(RR); |
---|
4136 | def GR=imap(@RP,G); |
---|
4137 | kill @P; kill @RP; kill @R; |
---|
4138 | list LL; |
---|
4139 | LL=rcgs(GR, #); |
---|
4140 | LL[1][6]="CRCGS"; |
---|
4141 | return(LL); |
---|
4142 | } |
---|
4143 | //example |
---|
4144 | //{ "EXAMPLE:"; echo = 2; |
---|
4145 | // ring R=(0,b,c,d,e,f),(x,y),dp; |
---|
4146 | // ideal F=x^2+b*y^2+2*c*x*y+2*d*x+2*e*y+f, 2*x+2*c*y+2*d, 2*b*y+2*c*x+2*e; |
---|
4147 | // def T=crcgs(F); |
---|
4148 | // T; |
---|
4149 | // cantreetoMaple(T,"Tc","Tc.txt"); |
---|
4150 | // cantodiffcgs(T); |
---|
4151 | //} |
---|
4152 | |
---|
4153 | //purpose ideal intersection called in @R and computed in @P |
---|
4154 | static proc idintR(ideal N, ideal M) |
---|
4155 | { |
---|
4156 | def RR=basering; |
---|
4157 | setring(@P); |
---|
4158 | def Np=imap(RR,N); |
---|
4159 | def Mp=imap(RR,M); |
---|
4160 | def Jp=idint(Np,Mp); |
---|
4161 | setring(RR); |
---|
4162 | return(imap(@P,Jp)); |
---|
4163 | } |
---|
4164 | |
---|
4165 | //purpose reduced groebner basis called in @R and computed in @P |
---|
4166 | static proc gbR(ideal N) |
---|
4167 | { |
---|
4168 | def RR=basering; |
---|
4169 | setring(@P); |
---|
4170 | def Np=imap(RR,N); |
---|
4171 | option(redSB); |
---|
4172 | Np=std(Np); |
---|
4173 | setring(RR); |
---|
4174 | return(imap(@P,Np)); |
---|
4175 | } |
---|
4176 | |
---|
4177 | // purpose: given the output of a locally closed CGS (i.e. from rcgs or crcgs) |
---|
4178 | // it returns the segments as difference of varieties. |
---|
4179 | static proc cantodiffcgs(list L) |
---|
4180 | //"USAGE: canttodiffcgs(T); |
---|
4181 | // T: is the list provided by mrcgs or crcgs or crcgs, |
---|
4182 | //RETURN: The list transforming the content of these routines to a simpler |
---|
4183 | // output where each segment corresponds to a single element of the list |
---|
4184 | // that is described as difference of two varieties. |
---|
4185 | // |
---|
4186 | // The first element of the list is identical to the first element |
---|
4187 | // of the list provided by the corresponding cgs algorithm, and |
---|
4188 | // contains general information on the call (see mrcgs). |
---|
4189 | // The remaining elements are lists of 4 elements, |
---|
4190 | // representing segments. These elements are |
---|
4191 | // [1]: the lpp of the segment |
---|
4192 | // [2]: the basis of the segment |
---|
4193 | // [3]; the ideal of the first variety (radical) |
---|
4194 | // [4]; the ideal of the second variety (radical) |
---|
4195 | // The segment is V([3]) \ V([4]). |
---|
4196 | // |
---|
4197 | //NOTE: It can be called from the output of mrcgs or rcgs of crcgs |
---|
4198 | //KEYWORDS: mrcgs, rcgs, crcgs, Maple |
---|
4199 | //EXAMPLE: cantodiffcgs; shows an example" |
---|
4200 | { |
---|
4201 | int i; int j; int k; int depth; list LL; list u; list v; list w; |
---|
4202 | ideal N; ideal Nn; ideal M; ideal Mn; ideal N0; ideal W0; |
---|
4203 | LL[1]=L[1]; |
---|
4204 | N0=L[1][5][1]; |
---|
4205 | W0=L[1][5][2]; |
---|
4206 | def RR=basering; |
---|
4207 | setring(@P); |
---|
4208 | def N0P=imap(RR,N0); |
---|
4209 | def W0P=imap(RR,N0); |
---|
4210 | ideal NP; |
---|
4211 | ideal MP; |
---|
4212 | setring(RR); |
---|
4213 | for (i=2;i<=size(L);i++) |
---|
4214 | { |
---|
4215 | depth=size(L[i][1]); |
---|
4216 | if (depth>3){ERROR("the given CGS has non locally closed segments");} |
---|
4217 | } |
---|
4218 | for (i=1;i<=L[1][2];i++) |
---|
4219 | { |
---|
4220 | N=ideal(1); |
---|
4221 | M=ideal(1); |
---|
4222 | u=tree(intvec(i),L); |
---|
4223 | for (j=1;j<=u[1][2];j++) |
---|
4224 | { |
---|
4225 | v=tree(intvec(i,j),L); |
---|
4226 | Nn=v[1][3]; |
---|
4227 | N=idintR(N,Nn); |
---|
4228 | for (k=1;k<=v[1][2];k++) |
---|
4229 | { |
---|
4230 | w=tree(intvec(i,j,k),L); |
---|
4231 | Mn=w[1][3]; |
---|
4232 | M=idintR(M,Mn); |
---|
4233 | } |
---|
4234 | } |
---|
4235 | setring(@P); |
---|
4236 | NP=imap(RR,N); |
---|
4237 | MP=imap(RR,M); |
---|
4238 | MP=MP+N0P; |
---|
4239 | for (j=1;j<=size(W0P);j++){MP=MP+ideal(W0P[j]);} |
---|
4240 | NP=NP+N0P; |
---|
4241 | NP=gbR(NP); |
---|
4242 | MP=gbR(MP); |
---|
4243 | setring(RR); |
---|
4244 | N=imap(@P,NP); |
---|
4245 | M=imap(@P,MP); |
---|
4246 | LL[i+1]=list(u[1][3],u[1][4],N,M); |
---|
4247 | } |
---|
4248 | return(LL); |
---|
4249 | } |
---|
4250 | //example |
---|
4251 | //{ "EXAMPLE:"; echo = 2; |
---|
4252 | // ring R=(0,b,c,d,e,f),(x,y),dp; |
---|
4253 | // ideal F=x^2+b*y^2+2*c*x*y+2*d*x+2*e*y+f, 2*x+2*c*y+2*d, 2*b*y+2*c*x+2*e; |
---|
4254 | // def T=crcgs(F); |
---|
4255 | // T; |
---|
4256 | // cantreetoMaple(T,"Tc","Tc.txt"); |
---|
4257 | // cantodiffcgs(T); |
---|
4258 | //} |
---|
4259 | |
---|
4260 | //**************End homogenizing************************ |
---|
4261 | |
---|
4262 | //**************End of redcgs************************ |
---|
4263 | |
---|
4264 | //**************Begin of Groebner Cover***************** |
---|
4265 | |
---|
4266 | // incquotient |
---|
4267 | // incremental quotient |
---|
4268 | // Input: ideal N: a Groebner basis of an ideal |
---|
4269 | // poly f: |
---|
4270 | // Output: Na = N:<f> |
---|
4271 | static proc incquotient(ideal N, poly f) |
---|
4272 | { |
---|
4273 | poly g; int i; |
---|
4274 | ideal Nb; ideal Na=N; |
---|
4275 | |
---|
4276 | // begins incquotient |
---|
4277 | if (size(Na)==1) |
---|
4278 | { |
---|
4279 | g=gcd(Na[1],f); |
---|
4280 | if (g!=1) |
---|
4281 | { |
---|
4282 | Na[1]=Na[1]/g; |
---|
4283 | } |
---|
4284 | attrib(Na,"IsSB",1); |
---|
4285 | return(Na); |
---|
4286 | } |
---|
4287 | def P=basering; |
---|
4288 | poly @t; |
---|
4289 | ring H=0,@t,lp; |
---|
4290 | def HP=H+P; |
---|
4291 | setring(HP); |
---|
4292 | def fh=imap(P,f); |
---|
4293 | def Nh=imap(P,N); |
---|
4294 | ideal Nht; |
---|
4295 | for (i=1;i<=size(Nh);i++) |
---|
4296 | { |
---|
4297 | Nht[i]=Nh[i]*@t; |
---|
4298 | } |
---|
4299 | attrib(Nht,"isSB",1); |
---|
4300 | def fht=(1-@t)*fh; |
---|
4301 | option(redSB); |
---|
4302 | Nht=std(Nht,fht); |
---|
4303 | ideal Nc; ideal v; |
---|
4304 | for (i=1;i<=size(Nht);i++) |
---|
4305 | { |
---|
4306 | v=variables(Nht[i]); |
---|
4307 | if(memberpos(@t,v)[1]==0) |
---|
4308 | { |
---|
4309 | Nc[size(Nc)+1]=Nht[i]/fh; |
---|
4310 | } |
---|
4311 | } |
---|
4312 | setring(P); |
---|
4313 | ideal HH; |
---|
4314 | def Nd=imap(HP,Nc); Nb=Nd; |
---|
4315 | option(redSB); |
---|
4316 | Nb=std(Nd); |
---|
4317 | return(Nb); |
---|
4318 | } |
---|
4319 | |
---|
4320 | // RrepNN: given a red-representation of a locally closed set and a new |
---|
4321 | // assumed non-null polynomial f, it returns the new R-representation. |
---|
4322 | // Called in any @P |
---|
4323 | // 13/09/2010 |
---|
4324 | // input: |
---|
4325 | // ideal N : the ideal of null-conditions |
---|
4326 | // ideal W : non-null set of polynomials. (N,W) is a R-representation of the |
---|
4327 | // initial locally closed set. |
---|
4328 | // poly f : A new assumed non-null polynomial |
---|
4329 | // returns: list (N1,W1), the new R-representation: |
---|
4330 | // N1 = new radical of the null conditions of the R-representation |
---|
4331 | // W1 = non-null list of polynomials of the new R-representation. |
---|
4332 | // If the given conditions are not compatible, then N1=ideal(1). This should not |
---|
4333 | // happen, because this has to be tested before using RrepNN. |
---|
4334 | |
---|
4335 | static proc RrepNN(ideal N, ideal W, poly f) |
---|
4336 | //"USAGE: RrepNN(N,W,f); |
---|
4337 | // N: null conditions ideal of the initial R-representation |
---|
4338 | // W: non-null list of polynomials of the initial R-representation |
---|
4339 | // f: new assumed non-null polynomial |
---|
4340 | //RETURN: a list (N1,W1) containing the new R-representation of the segment |
---|
4341 | // (N,W) adding the new non-null condition f. |
---|
4342 | //NOTE: Called from parameter ring (@P). |
---|
4343 | //KEYWORDS: representation |
---|
4344 | //EXAMPLE: RrepNN; shows an example" |
---|
4345 | { |
---|
4346 | ideal F=f; ideal W1=W; |
---|
4347 | def N1=incquotient(N,f); |
---|
4348 | option(redSB); |
---|
4349 | N1=std(N1); |
---|
4350 | //attrib(N1,"IsSB",1); |
---|
4351 | def H=sqrfree(f, 1); |
---|
4352 | int i; |
---|
4353 | for(i=1;i<=size(H);i++){W1[size(W1)+1]=reduce(H[i],N1);} |
---|
4354 | |
---|
4355 | W1=facvar(W1); |
---|
4356 | if (size(W1)==0){W1=1;} |
---|
4357 | return(list(N1,W1)); |
---|
4358 | } |
---|
4359 | //example |
---|
4360 | //{ "EXAMPLE:"; echo = 2; |
---|
4361 | // ring r=(0,a,b,c),(x,y),dp; |
---|
4362 | // setglobalrings(); |
---|
4363 | // ideal N=(ab-c)*(a-b),(a-bc)*(a-b); |
---|
4364 | // poly h=(a+b)bc; |
---|
4365 | // poly f=a-b; |
---|
4366 | //} |
---|
4367 | |
---|
4368 | // RrepN: given a red-representation of a locally closed set and a new |
---|
4369 | // assumed null polynomial f, that is not identically null, it returns |
---|
4370 | // the new red-representation. |
---|
4371 | // Called in ring @P |
---|
4372 | // 13/09/2010 |
---|
4373 | // input: |
---|
4374 | // ideal N : the ideal of null-conditions |
---|
4375 | // ideal W : non-null list of polynomials. (N,W) is a R-representation of the |
---|
4376 | // initial locally closed set. |
---|
4377 | // poly f : A new assumed null polynomial |
---|
4378 | // returns: list (N1,W1), the new R-representation: |
---|
4379 | // N1 = new radical of the null conditions of the R-representation |
---|
4380 | // W1 = non-null list of polynomials of the new R-representation. |
---|
4381 | // If the given conditions are not compatible, then N1=ideal(1). |
---|
4382 | static proc RrepN(ideal N, ideal W, poly f) |
---|
4383 | //"USAGE: RrepN(N,W,f); |
---|
4384 | // N: null conditions ideal of the initial R-representation |
---|
4385 | // W: non-null list of polynomials of the initial R-representation |
---|
4386 | // f: new assumed null polynomial |
---|
4387 | //RETURN: a list (N1,W1) containing the new R-representation of the segment |
---|
4388 | // (N,W) adding the new non-null condition f. |
---|
4389 | //NOTE: Called from parameter ring (@P). |
---|
4390 | //KEYWORDS: representation |
---|
4391 | //EXAMPLE: RrepN; shows an example" |
---|
4392 | { |
---|
4393 | attrib(N,"isSB",1); |
---|
4394 | def N1=std(N,f); |
---|
4395 | option(redSB); |
---|
4396 | N1=std(radical(N1)); |
---|
4397 | int i; |
---|
4398 | poly h; |
---|
4399 | for (i=1;i<=size(W);i++) |
---|
4400 | { |
---|
4401 | h=W[i]; |
---|
4402 | N1=incquotient(N1,h); |
---|
4403 | } |
---|
4404 | option(redSB); |
---|
4405 | N1=std(N1); |
---|
4406 | def W1=W; |
---|
4407 | if (size(W1)==0){W1=1;} |
---|
4408 | return(list(N1,W1)); |
---|
4409 | } |
---|
4410 | //example |
---|
4411 | //{ "EXAMPLE:"; echo = 2; |
---|
4412 | // ring r=(0,a,b,c),(x,y),dp; |
---|
4413 | // setglobalrings(); |
---|
4414 | // ideal N=(ab-c)*(a-b),(a-bc)*(a-b); |
---|
4415 | // poly h=(a+b)bc; |
---|
4416 | // poly f=a-b; |
---|
4417 | // RrepN(N,h,f); |
---|
4418 | //} |
---|
4419 | |
---|
4420 | // Rrep: generates a R-representation |
---|
4421 | // called from any ring |
---|
4422 | // it uses ring @P, thus the globalrings @P, @RP, @R must be |
---|
4423 | // active by a previous call to setglobalrings(); |
---|
4424 | // 13/09/2010 |
---|
4425 | // input: |
---|
4426 | // ideal N : the ideal of null-conditions (not necessarily radical nor canonical) |
---|
4427 | // ideal W : set of non-null polynomials: if W corresponds to no non null |
---|
4428 | // conditions then W=ideal(0) |
---|
4429 | // otherwise it should be given as an ideal. |
---|
4430 | // returns: list (Na,Wa) |
---|
4431 | // the R-representation of (N,W): |
---|
4432 | // ideal Na = radical of the R-representation (canonical) |
---|
4433 | // ideal Wa = set of non-null polynomials in the R-representation. |
---|
4434 | // if it corresponds to no non null conditions then it is ideal(0) |
---|
4435 | // otherwise the ideal is returned. |
---|
4436 | // If the given conditions are not compatible, then N=ideal(1). |
---|
4437 | static proc Rrep(ideal Ni, ideal Wi) |
---|
4438 | //"USAGE: Rrep(N,W); |
---|
4439 | // N: null conditions ideal |
---|
4440 | // W: set of non-null polynomials (ideal) |
---|
4441 | //RETURN: a list (N1,W1) containing the R-representation of the segment (N,W). |
---|
4442 | // N1 is the radical reduced ideal characterizing the segment. |
---|
4443 | // V(N1) is the Zarisky closure of the segment (N,W). |
---|
4444 | // The segment S=V(N1) \ V(h), where h=prod(w in W1) |
---|
4445 | // N1 is uniquely determined and no prime component of N1 contains none of |
---|
4446 | // the polynomials in W1. |
---|
4447 | //NOTE: Can be called from ring @R but it works in ring @P. Thus |
---|
4448 | // the globalrings @P, @RP, @R must be active by a previous call |
---|
4449 | // to setglobalrings(); |
---|
4450 | //KEYWORDS: R-representation |
---|
4451 | //EXAMPLE: Rrep shows an example" |
---|
4452 | { |
---|
4453 | def RR=basering; |
---|
4454 | setring(@P); |
---|
4455 | def N=imap(RR,Ni); |
---|
4456 | option(redSB); |
---|
4457 | N=std(radical(N)); |
---|
4458 | def W=imap(RR,Wi); |
---|
4459 | if(size(W)==0){W=ideal(0);} |
---|
4460 | //when there are no non-null conditions then W=ideal(1) |
---|
4461 | else |
---|
4462 | { |
---|
4463 | W=facvar(W); |
---|
4464 | } |
---|
4465 | if (size(W)==0) |
---|
4466 | { |
---|
4467 | setring(RR); |
---|
4468 | //def Wb=imap(@P,W); |
---|
4469 | return(list(imap(@P,N), ideal(1))); |
---|
4470 | } |
---|
4471 | else |
---|
4472 | { |
---|
4473 | int i; //ideal F; |
---|
4474 | for (i=1;i<=size(W);i++) |
---|
4475 | { |
---|
4476 | //F=W[i]; |
---|
4477 | N=incquotient(N,W[i]); |
---|
4478 | } |
---|
4479 | option(redSB); |
---|
4480 | N=std(N); |
---|
4481 | setring(RR); |
---|
4482 | def Nb=imap(@P,N); |
---|
4483 | def Wb=imap(@P,W); |
---|
4484 | if (equalideals(Wb,ideal(0))){Wb=ideal(1);} |
---|
4485 | return(list(Nb,Wb)); |
---|
4486 | } |
---|
4487 | } |
---|
4488 | //example |
---|
4489 | //{ "EXAMPLE:"; echo = 2; |
---|
4490 | // ring R=(0,a,b,c),(x,y),dp; |
---|
4491 | // setglobalrings(); |
---|
4492 | // ideal N=(ab-c)*(a-b),(a-bc)*(a-b); |
---|
4493 | // ideal W=a^2-b^2,bc; |
---|
4494 | // Rrep(N,W); |
---|
4495 | //} |
---|
4496 | |
---|
4497 | |
---|
4498 | // eliminate the ith element from a list |
---|
4499 | static proc elimfromlist(list l, int i) |
---|
4500 | { |
---|
4501 | list L; int j; |
---|
4502 | for(j=1;j<=i-1;j++) |
---|
4503 | {L[j]=l[j];} |
---|
4504 | for(j=i+1;j<=size(l);j++) |
---|
4505 | {L[j-1]=l[j];} |
---|
4506 | return(L); |
---|
4507 | } |
---|
4508 | |
---|
4509 | static proc idbefid(ideal a, ideal b) |
---|
4510 | { |
---|
4511 | poly fa; poly fb; poly la; poly lb; |
---|
4512 | int te=1; int i; int j; |
---|
4513 | int na=size(a); |
---|
4514 | int nb=size(b); |
---|
4515 | int nm; |
---|
4516 | if (na<=nb){nm=na;} else{nm=nb;} |
---|
4517 | for (i=1;i<=nm; i++) |
---|
4518 | { |
---|
4519 | fa=a[i]; fb=b[i]; |
---|
4520 | while((fa!=0) or (fb!=0)) |
---|
4521 | { |
---|
4522 | la=lead(fa); |
---|
4523 | lb=lead(fb); |
---|
4524 | fa=fa-la; |
---|
4525 | fb=fb-lb; |
---|
4526 | la=leadmonom(la); |
---|
4527 | lb=leadmonom(lb); |
---|
4528 | if(leadmonom(la+lb)!=la){return(1);} |
---|
4529 | else{if(leadmonom(la+lb)!=lb){return(2);}} |
---|
4530 | } |
---|
4531 | } |
---|
4532 | if(na<nb){return(1);} else{if(na>nb){return(2);} else{return(0);}} |
---|
4533 | } |
---|
4534 | |
---|
4535 | static proc sortlistideals(list L) |
---|
4536 | { |
---|
4537 | int i; int j; int n; |
---|
4538 | ideal a; ideal b; |
---|
4539 | list LL=L; |
---|
4540 | list NL; |
---|
4541 | int k; int te; |
---|
4542 | i=1; |
---|
4543 | while(size(LL)>0) |
---|
4544 | { |
---|
4545 | k=1; |
---|
4546 | for(j=2;j<=size(LL);j++) |
---|
4547 | { |
---|
4548 | te=idbefid(LL[k],LL[j]); |
---|
4549 | if (te==2){k=j;} |
---|
4550 | } |
---|
4551 | NL[size(NL)+1]=LL[k]; |
---|
4552 | n=size(LL); |
---|
4553 | if (n>1){LL=elimfromlist(LL,k);} else{LL=list();} |
---|
4554 | } |
---|
4555 | return(NL); |
---|
4556 | } |
---|
4557 | |
---|
4558 | // returns 1 if the two lists of ideals are equal and 0 if not |
---|
4559 | static proc equallistideals(list L, list M) |
---|
4560 | { |
---|
4561 | int t; int i; |
---|
4562 | if (size(L)!=size(M)){return(0);} |
---|
4563 | else |
---|
4564 | { |
---|
4565 | t=1; |
---|
4566 | if (size(L)>0) |
---|
4567 | { |
---|
4568 | i=1; |
---|
4569 | while ((t==1) and (i<=size(L))) |
---|
4570 | { |
---|
4571 | if (equalideals(L[i],M[i])==0){t=0;} |
---|
4572 | i++; |
---|
4573 | } |
---|
4574 | } |
---|
4575 | return(t); |
---|
4576 | } |
---|
4577 | } |
---|
4578 | |
---|
4579 | // RtoPrepNew |
---|
4580 | // Computes the P-representaion of a R-representaion (N,W) of a set |
---|
4581 | // input: |
---|
4582 | // ideal N (null conditions, must be radical) |
---|
4583 | // ideal W (non-null conditions ideal) |
---|
4584 | // list L must contain the radical decomposition of N. |
---|
4585 | // output: |
---|
4586 | // the ((p_1,(p_11,..,p_1k_1)),..,(p_r,(p_r1,..,p_rk_r))); |
---|
4587 | // the Prep of V(N) \ V(h), where h=prod(w in W). |
---|
4588 | static proc RtoPrepNew(ideal N, ideal W) |
---|
4589 | { |
---|
4590 | int i; int j; list L0; |
---|
4591 | if (N[1]==1) |
---|
4592 | { |
---|
4593 | L0[1]=list(ideal(1),list(ideal(1))); |
---|
4594 | return(L0); |
---|
4595 | } |
---|
4596 | def RR=basering; |
---|
4597 | setring(@P); |
---|
4598 | ideal Np=imap(RR,N); |
---|
4599 | ideal Wp=imap(RR,W); |
---|
4600 | list Lp=minGTZ(Np); |
---|
4601 | for(i=1;i<=size(Lp);i++) |
---|
4602 | { |
---|
4603 | option(redSB); |
---|
4604 | Lp[i]=std(Lp[i]); |
---|
4605 | } |
---|
4606 | //list Lp=imap(RR,L); |
---|
4607 | poly h=1; |
---|
4608 | for (i=1;i<=size(Wp);i++){h=h*Wp[i];} |
---|
4609 | list r; list Ti; list LL; |
---|
4610 | for (i=1;i<=size(Lp);i++) |
---|
4611 | { |
---|
4612 | Ti=minGTZ(Lp[i]+h); |
---|
4613 | for(j=1;j<=size(Ti);j++) |
---|
4614 | { |
---|
4615 | option(redSB); |
---|
4616 | Ti[j]=std(Ti[j]); |
---|
4617 | } |
---|
4618 | //list LL[i]; |
---|
4619 | LL[i]=list(Lp[i],Ti); |
---|
4620 | } |
---|
4621 | setring(RR); |
---|
4622 | return(imap(@P,LL)); |
---|
4623 | } |
---|
4624 | |
---|
4625 | // splitR: a new leading coefficient f is given to a R-representation |
---|
4626 | // then splitR computes the two new R-representation by |
---|
4627 | // considering it null, and non null. |
---|
4628 | // Can be called from any ring but it works in ring @P |
---|
4629 | // 14/09/2010 |
---|
4630 | // given the R-representation (N,W) and a new`polynomial f, |
---|
4631 | // it outputs the null and the non-null R-representations adding f. |
---|
4632 | // if the output R-representation (N0,W0) has N0==ideal(1) then |
---|
4633 | // there must be no split and recbtcgs must continue on |
---|
4634 | // the compatible (N1,W1) R-representation. |
---|
4635 | // input: |
---|
4636 | // ideal N: null-ideal of the R-representation |
---|
4637 | // ideal W: non-null list of polynomials of the R-representation |
---|
4638 | // poly f coefficient to split if needed |
---|
4639 | // output: |
---|
4640 | // list L = (list(ideal N0, ideal W0), list(ideal N1, ideal W1)) |
---|
4641 | static proc splitR(ideal Ni, ideal Wi, poly fi) |
---|
4642 | { |
---|
4643 | def RR=basering; |
---|
4644 | setring(@P); |
---|
4645 | def f=imap(RR,fi); |
---|
4646 | def N=imap(RR,Ni); |
---|
4647 | def W=imap(RR,Wi); |
---|
4648 | def L0=RrepN(N,W,f); |
---|
4649 | if(L0[1][1]==1) |
---|
4650 | { |
---|
4651 | setring(RR); |
---|
4652 | def LL0=list(ideal(1),ideal(1)); |
---|
4653 | list LL1=list(Ni,Wi); |
---|
4654 | return(list(LL0,LL1)); |
---|
4655 | } |
---|
4656 | else |
---|
4657 | { |
---|
4658 | def L1=RrepNN(N,W,f); |
---|
4659 | setring(RR); |
---|
4660 | def LL0=imap(@P,L0); |
---|
4661 | def LL1=imap(@P,L1); |
---|
4662 | return(list(LL0,LL1)); |
---|
4663 | } |
---|
4664 | } |
---|
4665 | |
---|
4666 | // Prep |
---|
4667 | // Computes the P-representation of V(N) \ V(M). |
---|
4668 | // input: |
---|
4669 | // ideal N (null ideal) (not necessarily radical nor maximal) |
---|
4670 | // ideal M (hole ideal) (not necessarily containing N) |
---|
4671 | // output: |
---|
4672 | // the ((p_1,(p_11,p_1k_1)),..,(p_r,(p_r1,p_rk_r))); |
---|
4673 | // the Prep of V(N)\V(M) |
---|
4674 | // Assumed to work in the ring @P of the parameters |
---|
4675 | static proc Prep(ideal N, ideal M) |
---|
4676 | { |
---|
4677 | if (N[1]==1) |
---|
4678 | { |
---|
4679 | //L0=list(list(ideal(1),list(ideal(1)))); |
---|
4680 | return(list(list(ideal(1),list(ideal(1))))); |
---|
4681 | } |
---|
4682 | def RR=basering; |
---|
4683 | setring(@P); |
---|
4684 | ideal Np=imap(RR,N); |
---|
4685 | ideal Mp=imap(RR,M); |
---|
4686 | int i; int j; list L0; |
---|
4687 | |
---|
4688 | list Ni=minGTZ(Np); |
---|
4689 | list prep; |
---|
4690 | for(j=1;j<=size(Ni);j++) |
---|
4691 | { |
---|
4692 | option(redSB); |
---|
4693 | Ni[j]=std(Ni[j]); |
---|
4694 | } |
---|
4695 | list Mij; |
---|
4696 | for (i=1;i<=size(Ni);i++) |
---|
4697 | { |
---|
4698 | Mij=minGTZ(Ni[i]+Mp); |
---|
4699 | for(j=1;j<=size(Mij);j++) |
---|
4700 | { |
---|
4701 | option(redSB); |
---|
4702 | Mij[j]=std(Mij[j]); |
---|
4703 | } |
---|
4704 | if ((size(Mij)==1) and (equalideals(Ni[i],Mij[1])==1)){;} |
---|
4705 | else |
---|
4706 | { |
---|
4707 | prep[size(prep)+1]=list(Ni[i],Mij); |
---|
4708 | } |
---|
4709 | } |
---|
4710 | if (size(prep)==0){prep=list(list(ideal(1),list(ideal(1))));} |
---|
4711 | setring(RR); |
---|
4712 | return(imap(@P,prep)); |
---|
4713 | } |
---|
4714 | |
---|
4715 | // PtoCrep |
---|
4716 | // Computes the C-representation from the P-representation. |
---|
4717 | // input: |
---|
4718 | // list ((p_1,(p_11,p_1k_1)),..,(p_r,(p_r1,p_rk_r))); |
---|
4719 | // the P-representation of V(N)\V(M) |
---|
4720 | // output: |
---|
4721 | // list (ideal ida, ideal idb) |
---|
4722 | // the C-represen taion of V(N)\V(M) = V(ida)\V(idb) |
---|
4723 | // Assumed to work in the ring @P of the parameters |
---|
4724 | static proc PtoCrep(list L) |
---|
4725 | { |
---|
4726 | def RR=basering; |
---|
4727 | setring(@P); |
---|
4728 | def Lp=imap(RR,L); |
---|
4729 | int i; int j; |
---|
4730 | ideal ida=ideal(1); ideal idb=ideal(1); list Lb; ideal N; |
---|
4731 | for (i=1;i<=size(Lp);i++) |
---|
4732 | { |
---|
4733 | option(returnSB); |
---|
4734 | N=Lp[i][1]; |
---|
4735 | ida=intersect(ida,N); |
---|
4736 | Lb=Lp[i][2]; |
---|
4737 | for(j=1;j<=size(Lb);j++) |
---|
4738 | { |
---|
4739 | idb=intersect(idb,Lb[j]); |
---|
4740 | } |
---|
4741 | } |
---|
4742 | def La=list(ida,idb); |
---|
4743 | setring(RR); |
---|
4744 | return(imap(@P,La)); |
---|
4745 | } |
---|
4746 | |
---|
4747 | // addnewpairs: |
---|
4748 | // 14/09/2010 |
---|
4749 | // input: |
---|
4750 | // ideal F, the given ideal |
---|
4751 | // list P: the list of existing pairs to be computed |
---|
4752 | // int l (the new index to add S-pols) |
---|
4753 | // output: list of ordered pairs (i,j,lcmij) of F in ascending order of lcmij |
---|
4754 | // adding the new (i,l,lcmil) and placing them in order of ascending lcm |
---|
4755 | // if a pair verifies Buchberger 1st criterion it is not stored |
---|
4756 | // ring @R |
---|
4757 | static proc addnewpairs(ideal F, list P, int l) |
---|
4758 | { |
---|
4759 | int i; |
---|
4760 | poly lm; |
---|
4761 | poly lpf; |
---|
4762 | poly lpg; |
---|
4763 | list P1=P; |
---|
4764 | list pair; |
---|
4765 | if (size(F)<=1){return(P);} |
---|
4766 | for (i=1;i<l;i++) |
---|
4767 | { |
---|
4768 | lm=lcmlmonoms(F[i],F[l]); |
---|
4769 | // Buchberger 1st criterion |
---|
4770 | lpf=leadmonom(F[i]); |
---|
4771 | lpg=leadmonom(F[l]); |
---|
4772 | if (lpf*lpg!=lm) |
---|
4773 | { |
---|
4774 | pair=(i,l,lm); |
---|
4775 | P1=placepairinlist(pair,P1); |
---|
4776 | } |
---|
4777 | } |
---|
4778 | return(P1); |
---|
4779 | } |
---|
4780 | |
---|
4781 | // DiscussPolys: given the data in a vertex of btcgs (BuildTree), it analyzes the |
---|
4782 | // leadcoef of the polynomials in B until it finds |
---|
4783 | // that one of them can be either null or non null. |
---|
4784 | // In that case, recbtcgs has to split into two branches, and then |
---|
4785 | // l < size(B) |
---|
4786 | // If not, and at the end only the non null option is compatible |
---|
4787 | // then the reduced B has all the leadcoef non null, and then l=size(B). |
---|
4788 | // 15/09/2010 |
---|
4789 | // ring @R |
---|
4790 | // input: |
---|
4791 | // B: (ideal) the actual basis |
---|
4792 | // N: (ideal) null conditions (R-rep) |
---|
4793 | // W: (ideal) non-null conditions set (R-rep) |
---|
4794 | // P: (list) of pairs of indices of S-polynomials that can and must be computed |
---|
4795 | // (its leading coefficients are non-null, and using Buchberger's |
---|
4796 | // criterions they are to be computed) |
---|
4797 | // l: (integer) representing the last polynomial in B for which the leading |
---|
4798 | // coefficient is already assumed non-null. |
---|
4799 | // output: list of (cond,lpp,B,N0,W0,P0,l0,N1,W1,P1,l1) |
---|
4800 | // cond (poly) is the polynomial responsible of the branch |
---|
4801 | // B is the new discussed basis. (It can contain less polynomials when |
---|
4802 | // some polynomial has been reduced to 0 by previous null-assumptions. |
---|
4803 | // (N0,W0,P0,l0) and (N1,W1,P1,l1) are respectively the R-representation, |
---|
4804 | // list of S-polys to be computed, and the last poly with assumed non-null |
---|
4805 | // coefficient in both the null side and the non-null side. |
---|
4806 | static proc DiscussPolys(ideal B, ideal N, ideal W, list P, int l) |
---|
4807 | { |
---|
4808 | list Pn=P; ideal Bn=B; int ln=l; ideal Nn=N; ideal Wn=W; |
---|
4809 | int testsplit=0; |
---|
4810 | poly f; poly lc; list L; int j; |
---|
4811 | int l0; int l1; list P0; list P1; ideal N0; ideal W0; ideal N1; ideal W1; |
---|
4812 | while((testsplit==0) and (ln<size(Bn))) |
---|
4813 | { |
---|
4814 | //f=redcoefs(Bn[ln+1],Nn); |
---|
4815 | f=pnormalform(Bn[ln+1],Nn,Wn); |
---|
4816 | if (f==0) |
---|
4817 | { |
---|
4818 | Bn=delfromideal(Bn,ln+1); //lppn=delfromideal(lppn,ln+1); |
---|
4819 | } |
---|
4820 | else |
---|
4821 | { |
---|
4822 | Bn[ln+1]=f; |
---|
4823 | lc=leadcoef(f); |
---|
4824 | L=splitR(Nn,Wn,lc); |
---|
4825 | N0=L[1][1]; |
---|
4826 | W0=L[1][2]; |
---|
4827 | N1=L[2][1]; |
---|
4828 | W1=L[2][2]; |
---|
4829 | P1=addnewpairs(Bn,Pn,ln+1); // uses Buchberger pair selection and standard order |
---|
4830 | if(N0[1]<>1) |
---|
4831 | { |
---|
4832 | testsplit=1; |
---|
4833 | l0=ln; l1=ln+1; |
---|
4834 | P0=Pn; |
---|
4835 | } |
---|
4836 | else |
---|
4837 | { |
---|
4838 | Pn=P1; P0=list(); ln=ln+1; Nn=N1; Wn=W1; l1=ln; |
---|
4839 | } |
---|
4840 | } |
---|
4841 | } |
---|
4842 | if(testsplit==0) |
---|
4843 | { |
---|
4844 | N1=Nn; W1=Wn; N0=ideal(1); W0=ideal(0); P0=list(); |
---|
4845 | l0=size(Bn); l1=size(Bn); P1=Pn; |
---|
4846 | } |
---|
4847 | return(list(lc,Bn,N0,W0,P0,l0,N1,W1,P1,l1)); |
---|
4848 | } |
---|
4849 | |
---|
4850 | // DiscussSPolys: given the data in a vertex of btcgs (BuildTree), |
---|
4851 | // and when DiscussPolys has already built a vertex where |
---|
4852 | // all the leadcoef are non-null in the R-representation, |
---|
4853 | // it computes and reduces the S-polys in the list P in order |
---|
4854 | // until it finds some non-reducing one. Then adds it to the |
---|
4855 | // basis and modifies the list P. |
---|
4856 | // Then it calls splitR and if the leadcoef non-null is, it |
---|
4857 | // continues with the next S-poly in the list. |
---|
4858 | // Else it finishes and recbtcgs will need to split. |
---|
4859 | // 15/09/2010 |
---|
4860 | // ring @R |
---|
4861 | // input: |
---|
4862 | // B: (ideal) the actual basis |
---|
4863 | // N: (ideal) null conditions (R-rep) |
---|
4864 | // W: (ideal) non-null conditions set (R-rep) |
---|
4865 | // P: (list) of pairs of indices of S-polynomials that can and must be computed |
---|
4866 | // (its leading coefficients are non-null, and using Buchberger's |
---|
4867 | // criterions they are to be computed) |
---|
4868 | // l: (integer) representing the last polynomial in B for which the leading |
---|
4869 | // coefficient is already assumed non-null. |
---|
4870 | // output: list of (cond,lpp,B,N0,W0,P0,l0,N1,W1,P1,l1) |
---|
4871 | // cond (poly) is the polynomial responsible of the branch |
---|
4872 | // B is the new discussed basis. (It can contain less polynomials when |
---|
4873 | // some polynomial has been reduced to 0 by previous null-assumptions. |
---|
4874 | // (N0,W0,P0,l0) and (N1,W1,P1,l1) are respectively the R-representation, |
---|
4875 | // list of S-polys to be computed, and the last poly with assumed non-null |
---|
4876 | // coefficient in both the null side and the non-null side. |
---|
4877 | static proc DiscussSPolys(ideal B,ideal N,ideal W,list P,int l) |
---|
4878 | { |
---|
4879 | def RR=basering; |
---|
4880 | list Pn=P; ideal Bn=B; int ln=l; ideal Nn=N; ideal Wn=W; |
---|
4881 | int testsplit=0; |
---|
4882 | poly lc; list L; int i; int j; poly S; list pair; |
---|
4883 | int l0; int l1; list P0; list P1; ideal N0; ideal W0; ideal N1; ideal W1; |
---|
4884 | // poly lc0; |
---|
4885 | while((testsplit==0) and (size(Pn)<>0)) |
---|
4886 | { |
---|
4887 | pair=Pn[1]; |
---|
4888 | i=pair[1]; j=pair[2]; |
---|
4889 | Pn=delete(Pn,1); |
---|
4890 | lc=1; N1=Nn; W1=Wn; |
---|
4891 | S=pspol(Bn[i],Bn[j]); |
---|
4892 | S=pdivi(S,Bn)[1]; |
---|
4893 | //S=redcoefs(S,Nn); |
---|
4894 | S=pnormalform(S,Nn,Wn); |
---|
4895 | if (S<>0) |
---|
4896 | { |
---|
4897 | Bn[size(Bn)+1]=S; |
---|
4898 | lc=leadcoef(S); |
---|
4899 | ln=ln+1; |
---|
4900 | L=splitR(Nn,Wn,lc); |
---|
4901 | N0=L[1][1]; |
---|
4902 | W0=L[1][2]; |
---|
4903 | N1=L[2][1]; |
---|
4904 | W1=L[2][2]; |
---|
4905 | P1=addnewpairs(Bn,Pn,ln); // uses Buchberger pair selection and standard order |
---|
4906 | if(N0[1]<>1) |
---|
4907 | { |
---|
4908 | testsplit=1; |
---|
4909 | l0=ln-1; l1=ln; |
---|
4910 | P0=Pn; |
---|
4911 | } |
---|
4912 | else |
---|
4913 | { |
---|
4914 | Pn=P1; Nn=N1; Wn=W1; P0=list(); W0=ideal(0); |
---|
4915 | } |
---|
4916 | } |
---|
4917 | } |
---|
4918 | if(testsplit==0) |
---|
4919 | { |
---|
4920 | N0=ideal(1); W0=ideal(0); P0=list(); l0=0; N1=Nn; W1=Wn; |
---|
4921 | l1=size(Bn); |
---|
4922 | } |
---|
4923 | return(list(lc,Bn,N0,W0,P0,l0,N1,W1,P1,l1)); |
---|
4924 | } |
---|
4925 | |
---|
4926 | // cgsdr |
---|
4927 | // 20/09/2010 |
---|
4928 | proc cgsdr(ideal F, list #) |
---|
4929 | "USAGE: cgsdr(F); To compute a disjoint, reduced CGS. |
---|
4930 | cgsdr is the starting point of the fundamental routine grobcov. |
---|
4931 | It is to be used if only a disjoint reduced CGS is required. |
---|
4932 | F: ideal in Q[a][x] (parameters and variables) to be discussed. |
---|
4933 | |
---|
4934 | Options: To modify the default options, pairs of arguments |
---|
4935 | -option name, value- of valid options must be added to the call. |
---|
4936 | |
---|
4937 | Options: |
---|
4938 | "null",ideal N: The default is "null",ideal(0). |
---|
4939 | "nonnull",ideal W: The default "nonnull",ideal(1). |
---|
4940 | When options "null" and/or "nonnull" are given, then |
---|
4941 | the parameter space is restricted to V(N) \ V(h), where |
---|
4942 | h is the product of the polynomials w in W. |
---|
4943 | "comment",0-1: The default is "comment",0. Setting "comments",1 |
---|
4944 | will provide information about the development of the |
---|
4945 | computation. |
---|
4946 | One can give none till 3 of these options. |
---|
4947 | RETURN: Returns a list T describing a reduced and disjoint comprehensive |
---|
4948 | Groebner system (CGS), and whose segments correspond to |
---|
4949 | constant leading power products (lpp) of the reduced Groebner |
---|
4950 | basis. The returned list is of the form: |
---|
4951 | ( |
---|
4952 | (lpp, (basis,segment),...,(basis,segment)), |
---|
4953 | ..,, |
---|
4954 | (lpp, (basis,segment),...,(basis,segment)) |
---|
4955 | ) |
---|
4956 | The bases are the reduced Groebner bases (after normalization) |
---|
4957 | for each point of the corresponding segment. |
---|
4958 | Each segment is given by a reduced representation (Ni,Wi), with |
---|
4959 | Ni radical and V(Ni)=Zariski closure of the segment Si=V(Ni)\V(hi), |
---|
4960 | where hi is the product of the polynomials w in Wi. |
---|
4961 | NOTE: The basering R, must be of the form Q[a][x], a=parameters, |
---|
4962 | x=variables, and should be defined previously, and the ideal |
---|
4963 | defined on R. |
---|
4964 | KEYWORDS: CGS, disjoint, reduced, comprehensive Groebner system |
---|
4965 | EXAMPLE: cgsdr; shows an example" |
---|
4966 | { |
---|
4967 | list @T; |
---|
4968 | exportto(Top,@T); |
---|
4969 | setglobalrings(); |
---|
4970 | int i; |
---|
4971 | ideal B; |
---|
4972 | poly f; |
---|
4973 | def N=ideal(0); |
---|
4974 | def W=ideal(1); |
---|
4975 | int comment=0; |
---|
4976 | list L=#; |
---|
4977 | for(i=1;i<=size(L) div 2;i++) |
---|
4978 | { |
---|
4979 | if(L[2*i-1]=="null"){N=L[2*i];} |
---|
4980 | else |
---|
4981 | { |
---|
4982 | if(L[2*i-1]=="nonnull"){W=L[2*i];} |
---|
4983 | else |
---|
4984 | { |
---|
4985 | if(L[2*i-1]=="comment"){comment=L[2*i];} |
---|
4986 | } |
---|
4987 | } |
---|
4988 | } |
---|
4989 | if(N!=0) |
---|
4990 | { |
---|
4991 | def LL=Rrep(N,W); |
---|
4992 | N=LL[1]; |
---|
4993 | W=LL[2]; |
---|
4994 | for (i=1;i<=size(F);i++) |
---|
4995 | { |
---|
4996 | f=pnormalform(F[i],N,W); |
---|
4997 | if (f!=0){B[size(B)+1]=f;} |
---|
4998 | } |
---|
4999 | } |
---|
5000 | else {B=F;} |
---|
5001 | reccgsdr(B,N,W,list(),0); |
---|
5002 | def T=@T; |
---|
5003 | if (comment==1) |
---|
5004 | {string("Number of segments in cgsdr (total) = ",size(T));} |
---|
5005 | kill @T; |
---|
5006 | kill @P; kill @RP; kill @R; |
---|
5007 | return(grsegments(T)); |
---|
5008 | } |
---|
5009 | example |
---|
5010 | { "EXAMPLE:"; echo = 2; |
---|
5011 | "Casas conjecture for degree 4"; |
---|
5012 | ring R=(0,a0,a1,a2,a3,a4),(x1,x2,x3),dp; |
---|
5013 | ideal F=x1^4+(4*a3)*x1^3+(6*a2)*x1^2+(4*a1)*x1+(a0), |
---|
5014 | x1^3+(3*a3)*x1^2+(3*a2)*x1+(a1), |
---|
5015 | x2^4+(4*a3)*x2^3+(6*a2)*x2^2+(4*a1)*x2+(a0), |
---|
5016 | x2^2+(2*a3)*x2+(a2), |
---|
5017 | x3^4+(4*a3)*x3^3+(6*a2)*x3^2+(4*a1)*x3+(a0), |
---|
5018 | x3+(a3); |
---|
5019 | cgsdr(F); |
---|
5020 | } |
---|
5021 | |
---|
5022 | //reccgsdr |
---|
5023 | // 20/09/2010 |
---|
5024 | static proc reccgsdr(ideal B, ideal N, ideal W, list P, int l) |
---|
5025 | { |
---|
5026 | ideal Bn=B; ideal Nn=N; ideal Wn=W; list Pn=P; int ln=l; ideal lppn; |
---|
5027 | list L; int i; |
---|
5028 | poly lc; ideal N0; ideal W0; list P0; int l0; |
---|
5029 | ideal N1=Nn; ideal W1=Wn; list P1=Pn; int l1=ln; |
---|
5030 | if (l>0) |
---|
5031 | { |
---|
5032 | if (size(variables(B[l]))==0) |
---|
5033 | { |
---|
5034 | lppn=1; Bn=1; |
---|
5035 | @T[size(@T)+1]=list(lppn,Bn,N,W); |
---|
5036 | return(); |
---|
5037 | } |
---|
5038 | } |
---|
5039 | if (ln<size(Bn)) |
---|
5040 | { |
---|
5041 | L=DiscussPolys(Bn, Nn, Wn, Pn, ln); |
---|
5042 | lc=L[1]; Bn=L[2]; N0=L[3]; W0=L[4]; P0=L[5]; l0=L[6]; |
---|
5043 | N1=L[7]; W1=L[8]; P1=L[9]; l1=L[10]; |
---|
5044 | ln=l0; |
---|
5045 | } |
---|
5046 | if ((ln==size(Bn)) and (size(Bn)<>0)) |
---|
5047 | { |
---|
5048 | L=DiscussSPolys(Bn, N1, W1, P1, l1); |
---|
5049 | lc=L[1]; Bn=L[2]; N0=L[3]; W0=L[4]; P0=L[5]; l0=L[6]; |
---|
5050 | N1=L[7]; W1=L[8]; P1=L[9]; l1=L[10]; |
---|
5051 | } |
---|
5052 | if (N0[1]<>1) |
---|
5053 | { |
---|
5054 | reccgsdr(Bn, N0,W0,P0,l0); |
---|
5055 | reccgsdr(Bn, N1,W1,P1,l1); |
---|
5056 | } |
---|
5057 | else |
---|
5058 | { |
---|
5059 | if (equalideals(N1,ideal(1))==0) |
---|
5060 | { |
---|
5061 | Bn=mingb(Bn); |
---|
5062 | Bn=redgb(Bn,N1,W1); |
---|
5063 | lppn=ideal(0); |
---|
5064 | for (i=1; i<=size(Bn);i++){lppn[i]=leadmonom(Bn[i]);} |
---|
5065 | @T[size(@T)+1]=list(lppn,Bn,N1,W1); |
---|
5066 | } |
---|
5067 | } |
---|
5068 | } |
---|
5069 | |
---|
5070 | // input: internal routine called by cgsdr at the end to improve the output |
---|
5071 | // output: grouped segments by lpp obtained in cgsdr |
---|
5072 | static proc grsegments(list T) |
---|
5073 | { |
---|
5074 | int i; |
---|
5075 | list L; |
---|
5076 | list lpp; |
---|
5077 | list lp; |
---|
5078 | list ls; |
---|
5079 | int n=size(T); |
---|
5080 | lpp[1]=T[n][1]; |
---|
5081 | L[1]=list(lpp[1],list(list(T[n][2],T[n][3],T[n][4]))); |
---|
5082 | if (n>1) |
---|
5083 | { |
---|
5084 | for (i=1;i<=size(T)-1;i++) |
---|
5085 | { |
---|
5086 | lp=memberpos(T[n-i][1],lpp); |
---|
5087 | if(lp[1]==1) |
---|
5088 | { |
---|
5089 | ls=L[lp[2]][2]; |
---|
5090 | ls[size(ls)+1]=list(T[n-i][2],T[n-i][3],T[n-i][4]); |
---|
5091 | L[lp[2]][2]=ls; |
---|
5092 | } |
---|
5093 | else |
---|
5094 | { |
---|
5095 | lpp[size(lpp)+1]=T[n-i][1]; |
---|
5096 | L[size(L)+1]=list(T[n-i][1],list(list(T[n-i][2],T[n-i][3],T[n-i][4]))); |
---|
5097 | } |
---|
5098 | } |
---|
5099 | } |
---|
5100 | //"L in groupsegments="; L; |
---|
5101 | return(L); |
---|
5102 | } |
---|
5103 | |
---|
5104 | // grRtoPrep |
---|
5105 | // input: L (list) is the output of cgsdr |
---|
5106 | // output: LL (list) the same list but the segments are expressed |
---|
5107 | // in canonical representations: |
---|
5108 | // ( (lpp, (basis, |
---|
5109 | // ((P_1),(P_{11},...,P_{1t1})) |
---|
5110 | // ... |
---|
5111 | // ((P_j),(P_{j1},...,P_{jtj})) |
---|
5112 | // ) |
---|
5113 | // ... |
---|
5114 | // (basis, |
---|
5115 | // ((P_1),(P_{11},...,P_{1t1})) |
---|
5116 | // ... |
---|
5117 | // ((P_j),(P_{j1},...,P_{jtj})) |
---|
5118 | // ) |
---|
5119 | // ) |
---|
5120 | // ... |
---|
5121 | // (lpp, (basis, |
---|
5122 | // ((P_1),(P_{11},...,P_{1t1})) |
---|
5123 | // ... |
---|
5124 | // ((P_j),(P_{j1},...,P_{jtj})) |
---|
5125 | // ) |
---|
5126 | // ... |
---|
5127 | // (basis, |
---|
5128 | // ((P_1),(P_{11},...,P_{1t1})) |
---|
5129 | // ... |
---|
5130 | // ((P_j),(P_{j1},...,P_{jtj})) |
---|
5131 | // ) |
---|
5132 | // ) |
---|
5133 | // ) |
---|
5134 | static proc grRtoPrep(list L) |
---|
5135 | { |
---|
5136 | int i; int j; |
---|
5137 | list LL; list ct; |
---|
5138 | // size(L)=number of lpp-segments |
---|
5139 | for (i=1;i<=size(L);i++) |
---|
5140 | { |
---|
5141 | LL[i]=list(); |
---|
5142 | LL[i][1]=L[i][1]; |
---|
5143 | // L[i][1]=lpp |
---|
5144 | LL[i][2]=list(); |
---|
5145 | for (j=1;j<=size(L[i][2]);j++) |
---|
5146 | { |
---|
5147 | ct=RtoPrepNew(L[i][2][j][2],L[i][2][j][3]); // ,L[i][2][j][5] |
---|
5148 | LL[i][2][j]=list(); |
---|
5149 | LL[i][2][j][1]=L[i][2][j][1]; |
---|
5150 | // L[i][2][j][1]=label |
---|
5151 | LL[i][2][j][2]=L[i][2][j][2]; |
---|
5152 | // L[i][2][j][2]=basis |
---|
5153 | LL[i][2][j][3]=ct; |
---|
5154 | } |
---|
5155 | } |
---|
5156 | return(LL); |
---|
5157 | } |
---|
5158 | |
---|
5159 | // idcontains |
---|
5160 | // input: ideal p, ideal q |
---|
5161 | // output: 1 if p contains q, 0 otherwise |
---|
5162 | static proc idcontains(ideal p, ideal q) |
---|
5163 | { |
---|
5164 | int t; int i; |
---|
5165 | t=1; i=1; |
---|
5166 | def RR=basering; |
---|
5167 | setring @P; |
---|
5168 | def P=imap(RR,p); |
---|
5169 | def Q=imap(RR,q); |
---|
5170 | attrib(P,"isSB",1); |
---|
5171 | poly r; |
---|
5172 | while ((t==1) and (i<=size(Q))) |
---|
5173 | { |
---|
5174 | r=reduce(Q[i],P); |
---|
5175 | if (r!=0){t=0;} |
---|
5176 | i++; |
---|
5177 | } |
---|
5178 | setring RR; |
---|
5179 | return(t); |
---|
5180 | } |
---|
5181 | |
---|
5182 | // selectminindeals |
---|
5183 | // given a list of ideals returns the list of integers corresponding |
---|
5184 | // to the minimal ideals in the list |
---|
5185 | // input: L (list of ideals) |
---|
5186 | // output: the list of integers corresponding to the minimal ideals in L |
---|
5187 | static proc selectminideals(list L) |
---|
5188 | { |
---|
5189 | if (size(L)==0){return(L);} |
---|
5190 | def RR=basering; |
---|
5191 | setring @P; |
---|
5192 | def Lp=imap(RR,L); |
---|
5193 | int i; int j; int t; intvec notsel; |
---|
5194 | list P; |
---|
5195 | for (i=1;i<=size(Lp);i++) |
---|
5196 | { |
---|
5197 | if(memberpos(i,notsel)[1]==1) |
---|
5198 | { |
---|
5199 | i++; |
---|
5200 | if(i>size(Lp)){break;} |
---|
5201 | } |
---|
5202 | t=1; |
---|
5203 | j=1; |
---|
5204 | while ((t==1) and (j<=size(Lp))) |
---|
5205 | { |
---|
5206 | if (i==j){j++;} |
---|
5207 | if ((j<=size(Lp)) and (memberpos(j,notsel)[1]==0)) |
---|
5208 | { |
---|
5209 | |
---|
5210 | if (idcontains(Lp[i],Lp[j])==1) |
---|
5211 | { |
---|
5212 | notsel[size(notsel)+1]=i; |
---|
5213 | t=0; |
---|
5214 | } |
---|
5215 | } |
---|
5216 | j++; |
---|
5217 | } |
---|
5218 | if (t==1){P[size(P)+1]=i;} |
---|
5219 | } |
---|
5220 | setring(RR); |
---|
5221 | return(P); |
---|
5222 | } |
---|
5223 | |
---|
5224 | // LCUnion |
---|
5225 | // Given a list of the P-representations of locally closed segments |
---|
5226 | // for which we know that the union is also locally closed |
---|
5227 | // it returns the P-representation of its union |
---|
5228 | // input: L list of segments in P-representation |
---|
5229 | // ((p_j^i,(p_j1^i,...,p_jk_j^i | j=1..t_i)) | i=1..s ) |
---|
5230 | // where i represents a segment |
---|
5231 | // output: P-representation of the union |
---|
5232 | // ((P_j,(P_j1,...,P_jk_j | j=1..t))) |
---|
5233 | static proc LCUnion(list LL) |
---|
5234 | { |
---|
5235 | def RR=basering; |
---|
5236 | setring(@P); |
---|
5237 | def L=imap(RR,LL); |
---|
5238 | int i; int j; int k; list H; list C; list T; |
---|
5239 | list L0; list P0; list P; list Q0; list Q; |
---|
5240 | for (i=1;i<=size(L);i++) |
---|
5241 | { |
---|
5242 | for (j=1;j<=size(L[i]);j++) |
---|
5243 | { |
---|
5244 | P0[size(P0)+1]=L[i][j][1]; |
---|
5245 | L0[size(L0)+1]=intvec(i,j); |
---|
5246 | } |
---|
5247 | } |
---|
5248 | Q0=selectminideals(P0); |
---|
5249 | for (i=1;i<=size(Q0);i++) |
---|
5250 | { |
---|
5251 | Q[i]=L0[Q0[i]]; |
---|
5252 | P[i]=L[Q[i][1]][Q[i][2]]; |
---|
5253 | } |
---|
5254 | // P is the list of the maximal components of the union |
---|
5255 | // with the corresponding initial holes. |
---|
5256 | // Q is the list of intvec positions in L of the first element of the P's |
---|
5257 | // Its elements give (num of segment, num of max component (=min ideal)) |
---|
5258 | for (k=1;k<=size(Q);k++) |
---|
5259 | { |
---|
5260 | H=P[k][2]; // holes of P[k][1] |
---|
5261 | for (i=1;i<=size(L);i++) |
---|
5262 | { |
---|
5263 | if (i!=Q[k][1]) |
---|
5264 | { |
---|
5265 | for (j=1;j<=size(L[i]);j++) |
---|
5266 | { |
---|
5267 | C[size(C)+1]=L[i][j]; |
---|
5268 | } |
---|
5269 | } |
---|
5270 | } |
---|
5271 | T[size(T)+1]=list(Q[k],P[k][1],addpart(H,C)); |
---|
5272 | } |
---|
5273 | setring(RR); |
---|
5274 | def TT=imap(@P,T); |
---|
5275 | return(TT); |
---|
5276 | } |
---|
5277 | |
---|
5278 | // Called by LCUnion to modify the holes of a primepart of the union |
---|
5279 | // by the addition of the segments that do not correspond to that part |
---|
5280 | // Works on @P ring. |
---|
5281 | // Input: |
---|
5282 | // H=(p_i1,..,p_is) the holes of a component to be transformed by the addition of |
---|
5283 | // the segments C that do not correspond to that component |
---|
5284 | // C=((q_1,(q_11,..,q_1l_1)),..,(q_k,(q_k1,..,q_kl_k))) |
---|
5285 | // the list of segments to be added to the holes |
---|
5286 | static proc addpart(list H, list C) |
---|
5287 | { |
---|
5288 | list Q; int i; int j; int k; int l; int t; int t1; |
---|
5289 | Q=H; intvec notQ; list QQ; list addq; |
---|
5290 | ideal q; |
---|
5291 | i=1; |
---|
5292 | while (i<=size(Q)) |
---|
5293 | { |
---|
5294 | if (memberpos(i,notQ)[1]==0) |
---|
5295 | { |
---|
5296 | q=Q[i]; |
---|
5297 | t=1; j=1; |
---|
5298 | while ((t==1) and (j<=size(C))) |
---|
5299 | { |
---|
5300 | if (equalideals(q,C[j][1])==1) |
---|
5301 | { |
---|
5302 | t=0; |
---|
5303 | for (k=1;k<=size(C[j][2]);k++) |
---|
5304 | { |
---|
5305 | t1=1; |
---|
5306 | //kill addq; |
---|
5307 | //list addq; |
---|
5308 | l=1; |
---|
5309 | while((t1==1) and (l<=size(Q))) |
---|
5310 | { |
---|
5311 | if ((l!=i) and (memberpos(l,notQ)[1]==0)) |
---|
5312 | { |
---|
5313 | if (idcontains(C[j][2][k],Q[l])==1) |
---|
5314 | { |
---|
5315 | t1=0; |
---|
5316 | } |
---|
5317 | } |
---|
5318 | l++; |
---|
5319 | } |
---|
5320 | if (t1==1) |
---|
5321 | { |
---|
5322 | addq[size(addq)+1]=C[j][2][k]; |
---|
5323 | } |
---|
5324 | } |
---|
5325 | if((size(notQ)==1) and (notQ[1]==0)){notQ[1]=i;} |
---|
5326 | else {notQ[size(notQ)+1]=i;} |
---|
5327 | } |
---|
5328 | j++; |
---|
5329 | } |
---|
5330 | if (size(addq)>0) |
---|
5331 | { |
---|
5332 | for (k=1;k<=size(addq);k++) |
---|
5333 | { |
---|
5334 | Q[size(Q)+1]=addq[k]; |
---|
5335 | } |
---|
5336 | kill addq; |
---|
5337 | list addq; |
---|
5338 | } |
---|
5339 | //print("Q="); Q; print("notQ="); notQ; |
---|
5340 | } |
---|
5341 | i++; |
---|
5342 | } |
---|
5343 | for (i=1;i<=size(Q);i++) |
---|
5344 | { |
---|
5345 | if(memberpos(i,notQ)[1]==0) |
---|
5346 | { |
---|
5347 | QQ[size(QQ)+1]=Q[i]; |
---|
5348 | } |
---|
5349 | } |
---|
5350 | if (size(QQ)==0){QQ[1]=ideal(1);} |
---|
5351 | return(addpartfine(QQ,C)); |
---|
5352 | } |
---|
5353 | |
---|
5354 | // Called by addpart to finish the modification of the holes of a primepart |
---|
5355 | // of the union by the addition of the segments that do not correspond to |
---|
5356 | // that part. |
---|
5357 | // Works on @P ring. |
---|
5358 | static proc addpartfine(list H, list C0) |
---|
5359 | { |
---|
5360 | int i; int j; int k; int te; intvec notQ; int l; list sel; int used; |
---|
5361 | intvec jtesC; |
---|
5362 | if ((size(H)==1) and (equalideals(H[1],ideal(1)))){return(H);} |
---|
5363 | if (size(C0)==0){return(H);} |
---|
5364 | def RR=basering; |
---|
5365 | setring(@P); |
---|
5366 | list newQ; list nQ; list Q; list nQ1; list Q0; |
---|
5367 | def Q1=imap(RR,H); |
---|
5368 | //Q1=sortlistideals(Q1); |
---|
5369 | def C=imap(RR,C0); |
---|
5370 | while(equallistideals(Q0,Q1)==0) |
---|
5371 | { |
---|
5372 | Q0=Q1; |
---|
5373 | i=0; |
---|
5374 | Q=Q1; |
---|
5375 | kill notQ; intvec notQ; |
---|
5376 | while(i<size(Q)) |
---|
5377 | { |
---|
5378 | i++; |
---|
5379 | for(j=1;j<=size(C);j++) |
---|
5380 | { |
---|
5381 | te=idcontains(Q[i],C[j][1]); |
---|
5382 | if(te==1) |
---|
5383 | { |
---|
5384 | for(k=1;k<=size(C[j][2]);k++) |
---|
5385 | { |
---|
5386 | if(idcontains(Q[i],C[j][2][k])==1) |
---|
5387 | { |
---|
5388 | te=0; break; |
---|
5389 | } |
---|
5390 | } |
---|
5391 | if (te==1) |
---|
5392 | { |
---|
5393 | used++; |
---|
5394 | if ((size(notQ)==1) and (notQ[1]==0)){notQ[1]=i;} |
---|
5395 | else{notQ[size(notQ)+1]=i;} |
---|
5396 | kill newQ; list newQ; |
---|
5397 | for(k=1;k<=size(C[j][2]);k++) |
---|
5398 | { |
---|
5399 | nQ=minGTZ(Q[i]+C[j][2][k]); |
---|
5400 | for(l=1;l<=size(nQ);l++) |
---|
5401 | { |
---|
5402 | option(redSB); |
---|
5403 | nQ[l]=std(nQ[l]); |
---|
5404 | newQ[size(newQ)+1]=nQ[l]; |
---|
5405 | } |
---|
5406 | } |
---|
5407 | sel=selectminideals(newQ); |
---|
5408 | kill nQ1; list nQ1; |
---|
5409 | for(l=1;l<=size(sel);l++) |
---|
5410 | { |
---|
5411 | nQ1[l]=newQ[sel[l]]; |
---|
5412 | } |
---|
5413 | newQ=nQ1; |
---|
5414 | for(l=1;l<=size(newQ);l++) |
---|
5415 | { |
---|
5416 | Q[size(Q)+1]=newQ[l]; |
---|
5417 | } |
---|
5418 | break; |
---|
5419 | } |
---|
5420 | } |
---|
5421 | } |
---|
5422 | } |
---|
5423 | kill Q1; list Q1; |
---|
5424 | for(i=1;i<=size(Q);i++) |
---|
5425 | { |
---|
5426 | if(memberpos(i,notQ)[1]==0) |
---|
5427 | { |
---|
5428 | Q1[size(Q1)+1]=Q[i]; |
---|
5429 | } |
---|
5430 | } |
---|
5431 | sel=selectminideals(Q1); |
---|
5432 | kill nQ1; list nQ1; |
---|
5433 | for(l=1;l<=size(sel);l++) |
---|
5434 | { |
---|
5435 | nQ1[l]=Q1[sel[l]]; |
---|
5436 | } |
---|
5437 | Q1=nQ1; |
---|
5438 | } |
---|
5439 | setring(RR); |
---|
5440 | //if(used>0){string("addpartfine was ", used, " times used");} |
---|
5441 | return(imap(@P,Q1)); |
---|
5442 | } |
---|
5443 | |
---|
5444 | //// specswell |
---|
5445 | //// used only in specswellonlpp (not used, can be deleted) |
---|
5446 | //// input: |
---|
5447 | //// given two corresponding polynomials g1 and g2 with the same lpp |
---|
5448 | //// g1 belonging to the basis in the segment N1,W1 |
---|
5449 | //// g2 belonging to the basis in the segment N2,W2 |
---|
5450 | //// output: |
---|
5451 | //// 1 if g1 spezializes well to g2 on the whole (N2,W2) segment |
---|
5452 | //// 0 if not |
---|
5453 | //proc specswell(poly g1, poly g2, ideal N2, ideal W2) |
---|
5454 | //{ |
---|
5455 | // poly S; |
---|
5456 | // S=leadcoef(g2)*g1-leadcoef(g1)*g2; |
---|
5457 | // def RR=basering; |
---|
5458 | // setring(@RPt); |
---|
5459 | // def SR=imap(RR,S); |
---|
5460 | // def N2R=imap(RR,N2); |
---|
5461 | // attrib(N2R,"isSB",1); |
---|
5462 | // poly S2R=reduce(SR,N2R); |
---|
5463 | // setring(RR); |
---|
5464 | // def S2=imap(@RPt,S2R); |
---|
5465 | // //if (S2==0) |
---|
5466 | // //if (nonnull(leadcoef(g1),N2,W2)==1) |
---|
5467 | // if ((S2==0) and (nonnull(leadcoef(g1),N2,W2))) |
---|
5468 | // {return(1);} |
---|
5469 | // else {return(0);} |
---|
5470 | //} |
---|
5471 | // |
---|
5472 | //// specswellonlpp |
---|
5473 | //// not used, can be deleted |
---|
5474 | //// input: |
---|
5475 | //// given a generic polynomial g with given lpp |
---|
5476 | //// and the list of tripets (p,N,W) of all the segments in |
---|
5477 | //// the same lpp-segment, where p is the correct image of g on (N,W) |
---|
5478 | //// output: |
---|
5479 | //// 1 if g spezializes well to p on the whole (N,W) segment for all segments |
---|
5480 | //// 0 if not |
---|
5481 | //proc specswellonlpp(poly g, list L) |
---|
5482 | //{ |
---|
5483 | // int i=1; int t=1; |
---|
5484 | // while ((t==1) and (i<=size(L))) |
---|
5485 | // { |
---|
5486 | // t=specswell(g, L[i][1],L[i][2],L[i][3]); |
---|
5487 | // i++; |
---|
5488 | // } |
---|
5489 | // return(t); |
---|
5490 | //} |
---|
5491 | |
---|
5492 | // specswellCrep |
---|
5493 | // input: |
---|
5494 | // given two corresponding polynomials g1 and g2 with the same lpp |
---|
5495 | // g1 belonging to the basis in the segment ida1,idb1 |
---|
5496 | // g2 belonging to the basis in the segment ida2,idb2 |
---|
5497 | // output: |
---|
5498 | // 1 if g1 spezializes well to g2 on the whole (ida2,idb2) segment |
---|
5499 | // 0 if not |
---|
5500 | static proc specswellCrep(poly g1, poly g2, ideal ida2) |
---|
5501 | { |
---|
5502 | poly S; |
---|
5503 | S=leadcoef(g2)*g1-leadcoef(g1)*g2; |
---|
5504 | def RR=basering; |
---|
5505 | setring(@RPt); |
---|
5506 | def SR=imap(RR,S); |
---|
5507 | def ida2R=imap(RR,ida2); |
---|
5508 | attrib(ida2R,"isSB",1); |
---|
5509 | poly S2R=reduce(SR,ida2R); |
---|
5510 | setring(RR); |
---|
5511 | def S2=imap(@RPt,S2R); |
---|
5512 | if (S2==0){return(1);} // and (nonnullCrep(leadcoef(g1),ida2,idb2)) |
---|
5513 | else {return(0);} |
---|
5514 | } |
---|
5515 | |
---|
5516 | |
---|
5517 | // gcover |
---|
5518 | // input: ideal F: a generating set of a homogeneous ideal in Q[a][x] |
---|
5519 | // list GenCase: Containing the generic case with basis 1 if it exists |
---|
5520 | // list #: optional |
---|
5521 | // output: the list |
---|
5522 | // S=((lpp, generic basis, Rrep, Crep),..,(lpp, generic basis, Rrep, Crep)) |
---|
5523 | // where a Rrep is ( (p1,(p11,..,p1k_1)),..,(pj,(pj1,..,p1k_j)) ) |
---|
5524 | // a Crep is ( ida, idb ) |
---|
5525 | static proc gcover(ideal F,list GenCase, list #) |
---|
5526 | { |
---|
5527 | int i; int j; int k; ideal lpp; list GPi2; list pairspP; ideal B; int ti; |
---|
5528 | int i1; int tes; int j1; int selind; int i2; int m; |
---|
5529 | list prep; list crep; list LCU; poly p; poly lcp; list L; ideal FF; |
---|
5530 | list NW=#; |
---|
5531 | int CGS=NW[3]; |
---|
5532 | int comment=NW[4]; |
---|
5533 | NW=NW[1],NW[2]; |
---|
5534 | list GS; list GP; |
---|
5535 | def RR=basering; |
---|
5536 | int start=timer; int start0=start; int start1=start; |
---|
5537 | if (CGS==0) |
---|
5538 | { |
---|
5539 | def BT=buildtree(F,list("null",NW[1],"nonnull",NW[2])); |
---|
5540 | setglobalrings(); |
---|
5541 | def FC=finalcases(BT); |
---|
5542 | GS=groupsegments(FC); |
---|
5543 | if(comment==1) |
---|
5544 | { |
---|
5545 | string("Number of segments in buildtree (total) = ",size(FC)); |
---|
5546 | string("Number of lpp segments in groupsegments = ",size(GS)); |
---|
5547 | string("Time in buildtree = ",timer-start," sec"); |
---|
5548 | } |
---|
5549 | start=timer; |
---|
5550 | GP=groupRtoPrep(GS); |
---|
5551 | if (comment==1){string("Time in groupRtoPrep = ",timer-start," sec");} |
---|
5552 | } |
---|
5553 | else |
---|
5554 | { |
---|
5555 | GS=cgsdr(F,list("null",NW[1],"nonnull",NW[2],"comment",comment)); |
---|
5556 | setglobalrings(); |
---|
5557 | if(comment==1) |
---|
5558 | { |
---|
5559 | string("Number of lpp segments in cgsdr = ",size(GS)); |
---|
5560 | string("Time in cgsdr = ",timer-start," sec"); |
---|
5561 | } |
---|
5562 | start=timer; |
---|
5563 | GP=grRtoPrep(GS); |
---|
5564 | if(comment==1){string("Time in grRtoPrep = ",timer-start," sec");} |
---|
5565 | } |
---|
5566 | for(i=1;i<=size(GP);i++) |
---|
5567 | { |
---|
5568 | if(size(GP[i][2])>1){GP[i][3]=1;} |
---|
5569 | else{GP[i][3]=0;} |
---|
5570 | } |
---|
5571 | int SizeGC=size(GenCase); |
---|
5572 | if (SizeGC>0) |
---|
5573 | { |
---|
5574 | int te=0; |
---|
5575 | list NewGen; list CH; |
---|
5576 | for (i=1;i<=size(GP);i++) |
---|
5577 | { |
---|
5578 | if(equalideals(GP[i][1],ideal(1))==1) |
---|
5579 | { |
---|
5580 | te=1; |
---|
5581 | NewGen[1]=GenCase; |
---|
5582 | for(j=1;j<=size(GP[i][2]);j++) |
---|
5583 | { |
---|
5584 | NewGen[j+1]=GP[i][2][j]; |
---|
5585 | } |
---|
5586 | GP[i][2]=NewGen; |
---|
5587 | if(i!=1) |
---|
5588 | { \\exchange cases i and 1 |
---|
5589 | CH=GP[i]; |
---|
5590 | GP[i]=GP[1]; |
---|
5591 | GP[1]=CH; |
---|
5592 | } |
---|
5593 | break; |
---|
5594 | } |
---|
5595 | } |
---|
5596 | if (te==0) // add GenCase as a new case |
---|
5597 | { |
---|
5598 | CH[1]=GenCase; |
---|
5599 | //CH[1]=list(ideal(1),list(GenCase)); |
---|
5600 | for (i=1;i<=size(GP);i++) |
---|
5601 | { |
---|
5602 | CH[i+1]=GP[i]; |
---|
5603 | } |
---|
5604 | GP=CH; |
---|
5605 | } |
---|
5606 | } |
---|
5607 | for(i=1;i<=size(GP);i++) |
---|
5608 | { |
---|
5609 | GP[i][3]=size(GP[i][2]); |
---|
5610 | } |
---|
5611 | list LL; |
---|
5612 | list S; |
---|
5613 | poly sp; |
---|
5614 | ideal BB; |
---|
5615 | start1=timer; |
---|
5616 | for (i=1;i<=size(GP);i++) |
---|
5617 | { |
---|
5618 | kill LL; |
---|
5619 | list LL; |
---|
5620 | lpp=GP[i][1]; |
---|
5621 | GPi2=GP[i][2]; |
---|
5622 | kill pairspP; list pairspP; |
---|
5623 | for(j=1;j<=size(GPi2);j++) |
---|
5624 | { |
---|
5625 | pairspP[size(pairspP)+1]=GPi2[j][3]; |
---|
5626 | } |
---|
5627 | LCU=LCUnion(pairspP); |
---|
5628 | kill prep; list prep; |
---|
5629 | for(k=1;k<=size(LCU);k++) |
---|
5630 | { |
---|
5631 | prep[k]=list(LCU[k][2],LCU[k][3]); |
---|
5632 | if (CGS==0) |
---|
5633 | { |
---|
5634 | B=GPi2[LCU[k][1][1]][2]; |
---|
5635 | } |
---|
5636 | else |
---|
5637 | { |
---|
5638 | B=GPi2[LCU[k][1][1]][1]; |
---|
5639 | } |
---|
5640 | LCU[k][1]=B; |
---|
5641 | } |
---|
5642 | // Deciding if combine is needed |
---|
5643 | kill BB; |
---|
5644 | ideal BB; |
---|
5645 | tes=1; m=1; |
---|
5646 | while((tes==1) and (m<=size(LCU[1][1]))) |
---|
5647 | { |
---|
5648 | j=1; |
---|
5649 | while((tes==1) and (j<=size(LCU))) |
---|
5650 | { |
---|
5651 | k=1; |
---|
5652 | while((tes==1) and (k<=size(LCU))) |
---|
5653 | { |
---|
5654 | if(j!=k) |
---|
5655 | { |
---|
5656 | sp=pnormalform(pspol(LCU[j][1][m],LCU[k][1][m]),LCU[k][2],NW[2]); |
---|
5657 | if(sp!=0){tes=0;} |
---|
5658 | } |
---|
5659 | k++; |
---|
5660 | } |
---|
5661 | if(tes==1) |
---|
5662 | { |
---|
5663 | BB[m]=LCU[j][1][m]; |
---|
5664 | } |
---|
5665 | j++; |
---|
5666 | } |
---|
5667 | if(tes==0){break;} |
---|
5668 | m++; |
---|
5669 | } |
---|
5670 | crep=PtoCrep(prep); |
---|
5671 | if(tes==0) |
---|
5672 | { |
---|
5673 | // combine is needed |
---|
5674 | kill B; ideal B; |
---|
5675 | for (j=1;j<=size(LCU);j++) |
---|
5676 | { |
---|
5677 | LL[j]=LCU[j][2]; |
---|
5678 | } |
---|
5679 | if (size(LCU)>1) |
---|
5680 | { |
---|
5681 | FF=precombint(LL); |
---|
5682 | } |
---|
5683 | for (k=1;k<=size(lpp);k++) |
---|
5684 | { |
---|
5685 | kill L; list L; |
---|
5686 | for (j=1;j<=size(LCU);j++) |
---|
5687 | { |
---|
5688 | L[j]=list(LCU[j][2],LCU[j][1][k]); |
---|
5689 | } |
---|
5690 | if (size(LCU)>1) |
---|
5691 | { |
---|
5692 | B[k]=combine(L,FF); |
---|
5693 | } |
---|
5694 | else{B[k]=L[1][2];} |
---|
5695 | } |
---|
5696 | } |
---|
5697 | else{B=BB;} |
---|
5698 | for(j=1;j<=size(B);j++) |
---|
5699 | { |
---|
5700 | B[j]=pnormalform(B[j],crep[1],NW[2]); |
---|
5701 | } |
---|
5702 | S[i]=list(lpp,B,prep,crep,GP[i][3]); |
---|
5703 | } |
---|
5704 | if(comment==1) |
---|
5705 | { |
---|
5706 | string("Time in LCUnion + combine = ",timer-start1," sec"); |
---|
5707 | } |
---|
5708 | kill @P; kill @RP; kill @R; |
---|
5709 | return(S); |
---|
5710 | } |
---|
5711 | |
---|
5712 | // grobcov |
---|
5713 | // input: |
---|
5714 | // ideal F: a parametric ideal in Q[a][x], where a are the parameters |
---|
5715 | // and x the variables |
---|
5716 | // list #: (options) list("null",N,"nonnull",W,"can",Method,"cgs",CGS), where |
---|
5717 | // N is the null conditions ideal (if desired) |
---|
5718 | // W is the ideal of non-null conditions (if desired) |
---|
5719 | // Method is 1 by default and can be set to 0 if we do not |
---|
5720 | // need to obtain the canonical GC, but only a GC. |
---|
5721 | // CGS is 1 by default and uses cgsdr. It can be set to 0 to |
---|
5722 | // use the old buildtree instead. |
---|
5723 | // output: |
---|
5724 | // list S: ((lpp,basis,(idp_1,(idp_11,..,idp_1s_1))), .. |
---|
5725 | // (lpp,basis,(idp_r,(idp_r1,..,idp_rs_r))) ) where |
---|
5726 | // each element of S corresponds to a lpp-segment |
---|
5727 | // given by the lpp, the basis, and the P-representation of the segment |
---|
5728 | proc grobcov(ideal F,list #) |
---|
5729 | "USAGE: grobcov(F); This is the fundamental routine of the |
---|
5730 | library. It computes the Groebner cover of a parametric ideal |
---|
5731 | (see (*) Montes A., Wibmer M., Groebner Bases for Polynomial |
---|
5732 | Systems with parameters. JSC 45 (2010) 1391-1425.) |
---|
5733 | The Groebner cover of a parametric ideal consist of a set of pairs |
---|
5734 | (S_i,B_i), where the S_i are disjoint locally closed segments |
---|
5735 | of the parameter space, and the B_i are the reduced Groebner |
---|
5736 | bases of the ideal on every point of S_i. |
---|
5737 | |
---|
5738 | The ideal F must be defined on a parametric ring Q[a][x]. |
---|
5739 | Options: To modify the default options, pair of arguments |
---|
5740 | -option name, value- of valid options must be added to the call. |
---|
5741 | |
---|
5742 | Options: |
---|
5743 | "null",ideal N: The default is "null",ideal(0). |
---|
5744 | "nonnull",ideal W: The default "nonnull",ideal(1). |
---|
5745 | When options "null" and/or "nonnull" are given, then |
---|
5746 | the parameter space is restricted to V(N) \ V(h), where |
---|
5747 | h is the product of the polynomials w in W. |
---|
5748 | "can",0-1: The default is "can",1. With the default option |
---|
5749 | the homogenized ideal is computed before obtaining the |
---|
5750 | Groebner cover, so that the result is the canonical |
---|
5751 | Groebner cover. Setting "can",0 only homogenizes the basis |
---|
5752 | so the result is not exactly canonical, but the computation |
---|
5753 | is more efficient. |
---|
5754 | "ext",0-1: The default is "ext",1. With the default option the |
---|
5755 | full representation of the bases is computed (possible |
---|
5756 | shaves) and often a simpler result is obtained. Setting |
---|
5757 | "ext",0 only the generic representation is computed |
---|
5758 | (single polynomials, but not specializing to non-zero at |
---|
5759 | each point of the segment. |
---|
5760 | "cgs",0-1: The default is "cgs",1. The default option uses the |
---|
5761 | cgsdr routine of the actual library to compute the initial |
---|
5762 | CGS (more efficient). Setting "cgs",0 it uses the routine |
---|
5763 | cgsdrold of the old library redcgs.lib. This option can be |
---|
5764 | tested if the default option does not terminate. |
---|
5765 | "comment",0-1: The default is "comment",0. Setting "comments",1 |
---|
5766 | will provide information about the development of the |
---|
5767 | computation. |
---|
5768 | One can give none till 6 of these options. |
---|
5769 | RETURN: The list |
---|
5770 | ( |
---|
5771 | (lpp_1,basis_1,P-representation_1), |
---|
5772 | ... |
---|
5773 | (lpp_s,basis_s,P-represntation_s) |
---|
5774 | ) |
---|
5775 | |
---|
5776 | The lpp are constant over a segment and correspond to the |
---|
5777 | set of lpp of the reduced Groebner basis for each point |
---|
5778 | of the segment. |
---|
5779 | |
---|
5780 | Basis: to each element of lpp corresponds an I-regular function given Groebner basis, and it is given in full representation (by |
---|
5781 | in full representation (by default option "ext",1) or in |
---|
5782 | generic representation (option "ext",0). The regular function is |
---|
5783 | the corresponding element of the reduced Groebner basis for |
---|
5784 | each point of the segment with the given lpp. |
---|
5785 | For each point in the segment, the polynomial or the set of |
---|
5786 | polynomials representing it, if they do not specialize to 0, |
---|
5787 | then after normalization, specialize to the corresponding |
---|
5788 | element of the reduced Groebner basis. |
---|
5789 | |
---|
5790 | The P-representation of a segment is of the form |
---|
5791 | ((p_1,(p_11,..,p_1k1)),..,(p_r,(p_r1,..,p_rkr)) |
---|
5792 | representing the segment U_i (V(p_i) \ U_j (V(p_ij))), where the |
---|
5793 | p's are prime ideals. |
---|
5794 | |
---|
5795 | NOTE: The basering R, must be of the form Q[a][x], a=parameters, |
---|
5796 | x=variables, and should be defined previously. The ideal must |
---|
5797 | be defined on R. |
---|
5798 | KEYWORDS: Groebner cover, parametric ideal, canonical, discussion of |
---|
5799 | parametric ideal, multigrobcov, gencase1. |
---|
5800 | EXAMPLE: grobcov; shows an example" |
---|
5801 | { |
---|
5802 | list S; int i; int ish=1; list GBR; list BR; int j; int k; |
---|
5803 | list NW; ideal idp; ideal idq; int s; ideal ext; list SS; |
---|
5804 | ideal N; ideal W; int canop; int extop; int CGS; int repop; |
---|
5805 | int gradorder; int comment=0; int m; |
---|
5806 | list L=#; |
---|
5807 | // default options |
---|
5808 | int start=timer; |
---|
5809 | def RR=basering; |
---|
5810 | list NW0; |
---|
5811 | W=ideal(1); |
---|
5812 | N=ideal(0); |
---|
5813 | canop=1; // canop=0 for homogenizing the basis but not the ideal (not canonical) |
---|
5814 | // canop=1 for working with the homogenized ideal |
---|
5815 | repop=0; // repop=0 for representing the segments in Prep |
---|
5816 | // repop=1 for representing the segments in Crep |
---|
5817 | // repop=2 for representing the segments in Prep and Crep |
---|
5818 | extop=1; // extop=1 if the full representation of the bases are to be computed |
---|
5819 | // extop=0 if only generic representation of the bases are to be computed |
---|
5820 | CGS=1; // CGS=1 if cgsdr is to be used (default) |
---|
5821 | // CGS=0 if buildtree is to be used instead |
---|
5822 | for(i=1;i<=size(L) div 2;i++) |
---|
5823 | { |
---|
5824 | if(L[2*i-1]=="can"){canop=L[2*i];} |
---|
5825 | else |
---|
5826 | { |
---|
5827 | if(L[2*i-1]=="ext"){extop=L[2*i];} |
---|
5828 | else |
---|
5829 | { |
---|
5830 | if(L[2*i-1]=="rep"){repop=L[2*i];} |
---|
5831 | else |
---|
5832 | { |
---|
5833 | if(L[2*i-1]=="null"){N=L[2*i];} |
---|
5834 | else |
---|
5835 | { |
---|
5836 | if(L[2*i-1]=="nonnull"){W=L[2*i];} |
---|
5837 | else |
---|
5838 | { |
---|
5839 | if (L[2*i-1]=="cgs"){CGS=L[2*i];} |
---|
5840 | else |
---|
5841 | { |
---|
5842 | if (L[2*i-1]=="comment"){comment=L[2*i];} |
---|
5843 | } |
---|
5844 | } |
---|
5845 | } |
---|
5846 | } |
---|
5847 | } |
---|
5848 | } |
---|
5849 | } |
---|
5850 | if (comment==1){string("Options: can = ",canop,", extend = ",extop,", cgs = ",CGS,", rep = ",repop);} |
---|
5851 | for (i=1;i<=size(F);i++){ish=ishomog(F[i]); if(ish==0){break;}} |
---|
5852 | NW0=list(N,W,CGS,comment); |
---|
5853 | if (ish==1) |
---|
5854 | { |
---|
5855 | kill S; |
---|
5856 | list gc; |
---|
5857 | def S=gcover(F,gc,NW0); |
---|
5858 | setglobalrings(); |
---|
5859 | } |
---|
5860 | else |
---|
5861 | { |
---|
5862 | list RRL=ringlist(RR); |
---|
5863 | if (RRL[3][1][1]=="dp"){gradorder=1;} else {gradorder=0;} |
---|
5864 | RRL[3][1][1]="dp"; |
---|
5865 | //RRL[1][3][1][1]="dp"; // COMMENTED GIVES ERROR IN S53. |
---|
5866 | def Pa=ring(RRL[1]); |
---|
5867 | list Lx; |
---|
5868 | Lx[1]=0; |
---|
5869 | Lx[2]=RRL[2]+RRL[1][2]; |
---|
5870 | Lx[3]=RRL[1][3]; |
---|
5871 | Lx[4]=RRL[1][4]; |
---|
5872 | RRL[1]=0; |
---|
5873 | def D=ring(RRL); |
---|
5874 | def RP=D+Pa; |
---|
5875 | setring(RP); |
---|
5876 | def F1=imap(RR,F); |
---|
5877 | def NW1=imap(RR,NW0); |
---|
5878 | int gcyes=0; |
---|
5879 | if (canop==1) |
---|
5880 | { |
---|
5881 | option(redSB); |
---|
5882 | def F11=std(F1); |
---|
5883 | setring(RR); |
---|
5884 | list gc; |
---|
5885 | def F2=imap(RP,F11); |
---|
5886 | def NW2=imap(RP,NW1); |
---|
5887 | if (size(NW2[1])==0) |
---|
5888 | { |
---|
5889 | gc=gencase1(F2,"compbas",0); |
---|
5890 | if (size(gc)>0) |
---|
5891 | { |
---|
5892 | gcyes=1; |
---|
5893 | NW2[1]=gc[4]; |
---|
5894 | //gc=delete(gc,4); |
---|
5895 | list gcn; |
---|
5896 | gcn[1]=ideal(1); // lpp |
---|
5897 | gcn[2]=list(list(ideal(1),ideal(0),list(gc[3]))); |
---|
5898 | gc=gcn; |
---|
5899 | } |
---|
5900 | } |
---|
5901 | } |
---|
5902 | else |
---|
5903 | { |
---|
5904 | setring(RR); |
---|
5905 | def NW2=NW0; |
---|
5906 | def F2=imap(RP,F1); |
---|
5907 | } |
---|
5908 | //setglobalrings(); |
---|
5909 | setring RR; // ja hi es ? |
---|
5910 | RRL=ringlist(RR); |
---|
5911 | //if (RRL[3][1][1]!="dp"){ERROR("the order must be dp");} |
---|
5912 | poly @t; |
---|
5913 | ring H=0,@t,dp; |
---|
5914 | def RH=RR+H; |
---|
5915 | setring(RH); |
---|
5916 | //kill @P; |
---|
5917 | //kill @RP; |
---|
5918 | //kill @R; |
---|
5919 | //setglobalrings(); |
---|
5920 | //setring(@Rt); |
---|
5921 | def FH=imap(RR,F2); |
---|
5922 | list gcH; |
---|
5923 | if (gcyes==1) |
---|
5924 | { |
---|
5925 | gcH=imap(RR,gc); |
---|
5926 | } |
---|
5927 | def NWH=imap(RR,NW2); |
---|
5928 | for (i=1;i<=size(FH);i++) |
---|
5929 | { |
---|
5930 | FH[i]=homog(FH[i],@t); |
---|
5931 | } |
---|
5932 | def G=gcover(FH,gcH,NWH); // list(NWH[1],NWH[2],CGS,comment)); |
---|
5933 | for (i=1;i<=size(G);i++) |
---|
5934 | { |
---|
5935 | G[i][1]=subst(G[i][1],@t,1); |
---|
5936 | G[i][2]=subst(G[i][2],@t,1); |
---|
5937 | } |
---|
5938 | setring(RR); |
---|
5939 | setglobalrings(); |
---|
5940 | S=imap(RH,G); |
---|
5941 | for (i=1;i<=size(S);i++) |
---|
5942 | { |
---|
5943 | S[i][2]=postredgb(mingb(S[i][2])); |
---|
5944 | S[i][1]=postredgb(mingb(S[i][1])); |
---|
5945 | } |
---|
5946 | } |
---|
5947 | // Now Extend; |
---|
5948 | poly leadc; |
---|
5949 | if (extop==1) |
---|
5950 | { |
---|
5951 | int start1=timer; |
---|
5952 | for (i=1;i<=size(S);i++) |
---|
5953 | { |
---|
5954 | m=size(S[i][2]); |
---|
5955 | for (j=1;j<=size(S[i][2]);j++) |
---|
5956 | { |
---|
5957 | idp=S[i][4][1]; |
---|
5958 | idq=S[i][4][2]; |
---|
5959 | if (size(idp)>0) |
---|
5960 | { |
---|
5961 | leadc=leadcoef(S[i][2][j]); |
---|
5962 | kill ext; |
---|
5963 | def ext=extend(S[i][2][j],idp,idq); |
---|
5964 | if (typeof(ext)=="poly") |
---|
5965 | { |
---|
5966 | S[i][2][j]=pnormalform(ext,idp,W); |
---|
5967 | //"T_Polynomial after extend="; S[i][2][j]; |
---|
5968 | } |
---|
5969 | else |
---|
5970 | { |
---|
5971 | if(size(ext)==1) |
---|
5972 | { |
---|
5973 | S[i][2][j]=ext[1]; |
---|
5974 | } |
---|
5975 | else |
---|
5976 | { |
---|
5977 | kill SS; list SS; |
---|
5978 | for(s=1;s<=size(ext);s++) |
---|
5979 | { |
---|
5980 | ext[s]=pnormalform(ext[s],idp,W); |
---|
5981 | } |
---|
5982 | for(s=1;s<=size(S[i][2]);s++) |
---|
5983 | { |
---|
5984 | if(s!=j){SS[s]=S[i][2][s];} |
---|
5985 | else{SS[s]=ext;} |
---|
5986 | } |
---|
5987 | S[i][2]=SS; |
---|
5988 | } |
---|
5989 | } |
---|
5990 | //"T_ poly or ideal after extend="; S[i][2][j]; |
---|
5991 | } |
---|
5992 | } |
---|
5993 | } |
---|
5994 | if(comment==1){string("Time in extend = ",timer-start1," sec");} |
---|
5995 | } |
---|
5996 | list Si; list nS; |
---|
5997 | if (repop==0) |
---|
5998 | { |
---|
5999 | for(i=1;i<=size(S);i++) |
---|
6000 | { |
---|
6001 | Si=list(S[i][1],S[i][2],S[i][3]); |
---|
6002 | nS[size(nS)+1]=Si; |
---|
6003 | } |
---|
6004 | S=nS; |
---|
6005 | } |
---|
6006 | else |
---|
6007 | { |
---|
6008 | if (repop==1) |
---|
6009 | { |
---|
6010 | for(i=1;i<=size(S);i++) |
---|
6011 | { |
---|
6012 | Si=list(S[i][1],S[i][2],S[i][4]); |
---|
6013 | nS[size(nS)+1]=Si; |
---|
6014 | } |
---|
6015 | S=nS; |
---|
6016 | } |
---|
6017 | } |
---|
6018 | kill @P; kill @RP; kill @R; |
---|
6019 | if (comment==1) |
---|
6020 | { |
---|
6021 | string("Time for grobcov = ", timer-start," sec"); |
---|
6022 | string("Number of segments of grobcov = ", size(S)); |
---|
6023 | } |
---|
6024 | return(S); |
---|
6025 | } |
---|
6026 | example |
---|
6027 | { "EXAMPLE:"; echo = 2; |
---|
6028 | "Casas conjecture for degree 4"; |
---|
6029 | ring R=(0,a0,a1,a2,a3,a4),(x1,x2,x3),dp; |
---|
6030 | ideal F=x1^4+(4*a3)*x1^3+(6*a2)*x1^2+(4*a1)*x1+(a0), |
---|
6031 | x1^3+(3*a3)*x1^2+(3*a2)*x1+(a1), |
---|
6032 | x2^4+(4*a3)*x2^3+(6*a2)*x2^2+(4*a1)*x2+(a0), |
---|
6033 | x2^2+(2*a3)*x2+(a2), |
---|
6034 | x3^4+(4*a3)*x3^3+(6*a2)*x3^2+(4*a1)*x3+(a0), |
---|
6035 | x3+(a3); |
---|
6036 | grobcov(F); |
---|
6037 | } |
---|
6038 | |
---|
6039 | |
---|
6040 | // input: |
---|
6041 | // poly g in K[a], |
---|
6042 | // list P=(p_1,..p_r) representing a minimal prime decomposition |
---|
6043 | // output |
---|
6044 | // poly f such taht f notin p_i forall i and |
---|
6045 | // g-f in p_i forall i such that g notin p_i |
---|
6046 | static proc nonzerodivisor(poly gr, list Pr) |
---|
6047 | { |
---|
6048 | def RR=basering; |
---|
6049 | setring(@P); |
---|
6050 | def g=imap(RR,gr); |
---|
6051 | def P=imap(RR,Pr); |
---|
6052 | int i; int k; list J; ideal F; |
---|
6053 | def f=g; |
---|
6054 | ideal Pi; |
---|
6055 | for (i=1;i<=size(P);i++) |
---|
6056 | { |
---|
6057 | option(redSB); |
---|
6058 | Pi=std(P[i]); |
---|
6059 | //attrib(Pi,"isST",1); |
---|
6060 | if (reduce(g,Pi,1)==0){J[size(J)+1]=i;} |
---|
6061 | } |
---|
6062 | for (i=1;i<=size(J);i++) |
---|
6063 | { |
---|
6064 | F=ideal(1); |
---|
6065 | for (k=1;k<=size(P);k++) |
---|
6066 | { |
---|
6067 | if (k!=J[i]) |
---|
6068 | { |
---|
6069 | F=idint(F,P[k]); |
---|
6070 | } |
---|
6071 | } |
---|
6072 | f=f+F[1]; |
---|
6073 | } |
---|
6074 | setring(RR); |
---|
6075 | def fr=imap(@P,f); |
---|
6076 | return(fr); |
---|
6077 | } |
---|
6078 | |
---|
6079 | // input: |
---|
6080 | // int i: |
---|
6081 | // list LPr: (p1,..,pr) of prime components of an ideal in K[a] |
---|
6082 | // output: |
---|
6083 | // list (fr,fnr) of two polynomials that are equal on V(pi) |
---|
6084 | // and fr=0 on V(P) \ V(pi), and fnr is nonzero on V(pj) for all j. |
---|
6085 | static proc deltai(int i, list LPr) |
---|
6086 | { |
---|
6087 | def RR=basering; |
---|
6088 | setring(@P); |
---|
6089 | def LP=imap(RR,LPr); |
---|
6090 | int j; poly p; |
---|
6091 | def F=ideal(1); |
---|
6092 | poly f; |
---|
6093 | poly fn; |
---|
6094 | ideal LPi; |
---|
6095 | for (j=1;j<=size(LP);j++) |
---|
6096 | { |
---|
6097 | if (j!=i) |
---|
6098 | { |
---|
6099 | F=idint(F,LP[j]); |
---|
6100 | } |
---|
6101 | } |
---|
6102 | p=0; j=1; |
---|
6103 | while ((p==0) and (j<=size(F))) |
---|
6104 | { |
---|
6105 | LPi=LP[i]; |
---|
6106 | attrib(LPi,"isSB",1); |
---|
6107 | p=reduce(F[j],LPi); |
---|
6108 | j++; |
---|
6109 | } |
---|
6110 | f=F[j-1]; |
---|
6111 | fn=nonzerodivisor(f,LP); |
---|
6112 | setring(RR); |
---|
6113 | def fr=imap(@P,f); |
---|
6114 | def fnr=imap(@P,fn); |
---|
6115 | return(list(fr,fnr)); |
---|
6116 | } |
---|
6117 | |
---|
6118 | // input: a list of pairs ((p1,P1),..,(pr,Pr)) where |
---|
6119 | // ideal pi is a prime component |
---|
6120 | // poly Pi is the polynomial in K[a][x] on V(pi)\ V(Mi) |
---|
6121 | // (p1,..,pr) are the prime decomposition of the lpp-segment |
---|
6122 | // list crep =(ideal ida,ideal idb): the Crep of the segment. |
---|
6123 | // list Pci of the intersecctions of all pj except the ith one |
---|
6124 | // output: |
---|
6125 | // poly P on an open and dense set of V(p_1 int ... p_r) |
---|
6126 | static proc combine(list L, ideal F) |
---|
6127 | { |
---|
6128 | // ATTENTION REVISE AND USE Pci and F |
---|
6129 | int i; poly f; |
---|
6130 | f=0; |
---|
6131 | for(i=1;i<=size(L);i++) |
---|
6132 | { |
---|
6133 | f=f+F[i]*L[i][2]; |
---|
6134 | } |
---|
6135 | f=elimconstfac(f); |
---|
6136 | return(f); |
---|
6137 | } |
---|
6138 | |
---|
6139 | // elimconstfac: eliminate the factors in the polynom f that are in K[a] |
---|
6140 | // input: |
---|
6141 | // poly f: |
---|
6142 | // list L: of components of the segment |
---|
6143 | // output: |
---|
6144 | // poly f2 where the factors of f in K[a] that are non-null on any component |
---|
6145 | // have been dropped from f |
---|
6146 | static proc elimconstfac(poly f) |
---|
6147 | { |
---|
6148 | int cond; int i; int j; int t; |
---|
6149 | if (f==0){return(f);} |
---|
6150 | def RR=basering; |
---|
6151 | setring(@R); |
---|
6152 | poly ff=imap(RR,f); |
---|
6153 | list l=factorize(ff,0); |
---|
6154 | poly f1=1; |
---|
6155 | for(i=2;i<=size(l[1]);i++) |
---|
6156 | { |
---|
6157 | f1=f1*(l[1][i])^(l[2][i]); |
---|
6158 | } |
---|
6159 | setring(RR); |
---|
6160 | def f2=imap(@R,f1); |
---|
6161 | return(f2); |
---|
6162 | } |
---|
6163 | |
---|
6164 | // input: |
---|
6165 | // poly f: a polynomial in K[a] |
---|
6166 | // ideal P: an ideal in K[a] |
---|
6167 | // called from ring @R |
---|
6168 | // output: |
---|
6169 | // t: with value 1 if f reduces modulo P, 0 if not. |
---|
6170 | static proc nullin(poly f,ideal P) |
---|
6171 | { |
---|
6172 | int t; |
---|
6173 | def RR=basering; |
---|
6174 | setring(@P); |
---|
6175 | poly f0=imap(RR,f); |
---|
6176 | ideal P0=imap(RR,P); |
---|
6177 | attrib(P0,"isSB",1); |
---|
6178 | if (reduce(f0,P0,1)==0){t=1;} |
---|
6179 | else{t=0;} |
---|
6180 | setring(RR); |
---|
6181 | return(t); |
---|
6182 | } |
---|
6183 | |
---|
6184 | static proc polyinparamsonly(poly f) |
---|
6185 | { |
---|
6186 | int t; |
---|
6187 | def RR=basering; |
---|
6188 | setring @R; |
---|
6189 | def f0=imap(RR,f); |
---|
6190 | if (size(variables(f0))==0){t=1;} |
---|
6191 | else{t=0;} |
---|
6192 | setring(RR); |
---|
6193 | return(t); |
---|
6194 | } |
---|
6195 | |
---|
6196 | // monoms |
---|
6197 | static proc monoms(poly f) |
---|
6198 | { |
---|
6199 | list L; |
---|
6200 | if (f!=0) { L[size(f)]=list();} |
---|
6201 | poly lm; poly lc; poly lp; poly Q; poly mQ; |
---|
6202 | def p=f; |
---|
6203 | int i=1; |
---|
6204 | while (p!=0) |
---|
6205 | { |
---|
6206 | lm=lead(p); |
---|
6207 | p=p-lm; |
---|
6208 | lc=leadcoef(lm); |
---|
6209 | lp=leadmonom(lm); |
---|
6210 | L[i]=list(lc,lp); |
---|
6211 | i++; |
---|
6212 | } |
---|
6213 | return(L); |
---|
6214 | } |
---|
6215 | |
---|
6216 | // input: |
---|
6217 | // poly f: a generic polynomial in the basis |
---|
6218 | // ideal idp: such that ideal(S)=idp |
---|
6219 | // ideal idq: such that S=V(idp)\V(idq) |
---|
6220 | //// NW the list of ((N1,W1),..,(Ns,Ws)) of red-rep of the grouped |
---|
6221 | //// segments in the lpp-segment NO MORE USED |
---|
6222 | // output: |
---|
6223 | static proc extend(poly f, ideal idp, ideal idq) |
---|
6224 | { |
---|
6225 | matrix CC; poly Q; list NewMonoms; |
---|
6226 | int i; int j; poly fout; ideal idout; |
---|
6227 | list L=monoms(f); |
---|
6228 | int nummonoms=size(L)-1; |
---|
6229 | Q=L[1][1]; |
---|
6230 | if (nummonoms==0){return(f);} |
---|
6231 | for (i=2;i<=size(L);i++) |
---|
6232 | { |
---|
6233 | CC=matrix(extendcoef(L[i][1],Q,idp,idq)); |
---|
6234 | NewMonoms[i-1]=list(CC,L[i][2]); |
---|
6235 | } |
---|
6236 | if (nummonoms==1) |
---|
6237 | { |
---|
6238 | for(j=1;j<=ncols(NewMonoms[1][1]);j++) |
---|
6239 | { |
---|
6240 | fout=NewMonoms[1][1][2,j]*L[1][2]+NewMonoms[1][1][1,j]*NewMonoms[1][2]; |
---|
6241 | //fout=pnormalform(fout,idp,W); |
---|
6242 | if(ncols(NewMonoms[1][1])>1){idout[j]=fout;} |
---|
6243 | } |
---|
6244 | if(ncols(NewMonoms[1][1])==1){return(fout);} else{return(idout);} |
---|
6245 | } |
---|
6246 | else |
---|
6247 | { |
---|
6248 | //int start=timer; |
---|
6249 | list cfi; |
---|
6250 | list coefs; |
---|
6251 | for (i=1;i<=nummonoms;i++) |
---|
6252 | { |
---|
6253 | kill cfi; list cfi; |
---|
6254 | for(j=1;j<=ncols(NewMonoms[i][1]);j++) |
---|
6255 | { |
---|
6256 | cfi[size(cfi)+1]=NewMonoms[i][1][2,j]; |
---|
6257 | } |
---|
6258 | coefs[i]=cfi; |
---|
6259 | } |
---|
6260 | def indexpolys=findindexpolys(coefs); |
---|
6261 | for(i=1;i<=size(indexpolys);i++) |
---|
6262 | { |
---|
6263 | fout=L[1][2]; |
---|
6264 | for(j=1;j<=nummonoms;j++) |
---|
6265 | { |
---|
6266 | fout=fout+(NewMonoms[j][1][1,indexpolys[i][j]])/(NewMonoms[j][1][2,indexpolys[i][j]])*NewMonoms[j][2]; |
---|
6267 | } |
---|
6268 | fout=cleardenom(fout); |
---|
6269 | if(size(indexpolys)>1){idout[i]=fout;} |
---|
6270 | } |
---|
6271 | if (size(indexpolys)==1){return(fout);} else{return(idout);} |
---|
6272 | } |
---|
6273 | } |
---|
6274 | |
---|
6275 | // input: |
---|
6276 | // list coefs=( (q11,..,q1r_1),..,(qs1,..,qsr_1) ) |
---|
6277 | // of denominators of the monoms |
---|
6278 | // output: |
---|
6279 | // list ind=(v_1,..,v_t) of intvec |
---|
6280 | // each intvec v=(i_1,..,is) corresponds to a polynomial in the sheaf |
---|
6281 | // that will be built from it in extend procedure. |
---|
6282 | static proc findindexpolys(list coefs) |
---|
6283 | { |
---|
6284 | int i; int j; intvec numdens; |
---|
6285 | for(i=1;i<=size(coefs);i++) |
---|
6286 | { |
---|
6287 | numdens[i]=size(coefs[i]); |
---|
6288 | } |
---|
6289 | def RR=basering; |
---|
6290 | setring(@P); |
---|
6291 | def coefsp=imap(RR,coefs); |
---|
6292 | ideal cof; list combpolys; intvec v; int te; list mp; |
---|
6293 | for(i=1;i<=size(coefsp);i++) |
---|
6294 | { |
---|
6295 | cof=ideal(0); |
---|
6296 | for(j=1;j<=size(coefsp[i]);j++) |
---|
6297 | { |
---|
6298 | cof[j]=factorize(coefsp[i][j],3); |
---|
6299 | } |
---|
6300 | coefsp[i]=cof; |
---|
6301 | } |
---|
6302 | for(j=1;j<=size(coefsp[1]);j++) |
---|
6303 | { |
---|
6304 | v[1]=j; |
---|
6305 | te=1; |
---|
6306 | for (i=2;i<=size(coefsp);i++) |
---|
6307 | { |
---|
6308 | mp=memberpos(coefsp[1][j],coefsp[i]); |
---|
6309 | if(mp[1]) |
---|
6310 | { |
---|
6311 | v[i]=mp[2]; |
---|
6312 | } |
---|
6313 | else{v[i]=0;} |
---|
6314 | } |
---|
6315 | combpolys[j]=v; |
---|
6316 | } |
---|
6317 | combpolys=reform(combpolys,numdens); |
---|
6318 | setring RR; |
---|
6319 | return(combpolys); |
---|
6320 | } |
---|
6321 | |
---|
6322 | |
---|
6323 | // extendcoef: given Q,P in K[a] where P/Q specializes on an open and dense subset |
---|
6324 | // of the whole V(p1 int...int pr), it returns a basis of the module |
---|
6325 | // of all syzygies equivalent to P/Q, |
---|
6326 | static proc extendcoef(poly P, poly Q, ideal idp, ideal idq) |
---|
6327 | { |
---|
6328 | def RR=basering; |
---|
6329 | setring(@P); |
---|
6330 | def PL=ringlist(@P); |
---|
6331 | PL[3][1][1]="dp"; |
---|
6332 | def P1=ring(PL); |
---|
6333 | setring(P1); |
---|
6334 | ideal idp0=imap(RR,idp); |
---|
6335 | option(redSB); |
---|
6336 | qring q=std(idp0); |
---|
6337 | poly P0=imap(RR,P); |
---|
6338 | poly Q0=imap(RR,Q); |
---|
6339 | ideal PQ=Q0,-P0; |
---|
6340 | module C=syz(PQ); |
---|
6341 | setring @P; |
---|
6342 | def idp1=imap(RR,idp); |
---|
6343 | def idq1=imap(RR,idq); |
---|
6344 | def C1=matrix(imap(q,C)); |
---|
6345 | def redC=selectregularfun(C1,idp1,idq1); |
---|
6346 | setring(RR); |
---|
6347 | def CC=imap(@P,redC); |
---|
6348 | return(CC); |
---|
6349 | } |
---|
6350 | |
---|
6351 | // input: |
---|
6352 | // list L of the polynomials matrix CC |
---|
6353 | // (we assume that one of them is non-null on V(N)\V(M)) |
---|
6354 | // ideal N, ideal M: ideals representing the locally closed set V(N)\V(M) |
---|
6355 | // assume to work in @P |
---|
6356 | static proc selectregularfun(matrix CC, ideal NN, ideal MM) |
---|
6357 | { |
---|
6358 | int numcombused; |
---|
6359 | def RR=basering; |
---|
6360 | setring @P; |
---|
6361 | def C=imap(RR,CC); |
---|
6362 | def N=imap(RR,NN); |
---|
6363 | def M=imap(RR,MM); |
---|
6364 | if (ncols(C)==1){return(C);} |
---|
6365 | |
---|
6366 | int i; int j; int k; list c; intvec ci; intvec c0; intvec c1; |
---|
6367 | list T; list T0; list T1; list LL; ideal N1;ideal M1; int te=0; |
---|
6368 | for(i=1;i<=ncols(C);i++) |
---|
6369 | { |
---|
6370 | if((C[1,i]!=0) and (C[2,i]!=0)) |
---|
6371 | { |
---|
6372 | if(c0==intvec(0)){c0[1]=i;} |
---|
6373 | else{c0[size(c0)+1]=i;} |
---|
6374 | } |
---|
6375 | } |
---|
6376 | def C1=submat(C,1..2,c0); |
---|
6377 | for (i=1;i<=ncols(C1);i++) |
---|
6378 | { |
---|
6379 | c=comb(ncols(C1),i); |
---|
6380 | for(j=1;j<=size(c);j++) |
---|
6381 | { |
---|
6382 | ci=c[j]; |
---|
6383 | numcombused++; |
---|
6384 | if(i==1){N1=N+C1[2,j]; M1=M;} |
---|
6385 | if(i>1) |
---|
6386 | { |
---|
6387 | kill c0; intvec c0 ; kill c1; intvec c1; |
---|
6388 | c1=ci[size(ci)]; |
---|
6389 | for(k=1;k<size(ci);k++){c0[k]=ci[k];} |
---|
6390 | T0=searchinlist(c0,LL); |
---|
6391 | T1=searchinlist(c1,LL); |
---|
6392 | N1=T0[1]+T1[1]; |
---|
6393 | M1=intersect(T0[2],T1[2]); |
---|
6394 | } |
---|
6395 | T=list(ci,PtoCrep(Prep(N1,M1))); |
---|
6396 | LL[size(LL)+1]=T; |
---|
6397 | if(equalideals(T[2][1],ideal(1))==1){te=1; break;} |
---|
6398 | } |
---|
6399 | if(te==1){break;} |
---|
6400 | } |
---|
6401 | ci=T[1]; |
---|
6402 | def Cs=submat(C1,1..2,ci); |
---|
6403 | setring RR; |
---|
6404 | return(imap(@P,Cs)); |
---|
6405 | } |
---|
6406 | |
---|
6407 | // input: |
---|
6408 | // intvec c: |
---|
6409 | // list L=( (c1,T1),..(ck,Tk) ) |
---|
6410 | // where the c's are assumed to be intvects |
---|
6411 | // output: |
---|
6412 | // object T with index c |
---|
6413 | static proc searchinlist(intvec c,list L) |
---|
6414 | { |
---|
6415 | int i; list T; |
---|
6416 | for(i=1;i<=size(L);i++) |
---|
6417 | { |
---|
6418 | if (L[i][1]==c) |
---|
6419 | { |
---|
6420 | T=L[i][2]; |
---|
6421 | break; |
---|
6422 | } |
---|
6423 | } |
---|
6424 | return(T); |
---|
6425 | } |
---|
6426 | |
---|
6427 | // Input: C0 the matrtix of (P1,..,Pr) |
---|
6428 | // (Q1,..,Qr) of the regular function of a coefficient (P,Q) |
---|
6429 | // NW0 the list of ((N1,W1),..(Ns,Ws)) of red-rep of the grouped |
---|
6430 | // segments in the lpp-segment |
---|
6431 | // Output: (B, T) where |
---|
6432 | // B is the submatrix of the selected minimal representants for the |
---|
6433 | // regular function |
---|
6434 | // T the matrix of ones and zeroes whose colums are associated |
---|
6435 | // to the colums of B, with 1 in the segments where the representant |
---|
6436 | // is nonnull and 0 if it can be. |
---|
6437 | static proc redext(matrix C0, list NW0) |
---|
6438 | { |
---|
6439 | def RR=basering; |
---|
6440 | setring(@P); |
---|
6441 | def C=imap(RR,C0); |
---|
6442 | def NW=imap(RR,NW0); |
---|
6443 | int nc=ncols(C); |
---|
6444 | int nr=size(NW); |
---|
6445 | intmat T[nr][nc]; |
---|
6446 | int i; int j; int k; int t; |
---|
6447 | for (i=1;i<=nc;i++) |
---|
6448 | { |
---|
6449 | for (j=1;j<=nr;j++) |
---|
6450 | { |
---|
6451 | t=nonnull(C[i][2],NW[j][1],NW[j][2]); // (Q,N,W) |
---|
6452 | T[j,i]=t; |
---|
6453 | } |
---|
6454 | } |
---|
6455 | int h; int tt=0; |
---|
6456 | intvec c; intvec r; |
---|
6457 | list cc; int l; |
---|
6458 | for (j=1;j<=2;j++){r[j]=j;} |
---|
6459 | i=1; |
---|
6460 | while((i<=nc) and (tt==0)) |
---|
6461 | { |
---|
6462 | cc=comb(nc,i); |
---|
6463 | tt=0; |
---|
6464 | l=1; |
---|
6465 | while((tt==0) and (l<=size(cc))) |
---|
6466 | { |
---|
6467 | tt=1; |
---|
6468 | c=cc[l]; |
---|
6469 | j=1; |
---|
6470 | while ((j<=nr) and (tt==1)) |
---|
6471 | { |
---|
6472 | h=0; |
---|
6473 | k=1; |
---|
6474 | while ((h==0) and (k<=i)) |
---|
6475 | { |
---|
6476 | if(T[j,c[k]]==1){h=1;} |
---|
6477 | k++; |
---|
6478 | } |
---|
6479 | if (h==0){tt=0;} |
---|
6480 | j++; |
---|
6481 | } |
---|
6482 | l++; |
---|
6483 | } |
---|
6484 | i++; |
---|
6485 | } |
---|
6486 | if (tt==0){"extendcoef does not extend to the whole S";} |
---|
6487 | intvec rr; |
---|
6488 | for (i=1;i<=nr;i++){rr[i]=i;} |
---|
6489 | def B=submat(C,r,c); |
---|
6490 | def TT=submat(T,rr,c); |
---|
6491 | setring(RR); |
---|
6492 | return(list(imap(@P,B),imap(@P,TT))); |
---|
6493 | } |
---|
6494 | |
---|
6495 | // comb: the list of combinations of elements (1,..n) of order p |
---|
6496 | static proc comb(int n, int p) |
---|
6497 | { |
---|
6498 | list L; list L0; |
---|
6499 | intvec c; intvec d; |
---|
6500 | int i; int j; int last; |
---|
6501 | if ((n<0) or (n<p)) |
---|
6502 | { |
---|
6503 | return(L); |
---|
6504 | } |
---|
6505 | if (p==1) |
---|
6506 | { |
---|
6507 | for (i=1;i<=n;i++) |
---|
6508 | { |
---|
6509 | c=i; |
---|
6510 | L[size(L)+1]=c; |
---|
6511 | } |
---|
6512 | return(L); |
---|
6513 | } |
---|
6514 | else |
---|
6515 | { |
---|
6516 | L0=comb(n,p-1); |
---|
6517 | for (i=1;i<=size(L0);i++) |
---|
6518 | { |
---|
6519 | c=L0[i]; d=c; |
---|
6520 | last=c[size(c)]; |
---|
6521 | for (j=last+1;j<=n;j++) |
---|
6522 | { |
---|
6523 | d[size(c)+1]=j; |
---|
6524 | L[size(L)+1]=d; |
---|
6525 | } |
---|
6526 | } |
---|
6527 | return(L); |
---|
6528 | } |
---|
6529 | } |
---|
6530 | |
---|
6531 | // selectminsheaves |
---|
6532 | // Input: L=((v_11,..,v_1k_1),..,(v_s1,..,v_sk_s)) |
---|
6533 | // where: |
---|
6534 | // The s lists correspond to the s coefficients of the polynomial f |
---|
6535 | // (v_i1,..,v_ik_i) correspond to the k_i intvec v_ij of the |
---|
6536 | // spezializations of the jth rekpresentant (Q,P) of the ith coefficient |
---|
6537 | // v_ij is an intvec of size equal to the number of little segments |
---|
6538 | // forming the lpp-segment of 0,1, where 1 represents that it specializes |
---|
6539 | // to non-zedro an the whole little segment and 0 if not. |
---|
6540 | // Output: S=(w_1,..,w_j) |
---|
6541 | // where the w_l=(n_l1,..,n_ls) are intvec of length size(L), where |
---|
6542 | // n_lt fixes which element of (v_t1,..,v_tk_t) is to be |
---|
6543 | // choosen to form the tth (Q,P) for the lth element of the sheaf |
---|
6544 | // representing the I-regular function. |
---|
6545 | // The selection is done to obtian the minimal number of elements |
---|
6546 | // of the sheaf that specializes to non-null everywhere. |
---|
6547 | static proc selectminsheaves(list L) |
---|
6548 | { |
---|
6549 | list C=allsheaves(L); |
---|
6550 | return(smsheaves(C[1],C[2])); |
---|
6551 | } |
---|
6552 | |
---|
6553 | // Input: |
---|
6554 | // list C of all the combrep |
---|
6555 | // list L of the intvec that correesponds to each element of C |
---|
6556 | // Output: |
---|
6557 | // list LL of the subsets of C that cover all the subsegments |
---|
6558 | // (the union of the corresponding L(C) has all 1). |
---|
6559 | static proc smsheaves(list C, list L) |
---|
6560 | { |
---|
6561 | int i; int i0; intvec W; |
---|
6562 | int nor; int norn; |
---|
6563 | intvec p; |
---|
6564 | int sp=size(L[1]); int j0=1; |
---|
6565 | for (i=1;i<=sp;i++){p[i]=1;} |
---|
6566 | while (p!=0) |
---|
6567 | { |
---|
6568 | i0=0; nor=0; |
---|
6569 | for (i=1; i<=size(L); i++) |
---|
6570 | { |
---|
6571 | norn=numones(L[i],pos(p)); |
---|
6572 | if (nor<norn){nor=norn; i0=i;} |
---|
6573 | } |
---|
6574 | W[j0]=i0; |
---|
6575 | j0++; |
---|
6576 | p=actualize(p,L[i0]); |
---|
6577 | } |
---|
6578 | list LL; |
---|
6579 | for (i=1;i<=size(W);i++) |
---|
6580 | { |
---|
6581 | LL[size(LL)+1]=C[W[i]]; |
---|
6582 | } |
---|
6583 | return(LL); |
---|
6584 | } |
---|
6585 | |
---|
6586 | // allsheaves |
---|
6587 | // Input: L=((v_11,..,v_1k_1),..,(v_s1,..,v_sk_s)) |
---|
6588 | // where: |
---|
6589 | // The s lists correspond to the s coefficients of the polynomial f |
---|
6590 | // (v_i1,..,v_ik_i) correspond to the k_i intvec v_ij of the |
---|
6591 | // spezializations of the jth rekpresentant (Q,P) of the ith coefficient |
---|
6592 | // v_ij is an intvec of size equal to the number of little segments |
---|
6593 | // forming the lpp-segment of 0,1, where 1 represents that it specializes |
---|
6594 | // to non-zero on the whole little segment and 1 if not. |
---|
6595 | // Output: |
---|
6596 | // (list LL, list LLS) where |
---|
6597 | // LL is the list of all combrep |
---|
6598 | // LLS is the list of intvec of the corresponding elements of LL |
---|
6599 | static proc allsheaves(list L) |
---|
6600 | { |
---|
6601 | intvec V; list LL; intvec W; int r; intvec U; |
---|
6602 | int i; int j; int k; |
---|
6603 | int s=size(L[1][1]); // s = number of little segments of the lpp-segment |
---|
6604 | list LLS; |
---|
6605 | for (i=1;i<=size(L);i++) |
---|
6606 | { |
---|
6607 | V[i]=size(L[i]); |
---|
6608 | } |
---|
6609 | LL=combrep(V); |
---|
6610 | for (i=1;i<=size(LL);i++) |
---|
6611 | { |
---|
6612 | W=LL[i]; // size(W)= number of coefficients of the polynomial |
---|
6613 | kill U; intvec U; |
---|
6614 | for (j=1;j<=s;j++) |
---|
6615 | { |
---|
6616 | k=1; r=1; U[j]=1; |
---|
6617 | while((r==1) and (k<=size(W))) |
---|
6618 | { |
---|
6619 | if(L[k][W[k]][j]==0){r=0; U[j]=0;} |
---|
6620 | k++; |
---|
6621 | } |
---|
6622 | } |
---|
6623 | LLS[i]=U; |
---|
6624 | } |
---|
6625 | return(list(LL,LLS)); |
---|
6626 | } |
---|
6627 | |
---|
6628 | // numones |
---|
6629 | // Input: |
---|
6630 | // intvec v of (0,1) in each position |
---|
6631 | // intvec pos: the positions to test |
---|
6632 | // Output: |
---|
6633 | // int nor: the nuber of 1 of v in the positions given by pos. |
---|
6634 | static proc numones(intvec v, intvec pos) |
---|
6635 | { |
---|
6636 | int i; int n; |
---|
6637 | for (i=1;i<=size(pos);i++) |
---|
6638 | { |
---|
6639 | if (v[pos[i]]==1){n++;} |
---|
6640 | } |
---|
6641 | return(n); |
---|
6642 | } |
---|
6643 | |
---|
6644 | // Input: intvec p of zeros and ones |
---|
6645 | // Output: intvec W of the positions where p has ones. |
---|
6646 | static proc pos(intvec p) |
---|
6647 | { |
---|
6648 | int i; |
---|
6649 | intvec W; int j=1; |
---|
6650 | for (i=1; i<=size(p); i++) |
---|
6651 | { |
---|
6652 | if (p[i]==1){W[j]=i; j++;} |
---|
6653 | } |
---|
6654 | return(W); |
---|
6655 | } |
---|
6656 | |
---|
6657 | // actualize: actualizes zeroes of p |
---|
6658 | // Input: |
---|
6659 | // intvec p: of zeroes and ones |
---|
6660 | // intvec c: of zeroes and ones (of the same length) |
---|
6661 | // Output; |
---|
6662 | // intvec pp: of zeroes and ones, where a 0 stays in pp[i] if either |
---|
6663 | // already p[i]==0 or c[i]==1. |
---|
6664 | static proc actualize(intvec p, intvec c) |
---|
6665 | { |
---|
6666 | int i; intvec pp=p; |
---|
6667 | for (i=1;i<=size(p);i++) |
---|
6668 | { |
---|
6669 | if ((pp[i]==1) and (c[i]==1)){pp[i]=0;} |
---|
6670 | } |
---|
6671 | return(pp); |
---|
6672 | } |
---|
6673 | |
---|
6674 | // combrep |
---|
6675 | // Input: V=(n_1,..,n_i) |
---|
6676 | // Output: L=(v_1,..,v_p) where p=prod_j=1^i (n_j) |
---|
6677 | // is the list of all intvec v_j=(v_j1,..,v_ji) where 1<=v_jk<=n_i |
---|
6678 | static proc combrep(intvec V) |
---|
6679 | { |
---|
6680 | list L; list LL; |
---|
6681 | int i; int j; int k; intvec W; |
---|
6682 | if (size(V)==1) |
---|
6683 | { |
---|
6684 | for (i=1;i<=V[1];i++) |
---|
6685 | { |
---|
6686 | L[i]=intvec(i); |
---|
6687 | } |
---|
6688 | return(L); |
---|
6689 | } |
---|
6690 | for (i=1;i<size(V);i++) |
---|
6691 | { |
---|
6692 | W[i]=V[i]; |
---|
6693 | } |
---|
6694 | LL=combrep(W); |
---|
6695 | for (i=1;i<=size(LL);i++) |
---|
6696 | { |
---|
6697 | W=LL[i]; |
---|
6698 | for (j=1;j<=V[size(V)];j++) |
---|
6699 | { |
---|
6700 | W[size(V)]=j; |
---|
6701 | L[size(L)+1]=W; |
---|
6702 | } |
---|
6703 | } |
---|
6704 | return(L); |
---|
6705 | } |
---|
6706 | |
---|
6707 | static proc reducemodN(poly f,ideal N) |
---|
6708 | { |
---|
6709 | def RR=basering; |
---|
6710 | setring(@RPt); |
---|
6711 | def fa=imap(RR,f); |
---|
6712 | def Na=imap(RR,N); |
---|
6713 | attrib(Na,"isSB",1); |
---|
6714 | // //option(redSB); |
---|
6715 | // Na=std(Na); |
---|
6716 | fa=reduce(fa,Na); |
---|
6717 | setring(RR); |
---|
6718 | def f1=imap(@RPt,fa); |
---|
6719 | return(f1); |
---|
6720 | } |
---|
6721 | |
---|
6722 | // computes the intersection of the ideals in S in @P |
---|
6723 | static proc intersp(list S) |
---|
6724 | { |
---|
6725 | def RR=basering; |
---|
6726 | setring(@P); |
---|
6727 | def SP=imap(RR,S); |
---|
6728 | option(returnSB); |
---|
6729 | def NP=intersect(SP[1..size(SP)]); |
---|
6730 | setring(RR); |
---|
6731 | return(imap(@P,NP)); |
---|
6732 | } |
---|
6733 | |
---|
6734 | static proc radicalmember(poly f,ideal ida) |
---|
6735 | { |
---|
6736 | int te; |
---|
6737 | def RR=basering; |
---|
6738 | setring(@P); |
---|
6739 | poly fp=imap(RR,f); |
---|
6740 | ideal idap=imap(RR,ida); |
---|
6741 | poly @t; |
---|
6742 | ring H=0,@t,dp; |
---|
6743 | def PH=@P+H; |
---|
6744 | setring(PH); |
---|
6745 | def fH=imap(@P,fp); |
---|
6746 | ideal idaH=imap(@P,idap); |
---|
6747 | idaH[ncols(idaH)+1]=1-@t*fH; |
---|
6748 | option(redSB); |
---|
6749 | ideal G=std(idaH); |
---|
6750 | //"G="; G; |
---|
6751 | if (G==1){te=1;} else {te=0;} |
---|
6752 | setring(RR); |
---|
6753 | return(te); |
---|
6754 | } |
---|
6755 | |
---|
6756 | // returns 1 if the poly f is nonnull on V(N)\V(M), 0 otherwise. |
---|
6757 | static proc NonNull(poly f, ideal N, ideal M) |
---|
6758 | { |
---|
6759 | int te=1; int i; |
---|
6760 | def RR=basering; |
---|
6761 | setring(@P); |
---|
6762 | poly fp=imap(RR,f); |
---|
6763 | ideal Np=imap(RR,N); |
---|
6764 | ideal Mp=imap(RR,M); |
---|
6765 | ideal H; |
---|
6766 | ideal Nf=Np+fp; |
---|
6767 | for (i=1;i<=ncols(Mp);i++) |
---|
6768 | { |
---|
6769 | te=radicalmember(Mp[i],Nf); |
---|
6770 | if (te==0) break; |
---|
6771 | } |
---|
6772 | setring RR; |
---|
6773 | return(te); |
---|
6774 | } |
---|
6775 | |
---|
6776 | // input: |
---|
6777 | // matrix CC: CC=(p_a1 .. p_ar_a) |
---|
6778 | // (q_a1 .. q_ar_a) |
---|
6779 | // the matrix of elements of a coefficient in oo[a]. |
---|
6780 | // (ideal ida, ideal idb): the canonical representation of the segment S. |
---|
6781 | // output: |
---|
6782 | // list caout |
---|
6783 | // the minimum set of elements of CC needed such that at least one |
---|
6784 | // of the q's is non-null on S, as well as the C-rep of of the |
---|
6785 | // points where the q's are null on S. |
---|
6786 | // The elements of caout are of the form (p,q,prep); |
---|
6787 | static proc selectextendcoef(matrix CC, ideal ida, ideal idb) |
---|
6788 | { |
---|
6789 | def RR=basering; |
---|
6790 | setring(@P); |
---|
6791 | def ca=imap(RR,CC); |
---|
6792 | def N0=imap(RR,ida); |
---|
6793 | ideal N; |
---|
6794 | def M=imap(RR,idb); |
---|
6795 | int r=ncols(ca); |
---|
6796 | int i; int te=1; list com; int j; int k; intvec c; list prep; |
---|
6797 | list cs; list caout; |
---|
6798 | i=1; |
---|
6799 | while ((i<=r) and (te==1)) |
---|
6800 | { |
---|
6801 | com=comb(r,i); |
---|
6802 | j=1; |
---|
6803 | while((j<=size(com)) and (te==1)) |
---|
6804 | { |
---|
6805 | N=N0; |
---|
6806 | c=com[j]; |
---|
6807 | for (k=1;k<=i;k++) |
---|
6808 | { |
---|
6809 | N=N+ca[2,c[k]]; |
---|
6810 | } |
---|
6811 | prep=Prep(N,M); |
---|
6812 | if (i==1) |
---|
6813 | { |
---|
6814 | cs[j]=list(ca[1,j],ca[2,j],prep); |
---|
6815 | } |
---|
6816 | if ((size(prep)==1) and (equalideals(prep[1][1],ideal(1)))) |
---|
6817 | { |
---|
6818 | te=0; |
---|
6819 | for(k=1;k<=size(c);k++) |
---|
6820 | { |
---|
6821 | caout[k]=cs[c[k]]; |
---|
6822 | } |
---|
6823 | } |
---|
6824 | j++; |
---|
6825 | } |
---|
6826 | i++; |
---|
6827 | } |
---|
6828 | if (te==1){"error: extendcoef does not extend to the whole S";} |
---|
6829 | setring(RR); |
---|
6830 | return(imap(@P,caout)); |
---|
6831 | } |
---|
6832 | |
---|
6833 | // input: |
---|
6834 | // ideal N1: in some basering (depends only on the parameters) |
---|
6835 | // ideal N2: in some basering (depends only on the parameters) |
---|
6836 | // output: |
---|
6837 | // ideal Np=N1+N2; computed in P |
---|
6838 | static proc plusP(ideal N1,ideal N2) |
---|
6839 | { |
---|
6840 | def RR=basering; |
---|
6841 | setring(@P); |
---|
6842 | def N1p=imap(RR,N1); |
---|
6843 | def N2p=imap(RR,N2); |
---|
6844 | def Np=N1p+N2p; |
---|
6845 | setring RR; |
---|
6846 | return(imap(@P,Np)); |
---|
6847 | } |
---|
6848 | |
---|
6849 | // input: |
---|
6850 | // list combpolys: (v1,..,vs) |
---|
6851 | // where vi are intvec. |
---|
6852 | // output outcomb: (w1,..,wt) |
---|
6853 | // whre wi are intvec. |
---|
6854 | // All the vi without zeroes are in outcomb, and those with zeroes are |
---|
6855 | // combined to form new intvec with the rest |
---|
6856 | static proc reform(list combpolys, intvec numdens) |
---|
6857 | { |
---|
6858 | list combp0; list combp1; int i; int j; int k; int l; list rest; intvec notfree; |
---|
6859 | list free; intvec free1; int te; intvec v; intvec w; |
---|
6860 | int nummonoms=size(combpolys[1]); |
---|
6861 | for(i=1;i<=size(combpolys);i++) |
---|
6862 | { |
---|
6863 | if(memberpos(0,combpolys[i])[1]==1) |
---|
6864 | { |
---|
6865 | combp0[size(combp0)+1]=combpolys[i]; |
---|
6866 | } |
---|
6867 | else {combp1[size(combp1)+1]=combpolys[i];} |
---|
6868 | } |
---|
6869 | for(i=1;i<=nummonoms;i++) |
---|
6870 | { |
---|
6871 | kill notfree; intvec notfree; |
---|
6872 | for(j=1;j<=size(combpolys);j++) |
---|
6873 | { |
---|
6874 | if(combpolys[j][i]<>0) |
---|
6875 | { |
---|
6876 | if(notfree[1]==0){notfree[1]=combpolys[j][i];} |
---|
6877 | else{notfree[size(notfree)+1]=combpolys[j][i];} |
---|
6878 | } |
---|
6879 | } |
---|
6880 | kill free1; intvec free1; |
---|
6881 | for(j=1;j<=numdens[i];j++) |
---|
6882 | { |
---|
6883 | if(memberpos(j,notfree)[1]==0) |
---|
6884 | { |
---|
6885 | if(free1[1]==0){free1[1]=j;} |
---|
6886 | else{free1[size(free1)+1]=j;} |
---|
6887 | } |
---|
6888 | free[i]=free1; |
---|
6889 | } |
---|
6890 | } |
---|
6891 | list amplcombp; list aux; |
---|
6892 | for(i=1;i<=size(combp0);i++) |
---|
6893 | { |
---|
6894 | v=combp0[i]; |
---|
6895 | kill amplcombp; list amplcombp; |
---|
6896 | amplcombp[1]=intvec(v[1]); |
---|
6897 | for(j=2;j<=size(v);j++) |
---|
6898 | { |
---|
6899 | if(v[j]!=0) |
---|
6900 | { |
---|
6901 | for(k=1;k<=size(amplcombp);k++) |
---|
6902 | { |
---|
6903 | w=amplcombp[k]; |
---|
6904 | w[size(w)+1]=v[j]; |
---|
6905 | amplcombp[k]=w; |
---|
6906 | } |
---|
6907 | } |
---|
6908 | else |
---|
6909 | { |
---|
6910 | kill aux; list aux; |
---|
6911 | for(k=1;k<=size(amplcombp);k++) |
---|
6912 | { |
---|
6913 | for(l=1;l<=size(free[j]);l++) |
---|
6914 | { |
---|
6915 | w=amplcombp[k]; |
---|
6916 | w[size(w)+1]=free[j][l]; |
---|
6917 | aux[size(aux)+1]=w; |
---|
6918 | } |
---|
6919 | } |
---|
6920 | amplcombp=aux; |
---|
6921 | } |
---|
6922 | } |
---|
6923 | for(j=1;j<=size(amplcombp);j++) |
---|
6924 | { |
---|
6925 | combp1[size(combp1)+1]=amplcombp[j]; |
---|
6926 | } |
---|
6927 | } |
---|
6928 | return(combp1); |
---|
6929 | } |
---|
6930 | |
---|
6931 | static proc nonnullCrep(poly f0,ideal ida0,ideal idb0) |
---|
6932 | { |
---|
6933 | int i; |
---|
6934 | def RR=basering; |
---|
6935 | setring(@P); |
---|
6936 | def f=imap(RR,f0); |
---|
6937 | def ida=imap(RR,ida0); |
---|
6938 | def idb=imap(RR,idb0); |
---|
6939 | def idaf=ida+f; |
---|
6940 | int te=1; |
---|
6941 | for(i=1;i<=size(idb);i++) |
---|
6942 | { |
---|
6943 | if(radicalmember(idb[i],idaf)==0) |
---|
6944 | { |
---|
6945 | te=0; break; |
---|
6946 | } |
---|
6947 | } |
---|
6948 | setring(RR); |
---|
6949 | return(te); |
---|
6950 | } |
---|
6951 | |
---|
6952 | // input: L: list of ideals (works in @P) |
---|
6953 | // output: F0: ideal of polys. F0[i] is a poly in the intersection of |
---|
6954 | // all ideals in L except in the ith one, where it is not. |
---|
6955 | // L=(p1,..,ps); F0=(f1,..,fs); |
---|
6956 | // F0[i] \in intersect_{j#i} p_i |
---|
6957 | static proc precombint(list L) |
---|
6958 | { |
---|
6959 | int i; int j; int tes; |
---|
6960 | def RR=basering; |
---|
6961 | setring(@P); |
---|
6962 | list L0; list L1; list L2; list L3; ideal F; |
---|
6963 | L0=imap(RR,L); |
---|
6964 | L1[1]=L0[1]; L2[1]=L0[size(L0)]; |
---|
6965 | for (i=2;i<=size(L0)-1;i++) |
---|
6966 | { |
---|
6967 | L1[i]=intersect(L1[i-1],L0[i]); |
---|
6968 | L2[i]=intersect(L2[i-1],L0[size(L0)-i+1]); |
---|
6969 | } |
---|
6970 | L3[1]=L2[size(L2)]; |
---|
6971 | for (i=2;i<=size(L0)-1;i++) |
---|
6972 | { |
---|
6973 | L3[i]=intersect(L1[i-1],L2[size(L0)-i]); |
---|
6974 | } |
---|
6975 | L3[size(L0)]=L1[size(L1)]; |
---|
6976 | for (i=1;i<=size(L3);i++) |
---|
6977 | { |
---|
6978 | option(redSB); L3[i]=std(L3[i]); |
---|
6979 | } |
---|
6980 | for (i=1;i<=size(L3);i++) |
---|
6981 | { |
---|
6982 | tes=1; j=0; |
---|
6983 | while((tes==1) and (j<size(L3[i]))) |
---|
6984 | { |
---|
6985 | j++; |
---|
6986 | option(redSB); |
---|
6987 | L0[i]=std(L0[i]); |
---|
6988 | if(reduce(L3[i][j],L0[i])!=0){tes=0; F[i]=L3[i][j];} |
---|
6989 | } |
---|
6990 | if (tes==1){"ERROR a polynomial in all p_j except p_i was not found";} |
---|
6991 | } |
---|
6992 | setring(RR); |
---|
6993 | def F0=imap(@P,F); |
---|
6994 | return(F0); |
---|
6995 | } |
---|
6996 | |
---|
6997 | // precombinediscussion |
---|
6998 | // not used, can be deleted |
---|
6999 | // input: list L: the LCU segment with bases for each pi component |
---|
7000 | // output: intvec vv: vv[1]=(1 if the generic polynomial of the vv[2] |
---|
7001 | // component already specializes well, |
---|
7002 | // 0 if combine is to be used) |
---|
7003 | // vv[2]=selind, the index for which the generic basis |
---|
7004 | // already specializes well if combine is not to be used (vv[1]=1). |
---|
7005 | static proc precombinediscussion(L,crep) |
---|
7006 | { |
---|
7007 | int tes=1; int selind; int i1; int j1; poly p; poly lcp; intvec vv; |
---|
7008 | if (size(L)==1){vv=1,1; return(vv);} |
---|
7009 | for (i1=1;i1<=size(L);i1++) |
---|
7010 | { |
---|
7011 | tes=1; |
---|
7012 | p=L[i1][2]; |
---|
7013 | lcp=leadcoef(p); |
---|
7014 | |
---|
7015 | |
---|
7016 | if(nonnullCrep(lcp,crep[1],crep[2])==1) |
---|
7017 | { |
---|
7018 | for(j1=1;j1<=size(L);j1++) |
---|
7019 | { |
---|
7020 | if(i1!=j1) |
---|
7021 | { |
---|
7022 | if(specswellCrep(p,L[j1][2],L[j1][1])==0){tes=0; break;} |
---|
7023 | } |
---|
7024 | } |
---|
7025 | } |
---|
7026 | else{tes=0;} |
---|
7027 | if(tes==1){selind=i1; break;} |
---|
7028 | } |
---|
7029 | vv=tes,selind; |
---|
7030 | return(vv); |
---|
7031 | } |
---|
7032 | |
---|
7033 | // only if N=0 and W=1 |
---|
7034 | proc gencase1(ideal F, list #) |
---|
7035 | "USAGE: gencase1(F); This routine determines the generic segment when |
---|
7036 | the generic case has basis 1, and returns the empty list if not. |
---|
7037 | It is useful, for example in automatic discovery of geometric |
---|
7038 | theorems, to determine the prime varieties over which solutions exist. |
---|
7039 | It can work, even if the complete grobcov does not finish. |
---|
7040 | It serves to obtain a partial result that can be sometimes very useful. |
---|
7041 | It is also used internally in the canonical computation grobcov, |
---|
7042 | but can be called by the user. Only the basering Q[a][x] needs |
---|
7043 | to be defined and the ideal given in this ring. |
---|
7044 | Options: It allows an option list("compbas",0-1), |
---|
7045 | If the routine is called with option |
---|
7046 | ("compbas",0), then the given ideal must be the reduced |
---|
7047 | Groebner basis of the ideal in the ring Q[x,a]. |
---|
7048 | If the routine is called by the user this option not to be used, |
---|
7049 | and the algorithm will compute internally the reduced Groebner |
---|
7050 | basis of the ideal in the ring Q[x,a]. |
---|
7051 | RETURN: The list of the generic case, when its basis is 1, or |
---|
7052 | the empty list if not. |
---|
7053 | The output is of the form |
---|
7054 | (lpp=1,basis=1,(null ideal=0,(p1,..ps)),N) |
---|
7055 | where (0,(p1,..,ps)) is the P-representation of the generic segment |
---|
7056 | (the pi's are the prime components) and N is its intersection |
---|
7057 | NOTE: The basering R, must be of the form Q[a][x], a=parameters, |
---|
7058 | x=variables, and should be defined previously. The ideal must |
---|
7059 | be defined on R. |
---|
7060 | KEYWORDS: generic segment, automatic discovery of geometric theorems, |
---|
7061 | EXAMPLE: gencase1; shows an example" |
---|
7062 | { |
---|
7063 | int compbas=1; list L=#; |
---|
7064 | // compbas==1 the gbasis wrt vars+param must be computed now |
---|
7065 | // compbas==0 the gbasis wrt vars+param is already computed |
---|
7066 | def RR=basering; list empty; int i; |
---|
7067 | setglobalrings(); |
---|
7068 | for(i=1;i<=size(L) div 2;i++) |
---|
7069 | { |
---|
7070 | if(L[2*i-1]=="compbas"){compbas=L[2*i];} |
---|
7071 | } |
---|
7072 | if (compbas==1) |
---|
7073 | { |
---|
7074 | setring(@RP); |
---|
7075 | def FP=imap(R,F); |
---|
7076 | option(redSB); |
---|
7077 | def G=std(FP); |
---|
7078 | setring(RR); |
---|
7079 | def F1=imap(@RP,G); |
---|
7080 | } |
---|
7081 | else {def F1=F;} |
---|
7082 | ideal Zero; |
---|
7083 | for(i=1;i<=size(F1);i++) |
---|
7084 | { |
---|
7085 | if (leadmonom(F1[i])==1) |
---|
7086 | { |
---|
7087 | Zero[size(Zero)+1]=F1[i]; |
---|
7088 | } |
---|
7089 | } |
---|
7090 | if (size(Zero)>0) |
---|
7091 | { |
---|
7092 | setring(@P); |
---|
7093 | def ZeroP=imap(RR,Zero); |
---|
7094 | //def N=radical(ZeroP); |
---|
7095 | def holes=minGTZ(ZeroP); |
---|
7096 | for(i=1;i<=size(holes);i++) |
---|
7097 | { |
---|
7098 | option(redSB); |
---|
7099 | holes[i]=std(holes[i]); |
---|
7100 | } |
---|
7101 | def N=holes[1]; |
---|
7102 | for(i=2;i<=size(holes);i++) |
---|
7103 | { |
---|
7104 | N=intersect(N,holes[i]); |
---|
7105 | } |
---|
7106 | option(redSB); |
---|
7107 | N=std(N); |
---|
7108 | setring(RR); |
---|
7109 | def hole=imap(@P,holes); |
---|
7110 | def Nn=imap(@P,N); |
---|
7111 | kill @P; kill @RP; kill @R; |
---|
7112 | return(ideal(1),ideal(1),list(ideal(0),hole),Nn); |
---|
7113 | } |
---|
7114 | else |
---|
7115 | { |
---|
7116 | kill @P; kill @RP; kill @R; |
---|
7117 | setring(RR); |
---|
7118 | return(empty); |
---|
7119 | } |
---|
7120 | } |
---|
7121 | example |
---|
7122 | { "EXAMPLE:"; echo = 2; |
---|
7123 | "Generic segment for the extended Steiner-Lehmus theorem"; |
---|
7124 | ring R=(0,x,y),(a,b,m,n,p,r),lp; |
---|
7125 | ideal S=p^2-(x^2+y^2), |
---|
7126 | -a*(y)+b*(x+p), |
---|
7127 | -a*y+b*(x-1)+y, |
---|
7128 | (r-1)^2-((x-1)^2+y^2), |
---|
7129 | -m*(y)+n*(x+r-2) +y, |
---|
7130 | -m*y+n*x, |
---|
7131 | (a^2+b^2)-((m-1)^2+n^2); |
---|
7132 | short=0; |
---|
7133 | gencase1(S); |
---|
7134 | } |
---|
7135 | |
---|
7136 | // minAssGTZ eliminating denominators |
---|
7137 | static proc minGTZ(ideal N); |
---|
7138 | { |
---|
7139 | int i; int j; |
---|
7140 | def L=minAssGTZ(N); |
---|
7141 | for(i=1;i<=size(L);i++) |
---|
7142 | { |
---|
7143 | for(j=1;j<=size(L[i]);j++) |
---|
7144 | { |
---|
7145 | L[i][j]=cleardenom(L[i][j]); |
---|
7146 | } |
---|
7147 | } |
---|
7148 | return(L); |
---|
7149 | } |
---|
7150 | |
---|
7151 | proc multigrobcov(ideal F, list #) |
---|
7152 | "USAGE: multigrobcov(F); This routine is to be used instead of grobcov |
---|
7153 | when grobcov does not finish, and the generic case is expected |
---|
7154 | to have basis 1. It can be useful for automating discovery of |
---|
7155 | geometric theorems. |
---|
7156 | The ideal F must be defined on a parametric ring Q[a][x]. |
---|
7157 | If the generic basis is not 1, then it returns the empty list, |
---|
7158 | but if the generic basis is one then it computes the |
---|
7159 | grobcov over each irreducible component of the complement of |
---|
7160 | the generic segment and returns the generic segment and the |
---|
7161 | different grobcov on each segment. From the result, the global |
---|
7162 | grobcov can be deduced eliminating convenablement the inter- |
---|
7163 | sections of the different grobcov computed over the components. |
---|
7164 | Options: A list of options of the form |
---|
7165 | ("comment",0-1,"can",0-1 can,"cgs",0-1,"ext",0-1), can be given. |
---|
7166 | One can give none till 4 of these options by giving the |
---|
7167 | name of the option and the value. Options "null" and "nonnull" are |
---|
7168 | avoided. |
---|
7169 | When option ("comment",1) is set, the routine provides information |
---|
7170 | about the development of the computation. The default option |
---|
7171 | is ("comment",0). |
---|
7172 | When option ("can",0) is given, then the computation is |
---|
7173 | done homogenizing the given basis but not computing the |
---|
7174 | whole homogenized ideal. Thus in this case the result is not |
---|
7175 | completely canonical but it is also useful. This option |
---|
7176 | facilitates the computation. The default option is ("can",1). |
---|
7177 | When option ("cgs",0) is set, then instead of using cgsdr |
---|
7178 | for computing the initial reduced disjoint CGS, then |
---|
7179 | cgsdrold is used. This can be tested when ("cgs",1) (the default |
---|
7180 | option) fails. When option ("ext",0) is set, only the generic |
---|
7181 | representation of the bases are computed instead of the |
---|
7182 | full representation (the default option is ("ext",1)). |
---|
7183 | RETURN: The list whose first element is the generic case, and the |
---|
7184 | remaining elements are the grobcov over the different irreducible |
---|
7185 | components in the complementary of the generic segment. |
---|
7186 | the empty list if the generic case does not have basis 1. |
---|
7187 | NOTE: The basering R, must be of the form Q[a][x], a=parameters, |
---|
7188 | x=variables, and should be defined previously. The ideal must |
---|
7189 | be defined on R. |
---|
7190 | KEYWORDS: grobcov, generic segment, automatic discovery of geometric theorems, |
---|
7191 | EXAMPLE: multigrobcov; shows an example." |
---|
7192 | { |
---|
7193 | int i; int comment=1; list L=#; ideal N; list gc; list GC; list GCA; |
---|
7194 | int start=timer; int ni; int nw; |
---|
7195 | for(i=1;i<=size(L) div 2;i++) |
---|
7196 | { |
---|
7197 | if (L[2*i-1]=="comment"){comment=L[2*i];} |
---|
7198 | else |
---|
7199 | { |
---|
7200 | if(L[2*i-1]=="null"){"multigrobcov does not allow null restriction"; ni=i;} |
---|
7201 | else |
---|
7202 | { |
---|
7203 | if(L[2*i-1]=="nonnull"){"multigrobcov does not allow nonnull restriction"; nw=i;} |
---|
7204 | } |
---|
7205 | } |
---|
7206 | } |
---|
7207 | if (ni>0) |
---|
7208 | { |
---|
7209 | L=delete(L,2*ni-1); L=delete(L,2*ni-1); |
---|
7210 | if(nw>0) |
---|
7211 | { |
---|
7212 | if(nw<ni) |
---|
7213 | { |
---|
7214 | L=delete(L,2*nw-1); L=delete(L,2*nw-1); |
---|
7215 | } |
---|
7216 | else |
---|
7217 | { |
---|
7218 | L=delete(L,2*nw-3); L=delete(L,2*nw-3); |
---|
7219 | } |
---|
7220 | } |
---|
7221 | } |
---|
7222 | else |
---|
7223 | { |
---|
7224 | if (nw>0){L=delete(L,2*nw-1);L=delete(L,2*nw-1);} |
---|
7225 | } |
---|
7226 | gc=gencase1(F); |
---|
7227 | if(size(gc)==0) |
---|
7228 | { |
---|
7229 | string("The generic case is not 1, thus multigrobcov is not useful"); |
---|
7230 | return(gc); |
---|
7231 | } |
---|
7232 | else |
---|
7233 | { |
---|
7234 | if(comment==1){"Generic case ="; gc; " ";} |
---|
7235 | def SS2=gc[3][2]; |
---|
7236 | GCA=list(list(list(gc[1],gc[2],list(gc[3])))); |
---|
7237 | if(comment==1){"Components to study="; SS2;} |
---|
7238 | for (i=1;i<=size(SS2);i++) |
---|
7239 | { |
---|
7240 | N=SS2[i]; |
---|
7241 | if(comment==1){" "; "Begin grobcov on the variety N ="; N;} |
---|
7242 | L[size(L)+1]="null"; L[size(L)+1]=N; |
---|
7243 | //"T_L=";L; |
---|
7244 | GC=grobcov(F,L); |
---|
7245 | GCA[size(GCA)+1]=GC; |
---|
7246 | } |
---|
7247 | if(comment==1){string("Time for multigrobcov = ",timer-start);} |
---|
7248 | return(GCA); |
---|
7249 | } |
---|
7250 | } |
---|
7251 | example |
---|
7252 | { |
---|
7253 | "Generalization of the Steiner-Lehmus theorem"; |
---|
7254 | ring R=(0,x,y),(a,b,m,n,p,r),lp; |
---|
7255 | ideal S=p^2-(x^2+y^2), |
---|
7256 | -a*(y)+b*(x+p), |
---|
7257 | -a*y+b*(x-1)+y, |
---|
7258 | (r-1)^2-((x-1)^2+y^2), |
---|
7259 | -m*(y)+n*(x+r-2) +y, |
---|
7260 | -m*y+n*x, |
---|
7261 | (a^2+b^2)-((m-1)^2+n^2); |
---|
7262 | short=0; |
---|
7263 | multigrobcov(S,list("can",0,"cgs",0,"comment",1)); |
---|
7264 | } |
---|
7265 | |
---|
7266 | proc cgsdrold(ideal F, list #) |
---|
7267 | "USAGE: cgsdrold(F); To compute a disjoint, reduced CGS. |
---|
7268 | From the old library redcgs.lib. |
---|
7269 | cgsdrold is the starting point of the fundamental routine |
---|
7270 | grobcovold of the library redcgs.lib. |
---|
7271 | Use instead cgsdr. cgsdrold is only recommended for comparison |
---|
7272 | with cgsdr or for didactic purposes to plot the tree (buildtree) |
---|
7273 | using the routine buildtreetoMaple. |
---|
7274 | F: ideal in Q[a][x] (parameters and variables) to be discussed. |
---|
7275 | |
---|
7276 | Options: To modify the default options, pairs of arguments |
---|
7277 | -option name, value- of valid options must be added to the call. |
---|
7278 | |
---|
7279 | Options: |
---|
7280 | "null",ideal N: The default is "null",ideal(0). |
---|
7281 | "nonnull",ideal W: The default "nonnull",ideal(1). |
---|
7282 | When options "null" and/or "nonnull" are given, then |
---|
7283 | the parameter space is restricted to V(N) \ V(h), where |
---|
7284 | h is the product of the polynomials w in W. |
---|
7285 | "old",0-1: The default option is "old",1 that gives an output |
---|
7286 | analogous to the one obtained by cgsdr. Setting "old",0 |
---|
7287 | provides an output representing a tree (buildtree), that |
---|
7288 | can be plotted using the routine buildtreetoMaple. |
---|
7289 | "comment",0-1: The default is "comment",0. Setting "comments",1 |
---|
7290 | will provide information about the development of the |
---|
7291 | computation. |
---|
7292 | One can give none till 4 of these options. |
---|
7293 | RETURN: With the default option "old",1, it returns a list T describing |
---|
7294 | a reduced and disjoint comprehensive Groebner system (CGS), |
---|
7295 | whose segments correspond to constant leading power products (lpp) |
---|
7296 | of the reduced Groebner basis. The returned list is of the form: |
---|
7297 | ( |
---|
7298 | (lpp, (basis,segment),...,(basis,segment)), |
---|
7299 | ..,, |
---|
7300 | (lpp, (basis,segment),...,(basis,segment)) |
---|
7301 | ) |
---|
7302 | The bases are the reduced Groebner bases (after normalization) |
---|
7303 | for each point of the corresponding segment. |
---|
7304 | Each segment is given by a reduced representation (Ni,Wi), with |
---|
7305 | Ni radical and V(Ni)=Zariski closure of the segment Si=V(Ni)\V(hi), |
---|
7306 | where hi is the product of the polynomials w in Wi. |
---|
7307 | Setting option "old",0 the output represents the tree and |
---|
7308 | can then be transformed to a plot structure using the routine |
---|
7309 | buildtreetoMaple. |
---|
7310 | Its structure in this case is: |
---|
7311 | The first element of the list is the root, and contains |
---|
7312 | [1] label: intvec(-1) |
---|
7313 | [2] number of children : int |
---|
7314 | [3] the ideal F |
---|
7315 | [4], [5], [6] the red-representation of the segment |
---|
7316 | (null, non-null conditions, prime components of the null |
---|
7317 | conditions) given (as option). |
---|
7318 | ideal (0), ideal (1), list(ideal(0)) is assumed if |
---|
7319 | no optional conditions are given. |
---|
7320 | [7] the set of lpp of ideal F |
---|
7321 | [8] condition that was taken to reach the vertex |
---|
7322 | (poly 1, for the root). |
---|
7323 | The remaining elements of the list represent vertices of the tree: |
---|
7324 | with the same structure: |
---|
7325 | [1] label: intvec (1,0,0,1,...) gives its position in the tree: |
---|
7326 | first branch condition is taken non-null, second null,... |
---|
7327 | [2] number of children (0 if it is a terminal vertex) |
---|
7328 | [3] the specialized ideal with the previous assumed conditions |
---|
7329 | to reach the vertex |
---|
7330 | [4],[5],[6] the red-representation of the segment corresponding |
---|
7331 | to the previous assumed conditions to reach the vertex |
---|
7332 | [7] the set of lpp of the specialized ideal at this stage |
---|
7333 | [8] condition that was taken to reach the vertex from the |
---|
7334 | father's vertex (that was taken non-null if the last |
---|
7335 | integer in the label is 1, and null if it is 0) |
---|
7336 | The terminal vertices form a disjoint partition of the parameter |
---|
7337 | space whose bases specialize to the reduced Groebner basis of the |
---|
7338 | specialized ideal on each point of the segment and preserve |
---|
7339 | the lpp. They form a disjoint reduced CGS, and is the only |
---|
7340 | vertices grouped and ordered by lpp that is returned with the |
---|
7341 | default option "old",1. |
---|
7342 | |
---|
7343 | NOTE: The basering R, must be of the form Q[a][x], a=parameters, |
---|
7344 | x=variables, and should be defined previously, and the ideal |
---|
7345 | defined on R. |
---|
7346 | KEYWORDS: CGS, cgsdr, buildtree, buildtreetoMaple, disjoint, reduced, |
---|
7347 | comprehensive Groebner system |
---|
7348 | EXAMPLE: cgsdrold; shows an example" |
---|
7349 | { |
---|
7350 | int i; list L=#; int oldop=1; |
---|
7351 | for(i=1;i<=size(L) div 2;i++) |
---|
7352 | { |
---|
7353 | if(L[2*i-1]=="old"){oldop=L[2*i];} |
---|
7354 | } |
---|
7355 | def bt=buildtree(F, #); |
---|
7356 | if (oldop==0){return(bt);} |
---|
7357 | else |
---|
7358 | { |
---|
7359 | setglobalrings(); |
---|
7360 | def gs=groupsegments(finalcases(bt)); |
---|
7361 | int j; |
---|
7362 | for (i=1;i<=size(gs);i++) |
---|
7363 | { |
---|
7364 | for (j=1;j<=size(gs[i][2]);j++) |
---|
7365 | { |
---|
7366 | gs[i][2][j]=delete(gs[i][2][j],1); |
---|
7367 | gs[i][2][j]=delete(gs[i][2][j],4); |
---|
7368 | if(equalideals(gs[i][2][j][3],ideal(0))){gs[i][2][j][3]=ideal(1);} |
---|
7369 | } |
---|
7370 | } |
---|
7371 | kill @P; kill @R; kill @RP; |
---|
7372 | return(gs); |
---|
7373 | } |
---|
7374 | } |
---|
7375 | example |
---|
7376 | { "EXAMPLE:"; echo = 2; |
---|
7377 | ring R=(0,a1,a2,a3,a4),(x1,x2,x3,x4),dp; |
---|
7378 | ideal F=x4-a4+a2, |
---|
7379 | x1+x2+x3+x4-a1-a3-a4, |
---|
7380 | x1*x3*x4-a1*a3*a4, |
---|
7381 | x1*x3+x1*x4+x2*x3+x3*x4-a1*a4-a1*a3-a3*a4; |
---|
7382 | cgsdrold(F); |
---|
7383 | cgsdrold(F,"old",0); |
---|
7384 | } |
---|
7385 | |
---|
7386 | proc grobcovold(ideal F,list #) |
---|
7387 | "USAGE: grobcovold(F); This is the fundamental routine of the |
---|
7388 | old library redcgs.lib. It is somewhat heuristic and does |
---|
7389 | not certify the obtention of the canonical Groebner cover of |
---|
7390 | a parametric ideal, as does grobcov, but usually it does or |
---|
7391 | provides a warning if not. It allows different options, recalling |
---|
7392 | all the different approaches of the old library redcgs.lib. |
---|
7393 | Use grobcov instead. The use of grobcovold is only recommended |
---|
7394 | to compare results or study alternatives. |
---|
7395 | |
---|
7396 | The ideal F must be defined on a parametric ring Q[a][x]. |
---|
7397 | Options: To modify the default options, pair of arguments |
---|
7398 | -option name, value- of valid options must be added to the call. |
---|
7399 | |
---|
7400 | Options: |
---|
7401 | "null",ideal N: The default is "null",ideal(0). |
---|
7402 | "nonnull",ideal W: The default "nonnull",ideal(1). |
---|
7403 | When options "null" and/or "nonnull" are given, then |
---|
7404 | the parameter space is restricted to V(N) \ V(h), where |
---|
7405 | h is the product of the polynomials w in W. |
---|
7406 | "can",0-2: The default is "can",1. With the default option |
---|
7407 | the homogenized ideal is computed before obtaining the |
---|
7408 | Groebner cover, so that the result is the canonical |
---|
7409 | Groebner cover. Setting "can",0 only homogenizes the basis |
---|
7410 | so the result is not exactly canonical, but the computation |
---|
7411 | is more efficient. Setting "can",2 no homogenization of |
---|
7412 | the ideal is carried out, and the segments with same lpp |
---|
7413 | are added so much as possible when a common basis is obtained. |
---|
7414 | The result, in this case is not canonical nor the segments |
---|
7415 | are always locally closed. Nevertheless it can have |
---|
7416 | less segments as the canonical result. |
---|
7417 | "out",0-1: The default is "out",0. With the default option the |
---|
7418 | output is analogous to that of grobcov. If option "can",2 |
---|
7419 | is also set, then this representation can be somewhat |
---|
7420 | confusing, because the segments are not always given in |
---|
7421 | P-representation, as they are not always locally closed. |
---|
7422 | With option "out",1 a representation in tree form is given |
---|
7423 | providing a canonical representation of the segments, even if |
---|
7424 | they are not locally closed. This representation can be transformed |
---|
7425 | by the routine cantreetoMaple into a file that can be read |
---|
7426 | in Maple and plotted with the plotcantree Maple routine of |
---|
7427 | the old dpgb library, showing the tree. |
---|
7428 | "comment",0-1: The default is "comment",0. Setting "comments",1 |
---|
7429 | will provide information about the development of the |
---|
7430 | computation. |
---|
7431 | One can give none till 5 of these options. |
---|
7432 | RETURN: With the default option ("out",0), the list |
---|
7433 | ( |
---|
7434 | (lpp_1,basis_1,P-representation_1) |
---|
7435 | ... |
---|
7436 | (lpp_s,basis_s,P-represntation_s) |
---|
7437 | ) |
---|
7438 | With option "out",1, a list T representing a rooted tree. |
---|
7439 | Each element of the list T has the two first entries with the |
---|
7440 | following content: |
---|
7441 | [1]: The label (intvec) representing the position in the rooted |
---|
7442 | tree: 0 for the root (and this is a special element) |
---|
7443 | i for the root of the segment i |
---|
7444 | (i,...) for the children of the segment i |
---|
7445 | [2]: the number of children (int) of the vertex. |
---|
7446 | There are three kind of vertices: |
---|
7447 | (1) the root (first element labelled 0), |
---|
7448 | (2) the vertices labelled with a single integer i, |
---|
7449 | (3) the rest of vertices labelled with more indices. |
---|
7450 | Description of the root. Vertex type (1) |
---|
7451 | There is a special vertex (the first one) whose content is |
---|
7452 | the following: |
---|
7453 | [3] lpp of the given ideal |
---|
7454 | [4] the given ideal |
---|
7455 | [5] the R-representation of the (optional) given null and |
---|
7456 | non-null conditions. |
---|
7457 | [6] CRCGS, RCGS, MRCGS depending on the "can" option (1,0,2). |
---|
7458 | Description of vertices type (2). These are the vertices that |
---|
7459 | initiate a segment, and are labelled with a single integer. |
---|
7460 | [3] lpp (ideal) of the reduced basis. If they are repeated lpp's this |
---|
7461 | will correspond to a sheaf. |
---|
7462 | [4] the reduced basis (ideal) of the segment. |
---|
7463 | Description of vertices type (3). These vertices have as first |
---|
7464 | label i and descend form vertex i in the position of the label |
---|
7465 | (i,...). They contain moreover a unique prime ideal in the parameters |
---|
7466 | and form ascending chains of ideals. |
---|
7467 | How is to be read the mrcgs tree? The vertices with an even number of |
---|
7468 | integers in the label are to be considered as additive and those |
---|
7469 | with an odd number of integers in the label are to be considered as |
---|
7470 | substraction. As an example consider the following vertices: |
---|
7471 | v1=((i),2,lpp,B), |
---|
7472 | v2=((i,1),2,P_(i,1)), |
---|
7473 | v3=((i,1,1),2,P_(i,1,1)), |
---|
7474 | v4=((i,1,1,1),1,P_(i,1,1,1)), |
---|
7475 | v5=((i,1,1,1,1),0,P_(i,1,1,1,1)), |
---|
7476 | v6=((i,1,1,2),1,P_(i,1,1,2)), |
---|
7477 | v7=((i,1,1,2,1),0,P_(i,1,1,2,1)), |
---|
7478 | v8=((i,1,2),0,P_(i,1,2)), |
---|
7479 | v9=((i,2),1,P_(i,2)), |
---|
7480 | v10=((i,2,1),0,P_(i,2,1)), |
---|
7481 | They represent the segment: |
---|
7482 | (V(i,1)\(((V(i,1,1) \ ((V(i,1,1,1) \ V(i,1,1,1,1)) u (V(i,1,1,2) \ V(i,1,1,2,1))))) |
---|
7483 | u V(i,1,2))) u (V(i,2) \ V(i,2,1)) |
---|
7484 | and can also be represented by |
---|
7485 | (V(i,1) \ (V(i,1,1) u V(i,1,2))) u |
---|
7486 | (V(i,1,1,1) \ V(i,1,1,1)) u |
---|
7487 | (V(i,1,1,2) \ V(i,1,1,2,1)) u |
---|
7488 | (V(i,2) \ V(i,2,1)) |
---|
7489 | where V(i,j,..) = V(P_(i,j,..)) |
---|
7490 | |
---|
7491 | The lpp are constant over a segment and correspond to the |
---|
7492 | set of lpp of the reduced Groebner basis for each point |
---|
7493 | of the segment. |
---|
7494 | |
---|
7495 | Basis: to each element of lpp corresponds an I-regular function given Groebner basis, and it is given in full representation (by |
---|
7496 | in full representation. The regular function is |
---|
7497 | the corresponding element of the reduced Groebner basis for |
---|
7498 | each point of the segment with the given lpp. |
---|
7499 | For each point in the segment, the polynomial or the set of |
---|
7500 | polynomials representing it, if they do not specialize to 0, |
---|
7501 | then after normalization, specialize to the corresponding |
---|
7502 | element of the reduced Groebner basis. |
---|
7503 | |
---|
7504 | The P-representation of a segment is of the form |
---|
7505 | ((p_1,(p_11,..,p_1k1)),..,(p_r,(p_r1,..,p_rkr)) |
---|
7506 | representing the segment U_i (V(p_i) \ U_j (V(p_ij))), where the |
---|
7507 | p's are prime ideals. |
---|
7508 | |
---|
7509 | NOTE: The basering R, must be of the form Q[a][x], a=parameters, |
---|
7510 | x=variables, and should be defined previously. The ideal must |
---|
7511 | be defined on R. |
---|
7512 | KEYWORDS: Groebner cover, grobcov, parametric ideal, canonical, discussion of |
---|
7513 | parametric ideal. |
---|
7514 | EXAMPLE: grobcovold; shows an example" |
---|
7515 | { |
---|
7516 | int i; |
---|
7517 | list LL=#; |
---|
7518 | list T; list NT; list NTe; |
---|
7519 | // default options |
---|
7520 | int comment=0; int canop=1; int outop=0; |
---|
7521 | int start=timer; |
---|
7522 | ideal W=ideal(1); |
---|
7523 | ideal N=ideal(0); |
---|
7524 | canop=1; // canop=0 for homogenizing the basis but not the ideal (not canonical) |
---|
7525 | // (old rcgs) |
---|
7526 | // canop=1 for homogenizing the ideal |
---|
7527 | // (old crcgs) |
---|
7528 | // canop=2 for not homogenizing and try to minimize the segments |
---|
7529 | // (old mrcgs) |
---|
7530 | outop=0; // outop=0 for an output analogous to grobcov (if canop<>2) |
---|
7531 | // outop=1 for an output as in the old library redcgs.lib |
---|
7532 | // in form of tree that can be transformed into Maple. |
---|
7533 | for(i=1;i<=size(LL) div 2;i++) |
---|
7534 | { |
---|
7535 | if(LL[2*i-1]=="can"){canop=LL[2*i];} |
---|
7536 | else |
---|
7537 | { |
---|
7538 | if(LL[2*i-1]=="out"){outop=LL[2*i];} |
---|
7539 | else |
---|
7540 | { |
---|
7541 | if (LL[2*i-1]=="comment"){comment=LL[2*i];} |
---|
7542 | } |
---|
7543 | } |
---|
7544 | } |
---|
7545 | if (comment>=1){string("can = ",canop," out = ", outop," comment = ",comment);} |
---|
7546 | if (canop==0){T=rcgs(F,LL);} |
---|
7547 | else |
---|
7548 | { |
---|
7549 | if (canop==1){T=crcgs(F,LL);} |
---|
7550 | else |
---|
7551 | { |
---|
7552 | if (canop==2){T=mrcgs(F,LL);} |
---|
7553 | } |
---|
7554 | } |
---|
7555 | if (comment>=1){string("Time in grobcovold = ",timer-start," sec");} |
---|
7556 | if (outop==0) |
---|
7557 | { |
---|
7558 | // transforming the output to the modern form |
---|
7559 | i=2; list Cap; int indCap; list Cua; ideal idp; list idq; |
---|
7560 | int tes; |
---|
7561 | while(i<=size(T)) |
---|
7562 | { |
---|
7563 | kill Cap; list Cap; |
---|
7564 | if(size(T[i][1])==1) |
---|
7565 | { |
---|
7566 | Cap=list(T[i][3],T[i][4]); |
---|
7567 | indCap=T[i][1][1]; |
---|
7568 | i++; |
---|
7569 | } |
---|
7570 | kill Cua; list Cua; |
---|
7571 | while(T[i][1][1]==indCap) |
---|
7572 | { |
---|
7573 | if(size(T[i][1]) mod 2 ==0) |
---|
7574 | { |
---|
7575 | if(size(idq)!=0){Cua[size(Cua)+1]=list(idp,idq);} |
---|
7576 | kill idq; list idq; |
---|
7577 | idp=T[i][3]; |
---|
7578 | } |
---|
7579 | else |
---|
7580 | { |
---|
7581 | idq[size(idq)+1]=T[i][3]; |
---|
7582 | } |
---|
7583 | i++; |
---|
7584 | if(i>size(T)){break;} |
---|
7585 | } |
---|
7586 | Cua[size(Cua)+1]=list(idp,idq); |
---|
7587 | Cap[3]=Cua; |
---|
7588 | NT[size(NT)+1]=Cap; |
---|
7589 | kill idp; ideal idp; kill idq; list idq; |
---|
7590 | } |
---|
7591 | if (comment==2){"rcgs="; T;} |
---|
7592 | return(NT); |
---|
7593 | } |
---|
7594 | else |
---|
7595 | { |
---|
7596 | return(T); |
---|
7597 | } |
---|
7598 | } |
---|
7599 | example |
---|
7600 | { |
---|
7601 | "EXAMPLE:"; echo = 2; |
---|
7602 | "Simple robot: A. Montes,"; |
---|
7603 | "New algorithm for discussing Groebner bases with parameters,"; |
---|
7604 | "JSC, 33: 183-208 (2002)."; |
---|
7605 | ring R=(0,r,z,l),(s1,c1,s2,c2), dp; |
---|
7606 | ideal S10=c1^2+s1^2-1, |
---|
7607 | c2^2+s2^2-1, |
---|
7608 | r-c1-l*c1*c2+l*s1*s2, |
---|
7609 | z-s1-l*c1*s2-l*s1*c2; |
---|
7610 | grobcovold(S10,"comment",1); |
---|
7611 | grobcovold(S10,"can",2,"comment",1); |
---|
7612 | } |
---|