1 | ////////////////////////////////////////////////////////////////////////////// |
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2 | version="version grobcov.lib 4-0-0-0 Dec_2013 "; |
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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 | Parametric Polynomial Systems. |
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8 | The library contains Montes-Wibmer's algorithms to compute the |
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9 | canonical Groebner cover of a parametric ideal as described in |
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10 | the paper: |
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11 | |
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12 | Montes A., Wibmer M., |
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13 | \"Groebner Bases for Polynomial Systems with parameters\". |
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14 | Journal of Symbolic Computation 45 (2010) 1391-1425. |
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15 | The locus algorithm and definitions will be |
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16 | published in a forthcoming paper by Abanades, Botana, Montes, Recio: |
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17 | \''An Algebraic Taxonomy for Locus Computation in Dynamic Geometry\''. |
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18 | |
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19 | The central routine is grobcov. Given a parametric |
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20 | ideal, grobcov outputs its Canonical Groebner Cover, consisting |
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21 | of a set of pairs of (basis, segment). The basis (after |
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22 | normalization) is the reduced Groebner basis for each point |
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23 | of the segment. The segments are disjoint, locally closed |
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24 | and correspond to constant lpp (leading power product) |
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25 | of the basis, and are represented in canonical prime |
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26 | representation. The segments are disjoint and cover the |
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27 | whole parameter space. The output is canonical, it only |
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28 | depends on the given parametric ideal and the monomial order. |
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29 | This is much more than a simple Comprehensive Groebner System. |
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30 | The algorithm grobcov allows options to solve partially the |
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31 | problem when the whole automatic algorithm does not finish |
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32 | in reasonable time. |
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33 | |
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34 | grobcov uses a first algorithm cgsdr that outputs a disjoint |
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35 | reduced Comprehensive Groebner System with constant lpp. |
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36 | For this purpose, in this library, the implemented algorithm is |
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37 | Kapur-Sun-Wang algorithm, because it is the most efficient |
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38 | algorithm known for this purpose. |
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39 | |
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40 | D. Kapur, Y. Sun, and D.K. Wang. |
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41 | \"A New Algorithm for Computing Comprehensive Groebner Systems\". |
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42 | Proceedings of ISSAC'2010, ACM Press, (2010), 29-36. |
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43 | |
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44 | cgsdr can be called directly if only a disjoint reduced |
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45 | Comprehensive Groebner System (CGS) is required. |
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46 | |
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47 | AUTHORS: Antonio Montes , Hans Schoenemann. |
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48 | OVERVIEW: see \"Groebner Bases for Polynomial Systems with parameters\" |
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49 | Montes A., Wibmer M., |
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50 | Journal of Symbolic Computation 45 (2010) 1391-1425. |
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51 | (http://www-ma2.upc.edu/~montes/). |
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52 | |
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53 | NOTATIONS: All given and determined polynomials and ideals are in the |
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54 | @* basering Q[a][x]; (a=parameters, x=variables) |
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55 | @* After defining the ring, the main routines |
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56 | @* grobcov, cgsdr, |
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57 | @* generate the global rings |
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58 | @* @R (Q[a][x]), |
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59 | @* @P (Q[a]), |
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60 | @* @RP (Q[x,a]) |
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61 | @* that are used inside and killed before the output. |
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62 | @* If you want to use some internal routine you must |
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63 | @* create before the above rings by calling setglobalrings(); |
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64 | @* because most of the internal routines use these rings. |
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65 | @* The call to the basic routines grobcov, cgsdr will |
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66 | @* kill these rings. |
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67 | |
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68 | PROCEDURES: |
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69 | |
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70 | grobcov(F); Is the basic routine giving the canonical |
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71 | Groebner cover of the parametric ideal F. |
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72 | This routine accepts many options, that |
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73 | allow to obtain results even when the canonical |
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74 | computation does not finish in reasonable time. |
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75 | |
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76 | cgsdr(F); Is the procedure for obtaining a first disjoint, |
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77 | reduced Comprehensive Groebner System that |
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78 | is used in grobcov, that can also be used |
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79 | independently if only the CGS is required. |
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80 | It is a more efficient routine than buildtree |
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81 | (the own routine that is no more used). It uses |
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82 | now KSW algorithm. |
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83 | |
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84 | setglobalrings(); Generates the global rings @R, @P and @PR that are |
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85 | respectively the rings Q[a][x], Q[a], Q[x,a]. |
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86 | It is called inside each of the fundamental routines |
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87 | of the library: grobcov, cgsdr, locus, locusdg and killed before |
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88 | the output. |
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89 | So, if the user want to use some other internal routine, |
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90 | then setglobalrings() is to be called before, as most of them use |
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91 | these rings. |
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92 | |
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93 | pdivi(f,F); Performs a pseudodivision of a parametric polynomial |
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94 | by a parametric ideal. |
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95 | Can be used without calling presiouly setglobalrings(), |
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96 | |
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97 | pnormalf(f,E,N); Reduces a parametric polynomial f over V(E) \ V(N) |
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98 | E is the null ideal and N the non-null ideal |
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99 | over the parameters. |
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100 | Can be used without calling presiouly setglobalrings(), |
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101 | |
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102 | Prep(N,M); Computes the P-representation of V(N) \ V(M). |
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103 | Can be used without calling previously setglobalrings(). |
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104 | |
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105 | extend(GC); When the grobcov of an ideal has been computed |
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106 | with the default option ('ext',0) and the explicit |
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107 | option ('rep',2) (which is not the default), then |
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108 | one can call extend (GC) (and options) to obtain the |
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109 | full representation of the bases. With the default |
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110 | option ('ext',0) only the generic representation of |
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111 | the bases are computed, and one can obtain the full |
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112 | representation using extend. |
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113 | Can be used without calling presiouly setglobalrings(), |
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114 | |
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115 | locus(G): Special routine for determining the locus of points |
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116 | of objects. Given a parametric ideal J with |
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117 | parameters (a_1,..a_m) and variables (x_1,..,xn), |
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118 | representing the system determining |
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119 | the locus of points (a_1,..,a_m)) who verify certain |
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120 | properties, computing the grobcov G of |
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121 | J and applying to it locus, determines the different |
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122 | classes of locus components. They can be |
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123 | Normal, Special, Accumulation point, Degenerate. |
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124 | The output are the components given in P-canonical form |
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125 | of two constructible sets: Normal, and Non-Normal |
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126 | The Normal Set has Normal and Special components |
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127 | The Non-Normal set has Accumulation and Degenerate components. |
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128 | The description of the algorithm and definitions will be |
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129 | given in a forthcoming paper by Abanades, Botana, Montes, Recio: |
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130 | ''An Algebraic Taxonomy for Locus Computation in Dynamic Geometry''. |
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131 | Can be used without calling presiouly setglobalrings(), |
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132 | |
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133 | locusdg(G): Is a special routine for computing the locus in dinamical geometry. |
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134 | It detects automatically a possible point that is to be avoided by the mover, |
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135 | whose coordinates must be the last two coordinates in the definition of the ring. |
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136 | If such a point is detected, then it eliminates the segments of the grobcov |
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137 | depending on the point that is to be avoided. |
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138 | Then it calls locus. |
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139 | Can be used without calling presiouly setglobalrings(), |
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140 | |
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141 | locusto(L): Transforms the output of locus to a string that |
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142 | can be reed from different computational systems. |
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143 | Can be used without calling presiouly setglobalrings(), |
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144 | |
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145 | addcons(L): Given a disjoint set of locally closed subsets in P-representation, |
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146 | it returns the canonical P-representation of the constructible set |
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147 | formed by the union of them. |
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148 | Can be used without calling presiouly setglobalrings(), |
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149 | |
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150 | SEE ALSO: compregb_lib |
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151 | "; |
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152 | |
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153 | LIB "primdec.lib"; |
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154 | LIB "qhmoduli.lib"; |
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155 | |
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156 | // ************ Begin of the grobcov library ********************* |
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157 | |
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158 | // Library grobcov.lib |
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159 | // (Groebner cover): |
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160 | // Release 1: (public) |
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161 | // Initial data: 21-1-2008 |
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162 | // Final data: 3-7-2008 |
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163 | // Release 2: (private) |
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164 | // Initial data: 6-9-2009 |
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165 | // Final data: 25-10-2011 |
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166 | // Release 3: |
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167 | // Initial data: 1-7-2012 |
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168 | // Final data: 4-9-2012 |
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169 | // Release 4: (this release, public) |
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170 | // Initial data: 4-9-2012 |
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171 | // Final data: 21-11-2013 |
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172 | // basering Q[a][x]; |
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173 | |
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174 | // ************ Begin of buildtree ****************************** |
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175 | |
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176 | proc setglobalrings() |
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177 | "USAGE: setglobalrings(); |
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178 | No arguments |
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179 | RETURN: After its call the rings @R=Q[a][x], @P=Q[a], @RP=Q[x,a] are |
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180 | defined as global variables. |
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181 | NOTE: It is called internally by the fundamental routines of the |
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182 | library grobcov, cgsdr, extend, pdivi, pnormalf, locus, locusdg, locusto, |
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183 | and killed before the output. |
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184 | The user does not need to call it, except when it is interested |
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185 | in using some internal routine of the library that |
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186 | uses these rings. |
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187 | The basering R, must be of the form Q[a][x], a=parameters, |
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188 | x=variables, and should be defined previously. |
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189 | KEYWORDS: ring, rings |
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190 | EXAMPLE: setglobalrings; shows an example" |
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191 | { |
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192 | if (defined(@P)) |
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193 | { |
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194 | kill @P; kill @R; kill @RP; |
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195 | } |
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196 | def RR=basering; |
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197 | def @R=basering; // must be of the form K[a][x], a=parameters, x=variables |
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198 | list Rx=ringlist(RR); |
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199 | def @P=ring(Rx[1]); |
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200 | list Lx; |
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201 | Lx[1]=0; |
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202 | Lx[2]=Rx[2]+Rx[1][2]; |
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203 | Lx[3]=Rx[1][3]; |
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204 | Lx[4]=Rx[1][4]; |
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205 | Rx[1]=0; |
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206 | def D=ring(Rx); |
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207 | def @RP=D+@P; |
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208 | exportto(Top,@R); // global ring K[a][x] |
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209 | exportto(Top,@P); // global ring K[a] |
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210 | exportto(Top,@RP); // global ring K[x,a] with product order |
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211 | setring(RR); |
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212 | } |
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213 | example |
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214 | { "EXAMPLE:"; echo = 2; |
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215 | ring R=(0,a,b),(x,y,z),dp; |
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216 | setglobalrings(); |
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217 | @R; |
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218 | @P; |
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219 | @RP; |
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220 | ringlist(@R); |
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221 | ringlist(@P); |
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222 | ringlist(@RP); |
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223 | } |
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224 | |
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225 | //*************Auxiliary routines************** |
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226 | |
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227 | // cld : clears denominators of an ideal and normalizes to content 1 |
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228 | // can be used in @R or @P or @RP |
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229 | // input: |
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230 | // ideal J (J can be also poly), but the output is an ideal; |
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231 | // output: |
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232 | // ideal Jc (the new form of ideal J without denominators and |
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233 | // normalized to content 1) |
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234 | proc cld(ideal J) |
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235 | { |
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236 | if (size(J)==0){return(ideal(0));} |
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237 | int te=0; |
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238 | def RR=basering; |
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239 | if(not(defined(@RP))) |
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240 | { |
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241 | te=1; |
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242 | setglobalrings(); |
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243 | } |
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244 | setring(@RP); |
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245 | def Ja=imap(RR,J); |
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246 | ideal Jb; |
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247 | if (size(Ja)==0){setring(RR); return(ideal(0));} |
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248 | int i; |
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249 | def j=0; |
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250 | for (i=1;i<=ncols(Ja);i++){if (size(Ja[i])!=0){j++; Jb[j]=cleardenom(Ja[i]);}} |
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251 | setring(RR); |
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252 | def Jc=imap(@RP,Jb); |
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253 | if(te){kill @R; kill @RP; kill @P;} |
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254 | return(Jc); |
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255 | } |
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256 | |
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257 | proc memberpos(f,J) |
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258 | //"USAGE: memberpos(f,J); |
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259 | // (f,J) expected (polynomial,ideal) |
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260 | // or (int,list(int)) |
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261 | // or (int,intvec) |
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262 | // or (intvec,list(intvec)) |
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263 | // or (list(int),list(list(int))) |
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264 | // or (ideal,list(ideal)) |
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265 | // or (list(intvec), list(list(intvec))). |
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266 | // The ring can be @R or @P or @RP or any other. |
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267 | //RETURN: The list (t,pos) t int; pos int; |
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268 | // t is 1 if f belongs to J and 0 if not. |
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269 | // pos gives the position in J (or 0 if f does not belong). |
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270 | //EXAMPLE: memberpos; shows an example" |
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271 | { |
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272 | int pos=0; |
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273 | int i=1; |
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274 | int j; |
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275 | int t=0; |
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276 | int nt; |
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277 | if (typeof(J)=="ideal"){nt=ncols(J);} |
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278 | else{nt=size(J);} |
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279 | if ((typeof(f)=="poly") or (typeof(f)=="int")) |
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280 | { // (poly,ideal) or |
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281 | // (poly,list(poly)) |
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282 | // (int,list(int)) or |
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283 | // (int,intvec) |
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284 | i=1; |
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285 | while(i<=nt) |
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286 | { |
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287 | if (f==J[i]){return(list(1,i));} |
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288 | i++; |
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289 | } |
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290 | return(list(0,0)); |
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291 | } |
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292 | else |
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293 | { |
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294 | if ((typeof(f)=="intvec") or ((typeof(f)=="list") and (typeof(f[1])=="int"))) |
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295 | { // (intvec,list(intvec)) or |
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296 | // (list(int),list(list(int))) |
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297 | i=1; |
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298 | t=0; |
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299 | pos=0; |
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300 | while((i<=nt) and (t==0)) |
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301 | { |
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302 | t=1; |
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303 | j=1; |
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304 | if (size(f)!=size(J[i])){t=0;} |
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305 | else |
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306 | { |
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307 | while ((j<=size(f)) and t) |
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308 | { |
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309 | if (f[j]!=J[i][j]){t=0;} |
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310 | j++; |
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311 | } |
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312 | } |
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313 | if (t){pos=i;} |
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314 | i++; |
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315 | } |
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316 | if (t){return(list(1,pos));} |
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317 | else{return(list(0,0));} |
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318 | } |
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319 | else |
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320 | { |
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321 | if (typeof(f)=="ideal") |
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322 | { // (ideal,list(ideal)) |
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323 | i=1; |
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324 | t=0; |
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325 | pos=0; |
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326 | while((i<=nt) and (t==0)) |
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327 | { |
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328 | t=1; |
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329 | j=1; |
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330 | if (ncols(f)!=ncols(J[i])){t=0;} |
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331 | else |
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332 | { |
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333 | while ((j<=ncols(f)) and t) |
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334 | { |
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335 | if (f[j]!=J[i][j]){t=0;} |
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336 | j++; |
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337 | } |
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338 | } |
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339 | if (t){pos=i;} |
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340 | i++; |
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341 | } |
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342 | if (t){return(list(1,pos));} |
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343 | else{return(list(0,0));} |
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344 | } |
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345 | else |
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346 | { |
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347 | if ((typeof(f)=="list") and (typeof(f[1])=="intvec")) |
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348 | { // (list(intvec),list(list(intvec))) |
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349 | i=1; |
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350 | t=0; |
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351 | pos=0; |
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352 | while((i<=nt) and (t==0)) |
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353 | { |
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354 | t=1; |
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355 | j=1; |
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356 | if (size(f)!=size(J[i])){t=0;} |
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357 | else |
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358 | { |
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359 | while ((j<=size(f)) and t) |
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360 | { |
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361 | if (f[j]!=J[i][j]){t=0;} |
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362 | j++; |
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363 | } |
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364 | } |
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365 | if (t){pos=i;} |
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366 | i++; |
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367 | } |
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368 | if (t){return(list(1,pos));} |
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369 | else{return(list(0,0));} |
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370 | } |
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371 | } |
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372 | } |
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373 | } |
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374 | } |
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375 | //example |
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376 | //{ "EXAMPLE:"; echo = 2; |
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377 | // list L=(7,4,5,1,1,4,9); |
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378 | // memberpos(1,L); |
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379 | //} |
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380 | |
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381 | proc subset(J,K) |
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382 | //"USAGE: subset(J,K); |
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383 | // (J,K) expected (ideal,ideal) |
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384 | // or (list, list) |
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385 | //RETURN: 1 if all the elements of J are in K, 0 if not. |
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386 | //EXAMPLE: subset; shows an example;" |
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387 | { |
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388 | int i=1; |
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389 | int nt; |
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390 | if (typeof(J)=="ideal"){nt=ncols(J);} |
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391 | else{nt=size(J);} |
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392 | if (size(J)==0){return(1);} |
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393 | while(i<=nt) |
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394 | { |
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395 | if (memberpos(J[i],K)[1]){i++;} |
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396 | else {return(0);} |
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397 | } |
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398 | return(1); |
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399 | } |
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400 | //example |
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401 | //{ "EXAMPLE:"; echo = 2; |
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402 | // list J=list(7,3,2); |
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403 | // list K=list(1,2,3,5,7,8); |
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404 | // subset(J,K); |
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405 | //} |
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406 | |
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407 | // elimintfromideal: elimine the constant numbers from an ideal |
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408 | // (designed for W, nonnull conditions) |
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409 | // input: ideal J |
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410 | // output:ideal K with the elements of J that are non constants, in the |
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411 | // ring @P |
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412 | proc elimintfromideal(ideal J) |
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413 | { |
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414 | int i; |
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415 | int j=0; |
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416 | ideal K; |
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417 | if (size(J)==0){return(ideal(0));} |
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418 | for (i=1;i<=ncols(J);i++){if (size(variables(J[i])) !=0){j++; K[j]=J[i];}} |
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419 | return(K); |
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420 | } |
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421 | |
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422 | // simpqcoeffs : simplifies a quotient of two polynomials |
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423 | // input: two coeficients (or terms), that are considered as a quotient |
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424 | // output: the two coeficients reduced without common factors |
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425 | proc simpqcoeffs(poly n,poly m) |
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426 | { |
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427 | def nc=content(n); |
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428 | def mc=content(m); |
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429 | def gc=gcd(nc,mc); |
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430 | ideal s=n/gc,m/gc; |
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431 | return (s); |
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432 | } |
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433 | |
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434 | // pdivi : pseudodivision of a parametric polynomial f by a parametric ideal F in Q[a][x]. |
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435 | // input: |
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436 | // poly f |
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437 | // ideal F |
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438 | // output: |
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439 | // list (poly r, ideal q, poly mu) |
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440 | proc pdivi(poly f,ideal F) |
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441 | "USAGE: pdivi(f,F); |
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442 | poly f: the polynomial to be divided |
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443 | ideal F: the divisor ideal |
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444 | RETURN: A list (poly r, ideal q, poly m). r is the remainder of the |
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445 | pseudodivision, q is the set of quotients, and m is the |
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446 | coefficient factor by which f is to be multiplied. |
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447 | NOTE: pseudodivision of a poly f by an ideal F in @R. Returns a |
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448 | list (r,q,m) such that m*f=r+sum(q.G), and no lpp of a divisor |
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449 | divides a pp of r. |
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450 | KEYWORDS: division, reduce |
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451 | EXAMPLE: pdivi; shows an example" |
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452 | { |
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453 | int i; |
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454 | int j; |
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455 | // string("T_f=",f); |
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456 | poly v=1; |
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457 | for(i=1;i<=nvars(basering);i++){v=v*var(i);} |
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458 | int te=0; |
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459 | def R=basering; |
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460 | if (defined(@P)==1){te=1;} |
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461 | else{setglobalrings();} |
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462 | poly r=0; |
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463 | poly mu=1; |
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464 | def p=f; |
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465 | ideal q; |
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466 | for (i=1; i<=size(F); i++){q[i]=0;}; |
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467 | ideal lpf; |
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468 | ideal lcf; |
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469 | for (i=1;i<=size(F);i++){lpf[i]=leadmonom(F[i]);} |
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470 | for (i=1;i<=size(F);i++){lcf[i]=leadcoef(F[i]);} |
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471 | poly lpp; |
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472 | poly lcp; |
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473 | poly qlm; |
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474 | poly nu; |
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475 | poly rho; |
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476 | int divoc=0; |
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477 | ideal qlc; |
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478 | while (p!=0) |
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479 | { |
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480 | i=1; |
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481 | divoc=0; |
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482 | lpp=leadmonom(p); |
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483 | lcp=leadcoef(p); |
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484 | while (divoc==0 and i<=size(F)) |
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485 | { |
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486 | qlm=lpp/lpf[i]; |
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487 | if (qlm!=0) |
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488 | { |
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489 | qlc=simpqcoeffs(lcp,lcf[i]); |
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490 | nu=qlc[2]; |
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491 | mu=mu*nu; |
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492 | rho=qlc[1]*qlm; |
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493 | p=nu*p-rho*F[i]; |
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494 | // string("i=",i," coef(p,v)=",coef(p,v)); |
---|
495 | r=nu*r; |
---|
496 | for (j=1;j<=size(F);j++){q[j]=nu*q[j];} |
---|
497 | q[i]=q[i]+rho; |
---|
498 | divoc=1; |
---|
499 | } |
---|
500 | else {i++;} |
---|
501 | } |
---|
502 | if (divoc==0) |
---|
503 | { |
---|
504 | r=r+lcp*lpp; |
---|
505 | p=p-lcp*lpp; |
---|
506 | // string("pasa al rem p=",p); |
---|
507 | } |
---|
508 | } |
---|
509 | list res=r,q,mu; |
---|
510 | if(te==0){kill @P; kill @R; kill @RP;} |
---|
511 | return(res); |
---|
512 | } |
---|
513 | example |
---|
514 | { "EXAMPLE:"; echo = 2; |
---|
515 | ring R=(0,a,b,c),(x,y),dp; |
---|
516 | "Divisor="; |
---|
517 | poly f=(ab-ac)*xy+(ab)*x+(5c); |
---|
518 | "Dividends="; |
---|
519 | ideal F=ax+b,cy+a; |
---|
520 | "(Remainder, quotients, factor)="; |
---|
521 | def r=pdivi(f,F); |
---|
522 | r; |
---|
523 | "Verifying the division: r[3]*f-(r[2][1]*F[1]+r[2][2]*F[2])-r[1] ="; |
---|
524 | r[3]*f-(r[2][1]*F[1]+r[2][2]*F[2])-r[1]; |
---|
525 | } |
---|
526 | |
---|
527 | // pspol : S-poly of two polynomials in @R |
---|
528 | // @R |
---|
529 | // input: |
---|
530 | // poly f (given in the ring @R) |
---|
531 | // poly g (given in the ring @R) |
---|
532 | // output: |
---|
533 | // list (S, red): S is the S-poly(f,g) and red is a Boolean variable |
---|
534 | // if red then S reduces by Buchberger 1st criterion |
---|
535 | // (not used) |
---|
536 | proc pspol(poly f,poly g) |
---|
537 | { |
---|
538 | def lcf=leadcoef(f); |
---|
539 | def lcg=leadcoef(g); |
---|
540 | def lpf=leadmonom(f); |
---|
541 | def lpg=leadmonom(g); |
---|
542 | def v=gcd(lpf,lpg); |
---|
543 | def s=simpqcoeffs(lcf,lcg); |
---|
544 | def vf=lpf/v; |
---|
545 | def vg=lpg/v; |
---|
546 | poly S=s[2]*vg*f-s[1]*vf*g; |
---|
547 | return(S); |
---|
548 | } |
---|
549 | |
---|
550 | // facvar: Returns all the free-square factors of the elements |
---|
551 | // of ideal J (non repeated). Integer factors are ignored, |
---|
552 | // even 0 is ignored. It can be called from ideal @R, but |
---|
553 | // the given ideal J must only contain poynomials in the |
---|
554 | // parameters. |
---|
555 | // Operates in the ring @P, but can be called from ring @R, |
---|
556 | // and the ideal @P must be defined calling first setglobalrings(); |
---|
557 | // input: ideal J |
---|
558 | // output: ideal Jc: Returns all the free-square factors of the elements |
---|
559 | // of ideal J (non repeated). Integer factors are ignored, |
---|
560 | // even 0 is ignored. It can be called from ideal @R. |
---|
561 | proc facvar(ideal J) |
---|
562 | //"USAGE: facvar(J); |
---|
563 | // J: an ideal in the parameters |
---|
564 | //RETURN: all the free-square factors of the elements |
---|
565 | // of ideal J (non repeated). Integer factors are ignored, |
---|
566 | // even 0 is ignored. It can be called from ideal @R, but |
---|
567 | // the given ideal J must only contain poynomials in the |
---|
568 | // parameters. |
---|
569 | //NOTE: Operates in the ring @P, and the ideal J must contain only |
---|
570 | // polynomials in the parameters, but can be called from ring @R. |
---|
571 | //KEYWORDS: factor |
---|
572 | //EXAMPLE: facvar; shows an example" |
---|
573 | { |
---|
574 | int i; |
---|
575 | def RR=basering; |
---|
576 | setring(@P); |
---|
577 | def Ja=imap(RR,J); |
---|
578 | if(size(Ja)==0){setring(RR); return(ideal(0));} |
---|
579 | Ja=elimintfromideal(Ja); // also in ideal @P |
---|
580 | ideal Jb; |
---|
581 | if (size(Ja)==0){Jb=ideal(0);} |
---|
582 | else |
---|
583 | { |
---|
584 | for (i=1;i<=ncols(Ja);i++){if(size(Ja[i])!=0){Jb=Jb,factorize(Ja[i],1);}} |
---|
585 | Jb=simplify(Jb,2+4+8); |
---|
586 | Jb=cld(Jb); |
---|
587 | Jb=elimintfromideal(Jb); // also in ideal @P |
---|
588 | } |
---|
589 | setring(RR); |
---|
590 | def Jc=imap(@P,Jb); |
---|
591 | return(Jc); |
---|
592 | } |
---|
593 | //example |
---|
594 | //{ "EXAMPLE:"; echo = 2; |
---|
595 | // ring R=(0,a,b,c),(x,y,z),dp; |
---|
596 | // setglobalrings(); |
---|
597 | // ideal J=a2-b2,a2-2ab+b2,abc-bc; |
---|
598 | // facvar(J); |
---|
599 | //} |
---|
600 | |
---|
601 | // Ered: eliminates the factors in the polynom f that are non-null. |
---|
602 | // In ring @R |
---|
603 | // input: |
---|
604 | // poly f: |
---|
605 | // ideal E of null-conditions |
---|
606 | // ideal N of non-null conditions |
---|
607 | // (E,N) represents V(E)\V(N), |
---|
608 | // Ered eliminates the non-null factors of f in V(E)\V(N) |
---|
609 | // output: |
---|
610 | // poly f2 where the non-null conditions have been dropped from f |
---|
611 | proc Ered(poly f,ideal E, ideal N) |
---|
612 | { |
---|
613 | def RR=basering; |
---|
614 | setring(@R); |
---|
615 | poly ff=imap(RR,f); |
---|
616 | ideal EE=imap(RR,E); |
---|
617 | ideal NN=imap(RR,N); |
---|
618 | if((ff==0) or (equalideals(NN,ideal(1)))){setring(RR); return(f);} |
---|
619 | def v=variables(ff); |
---|
620 | int i; |
---|
621 | poly X=1; |
---|
622 | for(i=1;i<=size(v);i++){X=X*v[i];} |
---|
623 | matrix M=coef(ff,X); |
---|
624 | setring(@P); |
---|
625 | def RPE=imap(@R,EE); |
---|
626 | def RPN=imap(@R,NN); |
---|
627 | matrix Mp=imap(@R,M); |
---|
628 | poly g=Mp[2,1]; |
---|
629 | if (size(Mp)!=2) |
---|
630 | { |
---|
631 | for(i=2;i<=size(Mp) div 2;i++) |
---|
632 | { |
---|
633 | g=gcd(g,Mp[2,i]); |
---|
634 | } |
---|
635 | } |
---|
636 | if (g==1){setring(RR); return(f);} |
---|
637 | else |
---|
638 | { |
---|
639 | def wg=factorize(g,2); |
---|
640 | if (wg[1][1]==1){setring(RR); return(f);} |
---|
641 | else |
---|
642 | { |
---|
643 | poly simp=1; |
---|
644 | int te; |
---|
645 | for(i=1;i<=size(wg[1]);i++) |
---|
646 | { |
---|
647 | te=inconsistent(RPE+wg[1][i],RPN); |
---|
648 | if(te) |
---|
649 | { |
---|
650 | simp=simp*(wg[1][i])^(wg[2][i]); |
---|
651 | } |
---|
652 | } |
---|
653 | } |
---|
654 | if (simp==1){setring(RR); return(f);} |
---|
655 | else |
---|
656 | { |
---|
657 | setring(RR); |
---|
658 | def simp0=imap(@P,simp); |
---|
659 | def f2=f/simp0; |
---|
660 | return(f2); |
---|
661 | } |
---|
662 | } |
---|
663 | } |
---|
664 | |
---|
665 | // pnormalf: reduces a polynomial f wrt a V(E)\V(N) |
---|
666 | // dividing by E and eliminating factors in N. |
---|
667 | // called in the ring @R, |
---|
668 | // operates in the ring @RP. |
---|
669 | // input: |
---|
670 | // poly f |
---|
671 | // ideal E (depends only on the parameters) |
---|
672 | // ideal N (depends only on the parameters) |
---|
673 | // (E,N) represents V(E)\V(N) |
---|
674 | // optional: |
---|
675 | // output: poly f2 reduced wrt to V(E)\V(N) |
---|
676 | proc pnormalf(poly f, ideal E, ideal N) |
---|
677 | "USAGE: pnormalf(f,E,N); |
---|
678 | f: the polynomial to be reduced modulo V(E)\V(N) |
---|
679 | of a segment in the parameters. |
---|
680 | E: the null conditions ideal |
---|
681 | N: the non-null conditions |
---|
682 | RETURN: a reduced polynomial g of f, whose coefficients are reduced |
---|
683 | modulo E and having no factor in N. |
---|
684 | NOTE: Should be called from ring Q[a][x]. |
---|
685 | Ideals E and N must be given by polynomials |
---|
686 | in the parameters. |
---|
687 | KEYWORDS: division, pdivi, reduce |
---|
688 | EXAMPLE: pnormalf; shows an example" |
---|
689 | { |
---|
690 | def RR=basering; |
---|
691 | int te=0; |
---|
692 | if (defined(@P)){te=1;} |
---|
693 | else{setglobalrings();} |
---|
694 | setring(@RP); |
---|
695 | def fa=imap(RR,f); |
---|
696 | def Ea=imap(RR,E); |
---|
697 | def Na=imap(RR,N); |
---|
698 | option(redSB); |
---|
699 | Ea=std(Ea); |
---|
700 | def r=cld(reduce(fa,Ea)); |
---|
701 | poly f1=r[1]; |
---|
702 | f1=Ered(r[1],Ea,Na); |
---|
703 | setring(RR); |
---|
704 | def f2=imap(@RP,f1); |
---|
705 | if(te==0){kill @R; kill @RP; kill @P;} |
---|
706 | return(f2) |
---|
707 | } |
---|
708 | example |
---|
709 | { "EXAMPLE:"; echo = 2; |
---|
710 | ring R=(0,a,b,c),(x,y),dp; |
---|
711 | poly f=(b^2-1)*x^3*y+(c^2-1)*x*y^2+(c^2*b-b)*x+(a-bc)*y; |
---|
712 | ideal E=(c-1); |
---|
713 | ideal N=a-b; |
---|
714 | pnormalf(f,E,N); |
---|
715 | } |
---|
716 | |
---|
717 | // idint: ideal intersection |
---|
718 | // in the ring @P. |
---|
719 | // it works in an extended ring |
---|
720 | // input: two ideals in the ring @P |
---|
721 | // output the intersection of both (is not a GB) |
---|
722 | proc idint(ideal I, ideal J) |
---|
723 | { |
---|
724 | def RR=basering; |
---|
725 | ring T=0,t,lp; |
---|
726 | def K=T+RR; |
---|
727 | setring(K); |
---|
728 | def Ia=imap(RR,I); |
---|
729 | def Ja=imap(RR,J); |
---|
730 | ideal IJ; |
---|
731 | int i; |
---|
732 | for(i=1;i<=size(Ia);i++){IJ[i]=t*Ia[i];} |
---|
733 | for(i=1;i<=size(Ja);i++){IJ[size(Ia)+i]=(1-t)*Ja[i];} |
---|
734 | ideal eIJ=eliminate(IJ,t); |
---|
735 | setring(RR); |
---|
736 | return(imap(K,eIJ)); |
---|
737 | } |
---|
738 | |
---|
739 | // lesspol: compare two polynomials by its leading power products |
---|
740 | // input: two polynomials f,g in the ring @R |
---|
741 | // output: 0 if f<g, 1 if f>=g |
---|
742 | proc lesspol(poly f, poly g) |
---|
743 | { |
---|
744 | if (leadmonom(f)<leadmonom(g)){return(1);} |
---|
745 | else |
---|
746 | { |
---|
747 | if (leadmonom(g)<leadmonom(f)){return(0);} |
---|
748 | else |
---|
749 | { |
---|
750 | if (leadcoef(f)<leadcoef(g)){return(1);} |
---|
751 | else {return(0);} |
---|
752 | } |
---|
753 | } |
---|
754 | } |
---|
755 | |
---|
756 | // delfromideal: deletes the i-th polynomial from the ideal F |
---|
757 | proc delfromideal(ideal F, int i) |
---|
758 | { |
---|
759 | int j; |
---|
760 | ideal G; |
---|
761 | if (size(F)<i){ERROR("delfromideal was called incorrect arguments");} |
---|
762 | if (size(F)<=1){return(ideal(0));} |
---|
763 | if (i==0){return(F)}; |
---|
764 | for (j=1;j<=size(F);j++) |
---|
765 | { |
---|
766 | if (j!=i){G[size(G)+1]=F[j];} |
---|
767 | } |
---|
768 | return(G); |
---|
769 | } |
---|
770 | |
---|
771 | // delidfromid: deletes the polynomials in J that are in I |
---|
772 | // input: ideals I,J |
---|
773 | // output: the ideal J without the polynomials in I |
---|
774 | proc delidfromid(ideal I, ideal J) |
---|
775 | { |
---|
776 | int i; list r; |
---|
777 | ideal JJ=J; |
---|
778 | for (i=1;i<=size(I);i++) |
---|
779 | { |
---|
780 | r=memberpos(I[i],JJ); |
---|
781 | if (r[1]) |
---|
782 | { |
---|
783 | JJ=delfromideal(JJ,r[2]); |
---|
784 | } |
---|
785 | } |
---|
786 | return(JJ); |
---|
787 | } |
---|
788 | |
---|
789 | // sortideal: sorts the polynomials in an ideal by lm in ascending order |
---|
790 | proc sortideal(ideal Fi) |
---|
791 | { |
---|
792 | def RR=basering; |
---|
793 | setring(@RP); |
---|
794 | def F=imap(RR,Fi); |
---|
795 | def H=F; |
---|
796 | ideal G; |
---|
797 | int i; |
---|
798 | int j; |
---|
799 | poly p; |
---|
800 | while (size(H)!=0) |
---|
801 | { |
---|
802 | j=1; |
---|
803 | p=H[1]; |
---|
804 | for (i=1;i<=size(H);i++) |
---|
805 | { |
---|
806 | if(lesspol(H[i],p)){j=i;p=H[j];} |
---|
807 | } |
---|
808 | G[size(G)+1]=p; |
---|
809 | H=delfromideal(H,j); |
---|
810 | } |
---|
811 | setring(RR); |
---|
812 | def GG=imap(@RP,G); |
---|
813 | return(GG); |
---|
814 | } |
---|
815 | |
---|
816 | // mingb: given a basis (gb reducing) it |
---|
817 | // order the polynomials is ascending order and |
---|
818 | // eliminates the polynomials whose lpp are divisible by some |
---|
819 | // smaller one |
---|
820 | proc mingb(ideal F) |
---|
821 | { |
---|
822 | int t; int i; int j; |
---|
823 | def H=sortideal(F); |
---|
824 | ideal G; |
---|
825 | if (ncols(H)<=1){return(H);} |
---|
826 | G=H[1]; |
---|
827 | for (i=2; i<=ncols(H); i++) |
---|
828 | { |
---|
829 | t=1; |
---|
830 | j=1; |
---|
831 | while (t and (j<i)) |
---|
832 | { |
---|
833 | if((leadmonom(H[i])/leadmonom(H[j]))!=0) {t=0;} |
---|
834 | j++; |
---|
835 | } |
---|
836 | if (t) {G[size(G)+1]=H[i];} |
---|
837 | } |
---|
838 | return(G); |
---|
839 | } |
---|
840 | |
---|
841 | // redgbn: given a minimal basis (gb reducing) it |
---|
842 | // reduces each polynomial wrt to V(E) \ V(N) |
---|
843 | proc redgbn(ideal F, ideal E, ideal N) |
---|
844 | { |
---|
845 | int te=0; |
---|
846 | if (defined(@P)==1){te=1;} |
---|
847 | ideal G=F; |
---|
848 | ideal H; |
---|
849 | int i; |
---|
850 | if (size(G)==0){return(ideal(0));} |
---|
851 | for (i=1;i<=size(G);i++) |
---|
852 | { |
---|
853 | H=delfromideal(G,i); |
---|
854 | G[i]=pnormalf(pdivi(G[i],H)[1],E,N); |
---|
855 | G[i]=primepartZ(G[i]); |
---|
856 | } |
---|
857 | if(te==1){setglobalrings();} |
---|
858 | return(G); |
---|
859 | } |
---|
860 | |
---|
861 | // eliminates repeated elements form an ideal or matrix or module or intmat or bigintmat |
---|
862 | proc elimrepeated(F) |
---|
863 | { |
---|
864 | int i; |
---|
865 | int nt; |
---|
866 | if (typeof(F)=="ideal"){nt=ncols(F);} |
---|
867 | else{nt=size(F);} |
---|
868 | |
---|
869 | def FF=F; |
---|
870 | FF=F[1]; |
---|
871 | for (i=2;i<=nt;i++) |
---|
872 | { |
---|
873 | if (not(memberpos(F[i],FF)[1])) |
---|
874 | { |
---|
875 | FF[size(FF)+1]=F[i]; |
---|
876 | } |
---|
877 | } |
---|
878 | return(FF); |
---|
879 | } |
---|
880 | |
---|
881 | proc elimrepeatedvl(F) |
---|
882 | { |
---|
883 | int i; |
---|
884 | def FF=F; |
---|
885 | FF=F[1]; |
---|
886 | for (i=2;i<=size(F);i++) |
---|
887 | { |
---|
888 | if (not(memberpos(F[i],FF)[1])) |
---|
889 | { |
---|
890 | FF[size(FF)+1]=F[i]; |
---|
891 | } |
---|
892 | } |
---|
893 | return(FF); |
---|
894 | } |
---|
895 | |
---|
896 | |
---|
897 | // equalideals |
---|
898 | // input: 2 ideals F and G; |
---|
899 | // output: 1 if they are identical (the same polynomials in the same order) |
---|
900 | // 0 else |
---|
901 | proc equalideals(ideal F, ideal G) |
---|
902 | { |
---|
903 | int i=1; int t=1; |
---|
904 | if (size(F)!=size(G)){return(0);} |
---|
905 | while ((i<=size(F)) and (t)) |
---|
906 | { |
---|
907 | if (F[i]!=G[i]){t=0;} |
---|
908 | i++; |
---|
909 | } |
---|
910 | return(t); |
---|
911 | } |
---|
912 | |
---|
913 | // delintvec |
---|
914 | // input: intvec V |
---|
915 | // int i |
---|
916 | // output: |
---|
917 | // intvec W (equal to V but the coordinate i is deleted |
---|
918 | proc delintvec(intvec V, int i) |
---|
919 | { |
---|
920 | int j; |
---|
921 | intvec W; |
---|
922 | for (j=1;j<i;j++){W[j]=V[j];} |
---|
923 | for (j=i+1;j<=size(V);j++){W[j-1]=V[j];} |
---|
924 | return(W); |
---|
925 | } |
---|
926 | |
---|
927 | //**************Begin homogenizing************************ |
---|
928 | |
---|
929 | // ishomog: |
---|
930 | // Purpose: test if a polynomial is homogeneous in the variables or not |
---|
931 | // input: poly f |
---|
932 | // output 1 if f is homogeneous, 0 if not |
---|
933 | proc ishomog(f) |
---|
934 | { |
---|
935 | int i; poly r; int d; int dr; |
---|
936 | if (f==0){return(1);} |
---|
937 | d=deg(f); dr=d; r=f; |
---|
938 | while ((d==dr) and (r!=0)) |
---|
939 | { |
---|
940 | r=r-lead(r); |
---|
941 | dr=deg(r); |
---|
942 | } |
---|
943 | if (r==0){return(1);} |
---|
944 | else{return(0);} |
---|
945 | } |
---|
946 | |
---|
947 | // postredgb: given a minimal basis (gb reducing) it |
---|
948 | // reduces each polynomial wrt to the others |
---|
949 | proc postredgb(ideal F) |
---|
950 | { |
---|
951 | int te=0; |
---|
952 | if(defined(@P)==1){te=1;} |
---|
953 | ideal G; |
---|
954 | ideal H; |
---|
955 | int i; |
---|
956 | if (size(F)==0){return(ideal(0));} |
---|
957 | for (i=1;i<=size(F);i++) |
---|
958 | { |
---|
959 | H=delfromideal(F,i); |
---|
960 | G[i]=pdivi(F[i],H)[1]; |
---|
961 | } |
---|
962 | if(te==1){setglobalrings();} |
---|
963 | return(G); |
---|
964 | } |
---|
965 | |
---|
966 | //purpose ideal intersection called in @R and computed in @P |
---|
967 | proc idintR(ideal N, ideal M) |
---|
968 | { |
---|
969 | def RR=basering; |
---|
970 | setring(@P); |
---|
971 | def Np=imap(RR,N); |
---|
972 | def Mp=imap(RR,M); |
---|
973 | def Jp=idint(Np,Mp); |
---|
974 | setring(RR); |
---|
975 | return(imap(@P,Jp)); |
---|
976 | } |
---|
977 | |
---|
978 | //purpose reduced Groebner basis called in @R and computed in @P |
---|
979 | proc gbR(ideal N) |
---|
980 | { |
---|
981 | def RR=basering; |
---|
982 | setring(@P); |
---|
983 | def Np=imap(RR,N); |
---|
984 | option(redSB); |
---|
985 | Np=std(Np); |
---|
986 | setring(RR); |
---|
987 | return(imap(@P,Np)); |
---|
988 | } |
---|
989 | |
---|
990 | //**************End homogenizing************************ |
---|
991 | |
---|
992 | //**************Begin of Groebner Cover***************** |
---|
993 | |
---|
994 | // incquotient |
---|
995 | // incremental quotient |
---|
996 | // Input: ideal N: a Groebner basis of an ideal |
---|
997 | // poly f: |
---|
998 | // Output: Na = N:<f> |
---|
999 | proc incquotient(ideal N, poly f) |
---|
1000 | { |
---|
1001 | poly g; int i; |
---|
1002 | ideal Nb; ideal Na=N; |
---|
1003 | if (size(Na)==1) |
---|
1004 | { |
---|
1005 | g=gcd(Na[1],f); |
---|
1006 | if (g!=1) |
---|
1007 | { |
---|
1008 | Na[1]=Na[1]/g; |
---|
1009 | } |
---|
1010 | attrib(Na,"IsSB",1); |
---|
1011 | return(Na); |
---|
1012 | } |
---|
1013 | def P=basering; |
---|
1014 | poly @t; |
---|
1015 | ring H=0,@t,lp; |
---|
1016 | def HP=H+P; |
---|
1017 | setring(HP); |
---|
1018 | def fh=imap(P,f); |
---|
1019 | def Nh=imap(P,N); |
---|
1020 | ideal Nht; |
---|
1021 | for (i=1;i<=size(Nh);i++) |
---|
1022 | { |
---|
1023 | Nht[i]=Nh[i]*@t; |
---|
1024 | } |
---|
1025 | attrib(Nht,"isSB",1); |
---|
1026 | def fht=(1-@t)*fh; |
---|
1027 | option(redSB); |
---|
1028 | Nht=std(Nht,fht); |
---|
1029 | ideal Nc; ideal v; |
---|
1030 | for (i=1;i<=size(Nht);i++) |
---|
1031 | { |
---|
1032 | v=variables(Nht[i]); |
---|
1033 | if(memberpos(@t,v)[1]==0) |
---|
1034 | { |
---|
1035 | Nc[size(Nc)+1]=Nht[i]/fh; |
---|
1036 | } |
---|
1037 | } |
---|
1038 | setring(P); |
---|
1039 | ideal HH; |
---|
1040 | def Nd=imap(HP,Nc); Nb=Nd; |
---|
1041 | option(redSB); |
---|
1042 | Nb=std(Nd); |
---|
1043 | return(Nb); |
---|
1044 | } |
---|
1045 | |
---|
1046 | // eliminates the ith element from a list or an intvec |
---|
1047 | proc elimfromlist(l, int i) |
---|
1048 | { |
---|
1049 | if(typeof(l)=="list"){list L;} |
---|
1050 | if (typeof(l)=="intvec"){intvec L;} |
---|
1051 | if (typeof(l)=="ideal"){ideal L;} |
---|
1052 | int j; |
---|
1053 | if((size(l)==0) or (size(l)==1 and i!=1)){return(l);} |
---|
1054 | if (size(l)==1 and i==1){return(L);} |
---|
1055 | // L=l[1]; |
---|
1056 | if(i==1) |
---|
1057 | { |
---|
1058 | for(j=2;j<=size(l);j++) |
---|
1059 | { |
---|
1060 | L[j-1]=l[j]; |
---|
1061 | } |
---|
1062 | } |
---|
1063 | else |
---|
1064 | { |
---|
1065 | for(j=1;j<=i-1;j++) |
---|
1066 | { |
---|
1067 | L[j]=l[j]; |
---|
1068 | } |
---|
1069 | for(j=i+1;j<=size(l);j++) |
---|
1070 | { |
---|
1071 | L[j-1]=l[j]; |
---|
1072 | } |
---|
1073 | } |
---|
1074 | return(L); |
---|
1075 | } |
---|
1076 | |
---|
1077 | // Auxiliary routine to define an order for ideals |
---|
1078 | // Returns 1 if the ideal a is shoud precede ideal b by sorting them in idbefid order |
---|
1079 | // 2 if the the contrary happen |
---|
1080 | // 0 if the order is not relevant |
---|
1081 | proc idbefid(ideal a, ideal b) |
---|
1082 | { |
---|
1083 | poly fa; poly fb; poly la; poly lb; |
---|
1084 | int te=1; int i; int j; |
---|
1085 | int na=size(a); |
---|
1086 | int nb=size(b); |
---|
1087 | int nm; |
---|
1088 | if (na<=nb){nm=na;} else{nm=nb;} |
---|
1089 | for (i=1;i<=nm; i++) |
---|
1090 | { |
---|
1091 | fa=a[i]; fb=b[i]; |
---|
1092 | while((fa!=0) or (fb!=0)) |
---|
1093 | { |
---|
1094 | la=lead(fa); |
---|
1095 | lb=lead(fb); |
---|
1096 | fa=fa-la; |
---|
1097 | fb=fb-lb; |
---|
1098 | la=leadmonom(la); |
---|
1099 | lb=leadmonom(lb); |
---|
1100 | if(leadmonom(la+lb)!=la){return(1);} |
---|
1101 | else{if(leadmonom(la+lb)!=lb){return(2);}} |
---|
1102 | } |
---|
1103 | } |
---|
1104 | if(na<nb){return(1);} |
---|
1105 | else |
---|
1106 | { |
---|
1107 | if(na>nb){return(2);} |
---|
1108 | else{return(0);} |
---|
1109 | } |
---|
1110 | } |
---|
1111 | |
---|
1112 | // sort a list of ideals using idbefid |
---|
1113 | proc sortlistideals(list L) |
---|
1114 | { |
---|
1115 | int i; int j; int n; |
---|
1116 | ideal a; ideal b; |
---|
1117 | list LL=L; |
---|
1118 | list NL; |
---|
1119 | int k; int te; |
---|
1120 | i=1; |
---|
1121 | while(size(LL)>0) |
---|
1122 | { |
---|
1123 | k=1; |
---|
1124 | for(j=2;j<=size(LL);j++) |
---|
1125 | { |
---|
1126 | te=idbefid(LL[k],LL[j]); |
---|
1127 | if (te==2){k=j;} |
---|
1128 | } |
---|
1129 | NL[size(NL)+1]=LL[k]; |
---|
1130 | n=size(LL); |
---|
1131 | if (n>1){LL=elimfromlist(LL,k);} else{LL=list();} |
---|
1132 | } |
---|
1133 | return(NL); |
---|
1134 | } |
---|
1135 | |
---|
1136 | // returns 1 if the two lists of ideals are equal and 0 if not |
---|
1137 | proc equallistideals(list L, list M) |
---|
1138 | { |
---|
1139 | int t; int i; |
---|
1140 | if (size(L)!=size(M)){return(0);} |
---|
1141 | else |
---|
1142 | { |
---|
1143 | t=1; |
---|
1144 | if (size(L)>0) |
---|
1145 | { |
---|
1146 | i=1; |
---|
1147 | while ((t) and (i<=size(L))) |
---|
1148 | { |
---|
1149 | if (equalideals(L[i],M[i])==0){t=0;} |
---|
1150 | i++; |
---|
1151 | } |
---|
1152 | } |
---|
1153 | return(t); |
---|
1154 | } |
---|
1155 | } |
---|
1156 | |
---|
1157 | // Prep |
---|
1158 | // Computes the P-representation of V(N) \ V(M). |
---|
1159 | // input: |
---|
1160 | // ideal N (null ideal) (not necessarily radical nor maximal) |
---|
1161 | // ideal M (hole ideal) (not necessarily containing N) |
---|
1162 | // output: |
---|
1163 | // the ((p_1,(p_11,p_1k_1)),..,(p_r,(p_r1,p_rk_r))); |
---|
1164 | // the Prep of V(N)\V(M) |
---|
1165 | // Assumed to work in the ring @P of the parameters |
---|
1166 | // Can be used without calling previously setglobalrings(). |
---|
1167 | proc Prep(ideal N, ideal M) |
---|
1168 | { |
---|
1169 | int te; |
---|
1170 | if (N[1]==1) |
---|
1171 | { |
---|
1172 | return(list(list(ideal(1),list(ideal(1))))); |
---|
1173 | } |
---|
1174 | def RR=basering; |
---|
1175 | if(defined(@P)){te=1;} |
---|
1176 | else{te=0; setglobalrings();} |
---|
1177 | setring(@P); |
---|
1178 | ideal Np=imap(RR,N); |
---|
1179 | ideal Mp=imap(RR,M); |
---|
1180 | int i; int j; list L0; |
---|
1181 | |
---|
1182 | list Ni=minGTZ(Np); |
---|
1183 | list prep; |
---|
1184 | for(j=1;j<=size(Ni);j++) |
---|
1185 | { |
---|
1186 | option(redSB); |
---|
1187 | Ni[j]=std(Ni[j]); |
---|
1188 | } |
---|
1189 | list Mij; |
---|
1190 | for (i=1;i<=size(Ni);i++) |
---|
1191 | { |
---|
1192 | Mij=minGTZ(Ni[i]+Mp); |
---|
1193 | for(j=1;j<=size(Mij);j++) |
---|
1194 | { |
---|
1195 | option(redSB); |
---|
1196 | Mij[j]=std(Mij[j]); |
---|
1197 | } |
---|
1198 | if ((size(Mij)==1) and (equalideals(Ni[i],Mij[1])==1)){;} |
---|
1199 | else |
---|
1200 | { |
---|
1201 | prep[size(prep)+1]=list(Ni[i],Mij); |
---|
1202 | } |
---|
1203 | } |
---|
1204 | if (size(prep)==0){prep=list(list(ideal(1),list(ideal(1))));} |
---|
1205 | setring(RR); |
---|
1206 | def pprep=imap(@P,prep); |
---|
1207 | if(te==0){kill @P; kill @R; kill @RP;} |
---|
1208 | return(pprep); |
---|
1209 | } |
---|
1210 | |
---|
1211 | // PtoCrep |
---|
1212 | // Computes the C-representation from the P-representation. |
---|
1213 | // input: |
---|
1214 | // list ((p_1,(p_11,p_1k_1)),..,(p_r,(p_r1,p_rk_r))); |
---|
1215 | // the P-representation of V(N)\V(M) |
---|
1216 | // output: |
---|
1217 | // list (ideal ida, ideal idb) |
---|
1218 | // the C-representaion of V(N)\V(M) = V(ida)\V(idb) |
---|
1219 | // Assumed to work in the ring @P of the parameters |
---|
1220 | // If the routine is to be called from the top, a previous call to |
---|
1221 | // setglobalrings() is needed. |
---|
1222 | proc PtoCrep(list L) |
---|
1223 | { |
---|
1224 | int te=0; |
---|
1225 | def RR=basering; |
---|
1226 | if(defined(@P)){te=1;} |
---|
1227 | else{te=0; setglobalrings();} |
---|
1228 | setring(@P); |
---|
1229 | def Lp=imap(RR,L); |
---|
1230 | int i; int j; |
---|
1231 | ideal ida=ideal(1); ideal idb=ideal(1); list Lb; ideal N; |
---|
1232 | for (i=1;i<=size(Lp);i++) |
---|
1233 | { |
---|
1234 | option(returnSB); |
---|
1235 | N=Lp[i][1]; |
---|
1236 | ida=intersect(ida,N); |
---|
1237 | Lb=Lp[i][2]; |
---|
1238 | for(j=1;j<=size(Lb);j++) |
---|
1239 | { |
---|
1240 | idb=intersect(idb,Lb[j]); |
---|
1241 | } |
---|
1242 | } |
---|
1243 | def La=list(ida,idb); |
---|
1244 | setring(RR); |
---|
1245 | def LL=imap(@P,La); |
---|
1246 | if(te==0){kill @P; kill @R; kill @RP;} |
---|
1247 | return(LL); |
---|
1248 | } |
---|
1249 | |
---|
1250 | // input: F a parametric ideal in Q[a][x] |
---|
1251 | // output: a rComprehensive Groebner System disjoint and reduced. |
---|
1252 | // It uses Kapur-Sun-Wang algorithm, and with the options |
---|
1253 | // can compute the homogenization before (('can',0) or ( 'can',1)) |
---|
1254 | // and dehomogenize the result. |
---|
1255 | proc cgsdr(ideal F, list #) |
---|
1256 | "USAGE: cgsdr(F); To compute a disjoint, reduced CGS. |
---|
1257 | cgsdr is the starting point of the fundamental routine grobcov. |
---|
1258 | Inside grobcov it is used only with options 'can' set to 0,1 and |
---|
1259 | not with options ('can',2). |
---|
1260 | It is to be used if only a disjoint reduced CGS is required. |
---|
1261 | F: ideal in Q[a][x] (parameters and variables) to be discussed. |
---|
1262 | |
---|
1263 | Options: To modify the default options, pairs of arguments |
---|
1264 | -option name, value- of valid options must be added to the call. |
---|
1265 | |
---|
1266 | Options: |
---|
1267 | \"can\",0-1-2: The default value is \"can\",2. In this case no |
---|
1268 | homogenization is done. With option (\"can\",0) the given |
---|
1269 | basis is homogenized, and with option (\"can\",1) the |
---|
1270 | whole given ideal is homogenized before computing the |
---|
1271 | cgs and dehomogenized after. |
---|
1272 | with option (\"can\",0) the homogenized basis is used |
---|
1273 | with option (\"can\",1) the homogenized ideal is used |
---|
1274 | with option (\"can\",2) the given basis is used |
---|
1275 | \"null\",ideal E: The default is (\"null\",ideal(0)). |
---|
1276 | \"nonnull\",ideal N: The default (\"nonnull\",ideal(1)). |
---|
1277 | When options 'null' and/or 'nonnull' are given, then |
---|
1278 | the parameter space is restricted to V(E)\V(N). |
---|
1279 | \"comment\",0-1: The default is (\"comment\",0). Setting (\"comment\",1) |
---|
1280 | will provide information about the development of the |
---|
1281 | computation. |
---|
1282 | \"out\",0-1: 1 (default) the output segments are given as |
---|
1283 | as difference of varieties. |
---|
1284 | 0: the output segments are given in P-representation |
---|
1285 | and the segments grouped by lpp |
---|
1286 | With options (\"can\",0) and (\"can\",1) the option (\"out\",1) |
---|
1287 | is set to (out,0) because it is not compatible. |
---|
1288 | One can give none or whatever of these options. |
---|
1289 | With the default options (\"can\",2,\"out\",1), only the |
---|
1290 | Kapur-Sun-Wang algorithm is computed. This is very effectif |
---|
1291 | but is only the starting point for the computation. |
---|
1292 | When grobcov is computed, the call to cgsdr inside uses |
---|
1293 | specific options that are more expensive ("can",0-1,"out",0). |
---|
1294 | RETURN: Returns a list T describing a reduced and disjoint |
---|
1295 | Comprehensive Groebner System (CGS), |
---|
1296 | With option (\"out\",0) |
---|
1297 | the segments are grouped by |
---|
1298 | leading power products (lpp) of the reduced Groebner |
---|
1299 | basis and given in P-representation. |
---|
1300 | The returned list is of the form: |
---|
1301 | ( |
---|
1302 | (lpp, (num,basis,segment),...,(num,basis,segment),lpp), |
---|
1303 | ..,, |
---|
1304 | (lpp, (num,basis,segment),...,(num,basis,segment),lpp) |
---|
1305 | ) |
---|
1306 | The bases are the reduced Groebner bases (after normalization) |
---|
1307 | for each point of the corresponding segment. |
---|
1308 | |
---|
1309 | The third element of each lpp segment is the lpp of the |
---|
1310 | used ideal in the CGS as a string: |
---|
1311 | with option (\"can\",0) the homogenized basis is used |
---|
1312 | with option (\"can\",1) the homogenized ideal is used |
---|
1313 | with option (\"can\",2) the given basis is used |
---|
1314 | |
---|
1315 | With option (\"out\",1) (default) |
---|
1316 | only KSW is applied and segments are given as |
---|
1317 | difference of varieties and are not grouped |
---|
1318 | The returned list is of the form: |
---|
1319 | ( |
---|
1320 | (E,N,B),..(E,N,B) |
---|
1321 | ) |
---|
1322 | E is the null variety |
---|
1323 | N is the nonnull variety |
---|
1324 | segment = V(E)\V(N) |
---|
1325 | B is the reduced Groebner basis |
---|
1326 | |
---|
1327 | NOTE: The basering R, must be of the form Q[a][x], a=parameters, |
---|
1328 | x=variables, and should be defined previously, and the ideal |
---|
1329 | defined on R. |
---|
1330 | KEYWORDS: CGS, disjoint, reduced, Comprehensive Groebner System |
---|
1331 | EXAMPLE: cgsdr; shows an example" |
---|
1332 | { |
---|
1333 | int te; |
---|
1334 | def RR=basering; |
---|
1335 | if(defined(@P)){te=1;} |
---|
1336 | else{te=0; setglobalrings();} |
---|
1337 | // INITIALIZING OPTIONS |
---|
1338 | int i; int j; |
---|
1339 | int can=2; |
---|
1340 | int out=1; |
---|
1341 | poly f; |
---|
1342 | ideal B; |
---|
1343 | def E=ideal(0); |
---|
1344 | def N=ideal(1); |
---|
1345 | int comment=0; |
---|
1346 | int start=timer; |
---|
1347 | list L=#; |
---|
1348 | for(i=1;i<=size(L) div 2;i++) |
---|
1349 | { |
---|
1350 | if(L[2*i-1]=="null"){E=L[2*i];} |
---|
1351 | else |
---|
1352 | { |
---|
1353 | if(L[2*i-1]=="nonnull"){N=L[2*i];} |
---|
1354 | else |
---|
1355 | { |
---|
1356 | if(L[2*i-1]=="comment"){comment=L[2*i];} |
---|
1357 | else |
---|
1358 | { |
---|
1359 | if(L[2*i-1]=="can"){can=L[2*i];} |
---|
1360 | else |
---|
1361 | { |
---|
1362 | if(L[2*i-1]=="out"){out=L[2*i];} |
---|
1363 | } |
---|
1364 | } |
---|
1365 | } |
---|
1366 | } |
---|
1367 | } |
---|
1368 | //if(can==2){out=1;} |
---|
1369 | B=F; |
---|
1370 | if ((printlevel) and (comment==0)){comment=printlevel;} |
---|
1371 | if((can<2) and (out>0)){"Option out,1 is not compatible with can,0,1"; out=0;} |
---|
1372 | // DEFINING OPTIONS |
---|
1373 | list LL; |
---|
1374 | LL[1]="can"; LL[2]=can; |
---|
1375 | LL[3]="comment"; LL[4]=comment; |
---|
1376 | LL[5]="out"; LL[6]=out; |
---|
1377 | LL[7]="null"; LL[8]=E; |
---|
1378 | LL[9]="nonnull"; LL[10]=N; |
---|
1379 | if(comment>=1) |
---|
1380 | { |
---|
1381 | string("Begin cgsdr with options: ",LL); |
---|
1382 | } |
---|
1383 | int ish; |
---|
1384 | for (i=1;i<=size(B);i++){ish=ishomog(B[i]); if(ish==0){break;};} |
---|
1385 | if (ish) |
---|
1386 | { |
---|
1387 | if(comment>0){string("The given system is homogneous");} |
---|
1388 | def GS=KSW(B,LL); |
---|
1389 | //can=0; |
---|
1390 | } |
---|
1391 | else |
---|
1392 | { |
---|
1393 | // ACTING DEPENDING ON OPTIONS |
---|
1394 | if(can==2) |
---|
1395 | { |
---|
1396 | // WITHOUT HOMOHGENIZING |
---|
1397 | if(comment>0){string("Option of cgsdr: do not homogenize");} |
---|
1398 | def GS=KSW(B,LL); |
---|
1399 | setglobalrings(); |
---|
1400 | } |
---|
1401 | else |
---|
1402 | { |
---|
1403 | if(can==1) |
---|
1404 | { |
---|
1405 | // COMPUTING THE HOMOGOENIZED IDEAL |
---|
1406 | if(comment>0){string("Homogenizing the whole ideal: option can=1");} |
---|
1407 | list RRL=ringlist(RR); |
---|
1408 | RRL[3][1][1]="dp"; |
---|
1409 | def Pa=ring(RRL[1]); |
---|
1410 | list Lx; |
---|
1411 | Lx[1]=0; |
---|
1412 | Lx[2]=RRL[2]+RRL[1][2]; |
---|
1413 | Lx[3]=RRL[1][3]; |
---|
1414 | Lx[4]=RRL[1][4]; |
---|
1415 | RRL[1]=0; |
---|
1416 | def D=ring(RRL); |
---|
1417 | def RP=D+Pa; |
---|
1418 | setring(RP); |
---|
1419 | def B1=imap(RR,B); |
---|
1420 | option(redSB); |
---|
1421 | B1=std(B1); |
---|
1422 | setring(RR); |
---|
1423 | def B2=imap(RP,B1); |
---|
1424 | } |
---|
1425 | else |
---|
1426 | { // (can=0) |
---|
1427 | if(comment>0){string("Homogenizing the basis: option can=0");} |
---|
1428 | def B2=B; |
---|
1429 | } |
---|
1430 | // COMPUTING HOMOGENIZED CGS |
---|
1431 | poly @t; |
---|
1432 | ring H=0,@t,dp; |
---|
1433 | def RH=RR+H; |
---|
1434 | setring(RH); |
---|
1435 | setglobalrings(); |
---|
1436 | def BH=imap(RR,B2); |
---|
1437 | def LH=imap(RR,LL); |
---|
1438 | for (i=1;i<=size(BH);i++) |
---|
1439 | { |
---|
1440 | BH[i]=homog(BH[i],@t); |
---|
1441 | } |
---|
1442 | if (comment>=1){string("Homogenized system = "); BH;} |
---|
1443 | def GSH=KSW(BH,LH); |
---|
1444 | setglobalrings(); |
---|
1445 | // DEHOMOGENIZING THE RESULT |
---|
1446 | if(out==0) |
---|
1447 | { |
---|
1448 | for (i=1;i<=size(GSH);i++) |
---|
1449 | { |
---|
1450 | GSH[i][1]=subst(GSH[i][1],@t,1); |
---|
1451 | for(j=1;j<=size(GSH[i][2]);j++) |
---|
1452 | { |
---|
1453 | GSH[i][2][j][2]=subst(GSH[i][2][j][2],@t,1); |
---|
1454 | } |
---|
1455 | } |
---|
1456 | } |
---|
1457 | else |
---|
1458 | { |
---|
1459 | for (i=1;i<=size(GSH);i++) |
---|
1460 | { |
---|
1461 | GSH[i][3]=subst(GSH[i][3],@t,1); |
---|
1462 | GSH[i][7]=subst(GSH[i][7],@t,1); |
---|
1463 | } |
---|
1464 | } |
---|
1465 | setring(RR); |
---|
1466 | def GS=imap(RH,GSH); |
---|
1467 | } |
---|
1468 | |
---|
1469 | |
---|
1470 | setglobalrings(); |
---|
1471 | if(out==0) |
---|
1472 | { |
---|
1473 | for (i=1;i<=size(GS);i++) |
---|
1474 | { |
---|
1475 | GS[i][1]=postredgb(mingb(GS[i][1])); |
---|
1476 | for(j=1;j<=size(GS[i][2]);j++) |
---|
1477 | { |
---|
1478 | GS[i][2][j][2]=postredgb(mingb(GS[i][2][j][2])); |
---|
1479 | } |
---|
1480 | } |
---|
1481 | } |
---|
1482 | else |
---|
1483 | { |
---|
1484 | for (i=1;i<=size(GS);i++) |
---|
1485 | { |
---|
1486 | if(GS[i][2]==1) |
---|
1487 | { |
---|
1488 | GS[i][3]=postredgb(mingb(GS[i][3])); |
---|
1489 | GS[i][7]=postredgb(mingb(GS[i][7])); |
---|
1490 | } |
---|
1491 | } |
---|
1492 | } |
---|
1493 | } |
---|
1494 | if(te==0){kill @P; kill @R; kill @RP;} |
---|
1495 | return(GS); |
---|
1496 | } |
---|
1497 | example |
---|
1498 | { "EXAMPLE:"; echo = 2; |
---|
1499 | "Casas conjecture for degree 4"; |
---|
1500 | ring R=(0,a0,a1,a2,a3,a4),(x1,x2,x3),dp; |
---|
1501 | ideal F=x1^4+(4*a3)*x1^3+(6*a2)*x1^2+(4*a1)*x1+(a0), |
---|
1502 | x1^3+(3*a3)*x1^2+(3*a2)*x1+(a1), |
---|
1503 | x2^4+(4*a3)*x2^3+(6*a2)*x2^2+(4*a1)*x2+(a0), |
---|
1504 | x2^2+(2*a3)*x2+(a2), |
---|
1505 | x3^4+(4*a3)*x3^3+(6*a2)*x3^2+(4*a1)*x3+(a0), |
---|
1506 | x3+(a3); |
---|
1507 | cgsdr(F); |
---|
1508 | } |
---|
1509 | |
---|
1510 | // input: internal routine called by cgsdr at the end to group the |
---|
1511 | // lpp segments and improve the output |
---|
1512 | // output: grouped segments by lpp obtained in cgsdr |
---|
1513 | proc grsegments(list T) |
---|
1514 | { |
---|
1515 | int i; |
---|
1516 | list L; |
---|
1517 | list lpp; |
---|
1518 | list lp; |
---|
1519 | list ls; |
---|
1520 | int n=size(T); |
---|
1521 | lpp[1]=T[n][1]; |
---|
1522 | L[1]=list(lpp[1],list(list(T[n][2],T[n][3],T[n][4]))); |
---|
1523 | if (n>1) |
---|
1524 | { |
---|
1525 | for (i=1;i<=size(T)-1;i++) |
---|
1526 | { |
---|
1527 | lp=memberpos(T[n-i][1],lpp); |
---|
1528 | if(lp[1]==1) |
---|
1529 | { |
---|
1530 | ls=L[lp[2]][2]; |
---|
1531 | ls[size(ls)+1]=list(T[n-i][2],T[n-i][3],T[n-i][4]); |
---|
1532 | L[lp[2]][2]=ls; |
---|
1533 | } |
---|
1534 | else |
---|
1535 | { |
---|
1536 | lpp[size(lpp)+1]=T[n-i][1]; |
---|
1537 | L[size(L)+1]=list(T[n-i][1],list(list(T[n-i][2],T[n-i][3],T[n-i][4]))); |
---|
1538 | } |
---|
1539 | } |
---|
1540 | } |
---|
1541 | return(L); |
---|
1542 | } |
---|
1543 | |
---|
1544 | // idcontains |
---|
1545 | // input: ideal p, ideal q |
---|
1546 | // output: 1 if p contains q, 0 otherwise |
---|
1547 | // If the routine is to be called from the top, a previous call to |
---|
1548 | // setglobalrings() is needed. |
---|
1549 | proc idcontains(ideal p, ideal q) |
---|
1550 | { |
---|
1551 | int t; int i; |
---|
1552 | t=1; i=1; |
---|
1553 | def RR=basering; |
---|
1554 | setring(@P); |
---|
1555 | def P=imap(RR,p); |
---|
1556 | def Q=imap(RR,q); |
---|
1557 | attrib(P,"isSB",1); |
---|
1558 | poly r; |
---|
1559 | while ((t) and (i<=size(Q))) |
---|
1560 | { |
---|
1561 | r=reduce(Q[i],P); |
---|
1562 | if (r!=0){t=0;} |
---|
1563 | i++; |
---|
1564 | } |
---|
1565 | setring(RR); |
---|
1566 | return(t); |
---|
1567 | } |
---|
1568 | |
---|
1569 | // selectminindeals |
---|
1570 | // given a list of ideals returns the list of integers corresponding |
---|
1571 | // to the minimal ideals in the list |
---|
1572 | // input: L (list of ideals) |
---|
1573 | // output: the list of integers corresponding to the minimal ideals in L |
---|
1574 | // If the routine is to be called from the top, a previous call to |
---|
1575 | // setglobalrings() is needed. |
---|
1576 | proc selectminideals(list L) |
---|
1577 | { |
---|
1578 | if (size(L)==0){return(L)}; |
---|
1579 | def RR=basering; |
---|
1580 | setring(@P); |
---|
1581 | def Lp=imap(RR,L); |
---|
1582 | int i; int j; int t; intvec notsel; |
---|
1583 | list P; |
---|
1584 | for (i=1;i<=size(Lp);i++) |
---|
1585 | { |
---|
1586 | if(memberpos(i,notsel)[1]) |
---|
1587 | { |
---|
1588 | i++; |
---|
1589 | if(i>size(Lp)){break;} |
---|
1590 | } |
---|
1591 | t=1; |
---|
1592 | j=1; |
---|
1593 | while ((t) and (j<=size(Lp))) |
---|
1594 | { |
---|
1595 | if (i==j){j++;} |
---|
1596 | if ((j<=size(Lp)) and (memberpos(j,notsel)[1]==0)) |
---|
1597 | { |
---|
1598 | |
---|
1599 | if (idcontains(Lp[i],Lp[j])) |
---|
1600 | { |
---|
1601 | notsel[size(notsel)+1]=i; |
---|
1602 | t=0; |
---|
1603 | } |
---|
1604 | } |
---|
1605 | j++; |
---|
1606 | } |
---|
1607 | if (t){P[size(P)+1]=i;} |
---|
1608 | } |
---|
1609 | setring(RR); |
---|
1610 | return(P); |
---|
1611 | } |
---|
1612 | |
---|
1613 | // LCUnion |
---|
1614 | // Given a list of the P-representations of disjoint locally closed segments |
---|
1615 | // for which we know that the union is also locally closed |
---|
1616 | // it returns the P-representation of its union |
---|
1617 | // input: L list of segments in P-representation |
---|
1618 | // ((p_j^i,(p_j1^i,...,p_jk_j^i | j=1..t_i)) | i=1..s ) |
---|
1619 | // where i represents a segment |
---|
1620 | // output: P-representation of the union |
---|
1621 | // ((P_j,(P_j1,...,P_jk_j | j=1..t))) |
---|
1622 | // If the routine is to be called from the top, a previous call to |
---|
1623 | // setglobalrings() is needed. |
---|
1624 | // Auxiliary routine called by grobcov and addcons |
---|
1625 | // A previous call to setglobarings() is needed if it is to be used at the top. |
---|
1626 | proc LCUnion(list LL) |
---|
1627 | { |
---|
1628 | def RR=basering; |
---|
1629 | setring(@P); |
---|
1630 | int care; |
---|
1631 | def L=imap(RR,LL); |
---|
1632 | int i; int j; int k; list H; list C; list T; |
---|
1633 | list L0; list P0; list P; list Q0; list Q; intvec Qi; |
---|
1634 | if(not(defined(@Q2))){list @Q2;} |
---|
1635 | else{kill @Q2; list @Q2;} |
---|
1636 | exportto(Top,@Q2); |
---|
1637 | for (i=1;i<=size(L);i++) |
---|
1638 | { |
---|
1639 | for (j=1;j<=size(L[i]);j++) |
---|
1640 | { |
---|
1641 | P0[size(P0)+1]=L[i][j][1]; |
---|
1642 | L0[size(L0)+1]=intvec(i,j); |
---|
1643 | } |
---|
1644 | } |
---|
1645 | Q0=selectminideals(P0); |
---|
1646 | //"T_3Q0="; Q0; |
---|
1647 | kill P; list P; |
---|
1648 | for (i=1;i<=size(Q0);i++) |
---|
1649 | { |
---|
1650 | Qi=L0[Q0[i]]; |
---|
1651 | Q[size(Q)+1]=Qi; |
---|
1652 | P[size(P)+1]=L[Qi[1]][Qi[2]]; |
---|
1653 | } |
---|
1654 | @Q2=Q; |
---|
1655 | if(size(P)==0) |
---|
1656 | { |
---|
1657 | setring(RR); |
---|
1658 | list TT; |
---|
1659 | return(TT); |
---|
1660 | } |
---|
1661 | // P is the list of the maximal components of the union |
---|
1662 | // with the corresponding initial holes. |
---|
1663 | // Q is the list of intvec positions in L of the first element of the P's |
---|
1664 | // Its elements give (num of segment, num of max component (=min ideal)) |
---|
1665 | for (k=1;k<=size(Q);k++) |
---|
1666 | { |
---|
1667 | H=P[k][2]; // holes of P[k][1] |
---|
1668 | for (i=1;i<=size(L);i++) |
---|
1669 | { |
---|
1670 | if (i!=Q[k][1]) |
---|
1671 | { |
---|
1672 | for (j=1;j<=size(L[i]);j++) |
---|
1673 | { |
---|
1674 | C[size(C)+1]=L[i][j]; |
---|
1675 | C[size(C)][3]=intvec(i,j); |
---|
1676 | } |
---|
1677 | } |
---|
1678 | } |
---|
1679 | if(size(P[k])>=3){T[size(T)+1]=list(Q[k],P[k][1],addpart(H,C),P[k][3]);} |
---|
1680 | else{T[size(T)+1]=list(Q[k],P[k][1],addpart(H,C));} |
---|
1681 | } |
---|
1682 | @Q2=sortpairs(@Q2); |
---|
1683 | setring(RR); |
---|
1684 | def TT=imap(@P,T); |
---|
1685 | return(TT); |
---|
1686 | } |
---|
1687 | |
---|
1688 | // Auxiliary routine |
---|
1689 | // called by LCUnion to modify the holes of a primepart of the union |
---|
1690 | // by the addition of the segments that do not correspond to that part |
---|
1691 | // Works on @P ring. |
---|
1692 | // Input: |
---|
1693 | // H=(p_i1,..,p_is) the holes of a component to be transformed by the addition of |
---|
1694 | // the segments C that do not correspond to that component |
---|
1695 | // C=((q_1,(q_11,..,q_1l_1),pos1),..,(q_k,(q_k1,..,q_kl_k),posk)) |
---|
1696 | // posi=(i,j) position of the component |
---|
1697 | // the list of segments to be added to the holes |
---|
1698 | proc addpart(list H, list C) |
---|
1699 | { |
---|
1700 | list Q; int i; int j; int k; int l; int t; int t1; |
---|
1701 | Q=H; intvec notQ; list QQ; list addq; |
---|
1702 | // @Q2=list of (i,j) positions of the components that have been aded to some hole of the maximal ideals |
---|
1703 | // plus those of the components added to the holes. |
---|
1704 | ideal q; |
---|
1705 | i=1; |
---|
1706 | while (i<=size(Q)) |
---|
1707 | { |
---|
1708 | if (memberpos(i,notQ)[1]==0) |
---|
1709 | { |
---|
1710 | q=Q[i]; |
---|
1711 | t=1; j=1; |
---|
1712 | while ((t) and (j<=size(C))) |
---|
1713 | { |
---|
1714 | if (equalideals(q,C[j][1])) |
---|
1715 | { |
---|
1716 | @Q2[size(@Q2)+1]=C[j][3]; |
---|
1717 | t=0; |
---|
1718 | for (k=1;k<=size(C[j][2]);k++) |
---|
1719 | { |
---|
1720 | t1=1; |
---|
1721 | //kill addq; |
---|
1722 | //list addq; |
---|
1723 | l=1; |
---|
1724 | while((t1) and (l<=size(Q))) |
---|
1725 | { |
---|
1726 | if ((l!=i) and (memberpos(l,notQ)[1]==0)) |
---|
1727 | { |
---|
1728 | if (idcontains(C[j][2][k],Q[l])) |
---|
1729 | { |
---|
1730 | t1=0; |
---|
1731 | } |
---|
1732 | } |
---|
1733 | l++; |
---|
1734 | } |
---|
1735 | if (t1) |
---|
1736 | { |
---|
1737 | addq[size(addq)+1]=C[j][2][k]; |
---|
1738 | @Q2[size(@Q2)+1]=C[j][3]; |
---|
1739 | } |
---|
1740 | } |
---|
1741 | if((size(notQ)==1) and (notQ[1]==0)){notQ[1]=i;} |
---|
1742 | else {notQ[size(notQ)+1]=i;} |
---|
1743 | } |
---|
1744 | j++; |
---|
1745 | } |
---|
1746 | if (size(addq)>0) |
---|
1747 | { |
---|
1748 | for (k=1;k<=size(addq);k++) |
---|
1749 | { |
---|
1750 | Q[size(Q)+1]=addq[k]; |
---|
1751 | } |
---|
1752 | kill addq; |
---|
1753 | list addq; |
---|
1754 | } |
---|
1755 | //print("Q="); Q; print("notQ="); notQ; |
---|
1756 | } |
---|
1757 | i++; |
---|
1758 | } |
---|
1759 | for (i=1;i<=size(Q);i++) |
---|
1760 | { |
---|
1761 | if(memberpos(i,notQ)[1]==0) |
---|
1762 | { |
---|
1763 | QQ[size(QQ)+1]=Q[i]; |
---|
1764 | } |
---|
1765 | } |
---|
1766 | if (size(QQ)==0){QQ[1]=ideal(1);} |
---|
1767 | return(addpartfine(QQ,C)); |
---|
1768 | } |
---|
1769 | |
---|
1770 | // Auxiliary routine called by addpart to finish the modification of the holes of a primepart |
---|
1771 | // of the union by the addition of the segments that do not correspond to |
---|
1772 | // that part. |
---|
1773 | // Works on @P ring. |
---|
1774 | proc addpartfine(list H, list C0) |
---|
1775 | { |
---|
1776 | int i; int j; int k; int te; intvec notQ; int l; list sel; int used; |
---|
1777 | intvec jtesC; |
---|
1778 | if ((size(H)==1) and (equalideals(H[1],ideal(1)))){return(H);} |
---|
1779 | if (size(C0)==0){return(H);} |
---|
1780 | def RR=basering; |
---|
1781 | setring(@P); |
---|
1782 | list newQ; list nQ; list Q; list nQ1; list Q0; |
---|
1783 | def Q1=imap(RR,H); |
---|
1784 | //Q1=sortlistideals(Q1); |
---|
1785 | def C=imap(RR,C0); |
---|
1786 | while(equallistideals(Q0,Q1)==0) |
---|
1787 | { |
---|
1788 | Q0=Q1; |
---|
1789 | i=0; |
---|
1790 | Q=Q1; |
---|
1791 | kill notQ; intvec notQ; |
---|
1792 | while(i<size(Q)) |
---|
1793 | { |
---|
1794 | i++; |
---|
1795 | for(j=1;j<=size(C);j++) |
---|
1796 | { |
---|
1797 | te=idcontains(Q[i],C[j][1]); |
---|
1798 | if(te) |
---|
1799 | { |
---|
1800 | for(k=1;k<=size(C[j][2]);k++) |
---|
1801 | { |
---|
1802 | if(idcontains(Q[i],C[j][2][k])) |
---|
1803 | { |
---|
1804 | te=0; break; |
---|
1805 | } |
---|
1806 | } |
---|
1807 | if (te) |
---|
1808 | { |
---|
1809 | used++; |
---|
1810 | if ((size(notQ)==1) and (notQ[1]==0)){notQ[1]=i;} |
---|
1811 | else{notQ[size(notQ)+1]=i;} |
---|
1812 | kill newQ; list newQ; |
---|
1813 | for(k=1;k<=size(C[j][2]);k++) |
---|
1814 | { |
---|
1815 | nQ=minGTZ(Q[i]+C[j][2][k]); |
---|
1816 | for(l=1;l<=size(nQ);l++) |
---|
1817 | { |
---|
1818 | option(redSB); |
---|
1819 | nQ[l]=std(nQ[l]); |
---|
1820 | newQ[size(newQ)+1]=nQ[l]; |
---|
1821 | } |
---|
1822 | } |
---|
1823 | sel=selectminideals(newQ); |
---|
1824 | kill nQ1; list nQ1; |
---|
1825 | for(l=1;l<=size(sel);l++) |
---|
1826 | { |
---|
1827 | nQ1[l]=newQ[sel[l]]; |
---|
1828 | } |
---|
1829 | newQ=nQ1; |
---|
1830 | for(l=1;l<=size(newQ);l++) |
---|
1831 | { |
---|
1832 | Q[size(Q)+1]=newQ[l]; |
---|
1833 | } |
---|
1834 | break; |
---|
1835 | } |
---|
1836 | } |
---|
1837 | } |
---|
1838 | } |
---|
1839 | kill Q1; list Q1; |
---|
1840 | for(i=1;i<=size(Q);i++) |
---|
1841 | { |
---|
1842 | if(memberpos(i,notQ)[1]==0) |
---|
1843 | { |
---|
1844 | Q1[size(Q1)+1]=Q[i]; |
---|
1845 | } |
---|
1846 | } |
---|
1847 | sel=selectminideals(Q1); |
---|
1848 | kill nQ1; list nQ1; |
---|
1849 | for(l=1;l<=size(sel);l++) |
---|
1850 | { |
---|
1851 | nQ1[l]=Q1[sel[l]]; |
---|
1852 | } |
---|
1853 | Q1=nQ1; |
---|
1854 | } |
---|
1855 | setring(RR); |
---|
1856 | //if(used>0){string("addpartfine was ", used, " times used");} |
---|
1857 | return(imap(@P,Q1)); |
---|
1858 | } |
---|
1859 | |
---|
1860 | // Auxiliary routine for grobcov: ideal F is assumed to be homogeneous |
---|
1861 | // gcover |
---|
1862 | // input: ideal F: a generating set of a homogeneous ideal in Q[a][x] |
---|
1863 | // list #: optional |
---|
1864 | // output: the list |
---|
1865 | // S=((lpp, generic basis, Prep, Crep),..,(lpp, generic basis, Prep, Crep)) |
---|
1866 | // where a Prep is ( (p1,(p11,..,p1k_1)),..,(pj,(pj1,..,p1k_j)) ) |
---|
1867 | // a Crep is ( ida, idb ) |
---|
1868 | proc gcover(ideal F,list #) |
---|
1869 | { |
---|
1870 | int i; int j; int k; ideal lpp; list GPi2; list pairspP; ideal B; int ti; |
---|
1871 | int i1; int tes; int j1; int selind; int i2; int m; |
---|
1872 | list prep; list crep; list LCU; poly p; poly lcp; ideal FF; |
---|
1873 | list lpi; |
---|
1874 | string lpph; |
---|
1875 | list L=#; |
---|
1876 | int canop=1; |
---|
1877 | int extop=1; |
---|
1878 | int repop=0; |
---|
1879 | ideal E=ideal(0);; |
---|
1880 | ideal N=ideal(1);; |
---|
1881 | int comment; |
---|
1882 | for(i=1;i<=size(L) div 2;i++) |
---|
1883 | { |
---|
1884 | if(L[2*i-1]=="can"){canop=L[2*i];} |
---|
1885 | else |
---|
1886 | { |
---|
1887 | if(L[2*i-1]=="ext"){extop=L[2*i];} |
---|
1888 | else |
---|
1889 | { |
---|
1890 | if(L[2*i-1]=="rep"){repop=L[2*i];} |
---|
1891 | else |
---|
1892 | { |
---|
1893 | if(L[2*i-1]=="null"){E=L[2*i];} |
---|
1894 | else |
---|
1895 | { |
---|
1896 | if(L[2*i-1]=="nonnull"){N=L[2*i];} |
---|
1897 | else |
---|
1898 | { |
---|
1899 | if (L[2*i-1]=="comment"){comment=L[2*i];} |
---|
1900 | } |
---|
1901 | } |
---|
1902 | } |
---|
1903 | } |
---|
1904 | } |
---|
1905 | } |
---|
1906 | list GS; list GP; |
---|
1907 | def RR=basering; |
---|
1908 | GS=cgsdr(F,L); // "null",NW[1],"nonnull",NW[2],"cgs",CGS,"comment",comment); |
---|
1909 | setglobalrings(); |
---|
1910 | int start=timer; |
---|
1911 | GP=GS; |
---|
1912 | ideal lppr; |
---|
1913 | list LL; |
---|
1914 | list S; |
---|
1915 | poly sp; |
---|
1916 | ideal BB; |
---|
1917 | for (i=1;i<=size(GP);i++) |
---|
1918 | { |
---|
1919 | kill LL; |
---|
1920 | list LL; |
---|
1921 | lpp=GP[i][1]; |
---|
1922 | GPi2=GP[i][2]; |
---|
1923 | lpph=GP[i][3]; |
---|
1924 | kill pairspP; list pairspP; |
---|
1925 | for(j=1;j<=size(GPi2);j++) |
---|
1926 | { |
---|
1927 | pairspP[size(pairspP)+1]=GPi2[j][3]; |
---|
1928 | } |
---|
1929 | LCU=LCUnion(pairspP); |
---|
1930 | kill prep; list prep; |
---|
1931 | kill crep; list crep; |
---|
1932 | for(k=1;k<=size(LCU);k++) |
---|
1933 | { |
---|
1934 | prep[k]=list(LCU[k][2],LCU[k][3]); |
---|
1935 | B=GPi2[LCU[k][1][1]][2]; // ATENTION last 1 has been changed to [2] |
---|
1936 | LCU[k][1]=B; |
---|
1937 | } |
---|
1938 | //"Deciding if combine is needed"; |
---|
1939 | kill BB; |
---|
1940 | ideal BB; |
---|
1941 | tes=1; m=1; |
---|
1942 | while((tes) and (m<=size(LCU[1][1]))) |
---|
1943 | { |
---|
1944 | j=1; |
---|
1945 | while((tes) and (j<=size(LCU))) |
---|
1946 | { |
---|
1947 | k=1; |
---|
1948 | while((tes) and (k<=size(LCU))) |
---|
1949 | { |
---|
1950 | if(j!=k) |
---|
1951 | { |
---|
1952 | sp=pnormalf(pspol(LCU[j][1][m],LCU[k][1][m]),LCU[k][2],N); |
---|
1953 | if(sp!=0){tes=0;} |
---|
1954 | } |
---|
1955 | k++; |
---|
1956 | } //setglobalrings(); |
---|
1957 | if(tes) |
---|
1958 | { |
---|
1959 | BB[m]=LCU[j][1][m]; |
---|
1960 | } |
---|
1961 | j++; |
---|
1962 | } |
---|
1963 | if(tes==0){break;} |
---|
1964 | m++; |
---|
1965 | } |
---|
1966 | crep=PtoCrep(prep); |
---|
1967 | if(tes==0) |
---|
1968 | { |
---|
1969 | // combine is needed |
---|
1970 | kill B; ideal B; |
---|
1971 | for (j=1;j<=size(LCU);j++) |
---|
1972 | { |
---|
1973 | LL[j]=LCU[j][2]; |
---|
1974 | } |
---|
1975 | if (size(LCU)>1) |
---|
1976 | { |
---|
1977 | FF=precombint(LL); |
---|
1978 | } |
---|
1979 | for (k=1;k<=size(lpp);k++) |
---|
1980 | { |
---|
1981 | kill L; list L; |
---|
1982 | for (j=1;j<=size(LCU);j++) |
---|
1983 | { |
---|
1984 | L[j]=list(LCU[j][2],LCU[j][1][k]); |
---|
1985 | } |
---|
1986 | if (size(LCU)>1) |
---|
1987 | { |
---|
1988 | B[k]=combine(L,FF); |
---|
1989 | } |
---|
1990 | else{B[k]=L[1][2];} |
---|
1991 | } |
---|
1992 | } |
---|
1993 | else{B=BB;} |
---|
1994 | for(j=1;j<=size(B);j++) |
---|
1995 | { |
---|
1996 | B[j]=pnormalf(B[j],crep[1],crep[2]); |
---|
1997 | } |
---|
1998 | S[i]=list(lpp,B,prep,crep,lpph); |
---|
1999 | if(comment>=1) |
---|
2000 | { |
---|
2001 | lpi[size(lpi)+1]=string("[",i,"]"); |
---|
2002 | lpi[size(lpi)+1]=S[i][1]; |
---|
2003 | } |
---|
2004 | } |
---|
2005 | if(comment>=1) |
---|
2006 | { |
---|
2007 | string("Time in LCUnion + combine = ",timer-start); |
---|
2008 | if(comment>=2){string("lpp=",lpi)}; |
---|
2009 | } |
---|
2010 | if(defined(@P)==1){kill @P; kill @RP; kill @R;} |
---|
2011 | return(S); |
---|
2012 | } |
---|
2013 | |
---|
2014 | // grobcov |
---|
2015 | // input: |
---|
2016 | // ideal F: a parametric ideal in Q[a][x], where a are the parameters |
---|
2017 | // and x the variables |
---|
2018 | // list #: (options) list("null",N,"nonnull",W,"can",0-1,ext",0-1, "rep",0-1-2) |
---|
2019 | // where |
---|
2020 | // N is the null conditions ideal (if desired) |
---|
2021 | // W is the ideal of non-null conditions (if desired) |
---|
2022 | // The value of \"can\"i s 1 by default and can be set to 0 if we do not |
---|
2023 | // need to obtain the canonical GC, but only a GC. |
---|
2024 | // The value of \"ext\" is 0 by default and so the generic representation |
---|
2025 | // of the bases is given. It can be set to 1, and then the full |
---|
2026 | // representation of the bases is given. |
---|
2027 | // The value of \"rep\" is 0 by default, and then the segments |
---|
2028 | // are given in canonical P-representation. It can be set to 1 |
---|
2029 | // and then they are given in canonical C-representation. |
---|
2030 | // If it is set to 2, then both representations are given. |
---|
2031 | // output: |
---|
2032 | // list S: ((lpp,basis,(idp_1,(idp_11,..,idp_1s_1))), .. |
---|
2033 | // (lpp,basis,(idp_r,(idp_r1,..,idp_rs_r))) ) where |
---|
2034 | // each element of S corresponds to a lpp-segment |
---|
2035 | // given by the lpp, the basis, and the P-representation of the segment |
---|
2036 | proc grobcov(ideal F,list #) |
---|
2037 | "USAGE: grobcov(F); This is the fundamental routine of the |
---|
2038 | library. It computes the Groebner cover of a parametric ideal |
---|
2039 | (see (*) Montes A., Wibmer M., Groebner Bases for Polynomial |
---|
2040 | Systems with parameters. JSC 45 (2010) 1391-1425.) |
---|
2041 | The Groebner cover of a parametric ideal consist of a set of |
---|
2042 | pairs(S_i,B_i), where the S_i are disjoint locally closed |
---|
2043 | segments of the parameter space, and the B_i are the reduced |
---|
2044 | Groebner bases of the ideal on every point of S_i. |
---|
2045 | |
---|
2046 | The ideal F must be defined on a parametric ring Q[a][x]. |
---|
2047 | Options: To modify the default options, pair of arguments |
---|
2048 | -option name, value- of valid options must be added to the call. |
---|
2049 | |
---|
2050 | Options: |
---|
2051 | \"null\",ideal E: The default is (\"null\",ideal(0)). |
---|
2052 | \"nonnull\",ideal N: The default (\"nonnull\",ideal(1)). |
---|
2053 | When options \"null\" and/or \"nonnull\" are given, then |
---|
2054 | the parameter space is restricted to V(E)\V(N). |
---|
2055 | \"can\",0-1: The default is (\"can\",1). With the default option |
---|
2056 | the homogenized ideal is computed before obtaining the |
---|
2057 | Groebner cover, so that the result is the canonical |
---|
2058 | Groebner cover. Setting (\"can\",0) only homogenizes the |
---|
2059 | basis so the result is not exactly canonical, but the |
---|
2060 | computation is shorter. |
---|
2061 | \"ext\",0-1: The default is (\"ext\",0). With the default |
---|
2062 | (\"ext\",0), only the generic representation is computed |
---|
2063 | (single polynomials, but not specializing to non-zero at |
---|
2064 | each point of the segment. With option (\"ext\",1) the |
---|
2065 | full representation of the bases is computed (possible |
---|
2066 | sheaves) and sometimes a simpler result is obtained. |
---|
2067 | \"rep\",0-1-2: The default is (\"rep\",0) and then the segments |
---|
2068 | are given in canonical P-representation. Option (\"rep\",1) |
---|
2069 | represents the segments in canonical C-representation, |
---|
2070 | and option (\"rep\",2) gives both representations. |
---|
2071 | \"comment\",0-3: The default is (\"comment\",0). Setting |
---|
2072 | \"comment\" higher will provide information about the |
---|
2073 | development of the computation. |
---|
2074 | One can give none or whatever of these options. |
---|
2075 | RETURN: The list |
---|
2076 | ( |
---|
2077 | (lpp_1,basis_1,segment_1,lpph_1), |
---|
2078 | ... |
---|
2079 | (lpp_s,basis_s,segment_s,lpph_s) |
---|
2080 | ) |
---|
2081 | |
---|
2082 | The lpp are constant over a segment and correspond to the |
---|
2083 | set of lpp of the reduced Groebner basis for each point |
---|
2084 | of the segment. |
---|
2085 | The lpph corresponds to the lpp of the homogenized ideal |
---|
2086 | and is different for each segment. It is given as a string. |
---|
2087 | |
---|
2088 | Basis: to each element of lpp corresponds an I-regular function given |
---|
2089 | in full representation (by option (\"ext\",1)) or in |
---|
2090 | generic representation (default option (\"ext\",0)). The |
---|
2091 | I-regular function is the corresponding element of the reduced |
---|
2092 | Groebner basis for each point of the segment with the given lpp. |
---|
2093 | For each point in the segment, the polynomial or the set of |
---|
2094 | polynomials representing it, if they do not specialize to 0, |
---|
2095 | then after normalization, specializes to the corresponding |
---|
2096 | element of the reduced Groebner basis. In the full representation |
---|
2097 | at least one of the polynomials representing the I-regular |
---|
2098 | function specializes to non-zero. |
---|
2099 | |
---|
2100 | With the default option (\"rep\",0) the representation of the |
---|
2101 | segment is the P-representation. |
---|
2102 | With option (\"rep\",1) the representation of the segment is |
---|
2103 | the C-representation. |
---|
2104 | With option (\"rep\",2) both representations of the segment are |
---|
2105 | given. |
---|
2106 | |
---|
2107 | The P-representation of a segment is of the form |
---|
2108 | ((p_1,(p_11,..,p_1k1)),..,(p_r,(p_r1,..,p_rkr)) |
---|
2109 | representing the segment U_i (V(p_i) \ U_j (V(p_ij))), |
---|
2110 | where the p's are prime ideals. |
---|
2111 | |
---|
2112 | The C-representation of a segment is of the form |
---|
2113 | (E,N) representing V(E)\V(N), and the ideals E and N are |
---|
2114 | radical and N contains E. |
---|
2115 | |
---|
2116 | NOTE: The basering R, must be of the form Q[a][x], a=parameters, |
---|
2117 | x=variables, and should be defined previously. The ideal must |
---|
2118 | be defined on R. |
---|
2119 | KEYWORDS: Groebner cover, parametric ideal, canonical, discussion of |
---|
2120 | parametric ideal. |
---|
2121 | EXAMPLE: grobcov; shows an example" |
---|
2122 | { |
---|
2123 | list S; int i; int ish=1; list GBR; list BR; int j; int k; |
---|
2124 | ideal idp; ideal idq; int s; ideal ext; list SS; |
---|
2125 | ideal E; ideal N; int canop; int extop; int repop; |
---|
2126 | int comment=0; int m; |
---|
2127 | def RR=basering; |
---|
2128 | setglobalrings(); |
---|
2129 | list L0=#; |
---|
2130 | int out=0; |
---|
2131 | L0[size(L0)+1]="res"; L0[size(L0)+1]=ideal(1); |
---|
2132 | // default options |
---|
2133 | int start=timer; |
---|
2134 | E=ideal(0); |
---|
2135 | N=ideal(1); |
---|
2136 | canop=1; // canop=0 for homogenizing the basis but not the ideal (not canonical) |
---|
2137 | // canop=1 for working with the homogenized ideal |
---|
2138 | repop=0; // repop=0 for representing the segments in Prep |
---|
2139 | // repop=1 for representing the segments in Crep |
---|
2140 | // repop=2 for representing the segments in Prep and Crep |
---|
2141 | extop=0; // extop=0 if only generic representation of the bases are to be computed |
---|
2142 | // extop=1 if the full representation of the bases are to be computed |
---|
2143 | for(i=1;i<=size(L0) div 2;i++) |
---|
2144 | { |
---|
2145 | if(L0[2*i-1]=="can"){canop=L0[2*i];} |
---|
2146 | else |
---|
2147 | { |
---|
2148 | if(L0[2*i-1]=="ext"){extop=L0[2*i];} |
---|
2149 | else |
---|
2150 | { |
---|
2151 | if(L0[2*i-1]=="rep"){repop=L0[2*i];} |
---|
2152 | else |
---|
2153 | { |
---|
2154 | if(L0[2*i-1]=="null"){E=L0[2*i];} |
---|
2155 | else |
---|
2156 | { |
---|
2157 | if(L0[2*i-1]=="nonnull"){N=L0[2*i];} |
---|
2158 | else |
---|
2159 | { |
---|
2160 | if (L0[2*i-1]=="comment"){comment=L0[2*i];} |
---|
2161 | } |
---|
2162 | } |
---|
2163 | } |
---|
2164 | } |
---|
2165 | } |
---|
2166 | } |
---|
2167 | if(not((canop==0) or (canop==1))) |
---|
2168 | { |
---|
2169 | string("Option can = ",canop," is not supported. It is changed to can = 1"); |
---|
2170 | canop=1; |
---|
2171 | } |
---|
2172 | for(i=1;i<=size(L0) div 2;i++) |
---|
2173 | { |
---|
2174 | if(L0[2*i-1]=="can"){L0[2*i]=canop;} |
---|
2175 | } |
---|
2176 | if ((printlevel) and (comment==0)){comment=printlevel;} |
---|
2177 | list LL; |
---|
2178 | LL[1]="can"; LL[2]=canop; |
---|
2179 | LL[3]="comment"; LL[4]=comment; |
---|
2180 | LL[5]="out"; LL[6]=0; |
---|
2181 | LL[7]="null"; LL[8]=E; |
---|
2182 | LL[9]="nonnull"; LL[10]=N; |
---|
2183 | LL[11]="ext"; LL[12]=extop; |
---|
2184 | LL[13]="rep"; LL[14]=repop; |
---|
2185 | if (comment>=1) |
---|
2186 | { |
---|
2187 | string("Begin grobcov with options: ",LL); |
---|
2188 | } |
---|
2189 | kill S; |
---|
2190 | def S=gcover(F,LL); |
---|
2191 | // NOW extend |
---|
2192 | if(extop) |
---|
2193 | { |
---|
2194 | S=extend(S,LL); |
---|
2195 | } |
---|
2196 | else |
---|
2197 | { |
---|
2198 | // NOW representation of the segments by option repop |
---|
2199 | list Si; list nS; |
---|
2200 | if(repop==0) |
---|
2201 | { |
---|
2202 | for(i=1;i<=size(S);i++) |
---|
2203 | { |
---|
2204 | Si=list(S[i][1],S[i][2],S[i][3],S[i][5]); |
---|
2205 | nS[size(nS)+1]=Si; |
---|
2206 | } |
---|
2207 | kill S; |
---|
2208 | def S=nS; |
---|
2209 | } |
---|
2210 | else |
---|
2211 | { |
---|
2212 | if(repop==1) |
---|
2213 | { |
---|
2214 | for(i=1;i<=size(S);i++) |
---|
2215 | { |
---|
2216 | Si=list(S[i][1],S[i][2],S[i][4],S[i][5]); |
---|
2217 | nS[size(nS)+1]=Si; |
---|
2218 | } |
---|
2219 | kill S; |
---|
2220 | def S=nS; |
---|
2221 | } |
---|
2222 | else |
---|
2223 | { |
---|
2224 | for(i=1;i<=size(S);i++) |
---|
2225 | { |
---|
2226 | Si=list(S[i][1],S[i][2],S[i][3],S[i][4],S[i][5]); |
---|
2227 | nS[size(nS)+1]=Si; |
---|
2228 | } |
---|
2229 | kill S; |
---|
2230 | def S=nS; |
---|
2231 | } |
---|
2232 | } |
---|
2233 | } |
---|
2234 | if (comment>=1) |
---|
2235 | { |
---|
2236 | string("Time in grobcov = ", timer-start); |
---|
2237 | string("Number of segments of grobcov = ", size(S)); |
---|
2238 | } |
---|
2239 | if(defined(@P)==1){kill @R; kill @P; kill @RP;} |
---|
2240 | return(S); |
---|
2241 | } |
---|
2242 | example |
---|
2243 | { "EXAMPLE:"; echo = 2; |
---|
2244 | "Casas conjecture for degree 4"; |
---|
2245 | ring R=(0,a0,a1,a2,a3,a4),(x1,x2,x3),dp; |
---|
2246 | ideal F=x1^4+(4*a3)*x1^3+(6*a2)*x1^2+(4*a1)*x1+(a0), |
---|
2247 | x1^3+(3*a3)*x1^2+(3*a2)*x1+(a1), |
---|
2248 | x2^4+(4*a3)*x2^3+(6*a2)*x2^2+(4*a1)*x2+(a0), |
---|
2249 | x2^2+(2*a3)*x2+(a2), |
---|
2250 | x3^4+(4*a3)*x3^3+(6*a2)*x3^2+(4*a1)*x3+(a0), |
---|
2251 | x3+(a3); |
---|
2252 | grobcov(F); |
---|
2253 | } |
---|
2254 | |
---|
2255 | // Input. GC the grobcov of an ideal in generic representation of the |
---|
2256 | // bases computed with option option ("rep",2). |
---|
2257 | // Output The grobcov in full representation. |
---|
2258 | // Option ("comment",1) shows the time. |
---|
2259 | // Can be called from the top |
---|
2260 | proc extend(list GC, list #); |
---|
2261 | "USAGE: extend(GC); When the grobcov of an ideal has been computed |
---|
2262 | with the default option (\"ext\",0) and the explicit option |
---|
2263 | (\"rep\",2) (which is not the default), then one can call |
---|
2264 | extend (GC) (and options) to obtain the full representation |
---|
2265 | of the bases. With the default option (\"ext\",0) only the |
---|
2266 | generic representation of the bases are computed, and one can |
---|
2267 | obtain the full representation using extend. |
---|
2268 | \"rep\",0-1-2: The default is (\"rep\",0) and then the segments |
---|
2269 | are given in canonical P-representation. Option (\"rep\",1) |
---|
2270 | represents the segments in canonical C-representation, |
---|
2271 | and option (\"rep\",2) gives both representations. |
---|
2272 | \"comment\",0-1: The default is (\"comment\",0). Setting |
---|
2273 | \"comment\" higher will provide information about the |
---|
2274 | time used in the computation. |
---|
2275 | One can give none or whatever of these options. |
---|
2276 | RETURN: The list |
---|
2277 | ( |
---|
2278 | (lpp_1,basis_1,segment_1,lpph_1), |
---|
2279 | ... |
---|
2280 | (lpp_s,basis_s,segment_s,lpph_s) |
---|
2281 | ) |
---|
2282 | |
---|
2283 | The lpp are constant over a segment and correspond to the |
---|
2284 | set of lpp of the reduced Groebner basis for each point |
---|
2285 | of the segment. |
---|
2286 | The lpph corresponds to the lpp of the homogenized ideal |
---|
2287 | and is different for each segment. It is given as a string. |
---|
2288 | |
---|
2289 | Basis: to each element of lpp corresponds an I-regular function given |
---|
2290 | in full representation. The |
---|
2291 | I-regular function is the corresponding element of the reduced |
---|
2292 | Groebner basis for each point of the segment with the given lpp. |
---|
2293 | For each point in the segment, the polynomial or the set of |
---|
2294 | polynomials representing it, if they do not specialize to 0, |
---|
2295 | then after normalization, specializes to the corresponding |
---|
2296 | element of the reduced Groebner basis. In the full representation |
---|
2297 | at least one of the polynomials representing the I-regular |
---|
2298 | function specializes to non-zero. |
---|
2299 | |
---|
2300 | With the default option (\"rep\",0) the segments are given |
---|
2301 | in P-representation. |
---|
2302 | With option (\"rep\",1) the segments are given |
---|
2303 | in C-representation. |
---|
2304 | With option (\"rep\",2) both representations of the segments are |
---|
2305 | given. |
---|
2306 | |
---|
2307 | The P-representation of a segment is of the form |
---|
2308 | ((p_1,(p_11,..,p_1k1)),..,(p_r,(p_r1,..,p_rkr)) |
---|
2309 | representing the segment U_i (V(p_i) \ U_j (V(p_ij))), |
---|
2310 | where the p's are prime ideals. |
---|
2311 | |
---|
2312 | The C-representation of a segment is of the form |
---|
2313 | (E,N) representing V(E)\V(N), and the ideals E and N are |
---|
2314 | radical and N contains E. |
---|
2315 | |
---|
2316 | NOTE: The basering R, must be of the form Q[a][x], a=parameters, |
---|
2317 | x=variables, and should be defined previously. The ideal must |
---|
2318 | be defined on R. |
---|
2319 | KEYWORDS: Groebner cover, parametric ideal, canonical, discussion of |
---|
2320 | parametric ideal, full representation. |
---|
2321 | EXAMPLE: extend; shows an example" |
---|
2322 | { |
---|
2323 | list L=#; |
---|
2324 | list S=GC; |
---|
2325 | ideal idp; |
---|
2326 | ideal idq; |
---|
2327 | int i; int j; int m; int s; |
---|
2328 | m=0; i=1; |
---|
2329 | while((i<=size(S)) and (m==0)) |
---|
2330 | { |
---|
2331 | if(typeof(S[i][2])=="list"){m=1;} |
---|
2332 | i++; |
---|
2333 | } |
---|
2334 | if(m==1){"Warning! grobcov has already extended bases"; return(S);} |
---|
2335 | if(size(GC[1])!=5){"Warning! extend make sense only when grobcov has been called with options 'rep',2,'ext',0"; " "; return();} |
---|
2336 | int repop=0; |
---|
2337 | int start3=timer; |
---|
2338 | int comment; |
---|
2339 | for(i=1;i<=size(L) div 2;i++) |
---|
2340 | { |
---|
2341 | if(L[2*i-1]=="comment"){comment=L[2*i];} |
---|
2342 | else |
---|
2343 | { |
---|
2344 | if(L[2*i-1]=="rep"){repop=L[2*i];} |
---|
2345 | } |
---|
2346 | } |
---|
2347 | poly leadc; |
---|
2348 | poly ext; |
---|
2349 | int te=0; |
---|
2350 | list SS; |
---|
2351 | def R=basering; |
---|
2352 | if (defined(@R)){te=1;} |
---|
2353 | else{setglobalrings();} |
---|
2354 | // Now extend |
---|
2355 | for (i=1;i<=size(S);i++) |
---|
2356 | { |
---|
2357 | m=size(S[i][2]); |
---|
2358 | for (j=1;j<=m;j++) |
---|
2359 | { |
---|
2360 | idp=S[i][4][1]; |
---|
2361 | idq=S[i][4][2]; |
---|
2362 | if (size(idp)>0) |
---|
2363 | { |
---|
2364 | leadc=leadcoef(S[i][2][j]); |
---|
2365 | kill ext; |
---|
2366 | def ext=extend0(S[i][2][j],idp,idq); |
---|
2367 | if (typeof(ext)=="poly") |
---|
2368 | { |
---|
2369 | S[i][2][j]=pnormalf(ext,idp,idq); |
---|
2370 | } |
---|
2371 | else |
---|
2372 | { |
---|
2373 | if(size(ext)==1) |
---|
2374 | { |
---|
2375 | S[i][2][j]=ext[1]; |
---|
2376 | } |
---|
2377 | else |
---|
2378 | { |
---|
2379 | kill SS; list SS; |
---|
2380 | for(s=1;s<=size(ext);s++) |
---|
2381 | { |
---|
2382 | ext[s]=pnormalf(ext[s],idp,idq); |
---|
2383 | } |
---|
2384 | for(s=1;s<=size(S[i][2]);s++) |
---|
2385 | { |
---|
2386 | if(s!=j){SS[s]=S[i][2][s];} |
---|
2387 | else{SS[s]=ext;} |
---|
2388 | } |
---|
2389 | S[i][2]=SS; |
---|
2390 | } |
---|
2391 | } |
---|
2392 | } |
---|
2393 | } |
---|
2394 | } |
---|
2395 | // NOW representation of the segments by option repop |
---|
2396 | list Si; list nS; |
---|
2397 | if (repop==0) |
---|
2398 | { |
---|
2399 | for(i=1;i<=size(S);i++) |
---|
2400 | { |
---|
2401 | Si=list(S[i][1],S[i][2],S[i][3],S[i][5]); |
---|
2402 | nS[size(nS)+1]=Si; |
---|
2403 | } |
---|
2404 | S=nS; |
---|
2405 | } |
---|
2406 | else |
---|
2407 | { |
---|
2408 | if (repop==1) |
---|
2409 | { |
---|
2410 | for(i=1;i<=size(S);i++) |
---|
2411 | { |
---|
2412 | Si=list(S[i][1],S[i][2],S[i][4],S[i][5]); |
---|
2413 | nS[size(nS)+1]=Si; |
---|
2414 | } |
---|
2415 | S=nS; |
---|
2416 | } |
---|
2417 | else |
---|
2418 | { |
---|
2419 | for(i=1;i<=size(S);i++) |
---|
2420 | { |
---|
2421 | Si=list(S[i][1],S[i][2],S[i][3],S[i][4],S[i][5]); |
---|
2422 | nS[size(nS)+1]=Si; |
---|
2423 | } |
---|
2424 | |
---|
2425 | } |
---|
2426 | } |
---|
2427 | if(comment>=1){string("Time in extend = ",timer-start3);} |
---|
2428 | if(te==0){kill @R; kill @RP; kill @P;} |
---|
2429 | return(S); |
---|
2430 | } |
---|
2431 | example |
---|
2432 | { |
---|
2433 | ring R=(0,a0,b0,c0,a1,b1,c1,a2,b2,c2),(x), dp; |
---|
2434 | short=0; |
---|
2435 | ideal S=a0*x^2+b0*x+c0, |
---|
2436 | a1*x^2+b1*x+c1, |
---|
2437 | a2*x^2+b2*x+c2; |
---|
2438 | "System S="; S; |
---|
2439 | |
---|
2440 | def GCS=grobcov(S,"rep",2,"comment",1); |
---|
2441 | "grobcov(S,'rep',2,'comment',1)="; GCS; |
---|
2442 | def FGC=extend(GCS,"rep",0,"comment",1); |
---|
2443 | "Full representation="; FGC; |
---|
2444 | } |
---|
2445 | |
---|
2446 | // Auxiliary routine |
---|
2447 | // nonzerodivisor |
---|
2448 | // input: |
---|
2449 | // poly g in K[a], |
---|
2450 | // list P=(p_1,..p_r) representing a minimal prime decomposition |
---|
2451 | // output |
---|
2452 | // poly f such that f notin p_i for all i and |
---|
2453 | // g-f in p_i for all i such that g notin p_i |
---|
2454 | proc nonzerodivisor(poly gr, list Pr) |
---|
2455 | { |
---|
2456 | def RR=basering; |
---|
2457 | setring(@P); |
---|
2458 | def g=imap(RR,gr); |
---|
2459 | def P=imap(RR,Pr); |
---|
2460 | int i; int k; list J; ideal F; |
---|
2461 | def f=g; |
---|
2462 | ideal Pi; |
---|
2463 | for (i=1;i<=size(P);i++) |
---|
2464 | { |
---|
2465 | option(redSB); |
---|
2466 | Pi=std(P[i]); |
---|
2467 | //attrib(Pi,"isSB",1); |
---|
2468 | if (reduce(g,Pi,1)==0){J[size(J)+1]=i;} |
---|
2469 | } |
---|
2470 | for (i=1;i<=size(J);i++) |
---|
2471 | { |
---|
2472 | F=ideal(1); |
---|
2473 | for (k=1;k<=size(P);k++) |
---|
2474 | { |
---|
2475 | if (k!=J[i]) |
---|
2476 | { |
---|
2477 | F=idint(F,P[k]); |
---|
2478 | } |
---|
2479 | } |
---|
2480 | f=f+F[1]; |
---|
2481 | } |
---|
2482 | setring(RR); |
---|
2483 | def fr=imap(@P,f); |
---|
2484 | return(fr); |
---|
2485 | } |
---|
2486 | |
---|
2487 | // Auxiliary routine |
---|
2488 | // deltai |
---|
2489 | // input: |
---|
2490 | // int i: |
---|
2491 | // list LPr: (p1,..,pr) of prime components of an ideal in K[a] |
---|
2492 | // output: |
---|
2493 | // list (fr,fnr) of two polynomials that are equal on V(pi) |
---|
2494 | // and fr=0 on V(P) \ V(pi), and fnr is nonzero on V(pj) for all j. |
---|
2495 | proc deltai(int i, list LPr) |
---|
2496 | { |
---|
2497 | def RR=basering; |
---|
2498 | setring(@P); |
---|
2499 | def LP=imap(RR,LPr); |
---|
2500 | int j; poly p; |
---|
2501 | ideal F=ideal(1); |
---|
2502 | poly f; |
---|
2503 | poly fn; |
---|
2504 | ideal LPi; |
---|
2505 | for (j=1;j<=size(LP);j++) |
---|
2506 | { |
---|
2507 | if (j!=i) |
---|
2508 | { |
---|
2509 | F=idint(F,LP[j]); |
---|
2510 | } |
---|
2511 | } |
---|
2512 | p=0; j=1; |
---|
2513 | while ((p==0) and (j<=size(F))) |
---|
2514 | { |
---|
2515 | LPi=LP[i]; |
---|
2516 | attrib(LPi,"isSB",1); |
---|
2517 | p=reduce(F[j],LPi); |
---|
2518 | j++; |
---|
2519 | } |
---|
2520 | f=F[j-1]; |
---|
2521 | fn=nonzerodivisor(f,LP); |
---|
2522 | setring(RR); |
---|
2523 | def fr=imap(@P,f); |
---|
2524 | def fnr=imap(@P,fn); |
---|
2525 | return(list(fr,fnr)); |
---|
2526 | } |
---|
2527 | |
---|
2528 | // Auxiliary routine |
---|
2529 | // combine |
---|
2530 | // input: a list of pairs ((p1,P1),..,(pr,Pr)) where |
---|
2531 | // ideal pi is a prime component |
---|
2532 | // poly Pi is the polynomial in Q[a][x] on V(pi)\ V(Mi) |
---|
2533 | // (p1,..,pr) are the prime decomposition of the lpp-segment |
---|
2534 | // list crep =(ideal ida,ideal idb): the Crep of the segment. |
---|
2535 | // list Pci of the intersecctions of all pj except the ith one |
---|
2536 | // output: |
---|
2537 | // poly P on an open and dense set of V(p_1 int ... p_r) |
---|
2538 | proc combine(list L, ideal F) |
---|
2539 | { |
---|
2540 | // ATTENTION REVISE AND USE Pci and F |
---|
2541 | int i; poly f; |
---|
2542 | f=0; |
---|
2543 | for(i=1;i<=size(L);i++) |
---|
2544 | { |
---|
2545 | f=f+F[i]*L[i][2]; |
---|
2546 | } |
---|
2547 | // f=elimconstfac(f); |
---|
2548 | f=primepartZ(f); |
---|
2549 | return(f); |
---|
2550 | } |
---|
2551 | |
---|
2552 | // Auxiliary routine |
---|
2553 | // elimconstfac: eliminate the factors in the polynom f that are in K[a] |
---|
2554 | // input: |
---|
2555 | // poly f: |
---|
2556 | // list L: of components of the segment |
---|
2557 | // output: |
---|
2558 | // poly f2 where the factors of f in K[a] that are non-null on any component |
---|
2559 | // have been dropped from f |
---|
2560 | proc elimconstfac(poly f) |
---|
2561 | { |
---|
2562 | int cond; int i; int j; int t; |
---|
2563 | if (f==0){return(f);} |
---|
2564 | def RR=basering; |
---|
2565 | setring(@R); |
---|
2566 | def ff=imap(RR,f); |
---|
2567 | list l=factorize(ff,0); |
---|
2568 | poly f1=1; |
---|
2569 | for(i=2;i<=size(l[1]);i++) |
---|
2570 | { |
---|
2571 | f1=f1*(l[1][i])^(l[2][i]); |
---|
2572 | } |
---|
2573 | setring(RR); |
---|
2574 | def f2=imap(@R,f1); |
---|
2575 | return(f2); |
---|
2576 | } |
---|
2577 | |
---|
2578 | //Auxiliary routine |
---|
2579 | // nullin |
---|
2580 | // input: |
---|
2581 | // poly f: a polynomial in Q[a] |
---|
2582 | // ideal P: an ideal in Q[a] |
---|
2583 | // called from ring @R |
---|
2584 | // output: |
---|
2585 | // t: with value 1 if f reduces modulo P, 0 if not. |
---|
2586 | proc nullin(poly f,ideal P) |
---|
2587 | { |
---|
2588 | int t; |
---|
2589 | def RR=basering; |
---|
2590 | setring(@P); |
---|
2591 | def f0=imap(RR,f); |
---|
2592 | def P0=imap(RR,P); |
---|
2593 | attrib(P0,"isSB",1); |
---|
2594 | if (reduce(f0,P0,1)==0){t=1;} |
---|
2595 | else{t=0;} |
---|
2596 | setring(RR); |
---|
2597 | return(t); |
---|
2598 | } |
---|
2599 | |
---|
2600 | // Auxiliary routine |
---|
2601 | // monoms |
---|
2602 | // Input: A polynomial f |
---|
2603 | // Output: The list of leading terms |
---|
2604 | proc monoms(poly f) |
---|
2605 | { |
---|
2606 | list L; |
---|
2607 | poly lm; poly lc; poly lp; poly Q; poly mQ; |
---|
2608 | def p=f; |
---|
2609 | int i=1; |
---|
2610 | while (p!=0) |
---|
2611 | { |
---|
2612 | lm=lead(p); |
---|
2613 | p=p-lm; |
---|
2614 | lc=leadcoef(lm); |
---|
2615 | lp=leadmonom(lm); |
---|
2616 | L[size(L)+1]=list(lc,lp); |
---|
2617 | i++; |
---|
2618 | } |
---|
2619 | return(L); |
---|
2620 | } |
---|
2621 | |
---|
2622 | // Auxiliary routine called by extend |
---|
2623 | // extend0 |
---|
2624 | // input: |
---|
2625 | // poly f: a generic polynomial in the basis |
---|
2626 | // ideal idp: such that ideal(S)=idp |
---|
2627 | // ideal idq: such that S=V(idp)\V(idq) |
---|
2628 | //// NW the list of ((N1,W1),..,(Ns,Ws)) of red-rep of the grouped |
---|
2629 | //// segments in the lpp-segment NO MORE USED |
---|
2630 | // output: |
---|
2631 | proc extend0(poly f, ideal idp, ideal idq) |
---|
2632 | { |
---|
2633 | matrix CC; poly Q; list NewMonoms; |
---|
2634 | int i; int j; poly fout; ideal idout; |
---|
2635 | list L=monoms(f); |
---|
2636 | int nummonoms=size(L)-1; |
---|
2637 | Q=L[1][1]; |
---|
2638 | if (nummonoms==0){return(f);} |
---|
2639 | for (i=2;i<=size(L);i++) |
---|
2640 | { |
---|
2641 | CC=matrix(extendcoef(L[i][1],Q,idp,idq)); |
---|
2642 | NewMonoms[i-1]=list(CC,L[i][2]); |
---|
2643 | } |
---|
2644 | if (nummonoms==1) |
---|
2645 | { |
---|
2646 | for(j=1;j<=ncols(NewMonoms[1][1]);j++) |
---|
2647 | { |
---|
2648 | fout=NewMonoms[1][1][2,j]*L[1][2]+NewMonoms[1][1][1,j]*NewMonoms[1][2]; |
---|
2649 | //fout=pnormalf(fout,idp,W); |
---|
2650 | if(ncols(NewMonoms[1][1])>1){idout[j]=fout;} |
---|
2651 | } |
---|
2652 | if(ncols(NewMonoms[1][1])==1){return(fout);} else{return(idout);} |
---|
2653 | } |
---|
2654 | else |
---|
2655 | { |
---|
2656 | list cfi; |
---|
2657 | list coefs; |
---|
2658 | for (i=1;i<=nummonoms;i++) |
---|
2659 | { |
---|
2660 | kill cfi; list cfi; |
---|
2661 | for(j=1;j<=ncols(NewMonoms[i][1]);j++) |
---|
2662 | { |
---|
2663 | cfi[size(cfi)+1]=NewMonoms[i][1][2,j]; |
---|
2664 | } |
---|
2665 | coefs[i]=cfi; |
---|
2666 | } |
---|
2667 | def indexpolys=findindexpolys(coefs); |
---|
2668 | for(i=1;i<=size(indexpolys);i++) |
---|
2669 | { |
---|
2670 | fout=L[1][2]; |
---|
2671 | for(j=1;j<=nummonoms;j++) |
---|
2672 | { |
---|
2673 | fout=fout+(NewMonoms[j][1][1,indexpolys[i][j]])/(NewMonoms[j][1][2,indexpolys[i][j]])*NewMonoms[j][2]; |
---|
2674 | } |
---|
2675 | fout=cleardenom(fout); |
---|
2676 | if(size(indexpolys)>1){idout[i]=fout;} |
---|
2677 | } |
---|
2678 | if (size(indexpolys)==1){return(fout);} else{return(idout);} |
---|
2679 | } |
---|
2680 | } |
---|
2681 | |
---|
2682 | // Auxiliary routine |
---|
2683 | // findindexpolys |
---|
2684 | // input: |
---|
2685 | // list coefs=( (q11,..,q1r_1),..,(qs1,..,qsr_1) ) |
---|
2686 | // of denominators of the monoms |
---|
2687 | // output: |
---|
2688 | // list ind=(v_1,..,v_t) of intvec |
---|
2689 | // each intvec v=(i_1,..,is) corresponds to a polynomial in the sheaf |
---|
2690 | // that will be built from it in extend procedure. |
---|
2691 | proc findindexpolys(list coefs) |
---|
2692 | { |
---|
2693 | int i; int j; intvec numdens; |
---|
2694 | for(i=1;i<=size(coefs);i++) |
---|
2695 | { |
---|
2696 | numdens[i]=size(coefs[i]); |
---|
2697 | } |
---|
2698 | def RR=basering; |
---|
2699 | setring(@P); |
---|
2700 | def coefsp=imap(RR,coefs); |
---|
2701 | ideal cof; list combpolys; intvec v; int te; list mp; |
---|
2702 | for(i=1;i<=size(coefsp);i++) |
---|
2703 | { |
---|
2704 | cof=ideal(0); |
---|
2705 | for(j=1;j<=size(coefsp[i]);j++) |
---|
2706 | { |
---|
2707 | cof[j]=factorize(coefsp[i][j],3); |
---|
2708 | } |
---|
2709 | coefsp[i]=cof; |
---|
2710 | } |
---|
2711 | for(j=1;j<=size(coefsp[1]);j++) |
---|
2712 | { |
---|
2713 | v[1]=j; |
---|
2714 | te=1; |
---|
2715 | for (i=2;i<=size(coefsp);i++) |
---|
2716 | { |
---|
2717 | mp=memberpos(coefsp[1][j],coefsp[i]); |
---|
2718 | if(mp[1]) |
---|
2719 | { |
---|
2720 | v[i]=mp[2]; |
---|
2721 | } |
---|
2722 | else{v[i]=0;} |
---|
2723 | } |
---|
2724 | combpolys[j]=v; |
---|
2725 | } |
---|
2726 | combpolys=reform(combpolys,numdens); |
---|
2727 | setring(RR); |
---|
2728 | return(combpolys); |
---|
2729 | } |
---|
2730 | |
---|
2731 | // Auxiliary routine |
---|
2732 | // extendcoef: given Q,P in K[a] where P/Q specializes on an open and dense subset |
---|
2733 | // of the whole V(p1 int...int pr), it returns a basis of the module |
---|
2734 | // of all syzygies equivalent to P/Q, |
---|
2735 | proc extendcoef(poly P, poly Q, ideal idp, ideal idq) |
---|
2736 | { |
---|
2737 | def RR=basering; |
---|
2738 | setring(@P); |
---|
2739 | def PL=ringlist(@P); |
---|
2740 | PL[3][1][1]="dp"; |
---|
2741 | def P1=ring(PL); |
---|
2742 | setring(P1); |
---|
2743 | ideal idp0=imap(RR,idp); |
---|
2744 | option(redSB); |
---|
2745 | qring q=std(idp0); |
---|
2746 | poly P0=imap(RR,P); |
---|
2747 | poly Q0=imap(RR,Q); |
---|
2748 | ideal PQ=Q0,-P0; |
---|
2749 | module C=syz(PQ); |
---|
2750 | setring(@P); |
---|
2751 | def idp1=imap(RR,idp); |
---|
2752 | def idq1=imap(RR,idq); |
---|
2753 | def C1=matrix(imap(q,C)); |
---|
2754 | def redC=selectregularfun(C1,idp1,idq1); |
---|
2755 | setring(RR); |
---|
2756 | def CC=imap(@P,redC); |
---|
2757 | return(CC); |
---|
2758 | } |
---|
2759 | |
---|
2760 | // Auxiliary routine |
---|
2761 | // selectregularfun |
---|
2762 | // input: |
---|
2763 | // list L of the polynomials matrix CC |
---|
2764 | // (we assume that one of them is non-null on V(N)\V(M)) |
---|
2765 | // ideal N, ideal M: ideals representing the locally closed set V(N)\V(M) |
---|
2766 | // assume to work in @P |
---|
2767 | proc selectregularfun(matrix CC, ideal NN, ideal MM) |
---|
2768 | { |
---|
2769 | int numcombused; |
---|
2770 | def RR=basering; |
---|
2771 | setring(@P); |
---|
2772 | def C=imap(RR,CC); |
---|
2773 | def N=imap(RR,NN); |
---|
2774 | def M=imap(RR,MM); |
---|
2775 | if (ncols(C)==1){return(C);} |
---|
2776 | |
---|
2777 | int i; int j; int k; list c; intvec ci; intvec c0; intvec c1; |
---|
2778 | list T; list T0; list T1; list LL; ideal N1;ideal M1; int te=0; |
---|
2779 | for(i=1;i<=ncols(C);i++) |
---|
2780 | { |
---|
2781 | if((C[1,i]!=0) and (C[2,i]!=0)) |
---|
2782 | { |
---|
2783 | if(c0==intvec(0)){c0[1]=i;} |
---|
2784 | else{c0[size(c0)+1]=i;} |
---|
2785 | } |
---|
2786 | } |
---|
2787 | def C1=submat(C,1..2,c0); |
---|
2788 | for (i=1;i<=ncols(C1);i++) |
---|
2789 | { |
---|
2790 | c=comb(ncols(C1),i); |
---|
2791 | for(j=1;j<=size(c);j++) |
---|
2792 | { |
---|
2793 | ci=c[j]; |
---|
2794 | numcombused++; |
---|
2795 | if(i==1){N1=N+C1[2,j]; M1=M;} |
---|
2796 | if(i>1) |
---|
2797 | { |
---|
2798 | kill c0; intvec c0 ; kill c1; intvec c1; |
---|
2799 | c1=ci[size(ci)]; |
---|
2800 | for(k=1;k<size(ci);k++){c0[k]=ci[k];} |
---|
2801 | T0=searchinlist(c0,LL); |
---|
2802 | T1=searchinlist(c1,LL); |
---|
2803 | N1=T0[1]+T1[1]; |
---|
2804 | M1=intersect(T0[2],T1[2]); |
---|
2805 | } |
---|
2806 | T=list(ci,PtoCrep(Prep(N1,M1))); |
---|
2807 | LL[size(LL)+1]=T; |
---|
2808 | if(equalideals(T[2][1],ideal(1))){te=1; break;} |
---|
2809 | } |
---|
2810 | if(te){break;} |
---|
2811 | } |
---|
2812 | ci=T[1]; |
---|
2813 | def Cs=submat(C1,1..2,ci); |
---|
2814 | setring(RR); |
---|
2815 | return(imap(@P,Cs)); |
---|
2816 | } |
---|
2817 | |
---|
2818 | // Auxiliary routine |
---|
2819 | // searchinlist |
---|
2820 | // input: |
---|
2821 | // intvec c: |
---|
2822 | // list L=( (c1,T1),..(ck,Tk) ) |
---|
2823 | // where the c's are assumed to be intvects |
---|
2824 | // output: |
---|
2825 | // object T with index c |
---|
2826 | proc searchinlist(intvec c,list L) |
---|
2827 | { |
---|
2828 | int i; list T; |
---|
2829 | for(i=1;i<=size(L);i++) |
---|
2830 | { |
---|
2831 | if (L[i][1]==c) |
---|
2832 | { |
---|
2833 | T=L[i][2]; |
---|
2834 | break; |
---|
2835 | } |
---|
2836 | } |
---|
2837 | return(T); |
---|
2838 | } |
---|
2839 | |
---|
2840 | // Auxiliary routine |
---|
2841 | // comb: the list of combinations of elements (1,..n) of order p |
---|
2842 | proc comb(int n, int p) |
---|
2843 | { |
---|
2844 | list L; list L0; |
---|
2845 | intvec c; intvec d; |
---|
2846 | int i; int j; int last; |
---|
2847 | if ((n<0) or (n<p)) |
---|
2848 | { |
---|
2849 | return(L); |
---|
2850 | } |
---|
2851 | if (p==1) |
---|
2852 | { |
---|
2853 | for (i=1;i<=n;i++) |
---|
2854 | { |
---|
2855 | c=i; |
---|
2856 | L[size(L)+1]=c; |
---|
2857 | } |
---|
2858 | return(L); |
---|
2859 | } |
---|
2860 | else |
---|
2861 | { |
---|
2862 | L0=comb(n,p-1); |
---|
2863 | for (i=1;i<=size(L0);i++) |
---|
2864 | { |
---|
2865 | c=L0[i]; d=c; |
---|
2866 | last=c[size(c)]; |
---|
2867 | for (j=last+1;j<=n;j++) |
---|
2868 | { |
---|
2869 | d[size(c)+1]=j; |
---|
2870 | L[size(L)+1]=d; |
---|
2871 | } |
---|
2872 | } |
---|
2873 | return(L); |
---|
2874 | } |
---|
2875 | } |
---|
2876 | |
---|
2877 | // Auxiliary routine |
---|
2878 | // selectminsheaves |
---|
2879 | // Input: L=((v_11,..,v_1k_1),..,(v_s1,..,v_sk_s)) |
---|
2880 | // where: |
---|
2881 | // The s lists correspond to the s coefficients of the polynomial f |
---|
2882 | // (v_i1,..,v_ik_i) correspond to the k_i intvec v_ij of the |
---|
2883 | // spezializations of the jth rekpresentant (Q,P) of the ith coefficient |
---|
2884 | // v_ij is an intvec of size equal to the number of little segments |
---|
2885 | // forming the lpp-segment of 0,1, where 1 represents that it specializes |
---|
2886 | // to non-zedro an the whole little segment and 0 if not. |
---|
2887 | // Output: S=(w_1,..,w_j) |
---|
2888 | // where the w_l=(n_l1,..,n_ls) are intvec of length size(L), where |
---|
2889 | // n_lt fixes which element of (v_t1,..,v_tk_t) is to be |
---|
2890 | // choosen to form the tth (Q,P) for the lth element of the sheaf |
---|
2891 | // representing the I-regular function. |
---|
2892 | // The selection is done to obtian the minimal number of elements |
---|
2893 | // of the sheaf that specializes to non-null everywhere. |
---|
2894 | proc selectminsheaves(list L) |
---|
2895 | { |
---|
2896 | list C=allsheaves(L); |
---|
2897 | return(smsheaves(C[1],C[2])); |
---|
2898 | } |
---|
2899 | |
---|
2900 | // Auxiliary routine |
---|
2901 | // smsheaves |
---|
2902 | // Input: |
---|
2903 | // list C of all the combrep |
---|
2904 | // list L of the intvec that correesponds to each element of C |
---|
2905 | // Output: |
---|
2906 | // list LL of the subsets of C that cover all the subsegments |
---|
2907 | // (the union of the corresponding L(C) has all 1). |
---|
2908 | proc smsheaves(list C, list L) |
---|
2909 | { |
---|
2910 | int i; int i0; intvec W; |
---|
2911 | int nor; int norn; |
---|
2912 | intvec p; |
---|
2913 | int sp=size(L[1]); int j0=1; |
---|
2914 | for (i=1;i<=sp;i++){p[i]=1;} |
---|
2915 | while (p!=0) |
---|
2916 | { |
---|
2917 | i0=0; nor=0; |
---|
2918 | for (i=1; i<=size(L); i++) |
---|
2919 | { |
---|
2920 | norn=numones(L[i],pos(p)); |
---|
2921 | if (nor<norn){nor=norn; i0=i;} |
---|
2922 | } |
---|
2923 | W[j0]=i0; |
---|
2924 | j0++; |
---|
2925 | p=actualize(p,L[i0]); |
---|
2926 | } |
---|
2927 | list LL; |
---|
2928 | for (i=1;i<=size(W);i++) |
---|
2929 | { |
---|
2930 | LL[size(LL)+1]=C[W[i]]; |
---|
2931 | } |
---|
2932 | return(LL); |
---|
2933 | } |
---|
2934 | |
---|
2935 | // Auxiliary routine |
---|
2936 | // allsheaves |
---|
2937 | // Input: L=((v_11,..,v_1k_1),..,(v_s1,..,v_sk_s)) |
---|
2938 | // where: |
---|
2939 | // The s lists correspond to the s coefficients of the polynomial f |
---|
2940 | // (v_i1,..,v_ik_i) correspond to the k_i intvec v_ij of the |
---|
2941 | // spezializations of the jth rekpresentant (Q,P) of the ith coefficient |
---|
2942 | // v_ij is an intvec of size equal to the number of little segments |
---|
2943 | // forming the lpp-segment of 0,1, where 1 represents that it specializes |
---|
2944 | // to non-zero on the whole little segment and 1 if not. |
---|
2945 | // Output: |
---|
2946 | // (list LL, list LLS) where |
---|
2947 | // LL is the list of all combrep |
---|
2948 | // LLS is the list of intvec of the corresponding elements of LL |
---|
2949 | proc allsheaves(list L) |
---|
2950 | { |
---|
2951 | intvec V; list LL; intvec W; int r; intvec U; |
---|
2952 | int i; int j; int k; |
---|
2953 | int s=size(L[1][1]); // s = number of little segments of the lpp-segment |
---|
2954 | list LLS; |
---|
2955 | for (i=1;i<=size(L);i++) |
---|
2956 | { |
---|
2957 | V[i]=size(L[i]); |
---|
2958 | } |
---|
2959 | LL=combrep(V); |
---|
2960 | for (i=1;i<=size(LL);i++) |
---|
2961 | { |
---|
2962 | W=LL[i]; // size(W)= number of coefficients of the polynomial |
---|
2963 | kill U; intvec U; |
---|
2964 | for (j=1;j<=s;j++) |
---|
2965 | { |
---|
2966 | k=1; r=1; U[j]=1; |
---|
2967 | while((r==1) and (k<=size(W))) |
---|
2968 | { |
---|
2969 | if(L[k][W[k]][j]==0){r=0; U[j]=0;} |
---|
2970 | k++; |
---|
2971 | } |
---|
2972 | } |
---|
2973 | LLS[i]=U; |
---|
2974 | } |
---|
2975 | return(list(LL,LLS)); |
---|
2976 | } |
---|
2977 | |
---|
2978 | // Auxiliary routine |
---|
2979 | // numones |
---|
2980 | // Input: |
---|
2981 | // intvec v of (0,1) in each position |
---|
2982 | // intvec pos: the positions to test |
---|
2983 | // Output: |
---|
2984 | // int nor: the nuber of 1 of v in the positions given by pos. |
---|
2985 | proc numones(intvec v, intvec pos) |
---|
2986 | { |
---|
2987 | int i; int n; |
---|
2988 | for (i=1;i<=size(pos);i++) |
---|
2989 | { |
---|
2990 | if (v[pos[i]]==1){n++;} |
---|
2991 | } |
---|
2992 | return(n); |
---|
2993 | } |
---|
2994 | |
---|
2995 | // Auxiliary routine |
---|
2996 | // pos |
---|
2997 | // Input: intvec p of zeros and ones |
---|
2998 | // Output: intvec W of the positions where p has ones. |
---|
2999 | proc pos(intvec p) |
---|
3000 | { |
---|
3001 | int i; |
---|
3002 | intvec W; int j=1; |
---|
3003 | for (i=1; i<=size(p); i++) |
---|
3004 | { |
---|
3005 | if (p[i]==1){W[j]=i; j++;} |
---|
3006 | } |
---|
3007 | return(W); |
---|
3008 | } |
---|
3009 | |
---|
3010 | // Auxiliary routine |
---|
3011 | // actualize: actualizes zeroes of p |
---|
3012 | // Input: |
---|
3013 | // intvec p: of zeroes and ones |
---|
3014 | // intvec c: of zeroes and ones (of the same length) |
---|
3015 | // Output; |
---|
3016 | // intvec pp: of zeroes and ones, where a 0 stays in pp[i] if either |
---|
3017 | // already p[i]==0 or c[i]==1. |
---|
3018 | proc actualize(intvec p, intvec c) |
---|
3019 | { |
---|
3020 | int i; intvec pp=p; |
---|
3021 | for (i=1;i<=size(p);i++) |
---|
3022 | { |
---|
3023 | if ((pp[i]==1) and (c[i]==1)){pp[i]=0;} |
---|
3024 | } |
---|
3025 | return(pp); |
---|
3026 | } |
---|
3027 | |
---|
3028 | // Auxiliary routine |
---|
3029 | // combrep |
---|
3030 | // Input: V=(n_1,..,n_i) |
---|
3031 | // Output: L=(v_1,..,v_p) where p=prod_j=1^i (n_j) |
---|
3032 | // is the list of all intvec v_j=(v_j1,..,v_ji) where 1<=v_jk<=n_i |
---|
3033 | proc combrep(intvec V) |
---|
3034 | { |
---|
3035 | list L; list LL; |
---|
3036 | int i; int j; int k; intvec W; |
---|
3037 | if (size(V)==1) |
---|
3038 | { |
---|
3039 | for (i=1;i<=V[1];i++) |
---|
3040 | { |
---|
3041 | L[i]=intvec(i); |
---|
3042 | } |
---|
3043 | return(L); |
---|
3044 | } |
---|
3045 | for (i=1;i<size(V);i++) |
---|
3046 | { |
---|
3047 | W[i]=V[i]; |
---|
3048 | } |
---|
3049 | LL=combrep(W); |
---|
3050 | for (i=1;i<=size(LL);i++) |
---|
3051 | { |
---|
3052 | W=LL[i]; |
---|
3053 | for (j=1;j<=V[size(V)];j++) |
---|
3054 | { |
---|
3055 | W[size(V)]=j; |
---|
3056 | L[size(L)+1]=W; |
---|
3057 | } |
---|
3058 | } |
---|
3059 | return(L); |
---|
3060 | } |
---|
3061 | |
---|
3062 | // Auxiliary routine |
---|
3063 | proc reducemodN(poly f,ideal E) |
---|
3064 | { |
---|
3065 | def RR=basering; |
---|
3066 | setring(@RPt); |
---|
3067 | def fa=imap(RR,f); |
---|
3068 | def Ea=imap(RR,E); |
---|
3069 | attrib(Ea,"isSB",1); |
---|
3070 | // option(redSB); |
---|
3071 | // Ea=std(Ea); |
---|
3072 | fa=reduce(fa,Ea); |
---|
3073 | setring(RR); |
---|
3074 | def f1=imap(@RPt,fa); |
---|
3075 | return(f1); |
---|
3076 | } |
---|
3077 | |
---|
3078 | // Auxiliary routine |
---|
3079 | // intersp: computes the intersection of the ideals in S in @P |
---|
3080 | proc intersp(list S) |
---|
3081 | { |
---|
3082 | def RR=basering; |
---|
3083 | setring(@P); |
---|
3084 | def SP=imap(RR,S); |
---|
3085 | option(returnSB); |
---|
3086 | def NP=intersect(SP[1..size(SP)]); |
---|
3087 | setring(RR); |
---|
3088 | return(imap(@P,NP)); |
---|
3089 | } |
---|
3090 | |
---|
3091 | // Auxiliary routine |
---|
3092 | // radicalmember |
---|
3093 | proc radicalmember(poly f,ideal ida) |
---|
3094 | { |
---|
3095 | int te; |
---|
3096 | def RR=basering; |
---|
3097 | setring(@P); |
---|
3098 | def fp=imap(RR,f); |
---|
3099 | def idap=imap(RR,ida); |
---|
3100 | poly @t; |
---|
3101 | ring H=0,@t,dp; |
---|
3102 | def PH=@P+H; |
---|
3103 | setring(PH); |
---|
3104 | def fH=imap(@P,fp); |
---|
3105 | def idaH=imap(@P,idap); |
---|
3106 | idaH[size(idaH)+1]=1-@t*fH; |
---|
3107 | option(redSB); |
---|
3108 | def G=std(idaH); |
---|
3109 | if (G==1){te=1;} else {te=0;} |
---|
3110 | setring(RR); |
---|
3111 | return(te); |
---|
3112 | } |
---|
3113 | |
---|
3114 | // Auxiliary routine |
---|
3115 | // NonNull: returns 1 if the poly f is nonnull on V(E)\V(N), 0 otherwise. |
---|
3116 | proc NonNull(poly f, ideal E, ideal N) |
---|
3117 | { |
---|
3118 | int te=1; int i; |
---|
3119 | def RR=basering; |
---|
3120 | setring(@P); |
---|
3121 | def fp=imap(RR,f); |
---|
3122 | def Ep=imap(RR,E); |
---|
3123 | def Np=imap(RR,N); |
---|
3124 | ideal H; |
---|
3125 | ideal Ef=Ep+fp; |
---|
3126 | for (i=1;i<=size(Np);i++) |
---|
3127 | { |
---|
3128 | te=radicalmember(Np[i],Ef); |
---|
3129 | if (te==0){break;} |
---|
3130 | } |
---|
3131 | setring(RR); |
---|
3132 | return(te); |
---|
3133 | } |
---|
3134 | |
---|
3135 | // Auxiliary routine |
---|
3136 | // selectextendcoef |
---|
3137 | // input: |
---|
3138 | // matrix CC: CC=(p_a1 .. p_ar_a) |
---|
3139 | // (q_a1 .. q_ar_a) |
---|
3140 | // the matrix of elements of a coefficient in oo[a]. |
---|
3141 | // (ideal ida, ideal idb): the canonical representation of the segment S. |
---|
3142 | // output: |
---|
3143 | // list caout |
---|
3144 | // the minimum set of elements of CC needed such that at least one |
---|
3145 | // of the q's is non-null on S, as well as the C-rep of of the |
---|
3146 | // points where the q's are null on S. |
---|
3147 | // The elements of caout are of the form (p,q,prep); |
---|
3148 | proc selectextendcoef(matrix CC, ideal ida, ideal idb) |
---|
3149 | { |
---|
3150 | def RR=basering; |
---|
3151 | setring(@P); |
---|
3152 | def ca=imap(RR,CC); |
---|
3153 | def E0=imap(RR,ida); |
---|
3154 | ideal E; |
---|
3155 | def N=imap(RR,idb); |
---|
3156 | int r=ncols(ca); |
---|
3157 | int i; int te=1; list com; int j; int k; intvec c; list prep; |
---|
3158 | list cs; list caout; |
---|
3159 | i=1; |
---|
3160 | while ((i<=r) and (te)) |
---|
3161 | { |
---|
3162 | com=comb(r,i); |
---|
3163 | j=1; |
---|
3164 | while((j<=size(com)) and (te)) |
---|
3165 | { |
---|
3166 | E=E0; |
---|
3167 | c=com[j]; |
---|
3168 | for (k=1;k<=i;k++) |
---|
3169 | { |
---|
3170 | E=E+ca[2,c[k]]; |
---|
3171 | } |
---|
3172 | prep=Prep(E,N); |
---|
3173 | if (i==1) |
---|
3174 | { |
---|
3175 | cs[j]=list(ca[1,j],ca[2,j],prep); |
---|
3176 | } |
---|
3177 | if ((size(prep)==1) and (equalideals(prep[1][1],ideal(1)))) |
---|
3178 | { |
---|
3179 | te=0; |
---|
3180 | for(k=1;k<=size(c);k++) |
---|
3181 | { |
---|
3182 | caout[k]=cs[c[k]]; |
---|
3183 | } |
---|
3184 | } |
---|
3185 | j++; |
---|
3186 | } |
---|
3187 | i++; |
---|
3188 | } |
---|
3189 | if (te){"error: extendcoef does not extend to the whole S";} |
---|
3190 | setring(RR); |
---|
3191 | return(imap(@P,caout)); |
---|
3192 | } |
---|
3193 | |
---|
3194 | // Auxiliary routine |
---|
3195 | // input: |
---|
3196 | // ideal E1: in some basering (depends only on the parameters) |
---|
3197 | // ideal E2: in some basering (depends only on the parameters) |
---|
3198 | // output: |
---|
3199 | // ideal Ep=E1+E2; computed in P |
---|
3200 | proc plusP(ideal E1,ideal E2) |
---|
3201 | { |
---|
3202 | def RR=basering; |
---|
3203 | setring(@P); |
---|
3204 | def E1p=imap(RR,E1); |
---|
3205 | def E2p=imap(RR,E2); |
---|
3206 | def Ep=E1p+E2p; |
---|
3207 | setring(RR); |
---|
3208 | return(imap(@P,Ep)); |
---|
3209 | } |
---|
3210 | |
---|
3211 | // Auxiliary routine |
---|
3212 | // reform |
---|
3213 | // input: |
---|
3214 | // list combpolys: (v1,..,vs) |
---|
3215 | // where vi are intvec. |
---|
3216 | // output outcomb: (w1,..,wt) |
---|
3217 | // whre wi are intvec. |
---|
3218 | // All the vi without zeroes are in outcomb, and those with zeroes are |
---|
3219 | // combined to form new intvec with the rest |
---|
3220 | proc reform(list combpolys, intvec numdens) |
---|
3221 | { |
---|
3222 | list combp0; list combp1; int i; int j; int k; int l; list rest; intvec notfree; |
---|
3223 | list free; intvec free1; int te; intvec v; intvec w; |
---|
3224 | int nummonoms=size(combpolys[1]); |
---|
3225 | for(i=1;i<=size(combpolys);i++) |
---|
3226 | { |
---|
3227 | if(memberpos(0,combpolys[i])[1]) |
---|
3228 | { |
---|
3229 | combp0[size(combp0)+1]=combpolys[i]; |
---|
3230 | } |
---|
3231 | else {combp1[size(combp1)+1]=combpolys[i];} |
---|
3232 | } |
---|
3233 | for(i=1;i<=nummonoms;i++) |
---|
3234 | { |
---|
3235 | kill notfree; intvec notfree; |
---|
3236 | for(j=1;j<=size(combpolys);j++) |
---|
3237 | { |
---|
3238 | if(combpolys[j][i]<>0) |
---|
3239 | { |
---|
3240 | if(notfree[1]==0){notfree[1]=combpolys[j][i];} |
---|
3241 | else{notfree[size(notfree)+1]=combpolys[j][i];} |
---|
3242 | } |
---|
3243 | } |
---|
3244 | kill free1; intvec free1; |
---|
3245 | for(j=1;j<=numdens[i];j++) |
---|
3246 | { |
---|
3247 | if(memberpos(j,notfree)[1]==0) |
---|
3248 | { |
---|
3249 | if(free1[1]==0){free1[1]=j;} |
---|
3250 | else{free1[size(free1)+1]=j;} |
---|
3251 | } |
---|
3252 | free[i]=free1; |
---|
3253 | } |
---|
3254 | } |
---|
3255 | list amplcombp; list aux; |
---|
3256 | for(i=1;i<=size(combp0);i++) |
---|
3257 | { |
---|
3258 | v=combp0[i]; |
---|
3259 | kill amplcombp; list amplcombp; |
---|
3260 | amplcombp[1]=intvec(v[1]); |
---|
3261 | for(j=2;j<=size(v);j++) |
---|
3262 | { |
---|
3263 | if(v[j]!=0) |
---|
3264 | { |
---|
3265 | for(k=1;k<=size(amplcombp);k++) |
---|
3266 | { |
---|
3267 | w=amplcombp[k]; |
---|
3268 | w[size(w)+1]=v[j]; |
---|
3269 | amplcombp[k]=w; |
---|
3270 | } |
---|
3271 | } |
---|
3272 | else |
---|
3273 | { |
---|
3274 | kill aux; list aux; |
---|
3275 | for(k=1;k<=size(amplcombp);k++) |
---|
3276 | { |
---|
3277 | for(l=1;l<=size(free[j]);l++) |
---|
3278 | { |
---|
3279 | w=amplcombp[k]; |
---|
3280 | w[size(w)+1]=free[j][l]; |
---|
3281 | aux[size(aux)+1]=w; |
---|
3282 | } |
---|
3283 | } |
---|
3284 | amplcombp=aux; |
---|
3285 | } |
---|
3286 | } |
---|
3287 | for(j=1;j<=size(amplcombp);j++) |
---|
3288 | { |
---|
3289 | combp1[size(combp1)+1]=amplcombp[j]; |
---|
3290 | } |
---|
3291 | } |
---|
3292 | return(combp1); |
---|
3293 | } |
---|
3294 | |
---|
3295 | // Auxiliary routine |
---|
3296 | // nonnullCrep |
---|
3297 | proc nonnullCrep(poly f0,ideal ida0,ideal idb0) |
---|
3298 | { |
---|
3299 | int i; |
---|
3300 | def RR=basering; |
---|
3301 | setring(@P); |
---|
3302 | def f=imap(RR,f0); |
---|
3303 | def ida=imap(RR,ida0); |
---|
3304 | def idb=imap(RR,idb0); |
---|
3305 | def idaf=ida+f; |
---|
3306 | int te=1; |
---|
3307 | for(i=1;i<=size(idb);i++) |
---|
3308 | { |
---|
3309 | if(radicalmember(idb[i],idaf)==0) |
---|
3310 | { |
---|
3311 | te=0; break; |
---|
3312 | } |
---|
3313 | } |
---|
3314 | setring(RR); |
---|
3315 | return(te); |
---|
3316 | } |
---|
3317 | |
---|
3318 | // Auxiliary routine |
---|
3319 | // precombint |
---|
3320 | // input: L: list of ideals (works in @P) |
---|
3321 | // output: F0: ideal of polys. F0[i] is a poly in the intersection of |
---|
3322 | // all ideals in L except in the ith one, where it is not. |
---|
3323 | // L=(p1,..,ps); F0=(f1,..,fs); |
---|
3324 | // F0[i] \in intersect_{j#i} p_i |
---|
3325 | proc precombint(list L) |
---|
3326 | { |
---|
3327 | int i; int j; int tes; |
---|
3328 | def RR=basering; |
---|
3329 | setring(@P); |
---|
3330 | list L0; list L1; list L2; list L3; ideal F; |
---|
3331 | L0=imap(RR,L); |
---|
3332 | L1[1]=L0[1]; L2[1]=L0[size(L0)]; |
---|
3333 | for (i=2;i<=size(L0)-1;i++) |
---|
3334 | { |
---|
3335 | L1[i]=intersect(L1[i-1],L0[i]); |
---|
3336 | L2[i]=intersect(L2[i-1],L0[size(L0)-i+1]); |
---|
3337 | } |
---|
3338 | L3[1]=L2[size(L2)]; |
---|
3339 | for (i=2;i<=size(L0)-1;i++) |
---|
3340 | { |
---|
3341 | L3[i]=intersect(L1[i-1],L2[size(L0)-i]); |
---|
3342 | } |
---|
3343 | L3[size(L0)]=L1[size(L1)]; |
---|
3344 | for (i=1;i<=size(L3);i++) |
---|
3345 | { |
---|
3346 | option(redSB); L3[i]=std(L3[i]); |
---|
3347 | } |
---|
3348 | for (i=1;i<=size(L3);i++) |
---|
3349 | { |
---|
3350 | tes=1; j=0; |
---|
3351 | while((tes) and (j<size(L3[i]))) |
---|
3352 | { |
---|
3353 | j++; |
---|
3354 | option(redSB); |
---|
3355 | L0[i]=std(L0[i]); |
---|
3356 | if(reduce(L3[i][j],L0[i])!=0){tes=0; F[i]=L3[i][j];} |
---|
3357 | } |
---|
3358 | if (tes){"ERROR a polynomial in all p_j except p_i was not found";} |
---|
3359 | } |
---|
3360 | setring(RR); |
---|
3361 | def F0=imap(@P,F); |
---|
3362 | return(F0); |
---|
3363 | } |
---|
3364 | |
---|
3365 | |
---|
3366 | // Auxiliary routine |
---|
3367 | // minAssGTZ eliminating denominators |
---|
3368 | proc minGTZ(ideal N); |
---|
3369 | { |
---|
3370 | int i; int j; |
---|
3371 | def L=minAssGTZ(N); |
---|
3372 | for(i=1;i<=size(L);i++) |
---|
3373 | { |
---|
3374 | for(j=1;j<=size(L[i]);j++) |
---|
3375 | { |
---|
3376 | L[i][j]=cleardenom(L[i][j]); |
---|
3377 | } |
---|
3378 | } |
---|
3379 | return(L); |
---|
3380 | } |
---|
3381 | |
---|
3382 | //********************* Begin KapurSunWang ************************* |
---|
3383 | |
---|
3384 | // Auxiliary routine |
---|
3385 | // inconsistent |
---|
3386 | // Input: |
---|
3387 | // ideal E: of null conditions |
---|
3388 | // ideal N: of non-null conditions representing V(E)\V(N) |
---|
3389 | // Output: |
---|
3390 | // 1 if V(E) \V(N) = empty |
---|
3391 | // 0 if not |
---|
3392 | proc inconsistent(ideal E, ideal N) |
---|
3393 | { |
---|
3394 | int j; |
---|
3395 | int te=1; |
---|
3396 | def R=basering; |
---|
3397 | setring(@P); |
---|
3398 | def EP=imap(R,E); |
---|
3399 | def NP=imap(R,N); |
---|
3400 | poly @t; |
---|
3401 | ring H=0,@t,dp; |
---|
3402 | def RH=@P+H; |
---|
3403 | setring(RH); |
---|
3404 | def EH=imap(@P,EP); |
---|
3405 | def NH=imap(@P,NP); |
---|
3406 | ideal G; |
---|
3407 | j=1; |
---|
3408 | while((te==1) and j<=size(NH)) |
---|
3409 | { |
---|
3410 | G=EH+(1-@t*NH[j]); |
---|
3411 | option(redSB); |
---|
3412 | G=std(G); |
---|
3413 | if (G[1]!=1){te=0;} |
---|
3414 | j++; |
---|
3415 | } |
---|
3416 | setring(R); |
---|
3417 | return(te); |
---|
3418 | } |
---|
3419 | |
---|
3420 | // Auxiliary routine |
---|
3421 | // MDBasis: Minimal Dickson Basis |
---|
3422 | proc MDBasis(ideal G) |
---|
3423 | { |
---|
3424 | int i; int j; int te=1; |
---|
3425 | G=sortideal(G); |
---|
3426 | ideal MD=G[1]; |
---|
3427 | poly lm; |
---|
3428 | for (i=2;i<=size(G);i++) |
---|
3429 | { |
---|
3430 | te=1; |
---|
3431 | lm=leadmonom(G[i]); |
---|
3432 | j=1; |
---|
3433 | while ((te==1) and (j<=size(MD))) |
---|
3434 | { |
---|
3435 | if (lm/leadmonom(MD[j])!=0){te=0;} |
---|
3436 | j++; |
---|
3437 | } |
---|
3438 | if (te==1) |
---|
3439 | { |
---|
3440 | MD[size(MD)+1]=(G[i]); |
---|
3441 | } |
---|
3442 | } |
---|
3443 | return(MD); |
---|
3444 | } |
---|
3445 | |
---|
3446 | // Auxiliary routine |
---|
3447 | // primepartZ |
---|
3448 | proc primepartZ(poly f); |
---|
3449 | { |
---|
3450 | def R=basering; |
---|
3451 | def cp=content(f); |
---|
3452 | def fp=f/cp; |
---|
3453 | return(fp); |
---|
3454 | } |
---|
3455 | |
---|
3456 | // LCMLC |
---|
3457 | proc LCMLC(ideal H) |
---|
3458 | { |
---|
3459 | int i; |
---|
3460 | def R=basering; |
---|
3461 | setring(@RP); |
---|
3462 | def HH=imap(R,H); |
---|
3463 | poly h=1; |
---|
3464 | for (i=1;i<=size(HH);i++) |
---|
3465 | { |
---|
3466 | h=lcm(h,HH[i]); |
---|
3467 | } |
---|
3468 | setring(R); |
---|
3469 | def hh=imap(@RP,h); |
---|
3470 | return(hh); |
---|
3471 | } |
---|
3472 | |
---|
3473 | // KSW: Kapur-Sun-Wang algorithm for computing a CGS |
---|
3474 | // Input: |
---|
3475 | // F: parametric ideal to be discussed |
---|
3476 | // Options: |
---|
3477 | // \"out\",0 Transforms the description of the segments into |
---|
3478 | // canonical P-representation form. |
---|
3479 | // \"out\",1 Original KSW routine describing the segments as |
---|
3480 | // difference of varieties |
---|
3481 | // The ideal must be defined on C[parameters][variables] |
---|
3482 | // Output: |
---|
3483 | // With option \"out\",0 : |
---|
3484 | // ((lpp, |
---|
3485 | // (1,B,((p_1,(p_11,..,p_1k_1)),..,(p_s,(p_s1,..,p_sk_s)))), |
---|
3486 | // string(lpp) |
---|
3487 | // ) |
---|
3488 | // ,.., |
---|
3489 | // (lpp, |
---|
3490 | // (k,B,((p_1,(p_11,..,p_1k_1)),..,(p_s,(p_s1,..,p_sk_s)))), |
---|
3491 | // string(lpp)) |
---|
3492 | // ) |
---|
3493 | // ) |
---|
3494 | // With option \"out\",1 ((default, original KSW) (shorter to be computed, |
---|
3495 | // but without canonical description of the segments. |
---|
3496 | // ((B,E,N),..,(B,E,N)) |
---|
3497 | proc KSW(ideal F, list #) |
---|
3498 | { |
---|
3499 | setglobalrings(); |
---|
3500 | int start=timer; |
---|
3501 | ideal E=ideal(0); |
---|
3502 | ideal N=ideal(1); |
---|
3503 | int comment=0; |
---|
3504 | int out=1; |
---|
3505 | int i; |
---|
3506 | def L=#; |
---|
3507 | if (size(L)>0) |
---|
3508 | { |
---|
3509 | for (i=1;i<=size(L) div 2;i++) |
---|
3510 | { |
---|
3511 | if (L[2*i-1]=="null"){E=L[2*i];} |
---|
3512 | else |
---|
3513 | { |
---|
3514 | if (L[2*i-1]=="nonnull"){N=L[2*i];} |
---|
3515 | else |
---|
3516 | { |
---|
3517 | if (L[2*i-1]=="comment"){comment=L[2*i];} |
---|
3518 | else |
---|
3519 | { |
---|
3520 | if (L[2*i-1]=="out"){out=L[2*i];} |
---|
3521 | } |
---|
3522 | } |
---|
3523 | } |
---|
3524 | } |
---|
3525 | } |
---|
3526 | if (comment>0){string("Begin KSW with null = ",E," nonnull = ",N);} |
---|
3527 | def CG=KSW0(F,E,N,comment); |
---|
3528 | if (comment>0) |
---|
3529 | { |
---|
3530 | string("Number of segments in KSW (total) = ",size(CG)); |
---|
3531 | string("Time in KSW = ",timer-start); |
---|
3532 | } |
---|
3533 | if(out==0) |
---|
3534 | { |
---|
3535 | CG=KSWtocgsdr(CG); |
---|
3536 | CG=groupKSWsegments(CG); |
---|
3537 | if (comment>0) |
---|
3538 | { |
---|
3539 | string("Number of lpp segments = ",size(CG)); |
---|
3540 | string("Time in KSW + group + Prep = ",timer-start); |
---|
3541 | } |
---|
3542 | } |
---|
3543 | if(defined(@P)){kill @P; kill @R; kill @RP;} |
---|
3544 | return(CG); |
---|
3545 | } |
---|
3546 | |
---|
3547 | // Auxiliary routine |
---|
3548 | // sqf |
---|
3549 | // This is for releases of Singular before 3-5-1 |
---|
3550 | // proc sqf(poly f) |
---|
3551 | // { |
---|
3552 | // def RR=basering; |
---|
3553 | // setring(@P); |
---|
3554 | // def ff=imap(RR,f); |
---|
3555 | // def G=sqrfree(ff); |
---|
3556 | // poly fff=1; |
---|
3557 | // int i; |
---|
3558 | // for (i=1;i<=size(G);i++) |
---|
3559 | // { |
---|
3560 | // fff=fff*G[i]; |
---|
3561 | // } |
---|
3562 | // setring(RR); |
---|
3563 | // def ffff=imap(@P,fff); |
---|
3564 | // return(ffff); |
---|
3565 | // } |
---|
3566 | |
---|
3567 | // Auxiliary routine |
---|
3568 | // sqf |
---|
3569 | proc sqf(poly f) |
---|
3570 | { |
---|
3571 | def RR=basering; |
---|
3572 | setring(@P); |
---|
3573 | def ff=imap(RR,f); |
---|
3574 | poly fff=sqrfree(ff,3); |
---|
3575 | setring(RR); |
---|
3576 | def ffff=imap(@P,fff); |
---|
3577 | return(ffff); |
---|
3578 | } |
---|
3579 | |
---|
3580 | |
---|
3581 | // Auxiliary routine |
---|
3582 | // KSW0: Kapur-Sun-Wang algorithm for computing a CGS, called by KSW |
---|
3583 | // Input: |
---|
3584 | // F: parametric ideal to be discussed |
---|
3585 | // Options: |
---|
3586 | // The ideal must be defined on C[parameters][variables] |
---|
3587 | // Output: |
---|
3588 | proc KSW0(ideal F, ideal E, ideal N, int comment) |
---|
3589 | { |
---|
3590 | def R=basering; |
---|
3591 | int i; int j; list emp; |
---|
3592 | list CGS; |
---|
3593 | ideal N0; |
---|
3594 | for (i=1;i<=size(N);i++) |
---|
3595 | { |
---|
3596 | N0[i]=sqf(N[i]); |
---|
3597 | } |
---|
3598 | ideal E0; |
---|
3599 | for (i=1;i<=size(E);i++) |
---|
3600 | { |
---|
3601 | E0[i]=sqf(leadcoef(E[i])); |
---|
3602 | } |
---|
3603 | setring(@P); |
---|
3604 | ideal E1=imap(R,E0); |
---|
3605 | E1=std(E1); |
---|
3606 | ideal N1=imap(R,N0); |
---|
3607 | N1=std(N1); |
---|
3608 | setring(R); |
---|
3609 | E0=imap(@P,E1); |
---|
3610 | N0=imap(@P,N1); |
---|
3611 | // E0=elimrepeated(E0); |
---|
3612 | // N0=elimrepeated(N0); |
---|
3613 | if (inconsistent(E0,N0)==1) |
---|
3614 | { |
---|
3615 | return(emp); |
---|
3616 | } |
---|
3617 | setring(@RP); |
---|
3618 | def FRP=imap(R,F); |
---|
3619 | def ERP=imap(R,E); |
---|
3620 | FRP=FRP+ERP; |
---|
3621 | option(redSB); |
---|
3622 | def GRP=std(FRP); |
---|
3623 | setring(R); |
---|
3624 | def G=imap(@RP,GRP); |
---|
3625 | if (memberpos(1,G)[1]==1) |
---|
3626 | { |
---|
3627 | if(comment>1){"Basis 1 is found"; E; N;} |
---|
3628 | list KK; KK[1]=list(E0,N0,ideal(1)); |
---|
3629 | return(KK); |
---|
3630 | } |
---|
3631 | ideal Gr; ideal Gm; ideal GM; |
---|
3632 | for (i=1;i<=size(G);i++) |
---|
3633 | { |
---|
3634 | if (variables(G[i])[1]==0){Gr[size(Gr)+1]=G[i];} |
---|
3635 | else{Gm[size(Gm)+1]=G[i];} |
---|
3636 | } |
---|
3637 | ideal Gr0; |
---|
3638 | for (i=1;i<=size(Gr);i++) |
---|
3639 | { |
---|
3640 | Gr0[i]=sqf(Gr[i]); |
---|
3641 | } |
---|
3642 | |
---|
3643 | |
---|
3644 | Gr=elimrepeated(Gr0); |
---|
3645 | ideal GrN; |
---|
3646 | for (i=1;i<=size(Gr);i++) |
---|
3647 | { |
---|
3648 | for (j=1;j<=size(N0);j++) |
---|
3649 | { |
---|
3650 | GrN[size(GrN)+1]=sqf(Gr[i]*N0[j]); |
---|
3651 | } |
---|
3652 | } |
---|
3653 | if (inconsistent(E,GrN)){;} |
---|
3654 | else |
---|
3655 | { |
---|
3656 | if(comment>1){"Basis 1 is found in a branch with arguments"; E; GrN;} |
---|
3657 | CGS[size(CGS)+1]=list(E,GrN,ideal(1)); |
---|
3658 | } |
---|
3659 | if (inconsistent(Gr,N0)){return(CGS);} |
---|
3660 | GM=Gm; |
---|
3661 | Gm=MDBasis(Gm); |
---|
3662 | ideal H; |
---|
3663 | for (i=1;i<=size(Gm);i++) |
---|
3664 | { |
---|
3665 | H[i]=sqf(leadcoef(Gm[i])); |
---|
3666 | } |
---|
3667 | H=facvar(H); |
---|
3668 | poly h=sqf(LCMLC(H)); |
---|
3669 | if(comment>1){"H = "; H; "h = "; h;} |
---|
3670 | ideal Nh=N0; |
---|
3671 | if(size(N0)==0){Nh=h;} |
---|
3672 | else |
---|
3673 | { |
---|
3674 | for (i=1;i<=size(N0);i++) |
---|
3675 | { |
---|
3676 | Nh[i]=sqf(N0[i]*h); |
---|
3677 | } |
---|
3678 | } |
---|
3679 | if (inconsistent(Gr,Nh)){;} |
---|
3680 | else |
---|
3681 | { |
---|
3682 | CGS[size(CGS)+1]=list(Gr,Nh,Gm); |
---|
3683 | } |
---|
3684 | poly hc=1; |
---|
3685 | list KS; |
---|
3686 | ideal GrHi; |
---|
3687 | for (i=1;i<=size(H);i++) |
---|
3688 | { |
---|
3689 | kill GrHi; |
---|
3690 | ideal GrHi; |
---|
3691 | Nh=N0; |
---|
3692 | if (i>1){hc=sqf(hc*H[i-1]);} |
---|
3693 | for (j=1;j<=size(N0);j++){Nh[j]=sqf(N0[j]*hc);} |
---|
3694 | if (equalideals(Gr,ideal(0))==1){GrHi=H[i];} |
---|
3695 | else {GrHi=Gr,H[i];} |
---|
3696 | // else {for (j=1;j<=size(Gr);j++){GrHi[size(GrHi)+1]=Gr[j]*H[i];}} |
---|
3697 | if(comment>1){"Call to KSW with arguments "; GM; GrHi; Nh;} |
---|
3698 | KS=KSW0(GM,GrHi,Nh,comment); |
---|
3699 | for (j=1;j<=size(KS);j++) |
---|
3700 | { |
---|
3701 | CGS[size(CGS)+1]=KS[j]; |
---|
3702 | } |
---|
3703 | if(comment>1){"CGS after KSW = "; CGS;} |
---|
3704 | } |
---|
3705 | return(CGS); |
---|
3706 | } |
---|
3707 | |
---|
3708 | // Auxiliary routine |
---|
3709 | // KSWtocgsdr |
---|
3710 | proc KSWtocgsdr(list L) |
---|
3711 | { |
---|
3712 | int i; list CG; ideal B; ideal lpp; int j; list NKrep; |
---|
3713 | for(i=1;i<=size(L);i++) |
---|
3714 | { |
---|
3715 | B=redgbn(L[i][3],L[i][1],L[i][2]); |
---|
3716 | lpp=ideal(0); |
---|
3717 | for(j=1;j<=size(B);j++) |
---|
3718 | { |
---|
3719 | lpp[j]=leadmonom(B[j]); |
---|
3720 | } |
---|
3721 | NKrep=KtoPrep(L[i][1],L[i][2]); |
---|
3722 | CG[i]=list(lpp,B,NKrep); |
---|
3723 | } |
---|
3724 | return(CG); |
---|
3725 | } |
---|
3726 | |
---|
3727 | // Auxiliary routine |
---|
3728 | // KtoPrep |
---|
3729 | // Computes the P-representaion of a K-representation (N,W) of a set |
---|
3730 | // input: |
---|
3731 | // ideal E (null conditions) |
---|
3732 | // ideal N (non-null conditions ideal) |
---|
3733 | // output: |
---|
3734 | // the ((p_1,(p_11,..,p_1k_1)),..,(p_r,(p_r1,..,p_rk_r))); |
---|
3735 | // the Prep of V(N) \ V(W) |
---|
3736 | proc KtoPrep(ideal N, ideal W) |
---|
3737 | { |
---|
3738 | int i; int j; |
---|
3739 | if (N[1]==1) |
---|
3740 | { |
---|
3741 | L0[1]=list(ideal(1),list(ideal(1))); |
---|
3742 | return(L0); |
---|
3743 | } |
---|
3744 | def RR=basering; |
---|
3745 | setring(@P); |
---|
3746 | ideal B; int te; poly f; |
---|
3747 | ideal Np=imap(RR,N); |
---|
3748 | ideal Wp=imap(RR,W); |
---|
3749 | list L; |
---|
3750 | list L0; list T0; |
---|
3751 | L0=minGTZ(Np); |
---|
3752 | for(j=1;j<=size(L0);j++) |
---|
3753 | { |
---|
3754 | option(redSB); |
---|
3755 | L0[j]=std(L0[j]); |
---|
3756 | } |
---|
3757 | for(i=1;i<=size(L0);i++) |
---|
3758 | { |
---|
3759 | if(inconsistent(L0[i],Wp)==0) |
---|
3760 | { |
---|
3761 | B=L0[i]+Wp; |
---|
3762 | T0=minGTZ(B); |
---|
3763 | option(redSB); |
---|
3764 | for(j=1;j<=size(T0);j++) |
---|
3765 | { |
---|
3766 | T0[j]=std(T0[j]); |
---|
3767 | } |
---|
3768 | L[size(L)+1]=list(L0[i],T0); |
---|
3769 | } |
---|
3770 | } |
---|
3771 | setring(RR); |
---|
3772 | def LL=imap(@P,L); |
---|
3773 | return(LL); |
---|
3774 | } |
---|
3775 | |
---|
3776 | // Auxiliary routine |
---|
3777 | // groupKSWsegments |
---|
3778 | // input: the list of vertices of KSW |
---|
3779 | // output: the same terminal vertices grouped by lpp |
---|
3780 | proc groupKSWsegments(list T) |
---|
3781 | { |
---|
3782 | //"T_T="; T; |
---|
3783 | int i; int j; |
---|
3784 | list L; |
---|
3785 | list lpp; list lppor; |
---|
3786 | list kk; |
---|
3787 | lpp[1]=T[1][1]; j=1; |
---|
3788 | lppor[1]=intvec(1); |
---|
3789 | for(i=2;i<=size(T);i++) |
---|
3790 | { |
---|
3791 | kk=memberpos(T[i][1],lpp); |
---|
3792 | if(kk[1]==0){j++; lpp[j]=T[i][1]; lppor[j]=intvec(i);} |
---|
3793 | else{lppor[kk[2]][size(lppor[kk[2]])+1]=i;} |
---|
3794 | } |
---|
3795 | list ll; |
---|
3796 | for (j=1;j<=size(lpp);j++) |
---|
3797 | { |
---|
3798 | kill ll; list ll; |
---|
3799 | for(i=1;i<=size(lppor[j]);i++) |
---|
3800 | { |
---|
3801 | ll[size(ll)+1]=list(i,T[lppor[j][i]][2],T[lppor[j][i]][3]); |
---|
3802 | } |
---|
3803 | L[j]=list(lpp[j],ll,string(lpp[j])); |
---|
3804 | } |
---|
3805 | return(L); |
---|
3806 | } |
---|
3807 | |
---|
3808 | //********************* End KapurSunWang ************************* |
---|
3809 | |
---|
3810 | //******************** Begin locus ****************************** |
---|
3811 | |
---|
3812 | // indepparameters |
---|
3813 | // Auxiliary routine to detect "Special" components of the locus |
---|
3814 | // Input: ideal B |
---|
3815 | // Output: |
---|
3816 | // 1 if the solutions of the ideal do not depend on the parameters |
---|
3817 | // 0 if they depend |
---|
3818 | proc indepparameters(ideal B) |
---|
3819 | { |
---|
3820 | def R=basering; |
---|
3821 | ideal v=variables(B); |
---|
3822 | setring @RP; |
---|
3823 | def BP=imap(R,B); |
---|
3824 | def vp=imap(R,v); |
---|
3825 | ideal varpar=variables(BP); |
---|
3826 | int te; |
---|
3827 | te=equalideals(vp,varpar); |
---|
3828 | setring(R); |
---|
3829 | if(te){return(1);} |
---|
3830 | else{return(0);} |
---|
3831 | } |
---|
3832 | |
---|
3833 | // dimP0: Auxiliary routine |
---|
3834 | // if the dimension in @P of an ideal in the parameters has dimension 0 then it returns 0 |
---|
3835 | // else it retuns 1 |
---|
3836 | proc dimP0(ideal N) |
---|
3837 | { |
---|
3838 | def R=basering; |
---|
3839 | setring(@P); |
---|
3840 | int te=1; |
---|
3841 | def NP=imap(R,N); |
---|
3842 | attrib(NP,"IsSB",1); |
---|
3843 | int d=dim(std(NP)); |
---|
3844 | if(d==0){te=0;} |
---|
3845 | setring(R); |
---|
3846 | return(te); |
---|
3847 | } |
---|
3848 | |
---|
3849 | // Auxiliary routine. |
---|
3850 | // input: ideals E and F (assumed in ring @P |
---|
3851 | // returns: 1 if ideal E is contained in ideal F (assumed F is std basis) |
---|
3852 | // 0 if not |
---|
3853 | proc containedideal(ideal E, ideal F) |
---|
3854 | { |
---|
3855 | int i; int t; poly f; |
---|
3856 | if(equalideals(F,ideal(0))) |
---|
3857 | { |
---|
3858 | if(equalideals(E,ideal(0))==0){return(0);} |
---|
3859 | else(return(1)); |
---|
3860 | } |
---|
3861 | t=1; i=1; |
---|
3862 | while((t==1) and (i<=size(E))) |
---|
3863 | { |
---|
3864 | attrib(F,"isSB",1); |
---|
3865 | f=reduce(E[i],F); |
---|
3866 | if(f!=0){t=0;} |
---|
3867 | i++; |
---|
3868 | } |
---|
3869 | return(t); |
---|
3870 | } |
---|
3871 | |
---|
3872 | // addcons: given a set of locally closed sets given in P-representation, |
---|
3873 | // (particularly a subset of components of a selection of segments |
---|
3874 | // of the Grobner cover), it builds the canonical P-representation |
---|
3875 | // of the corresponding constructible set, of its union, including levels it they are. |
---|
3876 | // input: a list L of disjoint segments (for exmple a selection of segments of the Groebner cover) |
---|
3877 | // of the form |
---|
3878 | // ( ( (p1_1,(p1_11,..,p1_1k_1).. (p1_s,(p1_s1,..,p1_sk_s)),.., |
---|
3879 | // ( (pn_1,(pn_11,..,pn_1j_1).. (pn_s,(pn_s1,..,pn_sj_s)) ) |
---|
3880 | // output: The canonical P-representation of the constructible set resulting by |
---|
3881 | // adding the given components. |
---|
3882 | // The first element of each component can be ignored. It is only for internal use. |
---|
3883 | // The fifth element of each component is the level of the constructible set |
---|
3884 | // of the component. |
---|
3885 | // It is no need a previous call to setglobalrings() |
---|
3886 | proc addcons(list L) |
---|
3887 | "USAGE: addcons( ( ( (p1_1,(p1_11,..,p1_1k_1).. (p1_s,(p1_s1,..,p1_sk_s)),.., |
---|
3888 | ( (pn_1,(pn_11,..,pn_1j_1).. (pn_s,(pn_s1,..,pn_sj_s)) ) ) |
---|
3889 | a list L of locally closed sets in P-representation |
---|
3890 | RETURN: the canonical P-representation of the constructible set of the union. |
---|
3891 | NOTE: It is called internally by the routines locus, locusdg, |
---|
3892 | |
---|
3893 | KEYWORDS: P-representation, constructible set |
---|
3894 | EXAMPLE: adccons; shows an example" |
---|
3895 | { |
---|
3896 | int te; |
---|
3897 | if(defined(@P)){te=1;} |
---|
3898 | else{setglobalrings();} |
---|
3899 | list notadded; list P0; |
---|
3900 | int i; int j; int k; int l; |
---|
3901 | intvec v; intvec v1; intvec v0; |
---|
3902 | ideal J; list K; list L1; |
---|
3903 | for(i=1;i<=size(L);i++) |
---|
3904 | { |
---|
3905 | if(equalideals(L[i][1][1],ideal(1))==0) |
---|
3906 | {L1[size(L1)+1]=L[i];} |
---|
3907 | } |
---|
3908 | L=L1; |
---|
3909 | for(i=1;i<=size(L);i++) |
---|
3910 | { |
---|
3911 | for(j=1;j<=size(L[i]);j++) |
---|
3912 | { |
---|
3913 | v=i,j; |
---|
3914 | notadded[size(notadded)+1]=v; |
---|
3915 | } |
---|
3916 | } |
---|
3917 | //"T_notadded="; notadded; |
---|
3918 | int level=1; |
---|
3919 | list P=L; |
---|
3920 | list LL; |
---|
3921 | list Ln; |
---|
3922 | //int cont=0; |
---|
3923 | while(size(notadded)>0) |
---|
3924 | { |
---|
3925 | kill notadded; list notadded; |
---|
3926 | for(i=1;i<=size(P);i++) |
---|
3927 | { |
---|
3928 | for(j=1;j<=size(P[i]);j++) |
---|
3929 | { |
---|
3930 | v=i,j; |
---|
3931 | notadded[size(notadded)+1]=v; |
---|
3932 | } |
---|
3933 | } |
---|
3934 | //"T_notadded="; notadded; |
---|
3935 | //cont++; |
---|
3936 | P0=P; |
---|
3937 | Ln=LCUnion(P); |
---|
3938 | //"T_Ln="; Ln; |
---|
3939 | notadded=minuselements(notadded,@Q2); |
---|
3940 | //"T_@Q2="; @Q2; |
---|
3941 | //"T_notadded="; notadded; |
---|
3942 | for(i=1;i<=size(Ln);i++) |
---|
3943 | { |
---|
3944 | //Ln[i][4]= ; JA HAURIA DE VENIR DE LCUnion |
---|
3945 | Ln[i][5]=level; |
---|
3946 | LL[size(LL)+1]=Ln[i]; |
---|
3947 | } |
---|
3948 | i=0;j=0; |
---|
3949 | v=0,0; |
---|
3950 | v0=0,0; |
---|
3951 | kill P; list P; |
---|
3952 | for(l=1;l<=size(notadded);l++) |
---|
3953 | { |
---|
3954 | v1=notadded[l]; |
---|
3955 | if(v1[1]<>v0[1]){i++;j=1; P[i]=K;} // list P[i]; |
---|
3956 | else |
---|
3957 | { |
---|
3958 | if(v1[2]<>v0[2]) |
---|
3959 | { |
---|
3960 | j=j+1; |
---|
3961 | } |
---|
3962 | } |
---|
3963 | v=i,j; |
---|
3964 | //"T_v1="; v1; |
---|
3965 | P[i][j]=K; P[i][j][1]=J; P[i][j][2]=K; |
---|
3966 | P[i][j][1]=P0[v1[1]][v1[2]][1]; |
---|
3967 | //"T_P0[v1[1]][v1[2]][2]="; P0[v1[1]][v1[2]][2]; |
---|
3968 | //"T_size(P0[v1[1]][v1[2]][2])="; size(P0[v1[1]][v1[2]][2]); |
---|
3969 | for(k=1;k<=size(P0[v1[1]][v1[2]][2]);k++) |
---|
3970 | { |
---|
3971 | P[i][j][2][k]=P0[v1[1]][v1[2]][2][k]; |
---|
3972 | //string("T_P[",i,"][",j,"][2][",k,"]="); P[i][j][2][k]; |
---|
3973 | } |
---|
3974 | v0=v1; |
---|
3975 | //"T_P_1="; P; |
---|
3976 | } |
---|
3977 | //"T_P="; P; |
---|
3978 | level++; |
---|
3979 | //if(cont>3){break;} |
---|
3980 | //kill notadded; list notadded; |
---|
3981 | } |
---|
3982 | if(defined(@Q2)){kill @Q2;} |
---|
3983 | if(te==0){kill @P; kill @R; kill @RP;} |
---|
3984 | //"T_LL_sortida addcons="; LL; "Fi sortida"; |
---|
3985 | return(LL); |
---|
3986 | } |
---|
3987 | example |
---|
3988 | { |
---|
3989 | "EXAMPLE:"; echo = 2; |
---|
3990 | ring R=(0,a,b),(x1,y1,x2,y2,p,r),dp; |
---|
3991 | ideal S93=(a+1)^2+b^2-(p+1)^2, |
---|
3992 | (a-1)^2+b^2-(1-r)^2, |
---|
3993 | a*y1-b*x1-y1+b, |
---|
3994 | a*y2-b*x2+y2-b, |
---|
3995 | -2*y1+b*x1-(a+p)*y1+b, |
---|
3996 | 2*y2+b*x2-(a+r)*y2-b, |
---|
3997 | (x1+1)^2+y1^2-(x2-1)^2-y2^2; |
---|
3998 | short=0; |
---|
3999 | "System S93="; S93; " "; |
---|
4000 | def GC93=grobcov(S93); |
---|
4001 | "grobcov(S93)="; GC93; " "; |
---|
4002 | int i; |
---|
4003 | list H; |
---|
4004 | for(i=1;i<=size(GC93);i++){H[i]=GC93[i][3];} |
---|
4005 | "H="; H; |
---|
4006 | "addcons(H)="; addcons(H); |
---|
4007 | } |
---|
4008 | |
---|
4009 | // Takes a list of intvec and sorts it and eliminates repeated elements. |
---|
4010 | // Auxiliary routine used in addcons |
---|
4011 | proc sortpairs(L) |
---|
4012 | { |
---|
4013 | def L1=sort(L); |
---|
4014 | def L2=elimrepeated(L1[1]); |
---|
4015 | return(L2); |
---|
4016 | } |
---|
4017 | |
---|
4018 | // Eliminates the pairs of L1 that are also in L2. |
---|
4019 | // Auxiliary routine used in addcons |
---|
4020 | proc minuselements(list L1,list L2) |
---|
4021 | { |
---|
4022 | int i; |
---|
4023 | list L3; |
---|
4024 | for (i=1;i<=size(L1);i++) |
---|
4025 | { |
---|
4026 | if(not(memberpos(L1[i],L2)[1])){L3[size(L3)+1]=L1[i];} |
---|
4027 | } |
---|
4028 | return(L3); |
---|
4029 | } |
---|
4030 | |
---|
4031 | // locus0(G): Private routine used by locus (the public routine), that |
---|
4032 | // builds the diferent components. |
---|
4033 | // input: The output G of the grobcov (in generic representation, which is the default option) |
---|
4034 | // output: |
---|
4035 | // list, the canonical P-representation of the Normal and Non-Normal locus: |
---|
4036 | // The Normal locus has two kind of components: Normal and Special. |
---|
4037 | // The Non-normal locus has two kind of components: Accumulation and Degenerate. |
---|
4038 | // This routine is compemented by locus that calls it in order to eliminate problems |
---|
4039 | // with degenerate points of the mover. |
---|
4040 | // The output components are given as |
---|
4041 | // ((p1,(p11,..p1s_1),type_1,level_1),..,(pk,(pk1,..pks_k),type_k,level_k) |
---|
4042 | // The components are given in canonical P-representation of the subset. |
---|
4043 | // If all levels of a class of locus are 1, then the set is locally closed. Otherwise the level |
---|
4044 | // gives the depth of the component. |
---|
4045 | proc locus0(list GG, list #) |
---|
4046 | { |
---|
4047 | int t1=1; int t2=1; |
---|
4048 | def R=basering; |
---|
4049 | int moverdim=nvars(R); |
---|
4050 | list HHH; |
---|
4051 | if (GG[1][1][1]==1 and GG[1][2][1]==1 and GG[1][3][1][1][1]==0 and GG[1][3][1][2][1]==1){return(HHH);} |
---|
4052 | list Lop=#; |
---|
4053 | if(size(Lop)>0){moverdim=Lop[1];} |
---|
4054 | setglobalrings(); |
---|
4055 | list G1; list G2; |
---|
4056 | def G=GG; |
---|
4057 | list Q1; list Q2; |
---|
4058 | int i; int d; int j; int k; |
---|
4059 | t1=1; |
---|
4060 | for(i=1;i<=size(G);i++) |
---|
4061 | { |
---|
4062 | attrib(G[i][1],"IsSB",1); |
---|
4063 | d=locusdim(G[i][2],moverdim); |
---|
4064 | if(d==0){G1[size(G1)+1]=G[i];} |
---|
4065 | else |
---|
4066 | { |
---|
4067 | if(d>0){G2[size(G2)+1]=G[i];} |
---|
4068 | } |
---|
4069 | } |
---|
4070 | if(size(G1)==0){t1=0;} |
---|
4071 | if(size(G2)==0){t2=0;} |
---|
4072 | setring(@RP); |
---|
4073 | if(t1) |
---|
4074 | { |
---|
4075 | list G1RP=imap(R,G1); |
---|
4076 | } |
---|
4077 | else {list G1RP;} |
---|
4078 | list P1RP; |
---|
4079 | ideal B; |
---|
4080 | for(i=1;i<=size(G1RP);i++) |
---|
4081 | { |
---|
4082 | kill B; |
---|
4083 | ideal B; |
---|
4084 | for(k=1;k<=size(G1RP[i][3]);k++) |
---|
4085 | { |
---|
4086 | attrib(G1RP[i][3][k][1],"IsSB",1); |
---|
4087 | G1RP[i][3][k][1]=std(G1RP[i][3][k][1]); |
---|
4088 | for(j=1;j<=size(G1RP[i][2]);j++) |
---|
4089 | { |
---|
4090 | B[j]=reduce(G1RP[i][2][j],G1RP[i][3][k][1]); |
---|
4091 | } |
---|
4092 | P1RP[size(P1RP)+1]=list(G1RP[i][3][k][1],G1RP[i][3][k][2],B); |
---|
4093 | } |
---|
4094 | } |
---|
4095 | setring(R); |
---|
4096 | ideal h; |
---|
4097 | if(t1) |
---|
4098 | { |
---|
4099 | def P1=imap(@RP,P1RP); |
---|
4100 | for(i=1;i<=size(P1);i++) |
---|
4101 | { |
---|
4102 | for(j=1;j<=size(P1[i][3]);j++) |
---|
4103 | { |
---|
4104 | h=factorize(P1[i][3][j],1); |
---|
4105 | P1[i][3][j]=h[1]; |
---|
4106 | for(k=2;k<=size(h);k++) |
---|
4107 | { |
---|
4108 | P1[i][3][j]=P1[i][3][j]*h[k]; |
---|
4109 | } |
---|
4110 | } |
---|
4111 | } |
---|
4112 | } |
---|
4113 | else{list P1;} |
---|
4114 | ideal BB; |
---|
4115 | for(i=1;i<=size(P1);i++) |
---|
4116 | { |
---|
4117 | if (indepparameters(P1[i][3])==1){P1[i][3]="Special";} |
---|
4118 | else{P1[i][3]="Normal";} |
---|
4119 | } |
---|
4120 | list P2; |
---|
4121 | for(i=1;i<=size(G2);i++) |
---|
4122 | { |
---|
4123 | for(k=1;k<=size(G2[i][3]);k++) |
---|
4124 | { |
---|
4125 | P2[size(P2)+1]=list(G2[i][3][k][1],G2[i][3][k][2]); |
---|
4126 | } |
---|
4127 | } |
---|
4128 | list l; |
---|
4129 | for(i=1;i<=size(P1);i++){Q1[i]=l; Q1[i][1]=P1[i];} P1=Q1; |
---|
4130 | for(i=1;i<=size(P2);i++){Q2[i]=l; Q2[i][1]=P2[i];} P2=Q2; |
---|
4131 | |
---|
4132 | setring @P; |
---|
4133 | ideal J; |
---|
4134 | if(t1==1) |
---|
4135 | { |
---|
4136 | def C1=imap(R,P1); |
---|
4137 | def L1=addcons(C1); |
---|
4138 | } |
---|
4139 | else{list C1; list L1; kill P1; list P1;} |
---|
4140 | if(t2==1) |
---|
4141 | { |
---|
4142 | def C2=imap(R,P2); |
---|
4143 | def L2=addcons(C2); |
---|
4144 | } |
---|
4145 | else{list L2; list C2; kill P2; list P2;} |
---|
4146 | for(i=1;i<=size(L2);i++) |
---|
4147 | { |
---|
4148 | J=std(L2[i][2]); |
---|
4149 | d=dim(J); // AQUI |
---|
4150 | if(d==0) |
---|
4151 | { |
---|
4152 | L2[i][4]=string("Accumulation",L2[i][4]); |
---|
4153 | } |
---|
4154 | else{L2[i][4]=string("Degenerate",L2[i][4]);} |
---|
4155 | } |
---|
4156 | list LN; |
---|
4157 | if(t1==1) |
---|
4158 | { |
---|
4159 | for(i=1;i<=size(L1);i++){LN[size(LN)+1]=L1[i];} |
---|
4160 | } |
---|
4161 | if(t2==1) |
---|
4162 | { |
---|
4163 | for(i=1;i<=size(L2);i++){LN[size(LN)+1]=L2[i];} |
---|
4164 | } |
---|
4165 | setring(R); |
---|
4166 | def L=imap(@P,LN); |
---|
4167 | for(i=1;i<=size(L);i++){if(size(L[i][2])==0){L[i][2]=ideal(1);}} |
---|
4168 | kill @R; kill @RP; kill @P; |
---|
4169 | list LL; |
---|
4170 | for(i=1;i<=size(L);i++) |
---|
4171 | { |
---|
4172 | LL[i]=l; |
---|
4173 | LL[i]=elimfromlist(L[i],1); |
---|
4174 | } |
---|
4175 | return(LL); |
---|
4176 | } |
---|
4177 | |
---|
4178 | // locus0dg(G): Private routine used by locusdg (the public routine), that |
---|
4179 | // builds the diferent components used in Dynamical Geometry |
---|
4180 | // input: The output G of the grobcov (in generic representation, which is the default option) |
---|
4181 | // output: |
---|
4182 | // list, the canonical P-representation of the Relevant components of the locus in Dynamical Geometry, |
---|
4183 | // i.e. the Normal and Accumulation components. |
---|
4184 | // This routine is compemented by locusdg that calls it in order to eliminate problems |
---|
4185 | // with degenerate points of the mover. |
---|
4186 | // The output components are given as |
---|
4187 | // ((p1,(p11,..p1s_1),type_1,level_1),..,(pk,(pk1,..pks_k),type_k,level_k) |
---|
4188 | // whwre type_i is always "Relevant". |
---|
4189 | // The components are given in canonical P-representation of the subset. |
---|
4190 | // If all levels of a class of locus are 1, then the set is locally closed. Otherwise the level |
---|
4191 | // gives the depth of the component. |
---|
4192 | proc locus0dg(list GG, list #) |
---|
4193 | { |
---|
4194 | int t1=1; int t2=1; int ojo; |
---|
4195 | def R=basering; |
---|
4196 | int moverdim=nvars(R); |
---|
4197 | list HHH; |
---|
4198 | if (GG[1][1][1]==1 and GG[1][2][1]==1 and GG[1][3][1][1][1]==0 and GG[1][3][1][2][1]==1){return(HHH);} |
---|
4199 | list Lop=#; |
---|
4200 | if(size(Lop)>0){moverdim=Lop[1];} |
---|
4201 | setglobalrings(); |
---|
4202 | list G1; list G2; |
---|
4203 | def G=GG; |
---|
4204 | list Q1; list Q2; |
---|
4205 | int i; int d; int j; int k; |
---|
4206 | t1=1; |
---|
4207 | for(i=1;i<=size(G);i++) |
---|
4208 | { |
---|
4209 | attrib(G[i][1],"IsSB",1); |
---|
4210 | d=locusdim(G[i][2],moverdim); |
---|
4211 | if(d==0){G1[size(G1)+1]=G[i];} |
---|
4212 | else |
---|
4213 | { |
---|
4214 | if(d>0){G2[size(G2)+1]=G[i];} |
---|
4215 | } |
---|
4216 | } |
---|
4217 | if(size(G1)==0){t1=0;} |
---|
4218 | if(size(G2)==0){t2=0;} |
---|
4219 | setring(@RP); |
---|
4220 | if(t1) |
---|
4221 | { |
---|
4222 | list G1RP=imap(R,G1); |
---|
4223 | } |
---|
4224 | else {list G1RP;} |
---|
4225 | list P1RP; |
---|
4226 | ideal B; |
---|
4227 | for(i=1;i<=size(G1RP);i++) |
---|
4228 | { |
---|
4229 | kill B; |
---|
4230 | ideal B; |
---|
4231 | for(k=1;k<=size(G1RP[i][3]);k++) |
---|
4232 | { |
---|
4233 | attrib(G1RP[i][3][k][1],"IsSB",1); |
---|
4234 | G1RP[i][3][k][1]=std(G1RP[i][3][k][1]); |
---|
4235 | for(j=1;j<=size(G1RP[i][2]);j++) |
---|
4236 | { |
---|
4237 | B[j]=reduce(G1RP[i][2][j],G1RP[i][3][k][1]); |
---|
4238 | } |
---|
4239 | P1RP[size(P1RP)+1]=list(G1RP[i][3][k][1],G1RP[i][3][k][2],B); |
---|
4240 | } |
---|
4241 | } |
---|
4242 | setring(R); |
---|
4243 | ideal h; |
---|
4244 | if(t1) |
---|
4245 | { |
---|
4246 | def P1=imap(@RP,P1RP); |
---|
4247 | for(i=1;i<=size(P1);i++) |
---|
4248 | { |
---|
4249 | for(j=1;j<=size(P1[i][3]);j++) |
---|
4250 | { |
---|
4251 | h=factorize(P1[i][3][j],1); |
---|
4252 | P1[i][3][j]=h[1]; |
---|
4253 | for(k=2;k<=size(h);k++) |
---|
4254 | { |
---|
4255 | P1[i][3][j]=P1[i][3][j]*h[k]; |
---|
4256 | } |
---|
4257 | } |
---|
4258 | } |
---|
4259 | } |
---|
4260 | else{list P1;} |
---|
4261 | ideal BB; |
---|
4262 | for(i=1;i<=size(P1);i++) |
---|
4263 | { |
---|
4264 | if (indepparameters(P1[i][3])==1){P1[i][3]="Special";} |
---|
4265 | else{P1[i][3]="Relevant";} |
---|
4266 | } |
---|
4267 | list P2; |
---|
4268 | for(i=1;i<=size(G2);i++) |
---|
4269 | { |
---|
4270 | for(k=1;k<=size(G2[i][3]);k++) |
---|
4271 | { |
---|
4272 | P2[size(P2)+1]=list(G2[i][3][k][1],G2[i][3][k][2]); |
---|
4273 | } |
---|
4274 | } |
---|
4275 | list l; |
---|
4276 | for(i=1;i<=size(P1);i++){Q1[i]=l; Q1[i][1]=P1[i];} P1=Q1; |
---|
4277 | for(i=1;i<=size(P2);i++){Q2[i]=l; Q2[i][1]=P2[i];} P2=Q2; |
---|
4278 | |
---|
4279 | setring @P; |
---|
4280 | ideal J; |
---|
4281 | if(t1==1) |
---|
4282 | { |
---|
4283 | def C1=imap(R,P1); |
---|
4284 | def L1=addcons(C1); |
---|
4285 | } |
---|
4286 | else{list C1; list L1; kill P1; list P1;} |
---|
4287 | if(t2==1) |
---|
4288 | { |
---|
4289 | def C2=imap(R,P2); |
---|
4290 | def L2=addcons(C2); |
---|
4291 | } |
---|
4292 | else{list L2; list C2; kill P2; list P2;} |
---|
4293 | for(i=1;i<=size(L2);i++) |
---|
4294 | { |
---|
4295 | J=std(L2[i][2]); |
---|
4296 | d=dim(J); // AQUI |
---|
4297 | if(d==0) |
---|
4298 | { |
---|
4299 | L2[i][4]=string("Relevant",L2[i][4]); |
---|
4300 | } |
---|
4301 | else{L2[i][4]=string("Degenerate",L2[i][4]);} |
---|
4302 | } |
---|
4303 | list LN; |
---|
4304 | list ll; list l0; |
---|
4305 | if(t1==1) |
---|
4306 | { |
---|
4307 | for(i=1;i<=size(L1);i++) |
---|
4308 | { |
---|
4309 | if(L1[i][4]=="Relevant") |
---|
4310 | { |
---|
4311 | LN[size(LN)+1]=l0; |
---|
4312 | LN[size(LN)][1]=L1[i]; |
---|
4313 | } |
---|
4314 | } |
---|
4315 | } |
---|
4316 | if(t2==1) |
---|
4317 | { |
---|
4318 | for(i=1;i<=size(L2);i++) |
---|
4319 | { |
---|
4320 | if(L2[i][4]=="Relevant") |
---|
4321 | { |
---|
4322 | LN[size(LN)+1]=l0; |
---|
4323 | LN[size(LN)][1]=L2[i]; |
---|
4324 | } |
---|
4325 | } |
---|
4326 | } |
---|
4327 | list LNN; |
---|
4328 | kill ll; list ll; list lll; list llll; |
---|
4329 | for(i=1;i<=size(LN);i++) |
---|
4330 | { |
---|
4331 | LNN[size(LNN)+1]=l0; |
---|
4332 | ll=LN[i][1]; lll[1]=ll[2]; lll[2]=ll[3]; lll[3]=ll[4]; LNN[size(LNN)][1]=lll; |
---|
4333 | } |
---|
4334 | kill LN; list LN; |
---|
4335 | LN=addcons(LNN); |
---|
4336 | if(size(LN)==0){ojo=1;} |
---|
4337 | setring(R); |
---|
4338 | if(ojo==1){list L;} |
---|
4339 | else |
---|
4340 | { |
---|
4341 | def L=imap(@P,LN); |
---|
4342 | for(i=1;i<=size(L);i++){if(size(L[i][2])==0){L[i][2]=ideal(1);}} |
---|
4343 | } |
---|
4344 | kill @R; kill @RP; kill @P; |
---|
4345 | list LL; |
---|
4346 | for(i=1;i<=size(L);i++) |
---|
4347 | { |
---|
4348 | LL[i]=l; |
---|
4349 | LL[i]=elimfromlist(L[i],1); |
---|
4350 | } |
---|
4351 | return(LL); |
---|
4352 | } |
---|
4353 | |
---|
4354 | |
---|
4355 | // locus(G): Special routine for determining the locus of points |
---|
4356 | // of geometrical constructions. Given a parametric ideal J with |
---|
4357 | // parameters (a_1,..a_m) and variables (x_1,..,xn), |
---|
4358 | // representing the system determining |
---|
4359 | // the locus of points (a_1,..,a_m) who verify certain |
---|
4360 | // properties, computing the grobcov G of |
---|
4361 | // J and applying to it locus, determines the different |
---|
4362 | // classes of locus components. The components can be |
---|
4363 | // Normal, Special, Accumulation, Degenerate. |
---|
4364 | // The output are the components given in P-canonical form |
---|
4365 | // of at most 4 constructible sets: Normal, Special, Accumulation, |
---|
4366 | // Degenerate. |
---|
4367 | // The description of the algorithm and definitions is |
---|
4368 | // given in a forthcoming paper by Abanades, Botana, Montes, Recio. |
---|
4369 | // Usually locus problems have mover coordinates, variables and tracer coordinates. |
---|
4370 | // The mover coordinates are to be placed as the last variables, and by default, |
---|
4371 | // its number is 2. If one consider locus problems in higer dimensions, the number of |
---|
4372 | // mover coordinates (placed as the last variables) is to be given as an option. |
---|
4373 | // |
---|
4374 | // input: The output G of the grobcov (in generic representation, which is the default option for grobcov) |
---|
4375 | // output: |
---|
4376 | // list, the canonical P-representation of the Normal and Non-Normal locus: |
---|
4377 | // The Normal locus has two kind of components: Normal and Special. |
---|
4378 | // The Non-normal locus has two kind of components: Accumulation and Degenerate. |
---|
4379 | // Normal components: for each point in the component, |
---|
4380 | // the number of solutions in the variables is finite, and |
---|
4381 | // the solutions depend on the point in the component if the component is not 0-dimensional. |
---|
4382 | // Special components: for each point in the component, |
---|
4383 | // the number of solutions in the variables is finite, |
---|
4384 | // the component is not 0-dimensional, but the solutions do not depend on the |
---|
4385 | // values of the parameters in the component. |
---|
4386 | // Accumlation points: are 0-dimensional components for which it exist |
---|
4387 | // an infinite number of solutions. |
---|
4388 | // Degenerate components: are components of dimension greater than 0 for which |
---|
4389 | // for every point in the component there exist infinite solutions. |
---|
4390 | // The output components are given as |
---|
4391 | // ((p1,(p11,..p1s_1),type_1,level_1),..,(pk,(pk1,..pks_k),type_k,level_k) |
---|
4392 | // The components are given in canonical P-representation of the subset. |
---|
4393 | // If all levels of a class of locus are 1, then the set is locally closed. Otherwise the level |
---|
4394 | // gives the depth of the component of the constructible set. |
---|
4395 | proc locus(list GG, list #) |
---|
4396 | "USAGE: locus(G); |
---|
4397 | The input must be the grobcov of a parametrical ideal |
---|
4398 | RETURN: The locus. |
---|
4399 | The output are the components of four constructible subsets of the locus |
---|
4400 | of the parametrical system: Normal , Special, Accumulation and Degenerate. |
---|
4401 | These components are |
---|
4402 | given as a list of (pi,(pi1,..pis_i),type_i,level_i) varying i, |
---|
4403 | where the p's are prime ideals, the type can be: Normal, Special, |
---|
4404 | Accumulation, Degenerate. |
---|
4405 | NOTE: It can only be called after computing the grobcov of the |
---|
4406 | parametrical ideal in generic representation ('ext',0), |
---|
4407 | which is the default. |
---|
4408 | The basering R, must be of the form Q[a_1,..,a_m][x_1,..,x_n]. |
---|
4409 | KEYWORDS: geometrical locus, locus, loci. |
---|
4410 | EXAMPLE: locus; shows an example" |
---|
4411 | { |
---|
4412 | def R=basering; |
---|
4413 | int dd=2; int comment=0; |
---|
4414 | list DD=#; |
---|
4415 | if (size(DD)>1){comment=1;} |
---|
4416 | if(size(DD)>0){dd=DD[1];} |
---|
4417 | int i; int j; int k; |
---|
4418 | def B0=GG[1][2]; |
---|
4419 | if (equalideals(B0,ideal(1))) |
---|
4420 | { |
---|
4421 | return(locus0(GG)); |
---|
4422 | } |
---|
4423 | else |
---|
4424 | { |
---|
4425 | int n=nvars(R); |
---|
4426 | ideal vmov=var(n-1),var(n); |
---|
4427 | ideal N; |
---|
4428 | intvec xw; intvec yw; |
---|
4429 | for(i=1;i<=n-1;i++){xw[i]=0;} |
---|
4430 | xw[n]=1; |
---|
4431 | for(i=1;i<=n;i++){yw[i]=0;} |
---|
4432 | yw[n-1]=1; |
---|
4433 | poly px; poly py; |
---|
4434 | int te=1; |
---|
4435 | i=1; |
---|
4436 | while( te and i<=size(B0)) |
---|
4437 | { |
---|
4438 | if((deg(B0[i],xw)==1) and (deg(B0[i])==1)){px=B0[i]; te=0;} |
---|
4439 | i++; |
---|
4440 | } |
---|
4441 | i=1; te=1; |
---|
4442 | while( te and i<=size(B0)) |
---|
4443 | { |
---|
4444 | if((deg(B0[i],yw)==1) and (deg(B0[i])==1)){py=B0[i]; te=0;} |
---|
4445 | i++; |
---|
4446 | } |
---|
4447 | N=px,py; |
---|
4448 | setglobalrings(); |
---|
4449 | te=indepparameters(N); |
---|
4450 | if(te) |
---|
4451 | { |
---|
4452 | string("locus detected that the mover must avoid point (",N,") in order to obtain the correct locus"); |
---|
4453 | // eliminates segments of GG where N is contained in the basis |
---|
4454 | list nGP; |
---|
4455 | def GP=GG; |
---|
4456 | ideal BP; |
---|
4457 | for(j=1;j<=size(GP);j++) |
---|
4458 | { |
---|
4459 | te=1; k=1; |
---|
4460 | BP=GP[j][2]; |
---|
4461 | while((te==1) and (k<=size(N))) |
---|
4462 | { |
---|
4463 | if(pdivi(N[k],BP)[1]!=0){te=0;} |
---|
4464 | k++; |
---|
4465 | } |
---|
4466 | if(te==0){nGP[size(nGP)+1]=GP[j];} |
---|
4467 | } |
---|
4468 | } |
---|
4469 | } |
---|
4470 | kill @RP; kill @P; kill @R; |
---|
4471 | return(locus0(nGP,dd)); |
---|
4472 | } |
---|
4473 | example |
---|
4474 | {"EXAMPLE:"; echo = 2; |
---|
4475 | ring R=(0,a,b),(x4,x3,x2,x1),dp; |
---|
4476 | ideal S=(x1-3)^2+(x2-1)^2-9, |
---|
4477 | (4-x2)*(x3-3)+(x1-3)*(x4-1), |
---|
4478 | (3-x1)*(x3-x1)+(4-x2)*(x4-x2), |
---|
4479 | (4-x4)*a+(x3-3)*b+3*x4-4*x3, |
---|
4480 | (a-x1)^2+(b-x2)^2-(x1-x3)^2-(x2-x4)^2; |
---|
4481 | short=0; |
---|
4482 | locus(grobcov(S)); " "; |
---|
4483 | kill R; |
---|
4484 | ring R=(0,a,b),(x,y),dp; |
---|
4485 | short=0; |
---|
4486 | "Concoid"; |
---|
4487 | ideal S96=x^2+y^2-4,(b-2)*x-a*y+2*a,(a-x)^2+(b-y)^2-1; |
---|
4488 | "System="; S96; " "; |
---|
4489 | locus(grobcov(S96)); |
---|
4490 | kill R; ring R=(0,x,y),(x1,x2),dp; |
---|
4491 | ideal S=-(x - 5)*(x1 - 1) - (x2 - 2)*(y - 2), |
---|
4492 | (x1 - 5)^2 + (x2 - 2)^2 - 4, |
---|
4493 | -2*(x - 5)*(x2 - 2) + 2*(x1 - 5)*(y - 2); |
---|
4494 | "System="; S; |
---|
4495 | locus(grobcov(S)); " "; |
---|
4496 | ideal S1=-(x - x1)*(x1 - 4) + (x2 - y)*(x2 - 4), |
---|
4497 | (x1 - 4)^2 + (x2 - 2)^2 - 4, |
---|
4498 | -2*(x1 - 4)*(2*x2 - y - 4) - 2*(x - 2*x1 + 4)*(x2 - 2); |
---|
4499 | "System="; S1; |
---|
4500 | locus(grobcov(S1)); |
---|
4501 | } |
---|
4502 | |
---|
4503 | // locusdg(G): Special routine for determining the locus of points |
---|
4504 | // of geometrical constructions. Given a parametric ideal J with |
---|
4505 | // parameters (a_1,..a_m) and variables (x_1,..,xn), |
---|
4506 | // representing the system determining |
---|
4507 | // the locus of points (a_1,..,a_m) who verify certain |
---|
4508 | // properties, computing the grobcov G of |
---|
4509 | // J and applying to it locus, determines the different |
---|
4510 | // classes of locus components. The components can be |
---|
4511 | // Normal, Special, Accumulation point, Degenerate. |
---|
4512 | // The output are the components given in P-canonical form |
---|
4513 | // of at most 4 constructible sets: Normal, Special, Accumulation, |
---|
4514 | // Degenerate. |
---|
4515 | // The description of the algorithm and definitions is |
---|
4516 | // given in a forthcoming paper by Abanades, Botana, Montes, Recio. |
---|
4517 | // Usually locus problems have mover coordinates, variables and tracer coordinates. |
---|
4518 | // The mover coordinates are to be placed as the last variables, and by default, |
---|
4519 | // its number is 2. If onw consider locus problems in higer dimensions, the number of |
---|
4520 | // mover coordinates (placed as the last variables) is to be given as an option. |
---|
4521 | // |
---|
4522 | // input: The output G of the grobcov (in generic representation, which is the default option for grobcov) |
---|
4523 | // output: |
---|
4524 | // list, the canonical P-representation of the Normal and Non-Normal locus: |
---|
4525 | // The Normal locus has two kind of components: Normal and Special. |
---|
4526 | // The Non-normal locus has two kind of components: Accumulation and Degenerate. |
---|
4527 | // Normal components: for each point in the component, |
---|
4528 | // the number of solutions in the variables is finite, and |
---|
4529 | // the solutions depend on the point in the component if the component is not 0-dimensional. |
---|
4530 | // Special components: for each point in the component, |
---|
4531 | // the number of solutions in the variables is finite, |
---|
4532 | // the component is not 0-dimensional, but the solutions do not depend on the |
---|
4533 | // values of the parameters in the component. |
---|
4534 | // Accumlation points: are 0-dimensional components for which it exist |
---|
4535 | // an infinite number of solutions. |
---|
4536 | // Degenerate components: are components of dimension greater than 0 for which |
---|
4537 | // for every point in the component there exist infinite solutions. |
---|
4538 | // The output components are given as |
---|
4539 | // ((p1,(p11,..p1s_1),type_1,level_1),..,(pk,(pk1,..pks_k),type_k,level_k) |
---|
4540 | // The components are given in canonical P-representation of the subset. |
---|
4541 | // If all levels of a class of locus are 1, then the set is locally closed. Otherwise the level |
---|
4542 | // gives the depth of the component of the constructible set. |
---|
4543 | proc locusdg(list GG, list #) |
---|
4544 | "USAGE: locus(G); |
---|
4545 | The input must be the grobcov of a parametrical ideal |
---|
4546 | RETURN: The locus. |
---|
4547 | The output are the components of two constructible subsets of the locus |
---|
4548 | of the parametrical system.: Normal and Non-normal. |
---|
4549 | These components are |
---|
4550 | given as a list of (pi,(pi1,..pis_i),type_i,level_i) varying i, |
---|
4551 | where the p's are prime ideals, the type can be: Normal, Special, |
---|
4552 | Accumulation, Degenerate. |
---|
4553 | NOTE: It can only be called after computing the grobcov of the |
---|
4554 | parametrical ideal in generic representation ('ext',0), |
---|
4555 | which is the default. |
---|
4556 | The basering R, must be of the form Q[a_1,..,a_m][x_1,..,x_n]. |
---|
4557 | KEYWORDS: geometrical locus, locus, loci. |
---|
4558 | EXAMPLE: locusdg; shows an example" |
---|
4559 | { |
---|
4560 | def R=basering; |
---|
4561 | int dd=2; int comment=0; |
---|
4562 | list DD=#; |
---|
4563 | if (size(DD)>1){comment=1;} |
---|
4564 | if(size(DD)>0){dd=DD[1];} |
---|
4565 | int i; int j; int k; |
---|
4566 | def B0=GG[1][2]; |
---|
4567 | if (equalideals(B0,ideal(1))) |
---|
4568 | { |
---|
4569 | return(locus0dg(GG)); |
---|
4570 | } |
---|
4571 | else |
---|
4572 | { |
---|
4573 | int n=nvars(R); |
---|
4574 | ideal vmov=var(n-1),var(n); |
---|
4575 | ideal N; |
---|
4576 | intvec xw; intvec yw; |
---|
4577 | for(i=1;i<=n-1;i++){xw[i]=0;} |
---|
4578 | xw[n]=1; |
---|
4579 | for(i=1;i<=n;i++){yw[i]=0;} |
---|
4580 | yw[n-1]=1; |
---|
4581 | poly px; poly py; |
---|
4582 | int te=1; |
---|
4583 | i=1; |
---|
4584 | while( te and i<=size(B0)) |
---|
4585 | { |
---|
4586 | if((deg(B0[i],xw)==1) and (deg(B0[i])==1)){px=B0[i]; te=0;} |
---|
4587 | i++; |
---|
4588 | } |
---|
4589 | i=1; te=1; |
---|
4590 | while( te and i<=size(B0)) |
---|
4591 | { |
---|
4592 | if((deg(B0[i],yw)==1) and (deg(B0[i])==1)){py=B0[i]; te=0;} |
---|
4593 | i++; |
---|
4594 | } |
---|
4595 | N=px,py; |
---|
4596 | setglobalrings(); |
---|
4597 | te=indepparameters(N); |
---|
4598 | if(te) |
---|
4599 | { |
---|
4600 | string("locus detected that the mover must avoid point (",N,") in order to obtain the correct locus"); |
---|
4601 | // eliminates segments of GG where N is contained in the basis |
---|
4602 | list nGP; |
---|
4603 | def GP=GG; |
---|
4604 | ideal BP; |
---|
4605 | for(j=1;j<=size(GP);j++) |
---|
4606 | { |
---|
4607 | te=1; k=1; |
---|
4608 | BP=GP[j][2]; |
---|
4609 | while((te==1) and (k<=size(N))) |
---|
4610 | { |
---|
4611 | if(pdivi(N[k],BP)[1]!=0){te=0;} |
---|
4612 | k++; |
---|
4613 | } |
---|
4614 | if(te==0){nGP[size(nGP)+1]=GP[j];} |
---|
4615 | } |
---|
4616 | } |
---|
4617 | } |
---|
4618 | //"T_nGP="; nGP; |
---|
4619 | kill @RP; kill @P; kill @R; |
---|
4620 | return(locus0dg(nGP,dd)); |
---|
4621 | } |
---|
4622 | example |
---|
4623 | {"EXAMPLE:"; echo = 2; |
---|
4624 | ring R=(0,a,b),(x4,x3,x2,x1),dp; |
---|
4625 | ideal S=(x1-3)^2+(x2-1)^2-9, |
---|
4626 | (4-x2)*(x3-3)+(x1-3)*(x4-1), |
---|
4627 | (3-x1)*(x3-x1)+(4-x2)*(x4-x2), |
---|
4628 | (4-x4)*a+(x3-3)*b+3*x4-4*x3, |
---|
4629 | (a-x1)^2+(b-x2)^2-(x1-x3)^2-(x2-x4)^2; |
---|
4630 | short=0; |
---|
4631 | locus(grobcov(S)); " "; |
---|
4632 | kill R; |
---|
4633 | ring R=(0,a,b),(x,y),dp; |
---|
4634 | short=0; |
---|
4635 | "Concoid"; |
---|
4636 | ideal S96=x^2+y^2-4,(b-2)*x-a*y+2*a,(a-x)^2+(b-y)^2-1; |
---|
4637 | "System="; S96; " "; |
---|
4638 | locusdg(grobcov(S96)); |
---|
4639 | kill R; ring R=(0,x,y),(x1,x2),dp; |
---|
4640 | ideal S=-(x - 5)*(x1 - 1) - (x2 - 2)*(y - 2), |
---|
4641 | (x1 - 5)^2 + (x2 - 2)^2 - 4, |
---|
4642 | -2*(x - 5)*(x2 - 2) + 2*(x1 - 5)*(y - 2); |
---|
4643 | "System="; S; |
---|
4644 | locusdg(grobcov(S)); " "; |
---|
4645 | ideal S1=-(x - x1)*(x1 - 4) + (x2 - y)*(x2 - 4), |
---|
4646 | (x1 - 4)^2 + (x2 - 2)^2 - 4, |
---|
4647 | -2*(x1 - 4)*(2*x2 - y - 4) - 2*(x - 2*x1 + 4)*(x2 - 2); |
---|
4648 | "System="; S1; |
---|
4649 | locusdg(grobcov(S1)); |
---|
4650 | } |
---|
4651 | |
---|
4652 | // locusto: Transforms the output of locus to a string that |
---|
4653 | // can be read by different computational systems. |
---|
4654 | // input: |
---|
4655 | // list L: The output of locus |
---|
4656 | // output: |
---|
4657 | // string s: The output of locus converted to a string readable by other programs |
---|
4658 | proc locusto(list L) |
---|
4659 | "USAGE: locusto(L); |
---|
4660 | The argument must be the output of locus of a parametrical ideal |
---|
4661 | It transforms the output into a string in standard form |
---|
4662 | readable in many languages (Geogebra). |
---|
4663 | RETURN: The locus in string standard form |
---|
4664 | NOTE: It can only be called after computing the locus(grobcov(F)) of the |
---|
4665 | parametrical ideal. |
---|
4666 | The basering R, must be of the form Q[a,b,..][x,y,..]. |
---|
4667 | KEYWORDS: geometrical locus, locus, loci. |
---|
4668 | EXAMPLE: locusto; shows an example" |
---|
4669 | { |
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4670 | int i; int j; int k; |
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4671 | string s="["; string sf="]"; string st=s+sf; |
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4672 | if(size(L)==0){return(st);} |
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4673 | ideal p; |
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4674 | ideal q; |
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4675 | for(i=1;i<=size(L);i++) |
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4676 | { |
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4677 | s=string(s,"[["); |
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4678 | for (j=1;j<=size(L[i][1]);j++) |
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4679 | { |
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4680 | s=string(s,L[i][1][j],","); |
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4681 | } |
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4682 | s[size(s)]="]"; |
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4683 | s=string(s,",["); |
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4684 | for(j=1;j<=size(L[i][2]);j++) |
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4685 | { |
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4686 | s=string(s,"["); |
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4687 | for(k=1;k<=size(L[i][2][j]);k++) |
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4688 | { |
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4689 | s=string(s,L[i][2][j][k],","); |
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4690 | } |
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4691 | s[size(s)]="]"; |
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4692 | s=string(s,","); |
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4693 | } |
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4694 | s[size(s)]="]"; |
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4695 | s=string(s,"]"); |
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4696 | if(size(L[i])>=3) |
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4697 | { |
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4698 | s[size(s)]=","; |
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4699 | s=string(s,string(L[i][3]),"]"); |
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4700 | } |
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4701 | if(size(L[i])>=4) |
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4702 | { |
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4703 | s[size(s)]=","; |
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4704 | s=string(s,string(L[i][4]),"],"); |
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4705 | } |
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4706 | s[size(s)]="]"; |
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4707 | s=string(s,","); |
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4708 | } |
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4709 | s[size(s)]="]"; |
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4710 | return(s); |
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4711 | } |
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4712 | example |
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4713 | {"EXAMPLE:"; echo = 2; |
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4714 | ring R=(0,a,b),(x,y),dp; |
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4715 | short=0; |
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4716 | ideal S96=x^2+y^2-4,(b-2)*x-a*y+2*a,(a-x)^2+(b-y)^2-1; |
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4717 | "System="; S96; " "; |
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4718 | locusto(locus(grobcov(S96))); |
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4719 | } |
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4720 | |
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4721 | // Auxiliary routione |
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4722 | // locusdim |
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4723 | // input: |
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4724 | // B: ideal, a basis of a segment of the grobcov |
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4725 | // dgdim: int, the dimension of the mover (for locusdg) |
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4726 | // by default dgdim is equal to the number of variables |
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4727 | proc locusdim(ideal B, list #) |
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4728 | { |
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4729 | def R=basering; |
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4730 | int dgdim; |
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4731 | int nv=nvars(R); |
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4732 | if (size(#)>0){dgdim=#[1];} |
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4733 | else {dgdim=nv;} |
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4734 | int d; |
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4735 | list v; |
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4736 | ideal vi; |
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4737 | int i; |
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4738 | for(i=1;i<=dgdim;i++) |
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4739 | { |
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4740 | v[size(v)+1]=varstr(nv-dgdim+i); |
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4741 | vi[size(v)+1]=var(nv-dgdim+i); |
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4742 | } |
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4743 | ideal B0; |
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4744 | for(i=1;i<=size(B);i++) |
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4745 | { |
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4746 | if(subset(variables(B[i]),vi)) |
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4747 | { |
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4748 | B0[size(B0)+1]=B[i]; |
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4749 | } |
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4750 | } |
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4751 | //"T_B0="; B0; |
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4752 | //"T_v="; v; |
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4753 | def RR=ringlist(R); |
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4754 | def RR0=RR; |
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4755 | RR0[2]=v; |
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4756 | def R0=ring(RR0); |
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4757 | //ringlist(R0); |
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4758 | setring(R0); |
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4759 | def B0r=imap(R,B0); |
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4760 | B0r=std(B0r); |
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4761 | d=dim(B0r); |
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4762 | setring R; |
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4763 | return(d); |
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4764 | } |
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4765 | |
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4766 | // locusdgto: Transforms the output of locus to a string that |
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4767 | // can be read by different computational systems. |
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4768 | // input: |
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4769 | // list L: The output of locus |
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4770 | // output: |
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4771 | // string s: The output of locus converted to a string readable by other programs |
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4772 | // Outputs only the relevant dynamical geometry components. |
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4773 | // Without unnecessary parenthesis |
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4774 | proc locusdgto(list LL) |
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4775 | { |
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4776 | def RR=basering; |
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4777 | int i; int j; int k; int short0=short; int ojo; |
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4778 | int te=0; |
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4779 | if(defined(@P)){te=1;} |
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4780 | else{setglobalrings();} |
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4781 | setring @P; |
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4782 | short=0; |
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4783 | if(size(LL)==0){ojo=1; list L;} |
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4784 | else |
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4785 | { |
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4786 | def L=imap(RR,LL); |
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4787 | } |
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4788 | string s="["; string sf="]"; string st=s+sf; |
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4789 | if(size(L)==0){return(st);} |
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4790 | ideal p; |
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4791 | ideal q; |
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4792 | for(i=1;i<=size(L);i++) |
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4793 | { |
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4794 | if(L[i][3]=="Relevant") |
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4795 | { |
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4796 | s=string(s,"[["); |
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4797 | for (j=1;j<=size(L[i][1]);j++) |
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4798 | { |
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4799 | s=string(s,L[i][1][j],","); |
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4800 | } |
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4801 | s[size(s)]="]"; |
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4802 | s=string(s,",["); |
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4803 | for(j=1;j<=size(L[i][2]);j++) |
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4804 | { |
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4805 | s=string(s,"["); |
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4806 | for(k=1;k<=size(L[i][2][j]);k++) |
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4807 | { |
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4808 | s=string(s,L[i][2][j][k],","); |
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4809 | } |
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4810 | s[size(s)]="]"; |
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4811 | s=string(s,","); |
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4812 | } |
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4813 | s[size(s)]="]"; |
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4814 | s=string(s,"]"); |
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4815 | s[size(s)]="]"; |
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4816 | s=string(s,","); |
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4817 | } |
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4818 | } |
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4819 | if(s=="["){s="[]";} |
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4820 | else{s[size(s)]="]";} |
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4821 | setring(RR); |
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4822 | short=short0; |
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4823 | if(te==0){kill @P; kill @R; kill @RP;} |
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4824 | return(s); |
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4825 | } |
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4826 | |
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4827 | //********************* End locus **************************** |
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4828 | ; |
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