1 | // |
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2 | version="version grobcov.lib 4.0.1.2 Jan_2015 "; // $Id$ |
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3 | category="General purpose"; |
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4 | info=" |
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5 | LIBRARY: grobcov.lib Groebner Cover for parametric ideals. |
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6 | |
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7 | Comprehensive Groebner Systems, Groebner Cover, Canonical Forms, |
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8 | Parametric Polynomial Systems, Dynamic Geometry, Loci, Envelop, |
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9 | Constructible sets. |
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10 | See |
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11 | |
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12 | A. Montes A, M. Wibmer, |
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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 | (http://www-ma2.upc.edu/~montes/). |
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16 | |
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17 | AUTHORS: Antonio Montes , Hans Schoenemann. |
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18 | |
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19 | OVERVIEW: |
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20 | In 2010, the library was designed to contain Montes-Wibmer's |
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21 | algorithms for compute the canonical Groebner Cover of a |
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22 | parametric ideal as described in the paper: |
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23 | |
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24 | Montes A., Wibmer M., |
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25 | \"Groebner Bases for Polynomial Systems with parameters\". |
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26 | Journal of Symbolic Computation 45 (2010) 1391-1425. |
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27 | |
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28 | The central routine is grobcov. Given a parametric |
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29 | ideal, grobcov outputs its Canonical Groebner Cover, consisting |
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30 | of a set of pairs of (basis, segment). The basis (after |
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31 | normalization) is the reduced Groebner basis for each point |
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32 | of the segment. The segments are disjoint, locally closed |
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33 | and correspond to constant lpp (leading power product) |
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34 | of the basis, and are represented in canonical prime |
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35 | representation. The segments are disjoint and cover the |
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36 | whole parameter space. The output is canonical, it only |
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37 | depends on the given parametric ideal and the monomial order. |
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38 | This is much more than a simple Comprehensive Groebner System. |
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39 | The algorithm grobcov allows options to solve partially the |
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40 | problem when the whole automatic algorithm does not finish |
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41 | in reasonable time. |
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42 | |
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43 | grobcov uses a first algorithm cgsdr that outputs a disjoint |
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44 | reduced Comprehensive Groebner System with constant lpp. |
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45 | For this purpose, in this library, the implemented algorithm is |
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46 | Kapur-Sun-Wang algorithm, because it is the most efficient |
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47 | algorithm known for this purpose. |
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48 | |
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49 | D. Kapur, Y. Sun, and D.K. Wang. |
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50 | \"A New Algorithm for Computing Comprehensive Groebner Systems\". |
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51 | Proceedings of ISSAC'2010, ACM Press, (2010), 29-36. |
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52 | |
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53 | The library has evolved to include new applications of the |
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54 | Groebner Cover, and new theoretical developments have been done. |
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55 | |
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56 | A new set of routines (locus, locusdg, locusto) has been included to |
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57 | compute loci of points. The routines are used in the Dynamic |
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58 | Geometry software Geogebra. They are described in: |
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59 | |
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60 | Abanades, Botana, Montes, Recio: |
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61 | \''An Algebraic Taxonomy for Locus Computation in Dynamic Geometry\''. |
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62 | Computer-Aided Design 56 (2014) 22-33. |
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63 | |
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64 | Recently also routines for computing the envelop of a family |
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65 | of curves (enverlop, envelopdg), to be used in Dynamic Geometry, |
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66 | has been included and will be described in a forthcoming paper: |
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67 | |
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68 | Abanades, Botana, Montes, Recio: |
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69 | \''Envelops in Dynamic Geometry using the Groebner cover\''. |
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70 | |
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71 | The actual version also includes two routines (AddCons and AddconsP) |
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72 | for computing the canonical form of a constructible set, given as a |
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73 | union of locally closed sets. They are used in the new version for the |
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74 | computation of loci and envelops. It determines the canonical locally closed |
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75 | level sets of a constructuble. They will be described in a forthcoming paper: |
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76 | |
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77 | A. Montes, J.M. Brunat, |
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78 | \"Canonical representations of constructible sets\". |
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79 | |
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80 | This version was finished on 31/01/2015 |
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81 | |
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82 | NOTATIONS: All given and determined polynomials and ideals are in the |
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83 | @* basering Q[a][x]; (a=parameters, x=variables) |
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84 | @* After defining the ring, the main routines |
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85 | @* grobcov, cgsdr, |
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86 | @* generate the global rings |
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87 | @* Grobcov::@R (Q[a][x]), |
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88 | @* Grobcov::@P (Q[a]), |
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89 | @* Grobcov::@RP (Q[x,a]) |
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90 | @* that are used inside and killed before the output. |
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91 | @* If you want to use some internal routine you must |
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92 | @* create before the above rings by calling setglobalrings(); |
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93 | @* because some of the internal routines use these rings. |
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94 | @* The call to the basic routines grobcov, cgsdr will |
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95 | @* kill these rings. |
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96 | |
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97 | PROCEDURES: |
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98 | grobcov(F); Is the basic routine giving the canonical |
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99 | Groebner Cover of the parametric ideal F. |
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100 | This routine accepts many options, that |
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101 | allow to obtain results even when the canonical |
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102 | computation does not finish in reasonable time. |
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103 | |
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104 | cgsdr(F); Is the procedure for obtaining a first disjoint, |
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105 | reduced Comprehensive Groebner System that |
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106 | is used in grobcov, that can also be used |
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107 | independently if only the CGS is required. |
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108 | It is a more efficient routine than buildtree |
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109 | (the own routine of 2010 that is no more used). |
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110 | Now, KSW algorithm is used. |
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111 | |
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112 | setglobalrings(); Generates the global rings @R, @P and @PR that |
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113 | are respectively the rings Q[a][x], Q[a], Q[x,a]. (a=parameters, |
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114 | x=variables) It is called inside each of the fundamental |
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115 | routines of the library: grobcov, cgsdr, locus, locusdg and |
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116 | killed before the output. |
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117 | If the user want to use some other internal routine, |
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118 | then setglobalrings() is to be called before, as most of |
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119 | them use these rings. |
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120 | |
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121 | pdivi(f,F); Performs a pseudodivision of a parametric polynomial |
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122 | by a parametric ideal. |
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123 | |
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124 | pnormalf(f,E,N); Reduces a parametric polynomial f over V(E) \ V(N) |
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125 | ( E is the null ideal and N the non-null ideal ) |
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126 | over the parameters. |
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127 | |
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128 | Crep(N,M); Computes the canonical C-representation of V(N) \ V(M). |
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129 | |
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130 | Prep(N,M); Computes the canonical P-representation of V(N) \ V(M). |
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131 | |
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132 | PtoCrep(L) Starting from the canonical Prep of a locally closed set |
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133 | computes its Crep. |
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134 | |
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135 | extend(GC); When the grobcov of an ideal has been computed |
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136 | with the default option ('ext',0) and the explicit |
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137 | option ('rep',2) (which is not the default), then |
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138 | one can call extend (GC) (and options) to obtain the |
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139 | full representation of the bases. With the default |
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140 | option ('ext',0) only the generic representation of |
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141 | the bases are computed, and one can obtain the full |
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142 | representation using extend. |
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143 | |
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144 | locus(G); Special routine for determining the geometrical locus of points |
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145 | verifying given conditions. Given a parametric ideal J with |
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146 | parameters (x,y) and variables (x_1,..,xn), representing the |
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147 | system determining the locus of points (x,y) who verify certain |
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148 | properties, one can apply locus to the output of grobcov(J), |
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149 | locus determines the different classes of locus components. |
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150 | described in the paper: |
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151 | |
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152 | \"An Algebraic Taxonomy for Locus Computation in Dynamic Geometry\", |
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153 | M. Abanades, F. Botana, A. Montes, T. Recio, |
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154 | Computer-Aided Design 56 (2014) 22-33. |
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155 | |
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156 | The components can be 'Normal', 'Special', 'Accumulation', 'Degenerate'. |
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157 | The output are the components is given in P-canonical form |
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158 | It also detects automatically a possible point that is to be |
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159 | avoided by the mover, whose coordinates must be the last two |
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160 | coordinates in the definition of the ring. If such a point is detected, |
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161 | then it eliminates the segments of the grobcov depending on the |
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162 | point that is to be avoided. |
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163 | |
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164 | locusdg(G); Is a special routine that determines the 'Relevant' components |
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165 | of the locus in dynamic geometry. It is to be called to the output of locus |
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166 | and selects from it the useful components. |
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167 | |
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168 | envelop(F,C); Special routine for determining the envelop of a family of curves |
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169 | F in Q[x,y][x_1,..xn] depending on a ideal of constraints C in Q[x_1,..,xn]. |
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170 | It detemines the different components as well as its type: |
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171 | 'Normal', 'Special', 'Accumulation', 'Degenerate'. And |
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172 | it also classifies the Special components, determining the |
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173 | zero dimensional antiimage of the component and verifying if |
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174 | the component is a special curve of the family or not. |
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175 | It calls internally first grobcov and then locus with special options |
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176 | to obtain the complete result. |
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177 | The taxonomy that it provides, as well as the algorithms involved |
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178 | will be described in a forthcoming paper: |
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179 | |
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180 | Abanades, Botana, Montes, Recio: |
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181 | \''Envelops in Dynamic Geometry using the Gr\"obner cover\''. |
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182 | |
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183 | |
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184 | envelopdg(ev); Is a special routine to determine the 'Relevant' components |
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185 | of the envelop of a family of curves to be used in Dynamic Geometry. |
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186 | It must be called to the output of envelop(F,C). |
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187 | |
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188 | locusto(L); Transforms the output of locus, locusdg, envelop and envelopdg |
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189 | into a string that can be reed from different computational systems. |
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190 | |
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191 | AddCons(L); Uses the routine AddConsP. Given a set of locally closed sets as |
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192 | difference of of varieties (does not need to be in C-representation) |
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193 | it returns the canonical P-representation of the constructible set |
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194 | formed by the union of them. The result is formed by a set of embedded, |
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195 | disjoint, locally closed sets (levels). |
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196 | |
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197 | AddConsP(L); Given a set of locally closed P-components, it returns the |
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198 | canonical P-representation of the constructible set |
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199 | formed by the union of them, consisting of a set of embedded, |
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200 | disjoint locally closed sets (levels). |
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201 | The routines AddCons and AddConsP and the canonical structure |
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202 | of the constructible sets will be described in a forthcoming paper. |
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203 | |
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204 | A. Montes, J.M. Brunat, |
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205 | \"Canonical representations of constructible sets\". |
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206 | |
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207 | SEE ALSO: compregb_lib |
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208 | "; |
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209 | |
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210 | LIB "primdec.lib"; |
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211 | LIB "qhmoduli.lib"; |
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212 | |
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213 | // ************ Begin of the grobcov library ********************* |
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214 | |
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215 | // Library grobcov.lib |
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216 | // (Groebner Cover): |
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217 | // Release 0: (public) |
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218 | // Initial data: 21-1-2008 |
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219 | // Uses buildtree for cgsdr |
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220 | // Final data: 3-7-2008 |
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221 | // Release 2: (private) |
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222 | // Initial data: 6-9-2009 |
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223 | // Last version using buildtree for cgsdr |
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224 | // Final data: 25-10-2011 |
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225 | // Release B: (private) |
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226 | // Initial data: 1-7-2012 |
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227 | // Uses both buildtree and KSW for cgsdr |
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228 | // Final data: 4-9-2012 |
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229 | // Release G: (public) |
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230 | // Initial data: 4-9-2012 |
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231 | // Uses KSW algorithm for cgsdr |
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232 | // Final data: 21-11-2013 |
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233 | // Release K: (public) |
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234 | // Updated locus: 8-1-2015 |
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235 | // Updated AddConsP and AddCons: 8-1-2015 |
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236 | // Reformed many routines, examples and helps: 8-1-2015 |
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237 | // New routines for computing the envelop of a family of curves: 22-1-2015 |
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238 | // Final data: 22-1-2015 |
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239 | |
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240 | //*************Auxiliary routines************** |
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241 | |
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242 | // elimintfromideal: elimine the constant numbers from an ideal |
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243 | // (designed for W, nonnull conditions) |
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244 | // Input: ideal J |
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245 | // Output:ideal K with the elements of J that are non constants, in the |
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246 | // ring K[x1,..,xm] |
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247 | static proc elimintfromideal(ideal J) |
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248 | { |
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249 | int i; |
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250 | int j=0; |
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251 | ideal K; |
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252 | if (size(J)==0){return(ideal(0));} |
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253 | for (i=1;i<=ncols(J);i++){if (size(variables(J[i])) !=0){j++; K[j]=J[i];}} |
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254 | return(K); |
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255 | } |
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256 | |
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257 | // delfromideal: deletes the i-th polynomial from the ideal F |
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258 | // Works in any kind of ideal |
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259 | static proc delfromideal(ideal F, int i) |
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260 | { |
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261 | int j; |
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262 | ideal G; |
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263 | if (size(F)<i){ERROR("delfromideal was called with incorrect arguments");} |
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264 | if (size(F)<=1){return(ideal(0));} |
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265 | if (i==0){return(F)}; |
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266 | for (j=1;j<=ncols(F);j++) |
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267 | { |
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268 | if (j!=i){G[ncols(G)+1]=F[j];} |
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269 | } |
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270 | return(G); |
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271 | } |
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272 | |
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273 | // delidfromid: deletes the polynomials in J that are in I |
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274 | // Input: ideals I, J |
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275 | // Output: the ideal J without the polynomials in I |
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276 | // Works in any kind of ideal |
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277 | static proc delidfromid(ideal I, ideal J) |
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278 | { |
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279 | int i; list r; |
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280 | ideal JJ=J; |
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281 | for (i=1;i<=size(I);i++) |
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282 | { |
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283 | r=memberpos(I[i],JJ); |
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284 | if (r[1]) |
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285 | { |
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286 | JJ=delfromideal(JJ,r[2]); |
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287 | } |
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288 | } |
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289 | return(JJ); |
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290 | } |
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291 | |
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292 | // eliminates the ith element from a list or an intvec |
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293 | static proc elimfromlist(l, int i) |
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294 | { |
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295 | if(typeof(l)=="list"){list L;} |
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296 | if (typeof(l)=="intvec"){intvec L;} |
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297 | if (typeof(l)=="ideal"){ideal L;} |
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298 | int j; |
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299 | if((size(l)==0) or (size(l)==1 and i!=1)){return(l);} |
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300 | if (size(l)==1 and i==1){return(L);} |
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301 | // L=l[1]; |
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302 | if(i==1) |
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303 | { |
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304 | for(j=2;j<=size(l);j++) |
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305 | { |
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306 | L[j-1]=l[j]; |
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307 | } |
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308 | } |
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309 | else |
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310 | { |
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311 | for(j=1;j<=i-1;j++) |
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312 | { |
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313 | L[j]=l[j]; |
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314 | } |
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315 | for(j=i+1;j<=size(l);j++) |
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316 | { |
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317 | L[j-1]=l[j]; |
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318 | } |
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319 | } |
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320 | return(L); |
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321 | } |
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322 | |
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323 | // eliminates repeated elements form an ideal or matrix or module or intmat or bigintmat |
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324 | static proc elimrepeated(F) |
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325 | { |
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326 | int i; |
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327 | int nt; |
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328 | if (typeof(F)=="ideal"){nt=ncols(F);} |
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329 | else{nt=size(F);} |
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330 | |
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331 | def FF=F; |
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332 | FF=F[1]; |
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333 | for (i=2;i<=nt;i++) |
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334 | { |
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335 | if (not(memberpos(F[i],FF)[1])) |
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336 | { |
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337 | FF[size(FF)+1]=F[i]; |
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338 | } |
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339 | } |
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340 | return(FF); |
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341 | } |
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342 | |
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343 | |
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344 | // equalideals |
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345 | // Input: ideals F and G; |
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346 | // Output: 1 if they are identical (the same polynomials in the same order) |
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347 | // 0 else |
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348 | static proc equalideals(ideal F, ideal G) |
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349 | { |
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350 | int i=1; int t=1; |
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351 | if (size(F)!=size(G)){return(0);} |
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352 | while ((i<=size(F)) and (t)) |
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353 | { |
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354 | if (F[i]!=G[i]){t=0;} |
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355 | i++; |
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356 | } |
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357 | return(t); |
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358 | } |
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359 | |
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360 | // returns 1 if the two lists of ideals are equal and 0 if not |
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361 | static proc equallistideals(list L, list M) |
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362 | { |
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363 | int t; int i; |
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364 | if (size(L)!=size(M)){return(0);} |
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365 | else |
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366 | { |
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367 | t=1; |
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368 | if (size(L)>0) |
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369 | { |
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370 | i=1; |
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371 | while ((t) and (i<=size(L))) |
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372 | { |
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373 | if (equalideals(L[i],M[i])==0){t=0;} |
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374 | i++; |
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375 | } |
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376 | } |
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377 | return(t); |
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378 | } |
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379 | } |
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380 | |
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381 | // idcontains |
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382 | // Input: ideal p, ideal q |
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383 | // Output: 1 if p contains q, 0 otherwise |
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384 | // If the routine is to be called from the top, a previous call to |
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385 | // setglobalrings() is needed. |
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386 | static proc idcontains(ideal p, ideal q) |
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387 | { |
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388 | int t; int i; |
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389 | t=1; i=1; |
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390 | def P=p; def Q=q; |
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391 | attrib(P,"isSB",1); |
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392 | poly r; |
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393 | while ((t) and (i<=size(Q))) |
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394 | { |
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395 | r=reduce(Q[i],P); |
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396 | if (r!=0){t=0;} |
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397 | i++; |
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398 | } |
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399 | return(t); |
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400 | } |
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401 | |
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402 | |
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403 | // selectminideals |
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404 | // given a list of ideals returns the list of integers corresponding |
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405 | // to the minimal ideals in the list |
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406 | // Input: L (list of ideals) |
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407 | // Output: the list of integers corresponding to the minimal ideals in L |
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408 | // Works in Q[u_1,..,u_m] |
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409 | static proc selectminideals(list L) |
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410 | { |
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411 | list P; int i; int j; int t; |
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412 | if(size(L)==0){return(L)}; |
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413 | if(size(L)==1){P[1]=1; return(P);} |
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414 | for (i=1;i<=size(L);i++) |
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415 | { |
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416 | t=1; |
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417 | j=1; |
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418 | while ((t) and (j<=size(L))) |
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419 | { |
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420 | if (i!=j) |
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421 | { |
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422 | if(idcontains(L[i],L[j])==1) |
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423 | { |
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424 | t=0; |
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425 | } |
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426 | } |
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427 | j++; |
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428 | } |
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429 | if (t){P[size(P)+1]=i;} |
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430 | } |
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431 | return(P); |
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432 | } |
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433 | |
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434 | |
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435 | // Auxiliary routine |
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436 | // elimconstfac: eliminate the factors in the polynom f that are in Q[a] |
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437 | // Input: |
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438 | // poly f: |
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439 | // list L: of components of the segment |
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440 | // Output: |
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441 | // poly f2 where the factors of f in Q[a] that are non-null on any component |
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442 | // have been dropped from f |
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443 | static proc elimconstfac(poly f) |
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444 | { |
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445 | int cond; int i; int j; int t; |
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446 | if (f==0){return(f);} |
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447 | def RR=basering; |
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448 | setring(@R); |
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449 | def ff=imap(RR,f); |
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450 | def l=factorize(ff,0); |
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451 | poly f1=1; |
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452 | for(i=2;i<=size(l[1]);i++) |
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453 | { |
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454 | f1=f1*(l[1][i])^(l[2][i]); |
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455 | } |
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456 | setring(RR); |
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457 | def f2=imap(@R,f1); |
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458 | return(f2); |
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459 | }; |
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460 | |
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461 | static proc memberpos(f,J) |
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462 | //"USAGE: memberpos(f,J); |
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463 | // (f,J) expected (polynomial,ideal) |
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464 | // or (int,list(int)) |
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465 | // or (int,intvec) |
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466 | // or (intvec,list(intvec)) |
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467 | // or (list(int),list(list(int))) |
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468 | // or (ideal,list(ideal)) |
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469 | // or (list(intvec), list(list(intvec))). |
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470 | // Works in any kind of ideals |
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471 | //RETURN: The list (t,pos) t int; pos int; |
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472 | // t is 1 if f belongs to J and 0 if not. |
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473 | // pos gives the position in J (or 0 if f does not belong). |
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474 | //EXAMPLE: memberpos; shows an example" |
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475 | { |
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476 | int pos=0; |
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477 | int i=1; |
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478 | int j; |
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479 | int t=0; |
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480 | int nt; |
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481 | if (typeof(J)=="ideal"){nt=ncols(J);} |
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482 | else{nt=size(J);} |
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483 | if ((typeof(f)=="poly") or (typeof(f)=="int")) |
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484 | { // (poly,ideal) or |
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485 | // (poly,list(poly)) |
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486 | // (int,list(int)) or |
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487 | // (int,intvec) |
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488 | i=1; |
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489 | while(i<=nt) |
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490 | { |
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491 | if (f==J[i]){return(list(1,i));} |
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492 | i++; |
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493 | } |
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494 | return(list(0,0)); |
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495 | } |
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496 | else |
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497 | { |
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498 | if ((typeof(f)=="intvec") or ((typeof(f)=="list") and (typeof(f[1])=="int"))) |
---|
499 | { // (intvec,list(intvec)) or |
---|
500 | // (list(int),list(list(int))) |
---|
501 | i=1; |
---|
502 | t=0; |
---|
503 | pos=0; |
---|
504 | while((i<=nt) and (t==0)) |
---|
505 | { |
---|
506 | t=1; |
---|
507 | j=1; |
---|
508 | if (size(f)!=size(J[i])){t=0;} |
---|
509 | else |
---|
510 | { |
---|
511 | while ((j<=size(f)) and t) |
---|
512 | { |
---|
513 | if (f[j]!=J[i][j]){t=0;} |
---|
514 | j++; |
---|
515 | } |
---|
516 | } |
---|
517 | if (t){pos=i;} |
---|
518 | i++; |
---|
519 | } |
---|
520 | if (t){return(list(1,pos));} |
---|
521 | else{return(list(0,0));} |
---|
522 | } |
---|
523 | else |
---|
524 | { |
---|
525 | if (typeof(f)=="ideal") |
---|
526 | { // (ideal,list(ideal)) |
---|
527 | i=1; |
---|
528 | t=0; |
---|
529 | pos=0; |
---|
530 | while((i<=nt) and (t==0)) |
---|
531 | { |
---|
532 | t=1; |
---|
533 | j=1; |
---|
534 | if (ncols(f)!=ncols(J[i])){t=0;} |
---|
535 | else |
---|
536 | { |
---|
537 | while ((j<=ncols(f)) and t) |
---|
538 | { |
---|
539 | if (f[j]!=J[i][j]){t=0;} |
---|
540 | j++; |
---|
541 | } |
---|
542 | } |
---|
543 | if (t){pos=i;} |
---|
544 | i++; |
---|
545 | } |
---|
546 | if (t){return(list(1,pos));} |
---|
547 | else{return(list(0,0));} |
---|
548 | } |
---|
549 | else |
---|
550 | { |
---|
551 | if ((typeof(f)=="list") and (typeof(f[1])=="intvec")) |
---|
552 | { // (list(intvec),list(list(intvec))) |
---|
553 | i=1; |
---|
554 | t=0; |
---|
555 | pos=0; |
---|
556 | while((i<=nt) and (t==0)) |
---|
557 | { |
---|
558 | t=1; |
---|
559 | j=1; |
---|
560 | if (size(f)!=size(J[i])){t=0;} |
---|
561 | else |
---|
562 | { |
---|
563 | while ((j<=size(f)) and t) |
---|
564 | { |
---|
565 | if (f[j]!=J[i][j]){t=0;} |
---|
566 | j++; |
---|
567 | } |
---|
568 | } |
---|
569 | if (t){pos=i;} |
---|
570 | i++; |
---|
571 | } |
---|
572 | if (t){return(list(1,pos));} |
---|
573 | else{return(list(0,0));} |
---|
574 | } |
---|
575 | } |
---|
576 | } |
---|
577 | } |
---|
578 | } |
---|
579 | //example |
---|
580 | //{ "EXAMPLE:"; echo = 2; |
---|
581 | // list L=(7,4,5,1,1,4,9); |
---|
582 | // memberpos(1,L); |
---|
583 | //} |
---|
584 | |
---|
585 | // Auxiliary routine |
---|
586 | // pos |
---|
587 | // Input: intvec p of zeros and ones |
---|
588 | // Output: intvec W of the positions where p has ones. |
---|
589 | static proc pos(intvec p) |
---|
590 | { |
---|
591 | int i; |
---|
592 | intvec W; int j=1; |
---|
593 | for (i=1; i<=size(p); i++) |
---|
594 | { |
---|
595 | if (p[i]==1){W[j]=i; j++;} |
---|
596 | } |
---|
597 | return(W); |
---|
598 | } |
---|
599 | |
---|
600 | // Input: |
---|
601 | // A,B: lists of ideals |
---|
602 | // Output: |
---|
603 | // 1 if both lists of ideals are equal, or 0 if not |
---|
604 | static proc equallistsofideals(list A, list B) |
---|
605 | { |
---|
606 | int i; |
---|
607 | int tes=0; |
---|
608 | if (size(A)!=size(B)){return(tes);} |
---|
609 | tes=1; i=1; |
---|
610 | while(tes==1 and i<=size(A)) |
---|
611 | { |
---|
612 | if (equalideals(A[i],B[i])==0){tes=0; return(tes);} |
---|
613 | i++; |
---|
614 | } |
---|
615 | return(tes); |
---|
616 | } |
---|
617 | |
---|
618 | // Input: |
---|
619 | // A,B: lists of P-rep, i.e. of the form [p_i,[p_{i1},..,p_{ij_i}]] |
---|
620 | // Output: |
---|
621 | // 1 if both lists of P-reps are equal, or 0 if not |
---|
622 | static proc equallistsA(list A, list B) |
---|
623 | { |
---|
624 | int tes=0; |
---|
625 | if(equalideals(A[1],B[1])==0){return(tes);} |
---|
626 | tes=equallistsofideals(A[2],B[2]); |
---|
627 | return(tes); |
---|
628 | } |
---|
629 | |
---|
630 | // Input: |
---|
631 | // A,B: lists lists of of P-rep, i.e. of the form [[p_1,[p_{11},..,p_{1j_1}]] .. [p_i,[p_{i1},..,p_{ij_i}]] |
---|
632 | // Output: |
---|
633 | // 1 if both lists of lists of P-rep are equal, or 0 if not |
---|
634 | static proc equallistsAall(list A,list B) |
---|
635 | { |
---|
636 | int i; int tes; |
---|
637 | if(size(A)!=size(B)){return(tes);} |
---|
638 | tes=1; i=1; |
---|
639 | while(tes and i<=size(A)) |
---|
640 | { |
---|
641 | tes=equallistsA(A[i],B[i]); |
---|
642 | i++; |
---|
643 | } |
---|
644 | return(tes); |
---|
645 | } |
---|
646 | |
---|
647 | // idint: ideal intersection |
---|
648 | // in the ring @P. |
---|
649 | // it works in an extended ring |
---|
650 | // input: two ideals in the ring @P |
---|
651 | // output the intersection of both (is not a GB) |
---|
652 | static proc idint(ideal I, ideal J) |
---|
653 | { |
---|
654 | def RR=basering; |
---|
655 | ring T=0,t,lp; |
---|
656 | def K=T+RR; |
---|
657 | setring(K); |
---|
658 | def Ia=imap(RR,I); |
---|
659 | def Ja=imap(RR,J); |
---|
660 | ideal IJ; |
---|
661 | int i; |
---|
662 | for(i=1;i<=size(Ia);i++){IJ[i]=t*Ia[i];} |
---|
663 | for(i=1;i<=size(Ja);i++){IJ[size(Ia)+i]=(1-t)*Ja[i];} |
---|
664 | ideal eIJ=eliminate(IJ,t); |
---|
665 | setring(RR); |
---|
666 | return(imap(K,eIJ)); |
---|
667 | } |
---|
668 | |
---|
669 | //purpose ideal intersection called in @R and computed in @P |
---|
670 | static proc idintR(ideal N, ideal M) |
---|
671 | { |
---|
672 | def RR=basering; |
---|
673 | setring(@P); |
---|
674 | def Np=imap(RR,N); |
---|
675 | def Mp=imap(RR,M); |
---|
676 | def Jp=idint(Np,Mp); |
---|
677 | setring(RR); |
---|
678 | return(imap(@P,Jp)); |
---|
679 | } |
---|
680 | |
---|
681 | // Auxiliary routine |
---|
682 | // comb: the list of combinations of elements (1,..n) of order p |
---|
683 | static proc comb(int n, int p) |
---|
684 | { |
---|
685 | list L; list L0; |
---|
686 | intvec c; intvec d; |
---|
687 | int i; int j; int last; |
---|
688 | if ((n<0) or (n<p)) |
---|
689 | { |
---|
690 | return(L); |
---|
691 | } |
---|
692 | if (p==1) |
---|
693 | { |
---|
694 | for (i=1;i<=n;i++) |
---|
695 | { |
---|
696 | c=i; |
---|
697 | L[size(L)+1]=c; |
---|
698 | } |
---|
699 | return(L); |
---|
700 | } |
---|
701 | else |
---|
702 | { |
---|
703 | L0=comb(n,p-1); |
---|
704 | for (i=1;i<=size(L0);i++) |
---|
705 | { |
---|
706 | c=L0[i]; d=c; |
---|
707 | last=c[size(c)]; |
---|
708 | for (j=last+1;j<=n;j++) |
---|
709 | { |
---|
710 | d[size(c)+1]=j; |
---|
711 | L[size(L)+1]=d; |
---|
712 | } |
---|
713 | } |
---|
714 | return(L); |
---|
715 | } |
---|
716 | } |
---|
717 | |
---|
718 | // Auxiliary routine |
---|
719 | // combrep |
---|
720 | // Input: V=(n_1,..,n_i) |
---|
721 | // Output: L=(v_1,..,v_p) where p=prod_j=1^i (n_j) |
---|
722 | // is the list of all intvec v_j=(v_j1,..,v_ji) where 1<=v_jk<=n_i |
---|
723 | static proc combrep(intvec V) |
---|
724 | { |
---|
725 | list L; list LL; |
---|
726 | int i; int j; int k; intvec W; |
---|
727 | if (size(V)==1) |
---|
728 | { |
---|
729 | for (i=1;i<=V[1];i++) |
---|
730 | { |
---|
731 | L[i]=intvec(i); |
---|
732 | } |
---|
733 | return(L); |
---|
734 | } |
---|
735 | for (i=1;i<size(V);i++) |
---|
736 | { |
---|
737 | W[i]=V[i]; |
---|
738 | } |
---|
739 | LL=combrep(W); |
---|
740 | for (i=1;i<=size(LL);i++) |
---|
741 | { |
---|
742 | W=LL[i]; |
---|
743 | for (j=1;j<=V[size(V)];j++) |
---|
744 | { |
---|
745 | W[size(V)]=j; |
---|
746 | L[size(L)+1]=W; |
---|
747 | } |
---|
748 | } |
---|
749 | return(L); |
---|
750 | } |
---|
751 | |
---|
752 | static proc subset(J,K) |
---|
753 | //"USAGE: subset(J,K); |
---|
754 | // (J,K) expected (ideal,ideal) |
---|
755 | // or (list, list) |
---|
756 | //RETURN: 1 if all the elements of J are in K, 0 if not. |
---|
757 | //EXAMPLE: subset; shows an example;" |
---|
758 | { |
---|
759 | int i=1; |
---|
760 | int nt; |
---|
761 | if (typeof(J)=="ideal"){nt=ncols(J);} |
---|
762 | else{nt=size(J);} |
---|
763 | if (size(J)==0){return(1);} |
---|
764 | while(i<=nt) |
---|
765 | { |
---|
766 | if (memberpos(J[i],K)[1]){i++;} |
---|
767 | else {return(0);} |
---|
768 | } |
---|
769 | return(1); |
---|
770 | } |
---|
771 | //example |
---|
772 | //{ "EXAMPLE:"; echo = 2; |
---|
773 | // list J=list(7,3,2); |
---|
774 | // list K=list(1,2,3,5,7,8); |
---|
775 | // subset(J,K); |
---|
776 | //} |
---|
777 | |
---|
778 | proc setglobalrings() |
---|
779 | "USAGE: setglobalrings(); |
---|
780 | No arguments |
---|
781 | RETURN: After its call the rings @R=Q[a][x], @P=Q[a], @RP=Q[x,a] are |
---|
782 | defined as global variables. (a=parameters, x=variables) |
---|
783 | NOTE: It is called internally by many basic routines of the |
---|
784 | library grobcov, cgsdr, extend, pdivi, pnormalf, locus, locusdg, |
---|
785 | envelop, envelopdg, and killed before the output. |
---|
786 | The user does not need to call it, except when it is interested |
---|
787 | in using some internal routine of the library that |
---|
788 | uses these rings. |
---|
789 | The basering R, must be of the form Q[a][x], (a=parameters, |
---|
790 | x=variables), and should be defined previously. |
---|
791 | KEYWORDS: ring, rings |
---|
792 | EXAMPLE: setglobalrings; shows an example" |
---|
793 | { |
---|
794 | if (defined(@P)) |
---|
795 | { |
---|
796 | kill @P; kill @R; kill @RP; |
---|
797 | } |
---|
798 | def RR=basering; |
---|
799 | def @R=basering; // must be of the form Q[a][x], (a=parameters, x=variables) |
---|
800 | def Rx=ringlist(RR); |
---|
801 | def @P=ring(Rx[1]); |
---|
802 | list Lx; |
---|
803 | Lx[1]=0; |
---|
804 | Lx[2]=Rx[2]+Rx[1][2]; |
---|
805 | Lx[3]=Rx[1][3]; |
---|
806 | Lx[4]=Rx[1][4]; |
---|
807 | Rx[1]=0; |
---|
808 | def D=ring(Rx); |
---|
809 | def @RP=D+@P; |
---|
810 | export(@R); // global ring Q[a][x] |
---|
811 | export(@P); // global ring Q[a] |
---|
812 | export(@RP); // global ring K[x,a] with product order |
---|
813 | setring(RR); |
---|
814 | }; |
---|
815 | example |
---|
816 | { |
---|
817 | "EXAMPLE:"; echo = 2; |
---|
818 | ring R=(0,a,b),(x,y,z),dp; |
---|
819 | setglobalrings(); |
---|
820 | R; |
---|
821 | Grobcov::@R; |
---|
822 | Grobcov::@P; |
---|
823 | Grobcov::@RP; |
---|
824 | ringlist(Grobcov::@R); |
---|
825 | ringlist(Grobcov::@P); |
---|
826 | ringlist(Grobcov::@RP); |
---|
827 | } |
---|
828 | |
---|
829 | // cld : clears denominators of an ideal and normalizes to content 1 |
---|
830 | // can be used in @R or @P or @RP |
---|
831 | // input: |
---|
832 | // ideal J (J can be also poly), but the output is an ideal; |
---|
833 | // output: |
---|
834 | // ideal Jc (the new form of ideal J without denominators and |
---|
835 | // normalized to content 1) |
---|
836 | static proc cld(ideal J) |
---|
837 | { |
---|
838 | if (size(J)==0){return(ideal(0));} |
---|
839 | int te=0; |
---|
840 | def RR=basering; |
---|
841 | if(not(defined(@RP))) |
---|
842 | { |
---|
843 | te=1; |
---|
844 | setglobalrings(); |
---|
845 | } |
---|
846 | setring(@RP); |
---|
847 | def Ja=imap(RR,J); |
---|
848 | ideal Jb; |
---|
849 | if (size(Ja)==0){setring(RR); return(ideal(0));} |
---|
850 | int i; |
---|
851 | def j=0; |
---|
852 | for (i=1;i<=ncols(Ja);i++){if (size(Ja[i])!=0){j++; Jb[j]=cleardenom(Ja[i]);}} |
---|
853 | setring(RR); |
---|
854 | def Jc=imap(@RP,Jb); |
---|
855 | if(te){kill @R; kill @RP; kill @P;} |
---|
856 | return(Jc); |
---|
857 | }; |
---|
858 | |
---|
859 | // simpqcoeffs : simplifies a quotient of two polynomials |
---|
860 | // input: two coeficients (or terms), that are considered as a quotient |
---|
861 | // output: the two coeficients reduced without common factors |
---|
862 | static proc simpqcoeffs(poly n,poly m) |
---|
863 | { |
---|
864 | def nc=content(n); |
---|
865 | def mc=content(m); |
---|
866 | def gc=gcd(nc,mc); |
---|
867 | ideal s=n/gc,m/gc; |
---|
868 | return (s); |
---|
869 | } |
---|
870 | |
---|
871 | // pdivi : pseudodivision of a parametric polynomial f by a parametric ideal F in Q[a][x]. |
---|
872 | // input: |
---|
873 | // poly f |
---|
874 | // ideal F |
---|
875 | // output: |
---|
876 | // list (poly r, ideal q, poly mu) |
---|
877 | proc pdivi(poly f,ideal F) |
---|
878 | "USAGE: pdivi(f,F); |
---|
879 | poly f: the polynomialin Q[a][x] to be divided |
---|
880 | ideal F: the divisor ideal in Q[a][x]. |
---|
881 | RETURN: A list (poly r, ideal q, poly m). r is the remainder of the |
---|
882 | pseudodivision, q is the set of quotients, and m is the |
---|
883 | coefficient factor by which f is to be multiplied. |
---|
884 | NOTE: pseudodivision of a poly f by an ideal F in Q[a][x]. Returns a |
---|
885 | list (r,q,m) such that m*f=r+sum(q.G), and no lpp of a divisor |
---|
886 | divides a pp of r. |
---|
887 | KEYWORDS: division, reduce |
---|
888 | EXAMPLE: pdivi; shows an example" |
---|
889 | { |
---|
890 | F=simplify(F,2); |
---|
891 | int i; |
---|
892 | int j; |
---|
893 | poly v=1; |
---|
894 | for(i=1;i<=nvars(basering);i++){v=v*var(i);} |
---|
895 | poly r=0; |
---|
896 | poly mu=1; |
---|
897 | def p=f; |
---|
898 | ideal q; |
---|
899 | for (i=1; i<=ncols(F); i++){q[i]=0;}; |
---|
900 | ideal lpf; |
---|
901 | ideal lcf; |
---|
902 | for (i=1;i<=ncols(F);i++){lpf[i]=leadmonom(F[i]);} |
---|
903 | for (i=1;i<=ncols(F);i++){lcf[i]=leadcoef(F[i]);} |
---|
904 | poly lpp; |
---|
905 | poly lcp; |
---|
906 | poly qlm; |
---|
907 | poly nu; |
---|
908 | poly rho; |
---|
909 | int divoc=0; |
---|
910 | ideal qlc; |
---|
911 | while (p!=0) |
---|
912 | { |
---|
913 | i=1; |
---|
914 | divoc=0; |
---|
915 | lpp=leadmonom(p); |
---|
916 | lcp=leadcoef(p); |
---|
917 | while (divoc==0 and i<=size(F)) |
---|
918 | { |
---|
919 | qlm=lpp/lpf[i]; |
---|
920 | if (qlm!=0) |
---|
921 | { |
---|
922 | qlc=simpqcoeffs(lcp,lcf[i]); |
---|
923 | nu=qlc[2]; |
---|
924 | mu=mu*nu; |
---|
925 | rho=qlc[1]*qlm; |
---|
926 | p=nu*p-rho*F[i]; |
---|
927 | r=nu*r; |
---|
928 | for (j=1;j<=size(F);j++){q[j]=nu*q[j];} |
---|
929 | q[i]=q[i]+rho; |
---|
930 | divoc=1; |
---|
931 | } |
---|
932 | else {i++;} |
---|
933 | } |
---|
934 | if (divoc==0) |
---|
935 | { |
---|
936 | r=r+lcp*lpp; |
---|
937 | p=p-lcp*lpp; |
---|
938 | } |
---|
939 | } |
---|
940 | list res=r,q,mu; |
---|
941 | return(res); |
---|
942 | } |
---|
943 | example |
---|
944 | { |
---|
945 | "EXAMPLE:"; echo = 2; |
---|
946 | ring R=(0,a,b,c),(x,y),dp; |
---|
947 | poly f=(ab-ac)*xy+(ab)*x+(5c); |
---|
948 | // Divisor="; |
---|
949 | f; |
---|
950 | ideal F=ax+b,cy+a; |
---|
951 | // Dividends="; |
---|
952 | F; |
---|
953 | def r=pdivi(f,F); |
---|
954 | // (Remainder, quotients, factor)="; |
---|
955 | r; |
---|
956 | // Verifying the division: |
---|
957 | r[3]*f-(r[2][1]*F[1]+r[2][2]*F[2]+r[1]); |
---|
958 | } |
---|
959 | |
---|
960 | // pspol : S-poly of two polynomials in @R |
---|
961 | // @R |
---|
962 | // input: |
---|
963 | // poly f (given in the ring @R) |
---|
964 | // poly g (given in the ring @R) |
---|
965 | // output: |
---|
966 | // list (S, red): S is the S-poly(f,g) and red is a Boolean variable |
---|
967 | // if red then S reduces by Buchberger 1st criterion |
---|
968 | // (not used) |
---|
969 | static proc pspol(poly f,poly g) |
---|
970 | { |
---|
971 | def lcf=leadcoef(f); |
---|
972 | def lcg=leadcoef(g); |
---|
973 | def lpf=leadmonom(f); |
---|
974 | def lpg=leadmonom(g); |
---|
975 | def v=gcd(lpf,lpg); |
---|
976 | def s=simpqcoeffs(lcf,lcg); |
---|
977 | def vf=lpf/v; |
---|
978 | def vg=lpg/v; |
---|
979 | poly S=s[2]*vg*f-s[1]*vf*g; |
---|
980 | return(S); |
---|
981 | } |
---|
982 | |
---|
983 | // facvar: Returns all the free-square factors of the elements |
---|
984 | // of ideal J (non repeated). Integer factors are ignored, |
---|
985 | // even 0 is ignored. It can be called from ideal @R, but |
---|
986 | // the given ideal J must only contain poynomials in the |
---|
987 | // parameters. |
---|
988 | // Operates in the ring @P, but can be called from ring @R, |
---|
989 | // and the ideal @P must be defined calling first setglobalrings(); |
---|
990 | // input: ideal J |
---|
991 | // output: ideal Jc: Returns all the free-square factors of the elements |
---|
992 | // of ideal J (non repeated). Integer factors are ignored, |
---|
993 | // even 0 is ignored. It can be called from ideal @R. |
---|
994 | static proc facvar(ideal J) |
---|
995 | //"USAGE: facvar(J); |
---|
996 | // J: an ideal in the parameters |
---|
997 | //RETURN: all the free-square factors of the elements |
---|
998 | // of ideal J (non repeated). Integer factors are ignored, |
---|
999 | // even 0 is ignored. It can be called from ideal @R, but |
---|
1000 | // the given ideal J must only contain poynomials in the |
---|
1001 | // parameters. |
---|
1002 | //NOTE: Operates in the ring @P, and the ideal J must contain only |
---|
1003 | // polynomials in the parameters, but can be called from ring @R. |
---|
1004 | //KEYWORDS: factor |
---|
1005 | //EXAMPLE: facvar; shows an example" |
---|
1006 | { |
---|
1007 | int i; |
---|
1008 | def RR=basering; |
---|
1009 | setring(@P); |
---|
1010 | def Ja=imap(RR,J); |
---|
1011 | if(size(Ja)==0){setring(RR); return(ideal(0));} |
---|
1012 | Ja=elimintfromideal(Ja); // also in ideal @P |
---|
1013 | ideal Jb; |
---|
1014 | if (size(Ja)==0){Jb=ideal(0);} |
---|
1015 | else |
---|
1016 | { |
---|
1017 | for (i=1;i<=ncols(Ja);i++){if(size(Ja[i])!=0){Jb=Jb,factorize(Ja[i],1);}} |
---|
1018 | Jb=simplify(Jb,2+4+8); |
---|
1019 | Jb=cld(Jb); |
---|
1020 | Jb=elimintfromideal(Jb); // also in ideal @P |
---|
1021 | } |
---|
1022 | setring(RR); |
---|
1023 | def Jc=imap(@P,Jb); |
---|
1024 | return(Jc); |
---|
1025 | } |
---|
1026 | //example |
---|
1027 | //{ "EXAMPLE:"; echo = 2; |
---|
1028 | // ring R=(0,a,b,c),(x,y,z),dp; |
---|
1029 | // setglobalrings(); |
---|
1030 | // ideal J=a2-b2,a2-2ab+b2,abc-bc; |
---|
1031 | // facvar(J); |
---|
1032 | //} |
---|
1033 | |
---|
1034 | // Ered: eliminates the factors in the polynom f that are non-null. |
---|
1035 | // In ring @R |
---|
1036 | // input: |
---|
1037 | // poly f: |
---|
1038 | // ideal E of null-conditions |
---|
1039 | // ideal N of non-null conditions |
---|
1040 | // (E,N) represents V(E) \ V(N), |
---|
1041 | // Ered eliminates the non-null factors of f in V(E) \ V(N) |
---|
1042 | // output: |
---|
1043 | // poly f2 where the non-null conditions have been dropped from f |
---|
1044 | static proc Ered(poly f,ideal E, ideal N) |
---|
1045 | { |
---|
1046 | def RR=basering; |
---|
1047 | setring(@R); |
---|
1048 | poly ff=imap(RR,f); |
---|
1049 | ideal EE=imap(RR,E); |
---|
1050 | ideal NN=imap(RR,N); |
---|
1051 | if((ff==0) or (equalideals(NN,ideal(1)))){setring(RR); return(f);} |
---|
1052 | def v=variables(ff); |
---|
1053 | int i; |
---|
1054 | poly X=1; |
---|
1055 | for(i=1;i<=size(v);i++){X=X*v[i];} |
---|
1056 | matrix M=coef(ff,X); |
---|
1057 | setring(@P); |
---|
1058 | def RPE=imap(@R,EE); |
---|
1059 | def RPN=imap(@R,NN); |
---|
1060 | matrix Mp=imap(@R,M); |
---|
1061 | poly g=Mp[2,1]; |
---|
1062 | if (size(Mp)!=2) |
---|
1063 | { |
---|
1064 | for(i=2;i<=size(Mp) div 2;i++) |
---|
1065 | { |
---|
1066 | g=gcd(g,Mp[2,i]); |
---|
1067 | } |
---|
1068 | } |
---|
1069 | if (g==1){setring(RR); return(f);} |
---|
1070 | else |
---|
1071 | { |
---|
1072 | def wg=factorize(g,2); |
---|
1073 | if (wg[1][1]==1){setring(RR); return(f);} |
---|
1074 | else |
---|
1075 | { |
---|
1076 | poly simp=1; |
---|
1077 | int te; |
---|
1078 | for(i=1;i<=size(wg[1]);i++) |
---|
1079 | { |
---|
1080 | te=inconsistent(RPE+wg[1][i],RPN); |
---|
1081 | if(te) |
---|
1082 | { |
---|
1083 | simp=simp*(wg[1][i])^(wg[2][i]); |
---|
1084 | } |
---|
1085 | } |
---|
1086 | } |
---|
1087 | if (simp==1){setring(RR); return(f);} |
---|
1088 | else |
---|
1089 | { |
---|
1090 | setring(RR); |
---|
1091 | def simp0=imap(@P,simp); |
---|
1092 | def f2=f/simp0; |
---|
1093 | return(f2); |
---|
1094 | } |
---|
1095 | } |
---|
1096 | } |
---|
1097 | |
---|
1098 | // pnormalf: reduces a polynomial f wrt a V(E) \ V(N) |
---|
1099 | // dividing by E and eliminating factors in N. |
---|
1100 | // called in the ring @R, |
---|
1101 | // operates in the ring @RP. |
---|
1102 | // input: |
---|
1103 | // poly f |
---|
1104 | // ideal E (depends only on the parameters) |
---|
1105 | // ideal N (depends only on the parameters) |
---|
1106 | // (E,N) represents V(E) \ V(N) |
---|
1107 | // optional: |
---|
1108 | // output: poly f2 reduced wrt to V(E) \ V(N) |
---|
1109 | proc pnormalf(poly f, ideal E, ideal N) |
---|
1110 | "USAGE: pnormalf(f,E,N); |
---|
1111 | f: the polynomial in Q[a][x] (a=parameters, x=variables) to be |
---|
1112 | reduced modulo V(E) \ V(N) of a segment in Q[a]. |
---|
1113 | E: the null conditions ideal in Q[a] |
---|
1114 | N: the non-null conditions in Q[a] |
---|
1115 | RETURN: a reduced polynomial g of f, whose coefficients are reduced |
---|
1116 | modulo E and having no factor in N. |
---|
1117 | NOTE: Should be called from ring Q[a][x]. |
---|
1118 | Ideals E and N must be given by polynomials in Q[a]. |
---|
1119 | KEYWORDS: division, pdivi, reduce |
---|
1120 | EXAMPLE: pnormalf; shows an example" |
---|
1121 | { |
---|
1122 | def RR=basering; |
---|
1123 | int te=0; |
---|
1124 | if (defined(@P)){te=1;} |
---|
1125 | else{setglobalrings();} |
---|
1126 | setring(@RP); |
---|
1127 | def fa=imap(RR,f); |
---|
1128 | def Ea=imap(RR,E); |
---|
1129 | def Na=imap(RR,N); |
---|
1130 | option(redSB); |
---|
1131 | Ea=std(Ea); |
---|
1132 | def r=cld(reduce(fa,Ea)); |
---|
1133 | poly f1=r[1]; |
---|
1134 | f1=Ered(r[1],Ea,Na); |
---|
1135 | setring(RR); |
---|
1136 | def f2=imap(@RP,f1); |
---|
1137 | if(te==0){kill @R; kill @RP; kill @P;} |
---|
1138 | return(f2) |
---|
1139 | }; |
---|
1140 | example |
---|
1141 | { |
---|
1142 | "EXAMPLE:"; echo = 2; |
---|
1143 | ring R=(0,a,b,c),(x,y),dp; |
---|
1144 | short=0; |
---|
1145 | poly f=(b^2-1)*x^3*y+(c^2-1)*x*y^2+(c^2*b-b)*x+(a-bc)*y; |
---|
1146 | ideal E=(c-1); |
---|
1147 | ideal N=a-b; |
---|
1148 | |
---|
1149 | pnormalf(f,E,N); |
---|
1150 | } |
---|
1151 | |
---|
1152 | // lesspol: compare two polynomials by its leading power products |
---|
1153 | // input: two polynomials f,g in the ring @R |
---|
1154 | // output: 0 if f<g, 1 if f>=g |
---|
1155 | static proc lesspol(poly f, poly g) |
---|
1156 | { |
---|
1157 | if (leadmonom(f)<leadmonom(g)){return(1);} |
---|
1158 | else |
---|
1159 | { |
---|
1160 | if (leadmonom(g)<leadmonom(f)){return(0);} |
---|
1161 | else |
---|
1162 | { |
---|
1163 | if (leadcoef(f)<leadcoef(g)){return(1);} |
---|
1164 | else {return(0);} |
---|
1165 | } |
---|
1166 | } |
---|
1167 | }; |
---|
1168 | |
---|
1169 | // sortideal: sorts the polynomials in an ideal by lm in ascending order |
---|
1170 | static proc sortideal(ideal Fi) |
---|
1171 | { |
---|
1172 | def RR=basering; |
---|
1173 | setring(@RP); |
---|
1174 | def F=imap(RR,Fi); |
---|
1175 | def H=F; |
---|
1176 | ideal G; |
---|
1177 | int i; |
---|
1178 | int j; |
---|
1179 | poly p; |
---|
1180 | while (size(H)!=0) |
---|
1181 | { |
---|
1182 | j=1; |
---|
1183 | p=H[1]; |
---|
1184 | for (i=1;i<=ncols(H);i++) |
---|
1185 | { |
---|
1186 | if(lesspol(H[i],p)){j=i;p=H[j];} |
---|
1187 | } |
---|
1188 | G[ncols(G)+1]=p; |
---|
1189 | H=delfromideal(H,j); |
---|
1190 | H=simplify(H,2); |
---|
1191 | } |
---|
1192 | setring(RR); |
---|
1193 | def GG=imap(@RP,G); |
---|
1194 | GG=simplify(GG,2); |
---|
1195 | return(GG); |
---|
1196 | } |
---|
1197 | |
---|
1198 | // mingb: given a basis (gb reducing) it |
---|
1199 | // order the polynomials in ascending order and |
---|
1200 | // eliminates the polynomials whose lpp are divisible by some |
---|
1201 | // smaller one |
---|
1202 | static proc mingb(ideal F) |
---|
1203 | { |
---|
1204 | int t; int i; int j; |
---|
1205 | def H=sortideal(F); |
---|
1206 | ideal G; |
---|
1207 | if (ncols(H)<=1){return(H);} |
---|
1208 | G=H[1]; |
---|
1209 | for (i=2; i<=ncols(H); i++) |
---|
1210 | { |
---|
1211 | t=1; |
---|
1212 | j=1; |
---|
1213 | while (t and (j<i)) |
---|
1214 | { |
---|
1215 | if((leadmonom(H[i])/leadmonom(H[j]))!=0) {t=0;} |
---|
1216 | j++; |
---|
1217 | } |
---|
1218 | if (t) {G[size(G)+1]=H[i];} |
---|
1219 | } |
---|
1220 | return(G); |
---|
1221 | } |
---|
1222 | |
---|
1223 | // redgbn: given a minimal basis (gb reducing) it |
---|
1224 | // reduces each polynomial wrt to V(E) \ V(N) |
---|
1225 | static proc redgbn(ideal F, ideal E, ideal N) |
---|
1226 | { |
---|
1227 | int te=0; |
---|
1228 | if (defined(@P)==1){te=1;} |
---|
1229 | ideal G=F; |
---|
1230 | ideal H; |
---|
1231 | int i; |
---|
1232 | if (size(G)==0){return(ideal(0));} |
---|
1233 | for (i=1;i<=size(G);i++) |
---|
1234 | { |
---|
1235 | H=delfromideal(G,i); |
---|
1236 | G[i]=pnormalf(pdivi(G[i],H)[1],E,N); |
---|
1237 | G[i]=primepartZ(G[i]); |
---|
1238 | } |
---|
1239 | if(te==1){setglobalrings();} |
---|
1240 | return(G); |
---|
1241 | } |
---|
1242 | |
---|
1243 | //**************Begin homogenizing************************ |
---|
1244 | |
---|
1245 | // ishomog: |
---|
1246 | // Purpose: test if a polynomial is homogeneous in the variables or not |
---|
1247 | // input: poly f |
---|
1248 | // output 1 if f is homogeneous, 0 if not |
---|
1249 | static proc ishomog(f) |
---|
1250 | { |
---|
1251 | int i; poly r; int d; int dr; |
---|
1252 | if (f==0){return(1);} |
---|
1253 | d=deg(f); dr=d; r=f; |
---|
1254 | while ((d==dr) and (r!=0)) |
---|
1255 | { |
---|
1256 | r=r-lead(r); |
---|
1257 | dr=deg(r); |
---|
1258 | } |
---|
1259 | if (r==0){return(1);} |
---|
1260 | else{return(0);} |
---|
1261 | } |
---|
1262 | |
---|
1263 | // postredgb: given a minimal basis (gb reducing) it |
---|
1264 | // reduces each polynomial wrt to the others |
---|
1265 | static proc postredgb(ideal F) |
---|
1266 | { |
---|
1267 | int te=0; |
---|
1268 | if(defined(@P)==1){te=1;} |
---|
1269 | ideal G; |
---|
1270 | ideal H; |
---|
1271 | int i; |
---|
1272 | if (size(F)==0){return(ideal(0));} |
---|
1273 | for (i=1;i<=size(F);i++) |
---|
1274 | { |
---|
1275 | H=delfromideal(F,i); |
---|
1276 | G[i]=pdivi(F[i],H)[1]; |
---|
1277 | } |
---|
1278 | if(te==1){setglobalrings();} |
---|
1279 | return(G); |
---|
1280 | } |
---|
1281 | |
---|
1282 | |
---|
1283 | //purpose reduced Groebner basis called in @R and computed in @P |
---|
1284 | static proc gbR(ideal N) |
---|
1285 | { |
---|
1286 | def RR=basering; |
---|
1287 | setring(@P); |
---|
1288 | def Np=imap(RR,N); |
---|
1289 | option(redSB); |
---|
1290 | Np=std(Np); |
---|
1291 | setring(RR); |
---|
1292 | return(imap(@P,Np)); |
---|
1293 | } |
---|
1294 | |
---|
1295 | //**************End homogenizing************************ |
---|
1296 | |
---|
1297 | //**************Begin of Groebner Cover***************** |
---|
1298 | |
---|
1299 | // incquotient |
---|
1300 | // incremental quotient |
---|
1301 | // Input: ideal N: a Groebner basis of an ideal |
---|
1302 | // poly f: |
---|
1303 | // Output: Na = N:<f> |
---|
1304 | static proc incquotient(ideal N, poly f) |
---|
1305 | { |
---|
1306 | poly g; int i; |
---|
1307 | ideal Nb; ideal Na=N; |
---|
1308 | if (size(Na)==1) |
---|
1309 | { |
---|
1310 | g=gcd(Na[1],f); |
---|
1311 | if (g!=1) |
---|
1312 | { |
---|
1313 | Na[1]=Na[1]/g; |
---|
1314 | } |
---|
1315 | attrib(Na,"IsSB",1); |
---|
1316 | return(Na); |
---|
1317 | } |
---|
1318 | def P=basering; |
---|
1319 | poly @t; |
---|
1320 | ring H=0,@t,lp; |
---|
1321 | def HP=H+P; |
---|
1322 | setring(HP); |
---|
1323 | def fh=imap(P,f); |
---|
1324 | def Nh=imap(P,N); |
---|
1325 | ideal Nht; |
---|
1326 | for (i=1;i<=size(Nh);i++) |
---|
1327 | { |
---|
1328 | Nht[i]=Nh[i]*@t; |
---|
1329 | } |
---|
1330 | attrib(Nht,"isSB",1); |
---|
1331 | def fht=(1-@t)*fh; |
---|
1332 | option(redSB); |
---|
1333 | Nht=std(Nht,fht); |
---|
1334 | ideal Nc; ideal v; |
---|
1335 | for (i=1;i<=size(Nht);i++) |
---|
1336 | { |
---|
1337 | v=variables(Nht[i]); |
---|
1338 | if(memberpos(@t,v)[1]==0) |
---|
1339 | { |
---|
1340 | Nc[size(Nc)+1]=Nht[i]/fh; |
---|
1341 | } |
---|
1342 | } |
---|
1343 | setring(P); |
---|
1344 | ideal HH; |
---|
1345 | def Nd=imap(HP,Nc); Nb=Nd; |
---|
1346 | option(redSB); |
---|
1347 | Nb=std(Nd); |
---|
1348 | return(Nb); |
---|
1349 | } |
---|
1350 | |
---|
1351 | // Auxiliary routine to define an order for ideals |
---|
1352 | // Returns 1 if the ideal a is shoud precede ideal b by sorting them in idbefid order |
---|
1353 | // 2 if the the contrary happen |
---|
1354 | // 0 if the order is not relevant |
---|
1355 | static proc idbefid(ideal a, ideal b) |
---|
1356 | { |
---|
1357 | poly fa; poly fb; poly la; poly lb; |
---|
1358 | int te=1; int i; int j; |
---|
1359 | int na=size(a); |
---|
1360 | int nb=size(b); |
---|
1361 | int nm; |
---|
1362 | if (na<=nb){nm=na;} else{nm=nb;} |
---|
1363 | for (i=1;i<=nm; i++) |
---|
1364 | { |
---|
1365 | fa=a[i]; fb=b[i]; |
---|
1366 | while((fa!=0) or (fb!=0)) |
---|
1367 | { |
---|
1368 | la=lead(fa); |
---|
1369 | lb=lead(fb); |
---|
1370 | fa=fa-la; |
---|
1371 | fb=fb-lb; |
---|
1372 | la=leadmonom(la); |
---|
1373 | lb=leadmonom(lb); |
---|
1374 | if(leadmonom(la+lb)!=la){return(1);} |
---|
1375 | else{if(leadmonom(la+lb)!=lb){return(2);}} |
---|
1376 | } |
---|
1377 | } |
---|
1378 | if(na<nb){return(1);} |
---|
1379 | else |
---|
1380 | { |
---|
1381 | if(na>nb){return(2);} |
---|
1382 | else{return(0);} |
---|
1383 | } |
---|
1384 | } |
---|
1385 | |
---|
1386 | // sort a list of ideals using idbefid |
---|
1387 | static proc sortlistideals(list L) |
---|
1388 | { |
---|
1389 | int i; int j; int n; |
---|
1390 | ideal a; ideal b; |
---|
1391 | list LL=L; |
---|
1392 | list NL; |
---|
1393 | int k; int te; |
---|
1394 | i=1; |
---|
1395 | while(size(LL)>0) |
---|
1396 | { |
---|
1397 | k=1; |
---|
1398 | for(j=2;j<=size(LL);j++) |
---|
1399 | { |
---|
1400 | te=idbefid(LL[k],LL[j]); |
---|
1401 | if (te==2){k=j;} |
---|
1402 | } |
---|
1403 | NL[size(NL)+1]=LL[k]; |
---|
1404 | n=size(LL); |
---|
1405 | if (n>1){LL=elimfromlist(LL,k);} else{LL=list();} |
---|
1406 | } |
---|
1407 | return(NL); |
---|
1408 | } |
---|
1409 | |
---|
1410 | // Crep |
---|
1411 | // Computes the C-representation of V(N) \ V(M). |
---|
1412 | // input: |
---|
1413 | // ideal N (null ideal) (not necessarily radical nor maximal) |
---|
1414 | // ideal M (hole ideal) (not necessarily containing N) |
---|
1415 | // output: |
---|
1416 | // the list (a,b) of the canonical ideals |
---|
1417 | // the Crep of V(N) \ V(M) |
---|
1418 | // Assumed to be called in the ring @R |
---|
1419 | // Works on the ring @P |
---|
1420 | proc Crep(ideal N, ideal M) |
---|
1421 | "USAGE: Crep(N,M); |
---|
1422 | Input: ideal N (null ideal) (not necessarily radical nor maximal) |
---|
1423 | ideal M (hole ideal) (not necessarily containing N) |
---|
1424 | RETURN: The canonical C-representation of the locally closed set. |
---|
1425 | [ P,Q ], a pair of radical ideals with P included in Q, |
---|
1426 | representing the set V(P) \ V(Q) = V(N) \ V(M) |
---|
1427 | NOTE: Operates in a ring R=Q[a] (a=parameters) |
---|
1428 | KEYWORDS: locally closed set, canoncial form |
---|
1429 | EXAMPLE: Crep; shows an example" |
---|
1430 | { |
---|
1431 | list l; |
---|
1432 | ideal Np=std(N); |
---|
1433 | if (equalideals(Np,ideal(1))) |
---|
1434 | { |
---|
1435 | l=ideal(1),ideal(1); |
---|
1436 | return(l); |
---|
1437 | } |
---|
1438 | int i; |
---|
1439 | list L; |
---|
1440 | ideal Q=Np+M; |
---|
1441 | ideal P=ideal(1); |
---|
1442 | L=minGTZ(Np); |
---|
1443 | for(i=1;i<=size(L);i++) |
---|
1444 | { |
---|
1445 | if(idcontains(L[i],Q)==0) |
---|
1446 | { |
---|
1447 | P=intersect(P,L[i]); |
---|
1448 | } |
---|
1449 | } |
---|
1450 | P=std(P); |
---|
1451 | Q=std(radical(Q+P)); |
---|
1452 | list T=P,Q; |
---|
1453 | return(T); |
---|
1454 | } |
---|
1455 | example |
---|
1456 | { |
---|
1457 | "EXAMPLE:"; echo = 2; |
---|
1458 | if(defined(Grobcov::@P)){kill Grobcov::@R; kill Grobcov::@P; kill Grobcov::@RP;} |
---|
1459 | ring R=0,(x,y,z),lp; |
---|
1460 | short=0; |
---|
1461 | ideal E=x*(x^2+y^2+z^2-25); |
---|
1462 | ideal N=x*(x-3),y-4; |
---|
1463 | def Cr=Crep(E,N); |
---|
1464 | Cr; |
---|
1465 | def L=Prep(E,N); |
---|
1466 | L; |
---|
1467 | def Cr1=PtoCrep(L); |
---|
1468 | Cr1; |
---|
1469 | } |
---|
1470 | |
---|
1471 | // Prep |
---|
1472 | // Computes the P-representation of V(N) \ V(M). |
---|
1473 | // input: |
---|
1474 | // ideal N (null ideal) (not necessarily radical nor maximal) |
---|
1475 | // ideal M (hole ideal) (not necessarily containing N) |
---|
1476 | // output: |
---|
1477 | // the ((p_1,(p_11,p_1k_1)),..,(p_r,(p_r1,p_rk_r))); |
---|
1478 | // the Prep of V(N) \ V(M) |
---|
1479 | // Assumed to be called in the ring @R |
---|
1480 | // Works on the ring @P |
---|
1481 | proc Prep(ideal N, ideal M) |
---|
1482 | "USAGE: Prep(N,M); |
---|
1483 | Input: ideal N (null ideal) (not necessarily radical nor maximal) |
---|
1484 | ideal M (hole ideal) (not necessarily containing N) |
---|
1485 | RETURN: The canonical P-representation of the locally closed set V(N) \ V(M) |
---|
1486 | Output: [ Comp_1, .. , Comp_s ] where |
---|
1487 | Comp_i=[p_i,[p_i1,..,p_is_i]] |
---|
1488 | NOTE: Operates in a ring R=Q[a] (a=parameters) |
---|
1489 | KEYWORDS: Locally closed set, Canoncial form |
---|
1490 | EXAMPLE: Prep; shows an example" |
---|
1491 | { |
---|
1492 | int te; |
---|
1493 | if (N[1]==1) |
---|
1494 | { |
---|
1495 | return(list(list(ideal(1),list(ideal(1))))); |
---|
1496 | } |
---|
1497 | int i; int j; list L0; |
---|
1498 | list Ni=minGTZ(N); |
---|
1499 | list prep; |
---|
1500 | for(j=1;j<=size(Ni);j++) |
---|
1501 | { |
---|
1502 | option(redSB); |
---|
1503 | Ni[j]=std(Ni[j]); |
---|
1504 | } |
---|
1505 | list Mij; |
---|
1506 | for (i=1;i<=size(Ni);i++) |
---|
1507 | { |
---|
1508 | Mij=minGTZ(Ni[i]+M); |
---|
1509 | for(j=1;j<=size(Mij);j++) |
---|
1510 | { |
---|
1511 | option(redSB); |
---|
1512 | Mij[j]=std(Mij[j]); |
---|
1513 | } |
---|
1514 | if ((size(Mij)==1) and (equalideals(Ni[i],Mij[1])==1)){;} |
---|
1515 | else |
---|
1516 | { |
---|
1517 | prep[size(prep)+1]=list(Ni[i],Mij); |
---|
1518 | } |
---|
1519 | } |
---|
1520 | if (size(prep)==0){prep=list(list(ideal(1),list(ideal(1))));} |
---|
1521 | def Lout=CompleteA(prep,prep); |
---|
1522 | return(Lout); |
---|
1523 | } |
---|
1524 | example |
---|
1525 | { |
---|
1526 | "EXAMPLE:"; echo = 2; |
---|
1527 | if(defined(Grobcov::@P)){kill Grobcov::@R; kill Grobcov::@P; kill Grobcov::@RP;} |
---|
1528 | short=0; |
---|
1529 | ring R=0,(x,y,z),lp; |
---|
1530 | ideal E=x*(x^2+y^2+z^2-25); |
---|
1531 | ideal N=x*(x-3),y-4; |
---|
1532 | Prep(E,N); |
---|
1533 | } |
---|
1534 | |
---|
1535 | // PtoCrep |
---|
1536 | // Computes the C-representation from the P-representation. |
---|
1537 | // input: |
---|
1538 | // list ((p_1,(p_11,p_1k_1)),..,(p_r,(p_r1,p_rk_r))); |
---|
1539 | // the P-representation of V(N) \ V(M) |
---|
1540 | // output: |
---|
1541 | // list (ideal ida, ideal idb) |
---|
1542 | // the C-representaion of V(N) \ V(M) = V(ida) \ V(idb) |
---|
1543 | // Assumed to be called in the ring @R |
---|
1544 | // Works on the ring @P |
---|
1545 | proc PtoCrep(list L) |
---|
1546 | "USAGE: PtoCrep(L) |
---|
1547 | Input L: list [ Comp_1, .. , Comp_s ] where |
---|
1548 | Comp_i=[p_i,[p_i1,..,p_is_i] ], is |
---|
1549 | the P-representation of a locally closed set V(N) \ V(M) |
---|
1550 | RETURN: The canonical C-representation of the locally closed set |
---|
1551 | [ P,Q ], a pair of radical ideals with P included in Q, |
---|
1552 | representing the set V(P) \ V(Q) |
---|
1553 | NOTE: Operates in a ring R=Q[a] (a=parameters) |
---|
1554 | KEYWORDS: locally closed set, canoncial form |
---|
1555 | EXAMPLE: PtoCrep; shows an example" |
---|
1556 | { |
---|
1557 | int te; |
---|
1558 | def RR=basering; |
---|
1559 | if(defined(@P)){te=1; setring(@P); list Lp=imap(RR,L);} |
---|
1560 | else { te=0; def Lp=L;} |
---|
1561 | def La=PtoCrep0(Lp); |
---|
1562 | if(te==1) {setring(RR); def LL=imap(@P,La);} |
---|
1563 | if(te==0){def LL=La;} |
---|
1564 | return(LL); |
---|
1565 | } |
---|
1566 | example |
---|
1567 | { |
---|
1568 | "EXAMPLE:"; echo = 2; |
---|
1569 | if(defined(Grobcov::@P)){kill Grobcov::@R; kill Grobcov::@P; kill Grobcov::@RP;} |
---|
1570 | short=0; |
---|
1571 | ring R=0,(x,y,z),lp; |
---|
1572 | |
---|
1573 | // (P,Q) represents a locally closed set |
---|
1574 | ideal P=x^3+x*y^2+x*z^2-25*x; |
---|
1575 | ideal Q=y-4,x*z,x^2-3*x; |
---|
1576 | |
---|
1577 | // Now compute the P-representation= |
---|
1578 | def L=Prep(P,Q); |
---|
1579 | L; |
---|
1580 | // Now compute the C-representation= |
---|
1581 | def J=PtoCrep(L); |
---|
1582 | J; |
---|
1583 | // Now come back recomputing the P-represetation of the C-representation= |
---|
1584 | Prep(J[1],J[2]); |
---|
1585 | } |
---|
1586 | |
---|
1587 | // PtoCrep0 |
---|
1588 | // Computes the C-representation from the P-representation. |
---|
1589 | // input: |
---|
1590 | // list ((p_1,(p_11,p_1k_1)),..,(p_r,(p_r1,p_rk_r))); |
---|
1591 | // the P-representation of V(N) \ V(M) |
---|
1592 | // output: |
---|
1593 | // list (ideal ida, ideal idb) |
---|
1594 | // the C-representation of V(N) \ V(M) = V(ida) \ V(idb) |
---|
1595 | // Works in a ring Q[u_1,..,u_m] and is called on it. |
---|
1596 | static proc PtoCrep0(list L) |
---|
1597 | { |
---|
1598 | int te=0; |
---|
1599 | def Lp=L; |
---|
1600 | int i; int j; |
---|
1601 | ideal ida=ideal(1); ideal idb=ideal(1); list Lb; ideal N; |
---|
1602 | for (i=1;i<=size(Lp);i++) |
---|
1603 | { |
---|
1604 | option(returnSB); |
---|
1605 | N=Lp[i][1]; |
---|
1606 | ida=intersect(ida,N); |
---|
1607 | Lb=Lp[i][2]; |
---|
1608 | for(j=1;j<=size(Lb);j++) |
---|
1609 | { |
---|
1610 | idb=intersect(idb,Lb[j]); |
---|
1611 | } |
---|
1612 | } |
---|
1613 | //idb=radical(idb); |
---|
1614 | def La=list(ida,idb); |
---|
1615 | return(La); |
---|
1616 | } |
---|
1617 | |
---|
1618 | // input: F a parametric ideal in Q[a][x] |
---|
1619 | // output: a rComprehensive Groebner System disjoint and reduced. |
---|
1620 | // It uses Kapur-Sun-Wang algorithm, and with the options |
---|
1621 | // can compute the homogenization before (('can',0) or ( 'can',1)) |
---|
1622 | // and dehomogenize the result. |
---|
1623 | proc cgsdr(ideal F, list #) |
---|
1624 | "USAGE: cgsdr(F); |
---|
1625 | F: ideal in Q[a][x] (a=parameters, x=variables) to be discussed. |
---|
1626 | To compute a disjoint, reduced Comprehensive Groebner System (CGS). |
---|
1627 | cgsdr is the starting point of the fundamental routine grobcov. |
---|
1628 | Inside grobcov it is used with options 'can' set to 0,1 and |
---|
1629 | not with options ('can',2). |
---|
1630 | It is to be used if only a disjoint reduced CGS is required. |
---|
1631 | |
---|
1632 | Options: To modify the default options, pairs of arguments |
---|
1633 | -option name, value- of valid options must be added to the call. |
---|
1634 | |
---|
1635 | Options: |
---|
1636 | \"can\",0-1-2: The default value is \"can\",2. In this case no |
---|
1637 | homogenization is done. With option (\"can\",0) the given |
---|
1638 | basis is homogenized, and with option (\"can\",1) the |
---|
1639 | whole given ideal is homogenized before computing the |
---|
1640 | cgs and dehomogenized after. |
---|
1641 | with option (\"can\",0) the homogenized basis is used |
---|
1642 | with option (\"can\",1) the homogenized ideal is used |
---|
1643 | with option (\"can\",2) the given basis is used |
---|
1644 | \"null\",ideal E: The default is (\"null\",ideal(0)). |
---|
1645 | \"nonnull\",ideal N: The default (\"nonnull\",ideal(1)). |
---|
1646 | When options \"null\" and/or \"nonnull\" are given, then |
---|
1647 | the parameter space is restricted to V(E) \ V(N). |
---|
1648 | \"comment\",0-1: The default is (\"comment\",0). Setting (\"comment\",1) |
---|
1649 | will provide information about the development of the |
---|
1650 | computation. |
---|
1651 | \"out\",0-1: 1 (default) the output segments are given as |
---|
1652 | as difference of varieties. |
---|
1653 | 0: the output segments are given in P-representation |
---|
1654 | and the segments grouped by lpp |
---|
1655 | With options (\"can\",0) and (\"can\",1) the option (\"out\",1) |
---|
1656 | is set to (\"out\",0) because it is not compatible. |
---|
1657 | One can give none or whatever of these options. |
---|
1658 | With the default options (\"can\",2,\"out\",1), only the |
---|
1659 | Kapur-Sun-Wang algorithm is computed. This is very efficient |
---|
1660 | but is only the starting point for the computation of grobcov. |
---|
1661 | When grobcov is computed, the call to cgsdr inside uses |
---|
1662 | specific options that are more expensive ("can",0-1,"out",0). |
---|
1663 | RETURN: Returns a list T describing a reduced and disjoint |
---|
1664 | Comprehensive Groebner System (CGS), |
---|
1665 | With option (\"out\",0) |
---|
1666 | the segments are grouped by |
---|
1667 | leading power products (lpp) of the reduced Groebner |
---|
1668 | basis and given in P-representation. |
---|
1669 | The returned list is of the form: |
---|
1670 | ( |
---|
1671 | (lpp, (num,basis,segment),...,(num,basis,segment),lpp), |
---|
1672 | ..,, |
---|
1673 | (lpp, (num,basis,segment),...,(num,basis,segment),lpp) |
---|
1674 | ) |
---|
1675 | The bases are the reduced Groebner bases (after normalization) |
---|
1676 | for each point of the corresponding segment. |
---|
1677 | |
---|
1678 | The third element of each lpp segment is the lpp of the |
---|
1679 | used ideal in the CGS as a string: |
---|
1680 | with option (\"can\",0) the homogenized basis is used |
---|
1681 | with option (\"can\",1) the homogenized ideal is used |
---|
1682 | with option (\"can\",2) the given basis is used |
---|
1683 | |
---|
1684 | With option (\"out\",1) (default) |
---|
1685 | only KSW is applied and segments are given as |
---|
1686 | difference of varieties and are not grouped |
---|
1687 | The returned list is of the form: |
---|
1688 | ( |
---|
1689 | (E,N,B),..(E,N,B) |
---|
1690 | ) |
---|
1691 | E is the null variety |
---|
1692 | N is the nonnull variety |
---|
1693 | segment = V(E) \ V(N) |
---|
1694 | B is the reduced Groebner basis |
---|
1695 | |
---|
1696 | NOTE: The basering R, must be of the form Q[a][x], (a=parameters, |
---|
1697 | x=variables), and should be defined previously, and the ideal |
---|
1698 | defined on R. |
---|
1699 | KEYWORDS: CGS, disjoint, reduced, Comprehensive Groebner System |
---|
1700 | EXAMPLE: cgsdr; shows an example" |
---|
1701 | { |
---|
1702 | int te; |
---|
1703 | def RR=basering; |
---|
1704 | if(defined(@P)){te=1;} |
---|
1705 | else{te=0; setglobalrings();} |
---|
1706 | // INITIALIZING OPTIONS |
---|
1707 | int i; int j; |
---|
1708 | def E=ideal(0); |
---|
1709 | def N=ideal(1); |
---|
1710 | int comment=0; |
---|
1711 | int can=2; |
---|
1712 | int out=1; |
---|
1713 | poly f; |
---|
1714 | ideal B; |
---|
1715 | int start=timer; |
---|
1716 | list L=#; |
---|
1717 | for(i=1;i<=size(L) div 2;i++) |
---|
1718 | { |
---|
1719 | if(L[2*i-1]=="null"){E=L[2*i];} |
---|
1720 | else |
---|
1721 | { |
---|
1722 | if(L[2*i-1]=="nonnull"){N=L[2*i];} |
---|
1723 | else |
---|
1724 | { |
---|
1725 | if(L[2*i-1]=="comment"){comment=L[2*i];} |
---|
1726 | else |
---|
1727 | { |
---|
1728 | if(L[2*i-1]=="can"){can=L[2*i];} |
---|
1729 | else |
---|
1730 | { |
---|
1731 | if(L[2*i-1]=="out"){out=L[2*i];} |
---|
1732 | } |
---|
1733 | } |
---|
1734 | } |
---|
1735 | } |
---|
1736 | } |
---|
1737 | //if(can==2){out=1;} |
---|
1738 | B=F; |
---|
1739 | if ((printlevel) and (comment==0)){comment=printlevel;} |
---|
1740 | if((can<2) and (out>0)){"Option out,1 is not compatible with can,0,1"; out=0;} |
---|
1741 | // DEFINING OPTIONS |
---|
1742 | list LL; |
---|
1743 | LL[1]="can"; LL[2]=can; |
---|
1744 | LL[3]="comment"; LL[4]=comment; |
---|
1745 | LL[5]="out"; LL[6]=out; |
---|
1746 | LL[7]="null"; LL[8]=E; |
---|
1747 | LL[9]="nonnull"; LL[10]=N; |
---|
1748 | if(comment>=1) |
---|
1749 | { |
---|
1750 | string("Begin cgsdr with options: ",LL); |
---|
1751 | } |
---|
1752 | int ish; |
---|
1753 | for (i=1;i<=size(B);i++){ish=ishomog(B[i]); if(ish==0){break;};} |
---|
1754 | if (ish) |
---|
1755 | { |
---|
1756 | if(comment>0){string("The given system is homogneous");} |
---|
1757 | def GS=KSW(B,LL); |
---|
1758 | //can=0; |
---|
1759 | } |
---|
1760 | else |
---|
1761 | { |
---|
1762 | // ACTING DEPENDING ON OPTIONS |
---|
1763 | if(can==2) |
---|
1764 | { |
---|
1765 | // WITHOUT HOMOHGENIZING |
---|
1766 | if(comment>0){string("Option of cgsdr: do not homogenize");} |
---|
1767 | def GS=KSW(B,LL); |
---|
1768 | setglobalrings(); |
---|
1769 | } |
---|
1770 | else |
---|
1771 | { |
---|
1772 | if(can==1) |
---|
1773 | { |
---|
1774 | // COMPUTING THE HOMOGOENIZED IDEAL |
---|
1775 | if(comment>=1){string("Homogenizing the whole ideal: option can=1");} |
---|
1776 | list RRL=ringlist(RR); |
---|
1777 | RRL[3][1][1]="dp"; |
---|
1778 | def Pa=ring(RRL[1]); |
---|
1779 | list Lx; |
---|
1780 | Lx[1]=0; |
---|
1781 | Lx[2]=RRL[2]+RRL[1][2]; |
---|
1782 | Lx[3]=RRL[1][3]; |
---|
1783 | Lx[4]=RRL[1][4]; |
---|
1784 | RRL[1]=0; |
---|
1785 | def D=ring(RRL); |
---|
1786 | def RP=D+Pa; |
---|
1787 | setring(RP); |
---|
1788 | def B1=imap(RR,B); |
---|
1789 | option(redSB); |
---|
1790 | B1=std(B1); |
---|
1791 | setring(RR); |
---|
1792 | def B2=imap(RP,B1); |
---|
1793 | } |
---|
1794 | else |
---|
1795 | { // (can=0) |
---|
1796 | if(comment>0){string("Homogenizing the basis: option can=0");} |
---|
1797 | def B2=B; |
---|
1798 | } |
---|
1799 | // COMPUTING HOMOGENIZED CGS |
---|
1800 | poly @t; |
---|
1801 | ring H=0,@t,dp; |
---|
1802 | def RH=RR+H; |
---|
1803 | setring(RH); |
---|
1804 | setglobalrings(); |
---|
1805 | def BH=imap(RR,B2); |
---|
1806 | def LH=imap(RR,LL); |
---|
1807 | for (i=1;i<=size(BH);i++) |
---|
1808 | { |
---|
1809 | BH[i]=homog(BH[i],@t); |
---|
1810 | } |
---|
1811 | if (comment>=2){string("Homogenized system = "); BH;} |
---|
1812 | def GSH=KSW(BH,LH); |
---|
1813 | setglobalrings(); |
---|
1814 | // DEHOMOGENIZING THE RESULT |
---|
1815 | if(out==0) |
---|
1816 | { |
---|
1817 | for (i=1;i<=size(GSH);i++) |
---|
1818 | { |
---|
1819 | GSH[i][1]=subst(GSH[i][1],@t,1); |
---|
1820 | for(j=1;j<=size(GSH[i][2]);j++) |
---|
1821 | { |
---|
1822 | GSH[i][2][j][2]=subst(GSH[i][2][j][2],@t,1); |
---|
1823 | } |
---|
1824 | } |
---|
1825 | } |
---|
1826 | else |
---|
1827 | { |
---|
1828 | for (i=1;i<=size(GSH);i++) |
---|
1829 | { |
---|
1830 | GSH[i][3]=subst(GSH[i][3],@t,1); |
---|
1831 | GSH[i][7]=subst(GSH[i][7],@t,1); |
---|
1832 | } |
---|
1833 | } |
---|
1834 | setring(RR); |
---|
1835 | def GS=imap(RH,GSH); |
---|
1836 | } |
---|
1837 | |
---|
1838 | |
---|
1839 | setglobalrings(); |
---|
1840 | if(out==0) |
---|
1841 | { |
---|
1842 | for (i=1;i<=size(GS);i++) |
---|
1843 | { |
---|
1844 | GS[i][1]=postredgb(mingb(GS[i][1])); |
---|
1845 | for(j=1;j<=size(GS[i][2]);j++) |
---|
1846 | { |
---|
1847 | GS[i][2][j][2]=postredgb(mingb(GS[i][2][j][2])); |
---|
1848 | } |
---|
1849 | } |
---|
1850 | } |
---|
1851 | else |
---|
1852 | { |
---|
1853 | for (i=1;i<=size(GS);i++) |
---|
1854 | { |
---|
1855 | if(GS[i][2]==1) |
---|
1856 | { |
---|
1857 | GS[i][3]=postredgb(mingb(GS[i][3])); |
---|
1858 | if (typeof(GS[i][7])=="ideal") |
---|
1859 | { GS[i][7]=postredgb(mingb(GS[i][7]));} |
---|
1860 | } |
---|
1861 | } |
---|
1862 | } |
---|
1863 | } |
---|
1864 | if(te==0){kill @P; kill @R; kill @RP;} |
---|
1865 | return(GS); |
---|
1866 | } |
---|
1867 | example |
---|
1868 | { |
---|
1869 | "EXAMPLE:"; echo = 2; |
---|
1870 | // Casas conjecture for degree 4: |
---|
1871 | ring R=(0,a0,a1,a2,a3,a4),(x1,x2,x3),dp; |
---|
1872 | short=0; |
---|
1873 | ideal F=x1^4+(4*a3)*x1^3+(6*a2)*x1^2+(4*a1)*x1+(a0), |
---|
1874 | x1^3+(3*a3)*x1^2+(3*a2)*x1+(a1), |
---|
1875 | x2^4+(4*a3)*x2^3+(6*a2)*x2^2+(4*a1)*x2+(a0), |
---|
1876 | x2^2+(2*a3)*x2+(a2), |
---|
1877 | x3^4+(4*a3)*x3^3+(6*a2)*x3^2+(4*a1)*x3+(a0), |
---|
1878 | x3+(a3); |
---|
1879 | cgsdr(F); |
---|
1880 | } |
---|
1881 | |
---|
1882 | // input: internal routine called by cgsdr at the end to group the |
---|
1883 | // lpp segments and improve the output |
---|
1884 | // output: grouped segments by lpp obtained in cgsdr |
---|
1885 | static proc grsegments(list T) |
---|
1886 | { |
---|
1887 | int i; |
---|
1888 | list L; |
---|
1889 | list lpp; |
---|
1890 | list lp; |
---|
1891 | list ls; |
---|
1892 | int n=size(T); |
---|
1893 | lpp[1]=T[n][1]; |
---|
1894 | L[1]=list(lpp[1],list(list(T[n][2],T[n][3],T[n][4]))); |
---|
1895 | if (n>1) |
---|
1896 | { |
---|
1897 | for (i=1;i<=size(T)-1;i++) |
---|
1898 | { |
---|
1899 | lp=memberpos(T[n-i][1],lpp); |
---|
1900 | if(lp[1]==1) |
---|
1901 | { |
---|
1902 | ls=L[lp[2]][2]; |
---|
1903 | ls[size(ls)+1]=list(T[n-i][2],T[n-i][3],T[n-i][4]); |
---|
1904 | L[lp[2]][2]=ls; |
---|
1905 | } |
---|
1906 | else |
---|
1907 | { |
---|
1908 | lpp[size(lpp)+1]=T[n-i][1]; |
---|
1909 | L[size(L)+1]=list(T[n-i][1],list(list(T[n-i][2],T[n-i][3],T[n-i][4]))); |
---|
1910 | } |
---|
1911 | } |
---|
1912 | } |
---|
1913 | return(L); |
---|
1914 | } |
---|
1915 | |
---|
1916 | // LCUnion |
---|
1917 | // Given a list of the P-representations of locally closed segments |
---|
1918 | // for which we know that the union is also locally closed |
---|
1919 | // it returns the P-representation of its union |
---|
1920 | // input: L list of segments in P-representation |
---|
1921 | // ((p_j^i,(p_j1^i,...,p_jk_j^i | j=1..t_i)) | i=1..s ) |
---|
1922 | // where i represents a segment |
---|
1923 | // output: P-representation of the union |
---|
1924 | // ((P_j,(P_j1,...,P_jk_j | j=1..t))) |
---|
1925 | static proc LCUnion(list LL) |
---|
1926 | { |
---|
1927 | def RR=basering; |
---|
1928 | setring(@P); |
---|
1929 | def L=imap(RR,LL); |
---|
1930 | int i; int j; int k; list H; list C; list T; |
---|
1931 | list L0; list P0; list P; list Q0; list Q; |
---|
1932 | for (i=1;i<=size(L);i++) |
---|
1933 | { |
---|
1934 | for (j=1;j<=size(L[i]);j++) |
---|
1935 | { |
---|
1936 | P0[size(P0)+1]=L[i][j][1]; |
---|
1937 | L0[size(L0)+1]=intvec(i,j); |
---|
1938 | } |
---|
1939 | } |
---|
1940 | Q0=selectminideals(P0); |
---|
1941 | for (i=1;i<=size(Q0);i++) |
---|
1942 | { |
---|
1943 | Q[i]=L0[Q0[i]]; |
---|
1944 | P[i]=L[Q[i][1]][Q[i][2]]; |
---|
1945 | } |
---|
1946 | // P is the list of the maximal components of the union |
---|
1947 | // with the corresponding initial holes. |
---|
1948 | // Q is the list of intvec positions in L of the first element of the P's |
---|
1949 | // Its elements give (num of segment, num of max component (=min ideal)) |
---|
1950 | for (k=1;k<=size(Q);k++) |
---|
1951 | { |
---|
1952 | H=P[k][2]; // holes of P[k][1] |
---|
1953 | for (i=1;i<=size(L);i++) |
---|
1954 | { |
---|
1955 | if (i!=Q[k][1]) |
---|
1956 | { |
---|
1957 | for (j=1;j<=size(L[i]);j++) |
---|
1958 | { |
---|
1959 | C[size(C)+1]=L[i][j]; |
---|
1960 | } |
---|
1961 | } |
---|
1962 | } |
---|
1963 | T[size(T)+1]=list(Q[k],P[k][1],addpart(H,C)); |
---|
1964 | } |
---|
1965 | setring(RR); |
---|
1966 | def TT=imap(@P,T); |
---|
1967 | return(TT); |
---|
1968 | } |
---|
1969 | |
---|
1970 | // Auxiliary routine |
---|
1971 | // called by LCUnion to modify the holes of a primepart of the union |
---|
1972 | // by the addition of the segments that do not correspond to that part |
---|
1973 | // Works on @P ring. |
---|
1974 | // Input: |
---|
1975 | // H=(p_i1,..,p_is) the holes of a component to be transformed by the addition of |
---|
1976 | // the segments C that do not correspond to that component |
---|
1977 | // C=((q_1,(q_11,..,q_1l_1),pos1),..,(q_k,(q_k1,..,q_kl_k),posk)) |
---|
1978 | // posi=(i,j) position of the component |
---|
1979 | // the list of segments to be added to the holes |
---|
1980 | static proc addpart(list H, list C) |
---|
1981 | { |
---|
1982 | list Q; int i; int j; int k; int l; int t; int t1; |
---|
1983 | Q=H; intvec notQ; list QQ; list addq; |
---|
1984 | // @Q2=list of (i,j) positions of the components that have been aded to some hole of the maximal ideals |
---|
1985 | // plus those of the components added to the holes. |
---|
1986 | ideal q; |
---|
1987 | i=1; |
---|
1988 | while (i<=size(Q)) |
---|
1989 | { |
---|
1990 | if (memberpos(i,notQ)[1]==0) |
---|
1991 | { |
---|
1992 | q=Q[i]; |
---|
1993 | t=1; j=1; |
---|
1994 | while ((t) and (j<=size(C))) |
---|
1995 | { |
---|
1996 | if (equalideals(q,C[j][1])) |
---|
1997 | { |
---|
1998 | // \\ @Q2[size(@Q2)+1]=C[j][3]; |
---|
1999 | t=0; |
---|
2000 | for (k=1;k<=size(C[j][2]);k++) |
---|
2001 | { |
---|
2002 | t1=1; |
---|
2003 | l=1; |
---|
2004 | while((t1) and (l<=size(Q))) |
---|
2005 | { |
---|
2006 | if ((l!=i) and (memberpos(l,notQ)[1]==0)) |
---|
2007 | { |
---|
2008 | if (idcontains(C[j][2][k],Q[l])) |
---|
2009 | { |
---|
2010 | t1=0; |
---|
2011 | } |
---|
2012 | } |
---|
2013 | l++; |
---|
2014 | } |
---|
2015 | if (t1) |
---|
2016 | { |
---|
2017 | addq[size(addq)+1]=C[j][2][k]; |
---|
2018 | // \\ @Q2[size(@Q2)+1]=C[j][3]; |
---|
2019 | } |
---|
2020 | } |
---|
2021 | if((size(notQ)==1) and (notQ[1]==0)){notQ[1]=i;} |
---|
2022 | else {notQ[size(notQ)+1]=i;} |
---|
2023 | } |
---|
2024 | j++; |
---|
2025 | } |
---|
2026 | if (size(addq)>0) |
---|
2027 | { |
---|
2028 | for (k=1;k<=size(addq);k++) |
---|
2029 | { |
---|
2030 | Q[size(Q)+1]=addq[k]; |
---|
2031 | } |
---|
2032 | kill addq; |
---|
2033 | list addq; |
---|
2034 | } |
---|
2035 | } |
---|
2036 | i++; |
---|
2037 | } |
---|
2038 | for (i=1;i<=size(Q);i++) |
---|
2039 | { |
---|
2040 | if(memberpos(i,notQ)[1]==0) |
---|
2041 | { |
---|
2042 | QQ[size(QQ)+1]=Q[i]; |
---|
2043 | } |
---|
2044 | } |
---|
2045 | if (size(QQ)==0){QQ[1]=ideal(1);} |
---|
2046 | return(addpartfine(QQ,C)); |
---|
2047 | } |
---|
2048 | |
---|
2049 | // Auxiliary routine called by addpart to finish the modification of the holes of a primepart |
---|
2050 | // of the union by the addition of the segments that do not correspond to |
---|
2051 | // that part. |
---|
2052 | // Works on @P ring. |
---|
2053 | static proc addpartfine(list H, list C0) |
---|
2054 | { |
---|
2055 | //"T_H="; H; |
---|
2056 | int i; int j; int k; int te; intvec notQ; int l; list sel; int used; |
---|
2057 | intvec jtesC; |
---|
2058 | if ((size(H)==1) and (equalideals(H[1],ideal(1)))){return(H);} |
---|
2059 | if (size(C0)==0){return(H);} |
---|
2060 | list newQ; list nQ; list Q; list nQ1; list Q0; |
---|
2061 | def Q1=H; |
---|
2062 | //Q1=sortlistideals(Q1,idbefid); |
---|
2063 | def C=C0; |
---|
2064 | while(equallistideals(Q0,Q1)==0) |
---|
2065 | { |
---|
2066 | Q0=Q1; |
---|
2067 | i=0; |
---|
2068 | Q=Q1; |
---|
2069 | kill notQ; intvec notQ; |
---|
2070 | while(i<size(Q)) |
---|
2071 | { |
---|
2072 | i++; |
---|
2073 | for(j=1;j<=size(C);j++) |
---|
2074 | { |
---|
2075 | te=idcontains(Q[i],C[j][1]); |
---|
2076 | if(te) |
---|
2077 | { |
---|
2078 | for(k=1;k<=size(C[j][2]);k++) |
---|
2079 | { |
---|
2080 | if(idcontains(Q[i],C[j][2][k])) |
---|
2081 | { |
---|
2082 | te=0; break; |
---|
2083 | } |
---|
2084 | } |
---|
2085 | if (te) |
---|
2086 | { |
---|
2087 | used++; |
---|
2088 | if ((size(notQ)==1) and (notQ[1]==0)){notQ[1]=i;} |
---|
2089 | else{notQ[size(notQ)+1]=i;} |
---|
2090 | kill newQ; list newQ; |
---|
2091 | for(k=1;k<=size(C[j][2]);k++) |
---|
2092 | { |
---|
2093 | nQ=minGTZ(Q[i]+C[j][2][k]); |
---|
2094 | for(l=1;l<=size(nQ);l++) |
---|
2095 | { |
---|
2096 | option(redSB); |
---|
2097 | nQ[l]=std(nQ[l]); |
---|
2098 | newQ[size(newQ)+1]=nQ[l]; |
---|
2099 | } |
---|
2100 | } |
---|
2101 | sel=selectminideals(newQ); |
---|
2102 | kill nQ1; list nQ1; |
---|
2103 | for(l=1;l<=size(sel);l++) |
---|
2104 | { |
---|
2105 | nQ1[l]=newQ[sel[l]]; |
---|
2106 | } |
---|
2107 | newQ=nQ1; |
---|
2108 | for(l=1;l<=size(newQ);l++) |
---|
2109 | { |
---|
2110 | Q[size(Q)+1]=newQ[l]; |
---|
2111 | } |
---|
2112 | break; |
---|
2113 | } |
---|
2114 | } |
---|
2115 | } |
---|
2116 | } |
---|
2117 | kill Q1; list Q1; |
---|
2118 | for(i=1;i<=size(Q);i++) |
---|
2119 | { |
---|
2120 | if(memberpos(i,notQ)[1]==0) |
---|
2121 | { |
---|
2122 | Q1[size(Q1)+1]=Q[i]; |
---|
2123 | } |
---|
2124 | } |
---|
2125 | //"T_Q1_0="; Q1; |
---|
2126 | sel=selectminideals(Q1); |
---|
2127 | kill nQ1; list nQ1; |
---|
2128 | for(l=1;l<=size(sel);l++) |
---|
2129 | { |
---|
2130 | nQ1[l]=Q1[sel[l]]; |
---|
2131 | } |
---|
2132 | Q1=nQ1; |
---|
2133 | } |
---|
2134 | if(size(Q1)==0){Q1=ideal(1),ideal(1);} |
---|
2135 | //"T_Q1_1="; Q1; |
---|
2136 | //if(used>0){string("addpartfine was ", used, " times used");} |
---|
2137 | return(Q1); |
---|
2138 | } |
---|
2139 | |
---|
2140 | |
---|
2141 | // Auxiliary routine for grobcov: ideal F is assumed to be homogeneous |
---|
2142 | // gcover |
---|
2143 | // input: ideal F: a generating set of a homogeneous ideal in Q[a][x] |
---|
2144 | // list #: optional |
---|
2145 | // output: the list |
---|
2146 | // S=((lpp, generic basis, Prep, Crep),..,(lpp, generic basis, Prep, Crep)) |
---|
2147 | // where a Prep is ( (p1,(p11,..,p1k_1)),..,(pj,(pj1,..,p1k_j)) ) |
---|
2148 | // a Crep is ( ida, idb ) |
---|
2149 | static proc gcover(ideal F,list #) |
---|
2150 | { |
---|
2151 | int i; int j; int k; ideal lpp; list GPi2; list pairspP; ideal B; int ti; |
---|
2152 | int i1; int tes; int j1; int selind; int i2; int m; |
---|
2153 | list prep; list crep; list LCU; poly p; poly lcp; ideal FF; |
---|
2154 | list lpi; |
---|
2155 | string lpph; |
---|
2156 | list L=#; |
---|
2157 | int canop=1; |
---|
2158 | int extop=1; |
---|
2159 | int repop=0; |
---|
2160 | ideal E=ideal(0);; |
---|
2161 | ideal N=ideal(1);; |
---|
2162 | int comment; |
---|
2163 | for(i=1;i<=size(L) div 2;i++) |
---|
2164 | { |
---|
2165 | if(L[2*i-1]=="can"){canop=L[2*i];} |
---|
2166 | else |
---|
2167 | { |
---|
2168 | if(L[2*i-1]=="ext"){extop=L[2*i];} |
---|
2169 | else |
---|
2170 | { |
---|
2171 | if(L[2*i-1]=="rep"){repop=L[2*i];} |
---|
2172 | else |
---|
2173 | { |
---|
2174 | if(L[2*i-1]=="null"){E=L[2*i];} |
---|
2175 | else |
---|
2176 | { |
---|
2177 | if(L[2*i-1]=="nonnull"){N=L[2*i];} |
---|
2178 | else |
---|
2179 | { |
---|
2180 | if (L[2*i-1]=="comment"){comment=L[2*i];} |
---|
2181 | } |
---|
2182 | } |
---|
2183 | } |
---|
2184 | } |
---|
2185 | } |
---|
2186 | } |
---|
2187 | list GS; list GP; |
---|
2188 | def RR=basering; |
---|
2189 | GS=cgsdr(F,L); // "null",NW[1],"nonnull",NW[2],"cgs",CGS,"comment",comment); |
---|
2190 | setglobalrings(); |
---|
2191 | int start=timer; |
---|
2192 | GP=GS; |
---|
2193 | ideal lppr; |
---|
2194 | list LL; |
---|
2195 | list S; |
---|
2196 | poly sp; |
---|
2197 | ideal BB; |
---|
2198 | for (i=1;i<=size(GP);i++) |
---|
2199 | { |
---|
2200 | kill LL; |
---|
2201 | list LL; |
---|
2202 | lpp=GP[i][1]; |
---|
2203 | GPi2=GP[i][2]; |
---|
2204 | lpph=GP[i][3]; |
---|
2205 | kill pairspP; list pairspP; |
---|
2206 | for(j=1;j<=size(GPi2);j++) |
---|
2207 | { |
---|
2208 | pairspP[size(pairspP)+1]=GPi2[j][3]; |
---|
2209 | } |
---|
2210 | LCU=LCUnion(pairspP); |
---|
2211 | kill prep; list prep; |
---|
2212 | kill crep; list crep; |
---|
2213 | for(k=1;k<=size(LCU);k++) |
---|
2214 | { |
---|
2215 | prep[k]=list(LCU[k][2],LCU[k][3]); |
---|
2216 | B=GPi2[LCU[k][1][1]][2]; // ATENTION last 1 has been changed to [2] |
---|
2217 | LCU[k][1]=B; |
---|
2218 | } |
---|
2219 | //"Deciding if combine is needed"; |
---|
2220 | kill BB; |
---|
2221 | ideal BB; |
---|
2222 | tes=1; m=1; |
---|
2223 | while((tes) and (m<=size(LCU[1][1]))) |
---|
2224 | { |
---|
2225 | j=1; |
---|
2226 | while((tes) and (j<=size(LCU))) |
---|
2227 | { |
---|
2228 | k=1; |
---|
2229 | while((tes) and (k<=size(LCU))) |
---|
2230 | { |
---|
2231 | if(j!=k) |
---|
2232 | { |
---|
2233 | sp=pnormalf(pspol(LCU[j][1][m],LCU[k][1][m]),LCU[k][2],N); |
---|
2234 | if(sp!=0){tes=0;} |
---|
2235 | } |
---|
2236 | k++; |
---|
2237 | } //setglobalrings(); |
---|
2238 | if(tes) |
---|
2239 | { |
---|
2240 | BB[m]=LCU[j][1][m]; |
---|
2241 | } |
---|
2242 | j++; |
---|
2243 | } |
---|
2244 | if(tes==0){break;} |
---|
2245 | m++; |
---|
2246 | } |
---|
2247 | crep=PtoCrep(prep); |
---|
2248 | if(tes==0) |
---|
2249 | { |
---|
2250 | // combine is needed |
---|
2251 | kill B; ideal B; |
---|
2252 | for (j=1;j<=size(LCU);j++) |
---|
2253 | { |
---|
2254 | LL[j]=LCU[j][2]; |
---|
2255 | } |
---|
2256 | if (size(LCU)>1) |
---|
2257 | { |
---|
2258 | FF=precombint(LL); |
---|
2259 | } |
---|
2260 | for (k=1;k<=size(lpp);k++) |
---|
2261 | { |
---|
2262 | kill L; list L; |
---|
2263 | for (j=1;j<=size(LCU);j++) |
---|
2264 | { |
---|
2265 | L[j]=list(LCU[j][2],LCU[j][1][k]); |
---|
2266 | } |
---|
2267 | if (size(LCU)>1) |
---|
2268 | { |
---|
2269 | B[k]=combine(L,FF); |
---|
2270 | } |
---|
2271 | else{B[k]=L[1][2];} |
---|
2272 | } |
---|
2273 | } |
---|
2274 | else{B=BB;} |
---|
2275 | for(j=1;j<=size(B);j++) |
---|
2276 | { |
---|
2277 | B[j]=pnormalf(B[j],crep[1],crep[2]); |
---|
2278 | } |
---|
2279 | S[i]=list(lpp,B,prep,crep,lpph); |
---|
2280 | if(comment>=1) |
---|
2281 | { |
---|
2282 | lpi[size(lpi)+1]=string("[",i,"]"); |
---|
2283 | lpi[size(lpi)+1]=S[i][1]; |
---|
2284 | } |
---|
2285 | } |
---|
2286 | if(comment>=1) |
---|
2287 | { |
---|
2288 | string("Time in LCUnion + combine = ",timer-start); |
---|
2289 | if(comment>=2){string("lpp=",lpi)}; |
---|
2290 | } |
---|
2291 | if(defined(@P)==1){kill @P; kill @RP; kill @R;} |
---|
2292 | return(S); |
---|
2293 | } |
---|
2294 | |
---|
2295 | // grobcov |
---|
2296 | // input: |
---|
2297 | // ideal F: a parametric ideal in Q[a][x], (a=parameters, x=variables). |
---|
2298 | // list #: (options) list("null",N,"nonnull",W,"can",0-1,ext",0-1, "rep",0-1-2) |
---|
2299 | // where |
---|
2300 | // N is the null conditions ideal (if desired) |
---|
2301 | // W is the ideal of non-null conditions (if desired) |
---|
2302 | // The value of \"can\"i s 1 by default and can be set to 0 if we do not |
---|
2303 | // need to obtain the canonical GC, but only a GC. |
---|
2304 | // The value of \"ext\" is 0 by default and so the generic representation |
---|
2305 | // of the bases is given. It can be set to 1, and then the full |
---|
2306 | // representation of the bases is given. |
---|
2307 | // The value of \"rep\" is 0 by default, and then the segments |
---|
2308 | // are given in canonical P-representation. It can be set to 1 |
---|
2309 | // and then they are given in canonical C-representation. |
---|
2310 | // If it is set to 2, then both representations are given. |
---|
2311 | // output: |
---|
2312 | // list S: ((lpp,basis,(idp_1,(idp_11,..,idp_1s_1))), .. |
---|
2313 | // (lpp,basis,(idp_r,(idp_r1,..,idp_rs_r))) ) where |
---|
2314 | // each element of S corresponds to a lpp-segment |
---|
2315 | // given by the lpp, the basis, and the P-representation of the segment |
---|
2316 | proc grobcov(ideal F,list #) |
---|
2317 | "USAGE: grobcov(F); This is the fundamental routine of the |
---|
2318 | library. It computes the Groebner cover of a parametric ideal. See |
---|
2319 | Montes A., Wibmer M., |
---|
2320 | \"Groebner Bases for Polynomial Systems with parameters\". |
---|
2321 | JSC 45 (2010) 1391-1425.) |
---|
2322 | The Groebner Cover of a parametric ideal consist of a set of |
---|
2323 | pairs(S_i,B_i), where the S_i are disjoint locally closed |
---|
2324 | segments of the parameter space, and the B_i are the reduced |
---|
2325 | Groebner bases of the ideal on every point of S_i. |
---|
2326 | |
---|
2327 | The ideal F must be defined on a parametric ring Q[a][x]. |
---|
2328 | (a=parameters, x=variables) |
---|
2329 | Options: To modify the default options, pair of arguments |
---|
2330 | -option name, value- of valid options must be added to the call. |
---|
2331 | |
---|
2332 | Options: |
---|
2333 | \"null\",ideal E: The default is (\"null\",ideal(0)). |
---|
2334 | \"nonnull\",ideal N: The default is (\"nonnull\",ideal(1)). |
---|
2335 | When options \"null\" and/or \"nonnull\" are given, then |
---|
2336 | the parameter space is restricted to V(E) \ V(N). |
---|
2337 | \"can\",0-1: The default is (\"can\",1). With the default option |
---|
2338 | the homogenized ideal is computed before obtaining the |
---|
2339 | Groebner Cover, so that the result is the canonical |
---|
2340 | Groebner Cover. Setting (\"can\",0) only homogenizes the |
---|
2341 | basis so the result is not exactly canonical, but the |
---|
2342 | computation is shorter. |
---|
2343 | \"ext\",0-1: The default is (\"ext\",0). With the default |
---|
2344 | (\"ext\",0), only the generic representation of the bases is |
---|
2345 | computed (single polynomials, but not specializing to non-zero |
---|
2346 | for every point of the segment. With option (\"ext\",1) the |
---|
2347 | full representation of the bases is computed (possible |
---|
2348 | sheaves) and sometimes a simpler result is obtained, |
---|
2349 | but the computation is more time consuming. |
---|
2350 | \"rep\",0-1-2: The default is (\"rep\",0) and then the segments |
---|
2351 | are given in canonical P-representation. Option (\"rep\",1) |
---|
2352 | represents the segments in canonical C-representation, |
---|
2353 | and option (\"rep\",2) gives both representations. |
---|
2354 | \"comment\",0-3: The default is (\"comment\",0). Setting |
---|
2355 | \"comment\" higher will provide information about the |
---|
2356 | development of the computation. |
---|
2357 | One can give none or whatever of these options. |
---|
2358 | RETURN: The list |
---|
2359 | ( |
---|
2360 | (lpp_1,basis_1,segment_1,lpph_1), |
---|
2361 | ... |
---|
2362 | (lpp_s,basis_s,segment_s,lpph_s) |
---|
2363 | ) |
---|
2364 | |
---|
2365 | The lpp are constant over a segment and correspond to the |
---|
2366 | set of lpp of the reduced Groebner basis for each point |
---|
2367 | of the segment. |
---|
2368 | The lpph corresponds to the lpp of the homogenized ideal |
---|
2369 | and is different for each segment. It is given as a string, |
---|
2370 | and shown only for information. With the default option |
---|
2371 | \"can\",1, the segments have different lpph. |
---|
2372 | |
---|
2373 | Basis: to each element of lpp corresponds an I-regular function |
---|
2374 | given in full representation (by option (\"ext\",1)) or in |
---|
2375 | generic representation (default option (\"ext\",0)). The |
---|
2376 | I-regular function is the corresponding element of the reduced |
---|
2377 | Groebner basis for each point of the segment with the given lpp. |
---|
2378 | For each point in the segment, the polynomial or the set of |
---|
2379 | polynomials representing it, if they do not specialize to 0, |
---|
2380 | then after normalization, specializes to the corresponding |
---|
2381 | element of the reduced Groebner basis. In the full representation |
---|
2382 | at least one of the polynomials representing the I-regular |
---|
2383 | function specializes to non-zero. |
---|
2384 | |
---|
2385 | With the default option (\"rep\",0) the representation of the |
---|
2386 | segment is the P-representation. |
---|
2387 | With option (\"rep\",1) the representation of the segment is |
---|
2388 | the C-representation. |
---|
2389 | With option (\"rep\",2) both representations of the segment are |
---|
2390 | given. |
---|
2391 | |
---|
2392 | The P-representation of a segment is of the form |
---|
2393 | ((p_1,(p_11,..,p_1k1)),..,(p_r,(p_r1,..,p_rkr)) |
---|
2394 | representing the segment Union_i ( V(p_i) \ ( Union_j V(p_ij) ) ), |
---|
2395 | where the p's are prime ideals. |
---|
2396 | |
---|
2397 | The C-representation of a segment is of the form |
---|
2398 | (E,N) representing V(E) \ V(N), and the ideals E and N are |
---|
2399 | radical and N contains E. |
---|
2400 | |
---|
2401 | NOTE: The basering R, must be of the form Q[a][x], (a=parameters, |
---|
2402 | x=variables), and should be defined previously. The ideal must |
---|
2403 | be defined on R. |
---|
2404 | KEYWORDS: Groebner cover, parametric ideal, canonical, discussion of |
---|
2405 | parametric ideal. |
---|
2406 | EXAMPLE: grobcov; shows an example" |
---|
2407 | { |
---|
2408 | list S; int i; int ish=1; list GBR; list BR; int j; int k; |
---|
2409 | ideal idp; ideal idq; int s; ideal ext; list SS; |
---|
2410 | ideal E; ideal N; int canop; int extop; int repop; |
---|
2411 | int comment=0; int m; |
---|
2412 | def RR=basering; |
---|
2413 | setglobalrings(); |
---|
2414 | list L0=#; |
---|
2415 | int out=0; |
---|
2416 | L0[size(L0)+1]="res"; L0[size(L0)+1]=ideal(1); |
---|
2417 | // default options |
---|
2418 | int start=timer; |
---|
2419 | E=ideal(0); |
---|
2420 | N=ideal(1); |
---|
2421 | canop=1; // canop=0 for homogenizing the basis but not the ideal (not canonical) |
---|
2422 | // canop=1 for working with the homogenized ideal |
---|
2423 | repop=0; // repop=0 for representing the segments in Prep |
---|
2424 | // repop=1 for representing the segments in Crep |
---|
2425 | // repop=2 for representing the segments in Prep and Crep |
---|
2426 | extop=0; // extop=0 if only generic representation of the bases are to be computed |
---|
2427 | // extop=1 if the full representation of the bases are to be computed |
---|
2428 | for(i=1;i<=size(L0) div 2;i++) |
---|
2429 | { |
---|
2430 | if(L0[2*i-1]=="can"){canop=L0[2*i];} |
---|
2431 | else |
---|
2432 | { |
---|
2433 | if(L0[2*i-1]=="ext"){extop=L0[2*i];} |
---|
2434 | else |
---|
2435 | { |
---|
2436 | if(L0[2*i-1]=="rep"){repop=L0[2*i];} |
---|
2437 | else |
---|
2438 | { |
---|
2439 | if(L0[2*i-1]=="null"){E=L0[2*i];} |
---|
2440 | else |
---|
2441 | { |
---|
2442 | if(L0[2*i-1]=="nonnull"){N=L0[2*i];} |
---|
2443 | else |
---|
2444 | { |
---|
2445 | if (L0[2*i-1]=="comment"){comment=L0[2*i];} |
---|
2446 | } |
---|
2447 | } |
---|
2448 | } |
---|
2449 | } |
---|
2450 | } |
---|
2451 | } |
---|
2452 | if(not((canop==0) or (canop==1))) |
---|
2453 | { |
---|
2454 | string("Option can = ",canop," is not supported. It is changed to can = 1"); |
---|
2455 | canop=1; |
---|
2456 | } |
---|
2457 | for(i=1;i<=size(L0) div 2;i++) |
---|
2458 | { |
---|
2459 | if(L0[2*i-1]=="can"){L0[2*i]=canop;} |
---|
2460 | } |
---|
2461 | if ((printlevel) and (comment==0)){comment=printlevel;} |
---|
2462 | list LL; |
---|
2463 | LL[1]="can"; LL[2]=canop; |
---|
2464 | LL[3]="comment"; LL[4]=comment; |
---|
2465 | LL[5]="out"; LL[6]=0; |
---|
2466 | LL[7]="null"; LL[8]=E; |
---|
2467 | LL[9]="nonnull"; LL[10]=N; |
---|
2468 | LL[11]="ext"; LL[12]=extop; |
---|
2469 | LL[13]="rep"; LL[14]=repop; |
---|
2470 | if (comment>=1) |
---|
2471 | { |
---|
2472 | string("Begin grobcov with options: ",LL); |
---|
2473 | } |
---|
2474 | kill S; |
---|
2475 | def S=gcover(F,LL); |
---|
2476 | // NOW extend |
---|
2477 | if(extop) |
---|
2478 | { |
---|
2479 | S=extend(S,LL); |
---|
2480 | } |
---|
2481 | else |
---|
2482 | { |
---|
2483 | // NOW representation of the segments by option repop |
---|
2484 | list Si; list nS; |
---|
2485 | if(repop==0) |
---|
2486 | { |
---|
2487 | for(i=1;i<=size(S);i++) |
---|
2488 | { |
---|
2489 | Si=list(S[i][1],S[i][2],S[i][3],S[i][5]); |
---|
2490 | nS[size(nS)+1]=Si; |
---|
2491 | } |
---|
2492 | kill S; |
---|
2493 | def S=nS; |
---|
2494 | } |
---|
2495 | else |
---|
2496 | { |
---|
2497 | if(repop==1) |
---|
2498 | { |
---|
2499 | for(i=1;i<=size(S);i++) |
---|
2500 | { |
---|
2501 | Si=list(S[i][1],S[i][2],S[i][4],S[i][5]); |
---|
2502 | nS[size(nS)+1]=Si; |
---|
2503 | } |
---|
2504 | kill S; |
---|
2505 | def S=nS; |
---|
2506 | } |
---|
2507 | else |
---|
2508 | { |
---|
2509 | for(i=1;i<=size(S);i++) |
---|
2510 | { |
---|
2511 | Si=list(S[i][1],S[i][2],S[i][3],S[i][4],S[i][5]); |
---|
2512 | nS[size(nS)+1]=Si; |
---|
2513 | } |
---|
2514 | kill S; |
---|
2515 | def S=nS; |
---|
2516 | } |
---|
2517 | } |
---|
2518 | } |
---|
2519 | if (comment>=1) |
---|
2520 | { |
---|
2521 | string("Time in grobcov = ", timer-start); |
---|
2522 | string("Number of segments of grobcov = ", size(S)); |
---|
2523 | } |
---|
2524 | if(defined(@P)==1){kill @R; kill @P; kill @RP;} |
---|
2525 | return(S); |
---|
2526 | } |
---|
2527 | example |
---|
2528 | { |
---|
2529 | "EXAMPLE:"; echo = 2; |
---|
2530 | // Casas conjecture for degree 4: |
---|
2531 | ring R=(0,a0,a1,a2,a3,a4),(x1,x2,x3),dp; |
---|
2532 | short=0; |
---|
2533 | ideal F=x1^4+(4*a3)*x1^3+(6*a2)*x1^2+(4*a1)*x1+(a0), |
---|
2534 | x1^3+(3*a3)*x1^2+(3*a2)*x1+(a1), |
---|
2535 | x2^4+(4*a3)*x2^3+(6*a2)*x2^2+(4*a1)*x2+(a0), |
---|
2536 | x2^2+(2*a3)*x2+(a2), |
---|
2537 | x3^4+(4*a3)*x3^3+(6*a2)*x3^2+(4*a1)*x3+(a0), |
---|
2538 | x3+(a3); |
---|
2539 | grobcov(F); |
---|
2540 | } |
---|
2541 | |
---|
2542 | // Input. GC the grobcov of an ideal in generic representation of the |
---|
2543 | // bases computed with option option ("rep",2). |
---|
2544 | // Output The grobcov in full representation. |
---|
2545 | // Option ("comment",1) shows the time. |
---|
2546 | // Can be called from the top |
---|
2547 | proc extend(list GC, list #); |
---|
2548 | "USAGE: extend(GC); The default option of grobcov provides |
---|
2549 | the bases in generic representation (the I-regular functions |
---|
2550 | of the bases ara given by a single polynomial. It can specialize |
---|
2551 | to zero for some points of the segments, but in general, it |
---|
2552 | is sufficient for many pouposes. Nevertheless the I-regular |
---|
2553 | functions allow a full representation given bey a set of |
---|
2554 | polynomials specializing to the value of the function (after normalization) |
---|
2555 | or to zero, but at least one of the polynomials specializes to non-zero. |
---|
2556 | The full representation can be obtained by computing the |
---|
2557 | grobcov with option \"ext\",1. The default option is \"ext\",0. |
---|
2558 | With option \"ext\",1 the computation can be much more |
---|
2559 | time consuming, even if the result can be simpler. |
---|
2560 | Alternatively, one can compute the full representation of the |
---|
2561 | bases after computing grobcov with the defaoult option \"ext\",0 |
---|
2562 | and the option \"rep\",2, that outputs both the Prep and the Crep |
---|
2563 | of the segments and then call \"extend\" to the output. |
---|
2564 | |
---|
2565 | RETURN: When calling extend(grobcov(S,\"rep\",2)) the result is of the form |
---|
2566 | ( |
---|
2567 | (lpp_1,basis_1,segment_1,lpph_1), |
---|
2568 | ... |
---|
2569 | (lpp_s,basis_s,segment_s,lpph_s) |
---|
2570 | ) |
---|
2571 | where each function of the basis can be given by an ideal |
---|
2572 | of representants. |
---|
2573 | |
---|
2574 | NOTE: The basering R, must be of the form Q[a][x], (a=parameters, |
---|
2575 | x=variables), and should be defined previously. The ideal must |
---|
2576 | be defined on R. |
---|
2577 | KEYWORDS: Groebner cover, parametric ideal, canonical, discussion of |
---|
2578 | parametric ideal, full representation. |
---|
2579 | EXAMPLE: extend; shows an example" |
---|
2580 | { |
---|
2581 | list L=#; |
---|
2582 | list S=GC; |
---|
2583 | ideal idp; |
---|
2584 | ideal idq; |
---|
2585 | int i; int j; int m; int s; |
---|
2586 | m=0; i=1; |
---|
2587 | while((i<=size(S)) and (m==0)) |
---|
2588 | { |
---|
2589 | if(typeof(S[i][2])=="list"){m=1;} |
---|
2590 | i++; |
---|
2591 | } |
---|
2592 | if(m==1){"Warning! grobcov has already extended bases"; return(S);} |
---|
2593 | if(size(GC[1])!=5){"Warning! extend make sense only when grobcov has been called with options 'rep',2,'ext',0"; " "; return();} |
---|
2594 | int repop=0; |
---|
2595 | int start3=timer; |
---|
2596 | int comment; |
---|
2597 | for(i=1;i<=size(L) div 2;i++) |
---|
2598 | { |
---|
2599 | if(L[2*i-1]=="comment"){comment=L[2*i];} |
---|
2600 | else |
---|
2601 | { |
---|
2602 | if(L[2*i-1]=="rep"){repop=L[2*i];} |
---|
2603 | } |
---|
2604 | } |
---|
2605 | poly leadc; |
---|
2606 | poly ext; |
---|
2607 | int te=0; |
---|
2608 | list SS; |
---|
2609 | def R=basering; |
---|
2610 | if (defined(@R)){te=1;} |
---|
2611 | else{setglobalrings();} |
---|
2612 | // Now extend |
---|
2613 | for (i=1;i<=size(S);i++) |
---|
2614 | { |
---|
2615 | m=size(S[i][2]); |
---|
2616 | for (j=1;j<=m;j++) |
---|
2617 | { |
---|
2618 | idp=S[i][4][1]; |
---|
2619 | idq=S[i][4][2]; |
---|
2620 | if (size(idp)>0) |
---|
2621 | { |
---|
2622 | leadc=leadcoef(S[i][2][j]); |
---|
2623 | kill ext; |
---|
2624 | def ext=extend0(S[i][2][j],idp,idq); |
---|
2625 | if (typeof(ext)=="poly") |
---|
2626 | { |
---|
2627 | S[i][2][j]=pnormalf(ext,idp,idq); |
---|
2628 | } |
---|
2629 | else |
---|
2630 | { |
---|
2631 | if(size(ext)==1) |
---|
2632 | { |
---|
2633 | S[i][2][j]=ext[1]; |
---|
2634 | } |
---|
2635 | else |
---|
2636 | { |
---|
2637 | kill SS; list SS; |
---|
2638 | for(s=1;s<=size(ext);s++) |
---|
2639 | { |
---|
2640 | ext[s]=pnormalf(ext[s],idp,idq); |
---|
2641 | } |
---|
2642 | for(s=1;s<=size(S[i][2]);s++) |
---|
2643 | { |
---|
2644 | if(s!=j){SS[s]=S[i][2][s];} |
---|
2645 | else{SS[s]=ext;} |
---|
2646 | } |
---|
2647 | S[i][2]=SS; |
---|
2648 | } |
---|
2649 | } |
---|
2650 | } |
---|
2651 | } |
---|
2652 | } |
---|
2653 | // NOW representation of the segments by option repop |
---|
2654 | list Si; list nS; |
---|
2655 | if (repop==0) |
---|
2656 | { |
---|
2657 | for(i=1;i<=size(S);i++) |
---|
2658 | { |
---|
2659 | Si=list(S[i][1],S[i][2],S[i][3],S[i][5]); |
---|
2660 | nS[size(nS)+1]=Si; |
---|
2661 | } |
---|
2662 | S=nS; |
---|
2663 | } |
---|
2664 | else |
---|
2665 | { |
---|
2666 | if (repop==1) |
---|
2667 | { |
---|
2668 | for(i=1;i<=size(S);i++) |
---|
2669 | { |
---|
2670 | Si=list(S[i][1],S[i][2],S[i][4],S[i][5]); |
---|
2671 | nS[size(nS)+1]=Si; |
---|
2672 | } |
---|
2673 | S=nS; |
---|
2674 | } |
---|
2675 | else |
---|
2676 | { |
---|
2677 | for(i=1;i<=size(S);i++) |
---|
2678 | { |
---|
2679 | Si=list(S[i][1],S[i][2],S[i][3],S[i][4],S[i][5]); |
---|
2680 | nS[size(nS)+1]=Si; |
---|
2681 | } |
---|
2682 | |
---|
2683 | } |
---|
2684 | } |
---|
2685 | if(comment>=1){string("Time in extend = ",timer-start3);} |
---|
2686 | if(te==0){kill @R; kill @RP; kill @P;} |
---|
2687 | return(S); |
---|
2688 | } |
---|
2689 | example |
---|
2690 | { |
---|
2691 | "EXAMPLE:"; echo = 2; |
---|
2692 | ring R=(0,a0,b0,c0,a1,b1,c1),(x), dp; |
---|
2693 | short=0; |
---|
2694 | ideal S=a0*x^2+b0*x+c0, |
---|
2695 | a1*x^2+b1*x+c1; |
---|
2696 | def GCS=grobcov(S,"rep",2); |
---|
2697 | GCS; |
---|
2698 | def FGC=extend(GCS,"rep",0); |
---|
2699 | // Full representation= |
---|
2700 | FGC; |
---|
2701 | } |
---|
2702 | |
---|
2703 | // Auxiliary routine |
---|
2704 | // nonzerodivisor |
---|
2705 | // input: |
---|
2706 | // poly g in Q[a], |
---|
2707 | // list P=(p_1,..p_r) representing a minimal prime decomposition |
---|
2708 | // output |
---|
2709 | // poly f such that f notin p_i for all i and |
---|
2710 | // g-f in p_i for all i such that g notin p_i |
---|
2711 | static proc nonzerodivisor(poly gr, list Pr) |
---|
2712 | { |
---|
2713 | def RR=basering; |
---|
2714 | setring(@P); |
---|
2715 | def g=imap(RR,gr); |
---|
2716 | def P=imap(RR,Pr); |
---|
2717 | int i; int k; list J; ideal F; |
---|
2718 | def f=g; |
---|
2719 | ideal Pi; |
---|
2720 | for (i=1;i<=size(P);i++) |
---|
2721 | { |
---|
2722 | option(redSB); |
---|
2723 | Pi=std(P[i]); |
---|
2724 | //attrib(Pi,"isSB",1); |
---|
2725 | if (reduce(g,Pi,1)==0){J[size(J)+1]=i;} |
---|
2726 | } |
---|
2727 | for (i=1;i<=size(J);i++) |
---|
2728 | { |
---|
2729 | F=ideal(1); |
---|
2730 | for (k=1;k<=size(P);k++) |
---|
2731 | { |
---|
2732 | if (k!=J[i]) |
---|
2733 | { |
---|
2734 | F=idint(F,P[k]); |
---|
2735 | } |
---|
2736 | } |
---|
2737 | f=f+F[1]; |
---|
2738 | } |
---|
2739 | setring(RR); |
---|
2740 | def fr=imap(@P,f); |
---|
2741 | return(fr); |
---|
2742 | } |
---|
2743 | |
---|
2744 | // Auxiliary routine |
---|
2745 | // deltai |
---|
2746 | // input: |
---|
2747 | // int i: |
---|
2748 | // list LPr: (p1,..,pr) of prime components of an ideal in Q[a] |
---|
2749 | // output: |
---|
2750 | // list (fr,fnr) of two polynomials that are equal on V(pi) |
---|
2751 | // and fr=0 on V(P) \ V(pi), and fnr is nonzero on V(pj) for all j. |
---|
2752 | static proc deltai(int i, list LPr) |
---|
2753 | { |
---|
2754 | def RR=basering; |
---|
2755 | setring(@P); |
---|
2756 | def LP=imap(RR,LPr); |
---|
2757 | int j; poly p; |
---|
2758 | def F=ideal(1); |
---|
2759 | poly f; |
---|
2760 | poly fn; |
---|
2761 | ideal LPi; |
---|
2762 | for (j=1;j<=size(LP);j++) |
---|
2763 | { |
---|
2764 | if (j!=i) |
---|
2765 | { |
---|
2766 | F=idint(F,LP[j]); |
---|
2767 | } |
---|
2768 | } |
---|
2769 | p=0; j=1; |
---|
2770 | while ((p==0) and (j<=size(F))) |
---|
2771 | { |
---|
2772 | LPi=LP[i]; |
---|
2773 | attrib(LPi,"isSB",1); |
---|
2774 | p=reduce(F[j],LPi); |
---|
2775 | j++; |
---|
2776 | } |
---|
2777 | f=F[j-1]; |
---|
2778 | fn=nonzerodivisor(f,LP); |
---|
2779 | setring(RR); |
---|
2780 | def fr=imap(@P,f); |
---|
2781 | def fnr=imap(@P,fn); |
---|
2782 | return(list(fr,fnr)); |
---|
2783 | } |
---|
2784 | |
---|
2785 | // Auxiliary routine |
---|
2786 | // combine |
---|
2787 | // input: a list of pairs ((p1,P1),..,(pr,Pr)) where |
---|
2788 | // ideal pi is a prime component |
---|
2789 | // poly Pi is the polynomial in Q[a][x] on V(pi)\ V(Mi) |
---|
2790 | // (p1,..,pr) are the prime decomposition of the lpp-segment |
---|
2791 | // list crep =(ideal ida,ideal idb): the Crep of the segment. |
---|
2792 | // list Pci of the intersecctions of all pj except the ith one |
---|
2793 | // output: |
---|
2794 | // poly P on an open and dense set of V(p_1 int ... p_r) |
---|
2795 | static proc combine(list L, ideal F) |
---|
2796 | { |
---|
2797 | // ATTENTION REVISE AND USE Pci and F |
---|
2798 | int i; poly f; |
---|
2799 | f=0; |
---|
2800 | for(i=1;i<=size(L);i++) |
---|
2801 | { |
---|
2802 | f=f+F[i]*L[i][2]; |
---|
2803 | } |
---|
2804 | // f=elimconstfac(f); |
---|
2805 | f=primepartZ(f); |
---|
2806 | return(f); |
---|
2807 | } |
---|
2808 | |
---|
2809 | |
---|
2810 | //Auxiliary routine |
---|
2811 | // nullin |
---|
2812 | // input: |
---|
2813 | // poly f: a polynomial in Q[a] |
---|
2814 | // ideal P: an ideal in Q[a] |
---|
2815 | // called from ring @R |
---|
2816 | // output: |
---|
2817 | // t: with value 1 if f reduces modulo P, 0 if not. |
---|
2818 | static proc nullin(poly f,ideal P) |
---|
2819 | { |
---|
2820 | int t; |
---|
2821 | def RR=basering; |
---|
2822 | setring(@P); |
---|
2823 | def f0=imap(RR,f); |
---|
2824 | def P0=imap(RR,P); |
---|
2825 | attrib(P0,"isSB",1); |
---|
2826 | if (reduce(f0,P0,1)==0){t=1;} |
---|
2827 | else{t=0;} |
---|
2828 | setring(RR); |
---|
2829 | return(t); |
---|
2830 | } |
---|
2831 | |
---|
2832 | // Auxiliary routine |
---|
2833 | // monoms |
---|
2834 | // Input: A polynomial f |
---|
2835 | // Output: The list of leading terms |
---|
2836 | static proc monoms(poly f) |
---|
2837 | { |
---|
2838 | list L; |
---|
2839 | poly lm; poly lc; poly lp; poly Q; poly mQ; |
---|
2840 | def p=f; |
---|
2841 | int i=1; |
---|
2842 | while (p!=0) |
---|
2843 | { |
---|
2844 | lm=lead(p); |
---|
2845 | p=p-lm; |
---|
2846 | lc=leadcoef(lm); |
---|
2847 | lp=leadmonom(lm); |
---|
2848 | L[size(L)+1]=list(lc,lp); |
---|
2849 | i++; |
---|
2850 | } |
---|
2851 | return(L); |
---|
2852 | } |
---|
2853 | |
---|
2854 | // Auxiliary routine called by extend |
---|
2855 | // extend0 |
---|
2856 | // input: |
---|
2857 | // poly f: a generic polynomial in the basis |
---|
2858 | // ideal idp: such that ideal(S)=idp |
---|
2859 | // ideal idq: such that S=V(idp) \ V(idq) |
---|
2860 | //// NW the list of ((N1,W1),..,(Ns,Ws)) of red-rep of the grouped |
---|
2861 | //// segments in the lpp-segment NO MORE USED |
---|
2862 | // output: |
---|
2863 | static proc extend0(poly f, ideal idp, ideal idq) |
---|
2864 | { |
---|
2865 | matrix CC; poly Q; list NewMonoms; |
---|
2866 | int i; int j; poly fout; ideal idout; |
---|
2867 | list L=monoms(f); |
---|
2868 | int nummonoms=size(L)-1; |
---|
2869 | Q=L[1][1]; |
---|
2870 | if (nummonoms==0){return(f);} |
---|
2871 | for (i=2;i<=size(L);i++) |
---|
2872 | { |
---|
2873 | CC=matrix(extendcoef(L[i][1],Q,idp,idq)); |
---|
2874 | NewMonoms[i-1]=list(CC,L[i][2]); |
---|
2875 | } |
---|
2876 | if (nummonoms==1) |
---|
2877 | { |
---|
2878 | for(j=1;j<=ncols(NewMonoms[1][1]);j++) |
---|
2879 | { |
---|
2880 | fout=NewMonoms[1][1][2,j]*L[1][2]+NewMonoms[1][1][1,j]*NewMonoms[1][2]; |
---|
2881 | //fout=pnormalf(fout,idp,W); |
---|
2882 | if(ncols(NewMonoms[1][1])>1){idout[j]=fout;} |
---|
2883 | } |
---|
2884 | if(ncols(NewMonoms[1][1])==1){return(fout);} else{return(idout);} |
---|
2885 | } |
---|
2886 | else |
---|
2887 | { |
---|
2888 | list cfi; |
---|
2889 | list coefs; |
---|
2890 | for (i=1;i<=nummonoms;i++) |
---|
2891 | { |
---|
2892 | kill cfi; list cfi; |
---|
2893 | for(j=1;j<=ncols(NewMonoms[i][1]);j++) |
---|
2894 | { |
---|
2895 | cfi[size(cfi)+1]=NewMonoms[i][1][2,j]; |
---|
2896 | } |
---|
2897 | coefs[i]=cfi; |
---|
2898 | } |
---|
2899 | def indexpolys=findindexpolys(coefs); |
---|
2900 | for(i=1;i<=size(indexpolys);i++) |
---|
2901 | { |
---|
2902 | fout=L[1][2]; |
---|
2903 | for(j=1;j<=nummonoms;j++) |
---|
2904 | { |
---|
2905 | fout=fout+(NewMonoms[j][1][1,indexpolys[i][j]])/(NewMonoms[j][1][2,indexpolys[i][j]])*NewMonoms[j][2]; |
---|
2906 | } |
---|
2907 | fout=cleardenom(fout); |
---|
2908 | if(size(indexpolys)>1){idout[i]=fout;} |
---|
2909 | } |
---|
2910 | if (size(indexpolys)==1){return(fout);} else{return(idout);} |
---|
2911 | } |
---|
2912 | } |
---|
2913 | |
---|
2914 | // Auxiliary routine |
---|
2915 | // findindexpolys |
---|
2916 | // input: |
---|
2917 | // list coefs=( (q11,..,q1r_1),..,(qs1,..,qsr_1) ) |
---|
2918 | // of denominators of the monoms |
---|
2919 | // output: |
---|
2920 | // list ind=(v_1,..,v_t) of intvec |
---|
2921 | // each intvec v=(i_1,..,is) corresponds to a polynomial in the sheaf |
---|
2922 | // that will be built from it in extend procedure. |
---|
2923 | static proc findindexpolys(list coefs) |
---|
2924 | { |
---|
2925 | int i; int j; intvec numdens; |
---|
2926 | for(i=1;i<=size(coefs);i++) |
---|
2927 | { |
---|
2928 | numdens[i]=size(coefs[i]); |
---|
2929 | } |
---|
2930 | def RR=basering; |
---|
2931 | setring(@P); |
---|
2932 | def coefsp=imap(RR,coefs); |
---|
2933 | ideal cof; list combpolys; intvec v; int te; list mp; |
---|
2934 | for(i=1;i<=size(coefsp);i++) |
---|
2935 | { |
---|
2936 | cof=ideal(0); |
---|
2937 | for(j=1;j<=size(coefsp[i]);j++) |
---|
2938 | { |
---|
2939 | cof[j]=factorize(coefsp[i][j],3); |
---|
2940 | } |
---|
2941 | coefsp[i]=cof; |
---|
2942 | } |
---|
2943 | for(j=1;j<=size(coefsp[1]);j++) |
---|
2944 | { |
---|
2945 | v[1]=j; |
---|
2946 | te=1; |
---|
2947 | for (i=2;i<=size(coefsp);i++) |
---|
2948 | { |
---|
2949 | mp=memberpos(coefsp[1][j],coefsp[i]); |
---|
2950 | if(mp[1]) |
---|
2951 | { |
---|
2952 | v[i]=mp[2]; |
---|
2953 | } |
---|
2954 | else{v[i]=0;} |
---|
2955 | } |
---|
2956 | combpolys[j]=v; |
---|
2957 | } |
---|
2958 | combpolys=reform(combpolys,numdens); |
---|
2959 | setring(RR); |
---|
2960 | return(combpolys); |
---|
2961 | } |
---|
2962 | |
---|
2963 | // Auxiliary routine |
---|
2964 | // extendcoef: given Q,P in Q[a] where P/Q specializes on an open and dense subset |
---|
2965 | // of the whole V(p1 int...int pr), it returns a basis of the module |
---|
2966 | // of all syzygies equivalent to P/Q, |
---|
2967 | static proc extendcoef(poly P, poly Q, ideal idp, ideal idq) |
---|
2968 | { |
---|
2969 | def RR=basering; |
---|
2970 | setring(@P); |
---|
2971 | def PL=ringlist(@P); |
---|
2972 | PL[3][1][1]="dp"; |
---|
2973 | def P1=ring(PL); |
---|
2974 | setring(P1); |
---|
2975 | ideal idp0=imap(RR,idp); |
---|
2976 | option(redSB); |
---|
2977 | qring q=std(idp0); |
---|
2978 | poly P0=imap(RR,P); |
---|
2979 | poly Q0=imap(RR,Q); |
---|
2980 | ideal PQ=Q0,-P0; |
---|
2981 | module C=syz(PQ); |
---|
2982 | setring(@P); |
---|
2983 | def idp1=imap(RR,idp); |
---|
2984 | def idq1=imap(RR,idq); |
---|
2985 | def C1=matrix(imap(q,C)); |
---|
2986 | def redC=selectregularfun(C1,idp1,idq1); |
---|
2987 | setring(RR); |
---|
2988 | def CC=imap(@P,redC); |
---|
2989 | return(CC); |
---|
2990 | } |
---|
2991 | |
---|
2992 | // Auxiliary routine |
---|
2993 | // selectregularfun |
---|
2994 | // input: |
---|
2995 | // list L of the polynomials matrix CC |
---|
2996 | // (we assume that one of them is non-null on V(N) \ V(M)) |
---|
2997 | // ideal N, ideal M: ideals representing the locally closed set V(N) \ V(M) |
---|
2998 | // assume to work in @P |
---|
2999 | static proc selectregularfun(matrix CC, ideal NN, ideal MM) |
---|
3000 | { |
---|
3001 | int numcombused; |
---|
3002 | def RR=basering; |
---|
3003 | setring(@P); |
---|
3004 | def C=imap(RR,CC); |
---|
3005 | def N=imap(RR,NN); |
---|
3006 | def M=imap(RR,MM); |
---|
3007 | if (ncols(C)==1){return(C);} |
---|
3008 | |
---|
3009 | int i; int j; int k; list c; intvec ci; intvec c0; intvec c1; |
---|
3010 | list T; list T0; list T1; list LL; ideal N1;ideal M1; int te=0; |
---|
3011 | for(i=1;i<=ncols(C);i++) |
---|
3012 | { |
---|
3013 | if((C[1,i]!=0) and (C[2,i]!=0)) |
---|
3014 | { |
---|
3015 | if(c0==intvec(0)){c0[1]=i;} |
---|
3016 | else{c0[size(c0)+1]=i;} |
---|
3017 | } |
---|
3018 | } |
---|
3019 | def C1=submat(C,1..2,c0); |
---|
3020 | for (i=1;i<=ncols(C1);i++) |
---|
3021 | { |
---|
3022 | c=comb(ncols(C1),i); |
---|
3023 | for(j=1;j<=size(c);j++) |
---|
3024 | { |
---|
3025 | ci=c[j]; |
---|
3026 | numcombused++; |
---|
3027 | if(i==1){N1=N+C1[2,j]; M1=M;} |
---|
3028 | if(i>1) |
---|
3029 | { |
---|
3030 | kill c0; intvec c0 ; kill c1; intvec c1; |
---|
3031 | c1=ci[size(ci)]; |
---|
3032 | for(k=1;k<size(ci);k++){c0[k]=ci[k];} |
---|
3033 | T0=searchinlist(c0,LL); |
---|
3034 | T1=searchinlist(c1,LL); |
---|
3035 | N1=T0[1]+T1[1]; |
---|
3036 | M1=intersect(T0[2],T1[2]); |
---|
3037 | } |
---|
3038 | T=list(ci,PtoCrep(Prep(N1,M1))); |
---|
3039 | LL[size(LL)+1]=T; |
---|
3040 | if(equalideals(T[2][1],ideal(1))){te=1; break;} |
---|
3041 | } |
---|
3042 | if(te){break;} |
---|
3043 | } |
---|
3044 | ci=T[1]; |
---|
3045 | def Cs=submat(C1,1..2,ci); |
---|
3046 | setring(RR); |
---|
3047 | return(imap(@P,Cs)); |
---|
3048 | } |
---|
3049 | |
---|
3050 | // Auxiliary routine |
---|
3051 | // searchinlist |
---|
3052 | // input: |
---|
3053 | // intvec c: |
---|
3054 | // list L=( (c1,T1),..(ck,Tk) ) |
---|
3055 | // where the c's are assumed to be intvects |
---|
3056 | // output: |
---|
3057 | // object T with index c |
---|
3058 | static proc searchinlist(intvec c,list L) |
---|
3059 | { |
---|
3060 | int i; list T; |
---|
3061 | for(i=1;i<=size(L);i++) |
---|
3062 | { |
---|
3063 | if (L[i][1]==c) |
---|
3064 | { |
---|
3065 | T=L[i][2]; |
---|
3066 | break; |
---|
3067 | } |
---|
3068 | } |
---|
3069 | return(T); |
---|
3070 | } |
---|
3071 | |
---|
3072 | |
---|
3073 | // Auxiliary routine |
---|
3074 | // selectminsheaves |
---|
3075 | // Input: L=((v_11,..,v_1k_1),..,(v_s1,..,v_sk_s)) |
---|
3076 | // where: |
---|
3077 | // The s lists correspond to the s coefficients of the polynomial f |
---|
3078 | // (v_i1,..,v_ik_i) correspond to the k_i intvec v_ij of the |
---|
3079 | // spezializations of the jth rekpresentant (Q,P) of the ith coefficient |
---|
3080 | // v_ij is an intvec of size equal to the number of little segments |
---|
3081 | // forming the lpp-segment of 0,1, where 1 represents that it specializes |
---|
3082 | // to non-zedro an the whole little segment and 0 if not. |
---|
3083 | // Output: S=(w_1,..,w_j) |
---|
3084 | // where the w_l=(n_l1,..,n_ls) are intvec of length size(L), where |
---|
3085 | // n_lt fixes which element of (v_t1,..,v_tk_t) is to be |
---|
3086 | // chosen to form the tth (Q,P) for the lth element of the sheaf |
---|
3087 | // representing the I-regular function. |
---|
3088 | // The selection is done to obtian the minimal number of elements |
---|
3089 | // of the sheaf that specializes to non-null everywhere. |
---|
3090 | static proc selectminsheaves(list L) |
---|
3091 | { |
---|
3092 | list C=allsheaves(L); |
---|
3093 | return(smsheaves(C[1],C[2])); |
---|
3094 | } |
---|
3095 | |
---|
3096 | // Auxiliary routine |
---|
3097 | // smsheaves |
---|
3098 | // Input: |
---|
3099 | // list C of all the combrep |
---|
3100 | // list L of the intvec that correesponds to each element of C |
---|
3101 | // Output: |
---|
3102 | // list LL of the subsets of C that cover all the subsegments |
---|
3103 | // (the union of the corresponding L(C) has all 1). |
---|
3104 | static proc smsheaves(list C, list L) |
---|
3105 | { |
---|
3106 | int i; int i0; intvec W; |
---|
3107 | int nor; int norn; |
---|
3108 | intvec p; |
---|
3109 | int sp=size(L[1]); int j0=1; |
---|
3110 | for (i=1;i<=sp;i++){p[i]=1;} |
---|
3111 | while (p!=0) |
---|
3112 | { |
---|
3113 | i0=0; nor=0; |
---|
3114 | for (i=1; i<=size(L); i++) |
---|
3115 | { |
---|
3116 | norn=numones(L[i],pos(p)); |
---|
3117 | if (nor<norn){nor=norn; i0=i;} |
---|
3118 | } |
---|
3119 | W[j0]=i0; |
---|
3120 | j0++; |
---|
3121 | p=actualize(p,L[i0]); |
---|
3122 | } |
---|
3123 | list LL; |
---|
3124 | for (i=1;i<=size(W);i++) |
---|
3125 | { |
---|
3126 | LL[size(LL)+1]=C[W[i]]; |
---|
3127 | } |
---|
3128 | return(LL); |
---|
3129 | } |
---|
3130 | |
---|
3131 | // Auxiliary routine |
---|
3132 | // allsheaves |
---|
3133 | // Input: L=((v_11,..,v_1k_1),..,(v_s1,..,v_sk_s)) |
---|
3134 | // where: |
---|
3135 | // The s lists correspond to the s coefficients of the polynomial f |
---|
3136 | // (v_i1,..,v_ik_i) correspond to the k_i intvec v_ij of the |
---|
3137 | // spezializations of the jth rekpresentant (Q,P) of the ith coefficient |
---|
3138 | // v_ij is an intvec of size equal to the number of little segments |
---|
3139 | // forming the lpp-segment of 0,1, where 1 represents that it specializes |
---|
3140 | // to non-zero on the whole little segment and 1 if not. |
---|
3141 | // Output: |
---|
3142 | // (list LL, list LLS) where |
---|
3143 | // LL is the list of all combrep |
---|
3144 | // LLS is the list of intvec of the corresponding elements of LL |
---|
3145 | static proc allsheaves(list L) |
---|
3146 | { |
---|
3147 | intvec V; list LL; intvec W; int r; intvec U; |
---|
3148 | int i; int j; int k; |
---|
3149 | int s=size(L[1][1]); // s = number of little segments of the lpp-segment |
---|
3150 | list LLS; |
---|
3151 | for (i=1;i<=size(L);i++) |
---|
3152 | { |
---|
3153 | V[i]=size(L[i]); |
---|
3154 | } |
---|
3155 | LL=combrep(V); |
---|
3156 | for (i=1;i<=size(LL);i++) |
---|
3157 | { |
---|
3158 | W=LL[i]; // size(W)= number of coefficients of the polynomial |
---|
3159 | kill U; intvec U; |
---|
3160 | for (j=1;j<=s;j++) |
---|
3161 | { |
---|
3162 | k=1; r=1; U[j]=1; |
---|
3163 | while((r==1) and (k<=size(W))) |
---|
3164 | { |
---|
3165 | if(L[k][W[k]][j]==0){r=0; U[j]=0;} |
---|
3166 | k++; |
---|
3167 | } |
---|
3168 | } |
---|
3169 | LLS[i]=U; |
---|
3170 | } |
---|
3171 | return(list(LL,LLS)); |
---|
3172 | } |
---|
3173 | |
---|
3174 | // Auxiliary routine |
---|
3175 | // numones |
---|
3176 | // Input: |
---|
3177 | // intvec v of (0,1) in each position |
---|
3178 | // intvec pos: the positions to test |
---|
3179 | // Output: |
---|
3180 | // int nor: the nuber of 1 of v in the positions given by pos. |
---|
3181 | static proc numones(intvec v, intvec pos) |
---|
3182 | { |
---|
3183 | int i; int n; |
---|
3184 | for (i=1;i<=size(pos);i++) |
---|
3185 | { |
---|
3186 | if (v[pos[i]]==1){n++;} |
---|
3187 | } |
---|
3188 | return(n); |
---|
3189 | } |
---|
3190 | |
---|
3191 | // Auxiliary routine |
---|
3192 | // actualize: actualizes zeroes of p |
---|
3193 | // Input: |
---|
3194 | // intvec p: of zeroes and ones |
---|
3195 | // intvec c: of zeroes and ones (of the same length) |
---|
3196 | // Output; |
---|
3197 | // intvec pp: of zeroes and ones, where a 0 stays in pp[i] if either |
---|
3198 | // already p[i]==0 or c[i]==1. |
---|
3199 | static proc actualize(intvec p, intvec c) |
---|
3200 | { |
---|
3201 | int i; intvec pp=p; |
---|
3202 | for (i=1;i<=size(p);i++) |
---|
3203 | { |
---|
3204 | if ((pp[i]==1) and (c[i]==1)){pp[i]=0;} |
---|
3205 | } |
---|
3206 | return(pp); |
---|
3207 | } |
---|
3208 | |
---|
3209 | |
---|
3210 | // Auxiliary routine |
---|
3211 | static proc reducemodN(poly f,ideal E) |
---|
3212 | { |
---|
3213 | def RR=basering; |
---|
3214 | setring(@RPt); |
---|
3215 | def fa=imap(RR,f); |
---|
3216 | def Ea=imap(RR,E); |
---|
3217 | attrib(Ea,"isSB",1); |
---|
3218 | // option(redSB); |
---|
3219 | // Ea=std(Ea); |
---|
3220 | fa=reduce(fa,Ea); |
---|
3221 | setring(RR); |
---|
3222 | def f1=imap(@RPt,fa); |
---|
3223 | return(f1); |
---|
3224 | } |
---|
3225 | |
---|
3226 | // Auxiliary routine |
---|
3227 | // intersp: computes the intersection of the ideals in S in @P |
---|
3228 | static proc intersp(list S) |
---|
3229 | { |
---|
3230 | def RR=basering; |
---|
3231 | setring(@P); |
---|
3232 | def SP=imap(RR,S); |
---|
3233 | option(returnSB); |
---|
3234 | def NP=intersect(SP[1..size(SP)]); |
---|
3235 | setring(RR); |
---|
3236 | return(imap(@P,NP)); |
---|
3237 | } |
---|
3238 | |
---|
3239 | // Auxiliary routine |
---|
3240 | // radicalmember |
---|
3241 | static proc radicalmember(poly f,ideal ida) |
---|
3242 | { |
---|
3243 | int te; |
---|
3244 | def RR=basering; |
---|
3245 | setring(@P); |
---|
3246 | def fp=imap(RR,f); |
---|
3247 | def idap=imap(RR,ida); |
---|
3248 | poly @t; |
---|
3249 | ring H=0,@t,dp; |
---|
3250 | def PH=@P+H; |
---|
3251 | setring(PH); |
---|
3252 | def fH=imap(@P,fp); |
---|
3253 | def idaH=imap(@P,idap); |
---|
3254 | idaH[size(idaH)+1]=1-@t*fH; |
---|
3255 | option(redSB); |
---|
3256 | def G=std(idaH); |
---|
3257 | if (G==1){te=1;} else {te=0;} |
---|
3258 | setring(RR); |
---|
3259 | return(te); |
---|
3260 | } |
---|
3261 | |
---|
3262 | // // Auxiliary routine |
---|
3263 | // // NonNull: returns 1 if the poly f is nonnull on V(E) \ V(N), 0 otherwise. |
---|
3264 | // // Input: |
---|
3265 | // // f: polynomial |
---|
3266 | // // E: null ideal |
---|
3267 | // // N: nonnull ideal |
---|
3268 | // // Output: |
---|
3269 | // // 1 if f is nonnul in V(P) \ V(Q), |
---|
3270 | // // 0 if it has zeroes inside. |
---|
3271 | // // Works in @P |
---|
3272 | // proc NonNull(poly f, ideal E, ideal N) |
---|
3273 | // { |
---|
3274 | // int te=1; int i; |
---|
3275 | // def RR=basering; |
---|
3276 | // setring(@P); |
---|
3277 | // def fp=imap(RR,f); |
---|
3278 | // def Ep=imap(RR,E); |
---|
3279 | // def Np=imap(RR,N); |
---|
3280 | // ideal H; |
---|
3281 | // ideal Ef=Ep+fp; |
---|
3282 | // for (i=1;i<=size(Np);i++) |
---|
3283 | // { |
---|
3284 | // te=radicalmember(Np[i],Ef); |
---|
3285 | // if (te==0){break;} |
---|
3286 | // } |
---|
3287 | // setring(RR); |
---|
3288 | // return(te); |
---|
3289 | // } |
---|
3290 | |
---|
3291 | // Auxiliary routine |
---|
3292 | // selectextendcoef |
---|
3293 | // input: |
---|
3294 | // matrix CC: CC=(p_a1 .. p_ar_a) |
---|
3295 | // (q_a1 .. q_ar_a) |
---|
3296 | // the matrix of elements of a coefficient in oo[a]. |
---|
3297 | // (ideal ida, ideal idb): the canonical representation of the segment S. |
---|
3298 | // output: |
---|
3299 | // list caout |
---|
3300 | // the minimum set of elements of CC needed such that at least one |
---|
3301 | // of the q's is non-null on S, as well as the C-rep of of the |
---|
3302 | // points where the q's are null on S. |
---|
3303 | // The elements of caout are of the form (p,q,prep); |
---|
3304 | static proc selectextendcoef(matrix CC, ideal ida, ideal idb) |
---|
3305 | { |
---|
3306 | def RR=basering; |
---|
3307 | setring(@P); |
---|
3308 | def ca=imap(RR,CC); |
---|
3309 | def E0=imap(RR,ida); |
---|
3310 | ideal E; |
---|
3311 | def N=imap(RR,idb); |
---|
3312 | int r=ncols(ca); |
---|
3313 | int i; int te=1; list com; int j; int k; intvec c; list prep; |
---|
3314 | list cs; list caout; |
---|
3315 | i=1; |
---|
3316 | while ((i<=r) and (te)) |
---|
3317 | { |
---|
3318 | com=comb(r,i); |
---|
3319 | j=1; |
---|
3320 | while((j<=size(com)) and (te)) |
---|
3321 | { |
---|
3322 | E=E0; |
---|
3323 | c=com[j]; |
---|
3324 | for (k=1;k<=i;k++) |
---|
3325 | { |
---|
3326 | E=E+ca[2,c[k]]; |
---|
3327 | } |
---|
3328 | prep=Prep(E,N); |
---|
3329 | if (i==1) |
---|
3330 | { |
---|
3331 | cs[j]=list(ca[1,j],ca[2,j],prep); |
---|
3332 | } |
---|
3333 | if ((size(prep)==1) and (equalideals(prep[1][1],ideal(1)))) |
---|
3334 | { |
---|
3335 | te=0; |
---|
3336 | for(k=1;k<=size(c);k++) |
---|
3337 | { |
---|
3338 | caout[k]=cs[c[k]]; |
---|
3339 | } |
---|
3340 | } |
---|
3341 | j++; |
---|
3342 | } |
---|
3343 | i++; |
---|
3344 | } |
---|
3345 | if (te){"error: extendcoef does not extend to the whole S";} |
---|
3346 | setring(RR); |
---|
3347 | return(imap(@P,caout)); |
---|
3348 | } |
---|
3349 | |
---|
3350 | // Auxiliary routine |
---|
3351 | // plusP |
---|
3352 | // Input: |
---|
3353 | // ideal E1: in some basering (depends only on the parameters) |
---|
3354 | // ideal E2: in some basering (depends only on the parameters) |
---|
3355 | // Output: |
---|
3356 | // ideal Ep=E1+E2; computed in @P |
---|
3357 | static proc plusP(ideal E1,ideal E2) |
---|
3358 | { |
---|
3359 | def RR=basering; |
---|
3360 | setring(@P); |
---|
3361 | def E1p=imap(RR,E1); |
---|
3362 | def E2p=imap(RR,E2); |
---|
3363 | def Ep=E1p+E2p; |
---|
3364 | setring(RR); |
---|
3365 | return(imap(@P,Ep)); |
---|
3366 | } |
---|
3367 | |
---|
3368 | // Auxiliary routine |
---|
3369 | // reform |
---|
3370 | // input: |
---|
3371 | // list combpolys: (v1,..,vs) |
---|
3372 | // where vi are intvec. |
---|
3373 | // output outcomb: (w1,..,wt) |
---|
3374 | // whre wi are intvec. |
---|
3375 | // All the vi without zeroes are in outcomb, and those with zeroes are |
---|
3376 | // combined to form new intvec with the rest |
---|
3377 | static proc reform(list combpolys, intvec numdens) |
---|
3378 | { |
---|
3379 | list combp0; list combp1; int i; int j; int k; int l; list rest; intvec notfree; |
---|
3380 | list free; intvec free1; int te; intvec v; intvec w; |
---|
3381 | int nummonoms=size(combpolys[1]); |
---|
3382 | for(i=1;i<=size(combpolys);i++) |
---|
3383 | { |
---|
3384 | if(memberpos(0,combpolys[i])[1]) |
---|
3385 | { |
---|
3386 | combp0[size(combp0)+1]=combpolys[i]; |
---|
3387 | } |
---|
3388 | else {combp1[size(combp1)+1]=combpolys[i];} |
---|
3389 | } |
---|
3390 | for(i=1;i<=nummonoms;i++) |
---|
3391 | { |
---|
3392 | kill notfree; intvec notfree; |
---|
3393 | for(j=1;j<=size(combpolys);j++) |
---|
3394 | { |
---|
3395 | if(combpolys[j][i]<>0) |
---|
3396 | { |
---|
3397 | if(notfree[1]==0){notfree[1]=combpolys[j][i];} |
---|
3398 | else{notfree[size(notfree)+1]=combpolys[j][i];} |
---|
3399 | } |
---|
3400 | } |
---|
3401 | kill free1; intvec free1; |
---|
3402 | for(j=1;j<=numdens[i];j++) |
---|
3403 | { |
---|
3404 | if(memberpos(j,notfree)[1]==0) |
---|
3405 | { |
---|
3406 | if(free1[1]==0){free1[1]=j;} |
---|
3407 | else{free1[size(free1)+1]=j;} |
---|
3408 | } |
---|
3409 | free[i]=free1; |
---|
3410 | } |
---|
3411 | } |
---|
3412 | list amplcombp; list aux; |
---|
3413 | for(i=1;i<=size(combp0);i++) |
---|
3414 | { |
---|
3415 | v=combp0[i]; |
---|
3416 | kill amplcombp; list amplcombp; |
---|
3417 | amplcombp[1]=intvec(v[1]); |
---|
3418 | for(j=2;j<=size(v);j++) |
---|
3419 | { |
---|
3420 | if(v[j]!=0) |
---|
3421 | { |
---|
3422 | for(k=1;k<=size(amplcombp);k++) |
---|
3423 | { |
---|
3424 | w=amplcombp[k]; |
---|
3425 | w[size(w)+1]=v[j]; |
---|
3426 | amplcombp[k]=w; |
---|
3427 | } |
---|
3428 | } |
---|
3429 | else |
---|
3430 | { |
---|
3431 | kill aux; list aux; |
---|
3432 | for(k=1;k<=size(amplcombp);k++) |
---|
3433 | { |
---|
3434 | for(l=1;l<=size(free[j]);l++) |
---|
3435 | { |
---|
3436 | w=amplcombp[k]; |
---|
3437 | w[size(w)+1]=free[j][l]; |
---|
3438 | aux[size(aux)+1]=w; |
---|
3439 | } |
---|
3440 | } |
---|
3441 | amplcombp=aux; |
---|
3442 | } |
---|
3443 | } |
---|
3444 | for(j=1;j<=size(amplcombp);j++) |
---|
3445 | { |
---|
3446 | combp1[size(combp1)+1]=amplcombp[j]; |
---|
3447 | } |
---|
3448 | } |
---|
3449 | return(combp1); |
---|
3450 | } |
---|
3451 | |
---|
3452 | |
---|
3453 | // Auxiliary routine |
---|
3454 | // precombint |
---|
3455 | // input: L: list of ideals (works in @P) |
---|
3456 | // output: F0: ideal of polys. F0[i] is a poly in the intersection of |
---|
3457 | // all ideals in L except in the ith one, where it is not. |
---|
3458 | // L=(p1,..,ps); F0=(f1,..,fs); |
---|
3459 | // F0[i] \in intersect_{j#i} p_i |
---|
3460 | static proc precombint(list L) |
---|
3461 | { |
---|
3462 | int i; int j; int tes; |
---|
3463 | def RR=basering; |
---|
3464 | setring(@P); |
---|
3465 | list L0; list L1; list L2; list L3; ideal F; |
---|
3466 | L0=imap(RR,L); |
---|
3467 | L1[1]=L0[1]; L2[1]=L0[size(L0)]; |
---|
3468 | for (i=2;i<=size(L0)-1;i++) |
---|
3469 | { |
---|
3470 | L1[i]=intersect(L1[i-1],L0[i]); |
---|
3471 | L2[i]=intersect(L2[i-1],L0[size(L0)-i+1]); |
---|
3472 | } |
---|
3473 | L3[1]=L2[size(L2)]; |
---|
3474 | for (i=2;i<=size(L0)-1;i++) |
---|
3475 | { |
---|
3476 | L3[i]=intersect(L1[i-1],L2[size(L0)-i]); |
---|
3477 | } |
---|
3478 | L3[size(L0)]=L1[size(L1)]; |
---|
3479 | for (i=1;i<=size(L3);i++) |
---|
3480 | { |
---|
3481 | option(redSB); L3[i]=std(L3[i]); |
---|
3482 | } |
---|
3483 | for (i=1;i<=size(L3);i++) |
---|
3484 | { |
---|
3485 | tes=1; j=0; |
---|
3486 | while((tes) and (j<size(L3[i]))) |
---|
3487 | { |
---|
3488 | j++; |
---|
3489 | option(redSB); |
---|
3490 | L0[i]=std(L0[i]); |
---|
3491 | if(reduce(L3[i][j],L0[i])!=0){tes=0; F[i]=L3[i][j];} |
---|
3492 | } |
---|
3493 | if (tes){"ERROR a polynomial in all p_j except p_i was not found";} |
---|
3494 | } |
---|
3495 | setring(RR); |
---|
3496 | def F0=imap(@P,F); |
---|
3497 | return(F0); |
---|
3498 | } |
---|
3499 | |
---|
3500 | |
---|
3501 | // Auxiliary routine |
---|
3502 | // minAssGTZ eliminating denominators |
---|
3503 | static proc minGTZ(ideal N); |
---|
3504 | { |
---|
3505 | int i; int j; |
---|
3506 | def L=minAssGTZ(N); |
---|
3507 | for(i=1;i<=size(L);i++) |
---|
3508 | { |
---|
3509 | for(j=1;j<=size(L[i]);j++) |
---|
3510 | { |
---|
3511 | L[i][j]=cleardenom(L[i][j]); |
---|
3512 | } |
---|
3513 | } |
---|
3514 | return(L); |
---|
3515 | } |
---|
3516 | |
---|
3517 | //********************* Begin KapurSunWang ************************* |
---|
3518 | |
---|
3519 | // Auxiliary routine |
---|
3520 | // inconsistent |
---|
3521 | // Input: |
---|
3522 | // ideal E: of null conditions |
---|
3523 | // ideal N: of non-null conditions representing V(E) \ V(N) |
---|
3524 | // Output: |
---|
3525 | // 1 if V(E) \ V(N) = empty |
---|
3526 | // 0 if not |
---|
3527 | static proc inconsistent(ideal E, ideal N) |
---|
3528 | { |
---|
3529 | int j; |
---|
3530 | int te=1; |
---|
3531 | def R=basering; |
---|
3532 | setring(@P); |
---|
3533 | def EP=imap(R,E); |
---|
3534 | def NP=imap(R,N); |
---|
3535 | poly @t; |
---|
3536 | ring H=0,@t,dp; |
---|
3537 | def RH=@P+H; |
---|
3538 | setring(RH); |
---|
3539 | def EH=imap(@P,EP); |
---|
3540 | def NH=imap(@P,NP); |
---|
3541 | ideal G; |
---|
3542 | j=1; |
---|
3543 | while((te==1) and j<=size(NH)) |
---|
3544 | { |
---|
3545 | G=EH+(1-@t*NH[j]); |
---|
3546 | option(redSB); |
---|
3547 | G=std(G); |
---|
3548 | if (G[1]!=1){te=0;} |
---|
3549 | j++; |
---|
3550 | } |
---|
3551 | setring(R); |
---|
3552 | return(te); |
---|
3553 | } |
---|
3554 | |
---|
3555 | // Auxiliary routine |
---|
3556 | // MDBasis: Minimal Dickson Basis |
---|
3557 | static proc MDBasis(ideal G) |
---|
3558 | { |
---|
3559 | int i; int j; int te=1; |
---|
3560 | G=sortideal(G); |
---|
3561 | ideal MD=G[1]; |
---|
3562 | poly lm; |
---|
3563 | for (i=2;i<=size(G);i++) |
---|
3564 | { |
---|
3565 | te=1; |
---|
3566 | lm=leadmonom(G[i]); |
---|
3567 | j=1; |
---|
3568 | while ((te==1) and (j<=size(MD))) |
---|
3569 | { |
---|
3570 | if (lm/leadmonom(MD[j])!=0){te=0;} |
---|
3571 | j++; |
---|
3572 | } |
---|
3573 | if (te==1) |
---|
3574 | { |
---|
3575 | MD[size(MD)+1]=(G[i]); |
---|
3576 | } |
---|
3577 | } |
---|
3578 | return(MD); |
---|
3579 | } |
---|
3580 | |
---|
3581 | // Auxiliary routine |
---|
3582 | // primepartZ |
---|
3583 | static proc primepartZ(poly f); |
---|
3584 | { |
---|
3585 | def cp=content(f); |
---|
3586 | def fp=f/cp; |
---|
3587 | return(fp); |
---|
3588 | } |
---|
3589 | |
---|
3590 | // LCMLC |
---|
3591 | static proc LCMLC(ideal H) |
---|
3592 | { |
---|
3593 | int i; |
---|
3594 | def R=basering; |
---|
3595 | setring(@RP); |
---|
3596 | def HH=imap(R,H); |
---|
3597 | poly h=1; |
---|
3598 | for (i=1;i<=size(HH);i++) |
---|
3599 | { |
---|
3600 | h=lcm(h,HH[i]); |
---|
3601 | } |
---|
3602 | setring(R); |
---|
3603 | def hh=imap(@RP,h); |
---|
3604 | return(hh); |
---|
3605 | } |
---|
3606 | |
---|
3607 | // KSW: Kapur-Sun-Wang algorithm for computing a CGS |
---|
3608 | // Input: |
---|
3609 | // F: parametric ideal to be discussed |
---|
3610 | // Options: |
---|
3611 | // \"out\",0 Transforms the description of the segments into |
---|
3612 | // canonical P-representation form. |
---|
3613 | // \"out\",1 Original KSW routine describing the segments as |
---|
3614 | // difference of varieties |
---|
3615 | // The ideal must be defined on C[parameters][variables] |
---|
3616 | // Output: |
---|
3617 | // With option \"out\",0 : |
---|
3618 | // ((lpp, |
---|
3619 | // (1,B,((p_1,(p_11,..,p_1k_1)),..,(p_s,(p_s1,..,p_sk_s)))), |
---|
3620 | // string(lpp) |
---|
3621 | // ) |
---|
3622 | // ,.., |
---|
3623 | // (lpp, |
---|
3624 | // (k,B,((p_1,(p_11,..,p_1k_1)),..,(p_s,(p_s1,..,p_sk_s)))), |
---|
3625 | // string(lpp)) |
---|
3626 | // ) |
---|
3627 | // ) |
---|
3628 | // With option \"out\",1 ((default, original KSW) (shorter to be computed, |
---|
3629 | // but without canonical description of the segments. |
---|
3630 | // ((B,E,N),..,(B,E,N)) |
---|
3631 | static proc KSW(ideal F, list #) |
---|
3632 | { |
---|
3633 | setglobalrings(); |
---|
3634 | int start=timer; |
---|
3635 | ideal E=ideal(0); |
---|
3636 | ideal N=ideal(1); |
---|
3637 | int comment=0; |
---|
3638 | int out=1; |
---|
3639 | int i; |
---|
3640 | def L=#; |
---|
3641 | if (size(L)>0) |
---|
3642 | { |
---|
3643 | for (i=1;i<=size(L) div 2;i++) |
---|
3644 | { |
---|
3645 | if (L[2*i-1]=="null"){E=L[2*i];} |
---|
3646 | else |
---|
3647 | { |
---|
3648 | if (L[2*i-1]=="nonnull"){N=L[2*i];} |
---|
3649 | else |
---|
3650 | { |
---|
3651 | if (L[2*i-1]=="comment"){comment=L[2*i];} |
---|
3652 | else |
---|
3653 | { |
---|
3654 | if (L[2*i-1]=="out"){out=L[2*i];} |
---|
3655 | } |
---|
3656 | } |
---|
3657 | } |
---|
3658 | } |
---|
3659 | } |
---|
3660 | if (comment>0){string("Begin KSW with null = ",E," nonnull = ",N);} |
---|
3661 | def CG=KSW0(F,E,N,comment); |
---|
3662 | if (comment>0) |
---|
3663 | { |
---|
3664 | string("Number of segments in KSW (total) = ",size(CG)); |
---|
3665 | string("Time in KSW = ",timer-start); |
---|
3666 | } |
---|
3667 | if(out==0) |
---|
3668 | { |
---|
3669 | CG=KSWtocgsdr(CG); |
---|
3670 | CG=groupKSWsegments(CG); |
---|
3671 | if (comment>0) |
---|
3672 | { |
---|
3673 | string("Number of lpp segments = ",size(CG)); |
---|
3674 | string("Time in KSW + group + Prep = ",timer-start); |
---|
3675 | } |
---|
3676 | } |
---|
3677 | if(defined(@P)){kill @P; kill @R; kill @RP;} |
---|
3678 | return(CG); |
---|
3679 | } |
---|
3680 | |
---|
3681 | // Auxiliary routine |
---|
3682 | // sqf |
---|
3683 | // This is for releases of Singular before 3-5-1 |
---|
3684 | // proc sqf(poly f) |
---|
3685 | // { |
---|
3686 | // def RR=basering; |
---|
3687 | // setring(@P); |
---|
3688 | // def ff=imap(RR,f); |
---|
3689 | // def G=sqrfree(ff); |
---|
3690 | // poly fff=1; |
---|
3691 | // int i; |
---|
3692 | // for (i=1;i<=size(G);i++) |
---|
3693 | // { |
---|
3694 | // fff=fff*G[i]; |
---|
3695 | // } |
---|
3696 | // setring(RR); |
---|
3697 | // def ffff=imap(@P,fff); |
---|
3698 | // return(ffff); |
---|
3699 | // } |
---|
3700 | |
---|
3701 | // Auxiliary routine |
---|
3702 | // sqf |
---|
3703 | static proc sqf(poly f) |
---|
3704 | { |
---|
3705 | def RR=basering; |
---|
3706 | setring(@P); |
---|
3707 | def ff=imap(RR,f); |
---|
3708 | poly fff=sqrfree(ff,3); |
---|
3709 | setring(RR); |
---|
3710 | def ffff=imap(@P,fff); |
---|
3711 | return(ffff); |
---|
3712 | } |
---|
3713 | |
---|
3714 | |
---|
3715 | // Auxiliary routine |
---|
3716 | // KSW0: Kapur-Sun-Wang algorithm for computing a CGS, called by KSW |
---|
3717 | // Input: |
---|
3718 | // F: parametric ideal to be discussed |
---|
3719 | // Options: |
---|
3720 | // The ideal must be defined on C[parameters][variables] |
---|
3721 | // Output: |
---|
3722 | static proc KSW0(ideal F, ideal E, ideal N, int comment) |
---|
3723 | { |
---|
3724 | def R=basering; |
---|
3725 | int i; int j; list emp; |
---|
3726 | list CGS; |
---|
3727 | ideal N0; |
---|
3728 | for (i=1;i<=size(N);i++) |
---|
3729 | { |
---|
3730 | N0[i]=sqf(N[i]); |
---|
3731 | } |
---|
3732 | ideal E0; |
---|
3733 | for (i=1;i<=size(E);i++) |
---|
3734 | { |
---|
3735 | E0[i]=sqf(leadcoef(E[i])); |
---|
3736 | } |
---|
3737 | setring(@P); |
---|
3738 | ideal E1=imap(R,E0); |
---|
3739 | E1=std(E1); |
---|
3740 | ideal N1=imap(R,N0); |
---|
3741 | N1=std(N1); |
---|
3742 | setring(R); |
---|
3743 | E0=imap(@P,E1); |
---|
3744 | N0=imap(@P,N1); |
---|
3745 | if (inconsistent(E0,N0)==1) |
---|
3746 | { |
---|
3747 | return(emp); |
---|
3748 | } |
---|
3749 | setring(@RP); |
---|
3750 | def FRP=imap(R,F); |
---|
3751 | def ERP=imap(R,E); |
---|
3752 | FRP=FRP+ERP; |
---|
3753 | option(redSB); |
---|
3754 | def GRP=std(FRP); |
---|
3755 | setring(R); |
---|
3756 | def G=imap(@RP,GRP); |
---|
3757 | if (memberpos(1,G)[1]==1) |
---|
3758 | { |
---|
3759 | if(comment>1){"Basis 1 is found"; E; N;} |
---|
3760 | list KK; KK[1]=list(E0,N0,ideal(1)); |
---|
3761 | return(KK); |
---|
3762 | } |
---|
3763 | ideal Gr; ideal Gm; ideal GM; |
---|
3764 | for (i=1;i<=size(G);i++) |
---|
3765 | { |
---|
3766 | if (variables(G[i])[1]==0){Gr[size(Gr)+1]=G[i];} |
---|
3767 | else{Gm[size(Gm)+1]=G[i];} |
---|
3768 | } |
---|
3769 | ideal Gr0; |
---|
3770 | for (i=1;i<=size(Gr);i++) |
---|
3771 | { |
---|
3772 | Gr0[i]=sqf(Gr[i]); |
---|
3773 | } |
---|
3774 | |
---|
3775 | |
---|
3776 | Gr=elimrepeated(Gr0); |
---|
3777 | ideal GrN; |
---|
3778 | for (i=1;i<=size(Gr);i++) |
---|
3779 | { |
---|
3780 | for (j=1;j<=size(N0);j++) |
---|
3781 | { |
---|
3782 | GrN[size(GrN)+1]=sqf(Gr[i]*N0[j]); |
---|
3783 | } |
---|
3784 | } |
---|
3785 | if (inconsistent(E,GrN)){;} |
---|
3786 | else |
---|
3787 | { |
---|
3788 | if(comment>1){"Basis 1 is found in a branch with arguments"; E; GrN;} |
---|
3789 | CGS[size(CGS)+1]=list(E,GrN,ideal(1)); |
---|
3790 | } |
---|
3791 | if (inconsistent(Gr,N0)){return(CGS);} |
---|
3792 | GM=Gm; |
---|
3793 | Gm=MDBasis(Gm); |
---|
3794 | ideal H; |
---|
3795 | for (i=1;i<=size(Gm);i++) |
---|
3796 | { |
---|
3797 | H[i]=sqf(leadcoef(Gm[i])); |
---|
3798 | } |
---|
3799 | H=facvar(H); |
---|
3800 | poly h=sqf(LCMLC(H)); |
---|
3801 | if(comment>1){"H = "; H; "h = "; h;} |
---|
3802 | ideal Nh=N0; |
---|
3803 | if(size(N0)==0){Nh=h;} |
---|
3804 | else |
---|
3805 | { |
---|
3806 | for (i=1;i<=size(N0);i++) |
---|
3807 | { |
---|
3808 | Nh[i]=sqf(N0[i]*h); |
---|
3809 | } |
---|
3810 | } |
---|
3811 | if (inconsistent(Gr,Nh)){;} |
---|
3812 | else |
---|
3813 | { |
---|
3814 | CGS[size(CGS)+1]=list(Gr,Nh,Gm); |
---|
3815 | } |
---|
3816 | poly hc=1; |
---|
3817 | list KS; |
---|
3818 | ideal GrHi; |
---|
3819 | for (i=1;i<=size(H);i++) |
---|
3820 | { |
---|
3821 | kill GrHi; |
---|
3822 | ideal GrHi; |
---|
3823 | Nh=N0; |
---|
3824 | if (i>1){hc=sqf(hc*H[i-1]);} |
---|
3825 | for (j=1;j<=size(N0);j++){Nh[j]=sqf(N0[j]*hc);} |
---|
3826 | if (equalideals(Gr,ideal(0))==1){GrHi=H[i];} |
---|
3827 | else {GrHi=Gr,H[i];} |
---|
3828 | // else {for (j=1;j<=size(Gr);j++){GrHi[size(GrHi)+1]=Gr[j]*H[i];}} |
---|
3829 | if(comment>1){"Call to KSW with arguments "; GM; GrHi; Nh;} |
---|
3830 | KS=KSW0(GM,GrHi,Nh,comment); |
---|
3831 | for (j=1;j<=size(KS);j++) |
---|
3832 | { |
---|
3833 | CGS[size(CGS)+1]=KS[j]; |
---|
3834 | } |
---|
3835 | if(comment>1){"CGS after KSW = "; CGS;} |
---|
3836 | } |
---|
3837 | return(CGS); |
---|
3838 | } |
---|
3839 | |
---|
3840 | // Auxiliary routine |
---|
3841 | // KSWtocgsdr |
---|
3842 | static proc KSWtocgsdr(list L) |
---|
3843 | { |
---|
3844 | int i; list CG; ideal B; ideal lpp; int j; list NKrep; |
---|
3845 | for(i=1;i<=size(L);i++) |
---|
3846 | { |
---|
3847 | B=redgbn(L[i][3],L[i][1],L[i][2]); |
---|
3848 | lpp=ideal(0); |
---|
3849 | for(j=1;j<=size(B);j++) |
---|
3850 | { |
---|
3851 | lpp[j]=leadmonom(B[j]); |
---|
3852 | } |
---|
3853 | NKrep=KtoPrep(L[i][1],L[i][2]); |
---|
3854 | CG[i]=list(lpp,B,NKrep); |
---|
3855 | } |
---|
3856 | return(CG); |
---|
3857 | } |
---|
3858 | |
---|
3859 | // Auxiliary routine |
---|
3860 | // KtoPrep |
---|
3861 | // Computes the P-representaion of a K-representation (N,W) of a set |
---|
3862 | // input: |
---|
3863 | // ideal E (null conditions) |
---|
3864 | // ideal N (non-null conditions ideal) |
---|
3865 | // output: |
---|
3866 | // the ((p_1,(p_11,..,p_1k_1)),..,(p_r,(p_r1,..,p_rk_r))); |
---|
3867 | // the Prep of V(N) \ V(W) |
---|
3868 | static proc KtoPrep(ideal N, ideal W) |
---|
3869 | { |
---|
3870 | int i; int j; |
---|
3871 | if (N[1]==1) |
---|
3872 | { |
---|
3873 | L0[1]=list(ideal(1),list(ideal(1))); |
---|
3874 | return(L0); |
---|
3875 | } |
---|
3876 | def RR=basering; |
---|
3877 | setring(@P); |
---|
3878 | ideal B; int te; poly f; |
---|
3879 | ideal Np=imap(RR,N); |
---|
3880 | ideal Wp=imap(RR,W); |
---|
3881 | list L; |
---|
3882 | list L0; list T0; |
---|
3883 | L0=minGTZ(Np); |
---|
3884 | for(j=1;j<=size(L0);j++) |
---|
3885 | { |
---|
3886 | option(redSB); |
---|
3887 | L0[j]=std(L0[j]); |
---|
3888 | } |
---|
3889 | for(i=1;i<=size(L0);i++) |
---|
3890 | { |
---|
3891 | if(inconsistent(L0[i],Wp)==0) |
---|
3892 | { |
---|
3893 | B=L0[i]+Wp; |
---|
3894 | T0=minGTZ(B); |
---|
3895 | option(redSB); |
---|
3896 | for(j=1;j<=size(T0);j++) |
---|
3897 | { |
---|
3898 | T0[j]=std(T0[j]); |
---|
3899 | } |
---|
3900 | L[size(L)+1]=list(L0[i],T0); |
---|
3901 | } |
---|
3902 | } |
---|
3903 | setring(RR); |
---|
3904 | def LL=imap(@P,L); |
---|
3905 | return(LL); |
---|
3906 | } |
---|
3907 | |
---|
3908 | // Auxiliary routine |
---|
3909 | // groupKSWsegments |
---|
3910 | // input: the list of vertices of KSW |
---|
3911 | // output: the same terminal vertices grouped by lpp |
---|
3912 | static proc groupKSWsegments(list T) |
---|
3913 | { |
---|
3914 | int i; int j; |
---|
3915 | list L; |
---|
3916 | list lpp; list lppor; |
---|
3917 | list kk; |
---|
3918 | lpp[1]=T[1][1]; j=1; |
---|
3919 | lppor[1]=intvec(1); |
---|
3920 | for(i=2;i<=size(T);i++) |
---|
3921 | { |
---|
3922 | kk=memberpos(T[i][1],lpp); |
---|
3923 | if(kk[1]==0){j++; lpp[j]=T[i][1]; lppor[j]=intvec(i);} |
---|
3924 | else{lppor[kk[2]][size(lppor[kk[2]])+1]=i;} |
---|
3925 | } |
---|
3926 | list ll; |
---|
3927 | for (j=1;j<=size(lpp);j++) |
---|
3928 | { |
---|
3929 | kill ll; list ll; |
---|
3930 | for(i=1;i<=size(lppor[j]);i++) |
---|
3931 | { |
---|
3932 | ll[size(ll)+1]=list(i,T[lppor[j][i]][2],T[lppor[j][i]][3]); |
---|
3933 | } |
---|
3934 | L[j]=list(lpp[j],ll,string(lpp[j])); |
---|
3935 | } |
---|
3936 | return(L); |
---|
3937 | } |
---|
3938 | |
---|
3939 | //********************* End KapurSunWang ************************* |
---|
3940 | |
---|
3941 | //******************** Begin locus ****************************** |
---|
3942 | |
---|
3943 | // indepparameters |
---|
3944 | // Auxiliary routine to detect 'Special' components of the locus |
---|
3945 | // Input: ideal B |
---|
3946 | // Output: |
---|
3947 | // 1 if the solutions of the ideal do not depend on the parameters |
---|
3948 | // 0 if they depend |
---|
3949 | static proc indepparameters(ideal B) |
---|
3950 | { |
---|
3951 | def R=basering; |
---|
3952 | ideal v=variables(B); |
---|
3953 | setring @RP; |
---|
3954 | def BP=imap(R,B); |
---|
3955 | def vp=imap(R,v); |
---|
3956 | ideal varpar=variables(BP); |
---|
3957 | int te; |
---|
3958 | te=equalideals(vp,varpar); |
---|
3959 | setring(R); |
---|
3960 | if(te){return(1);} |
---|
3961 | else{return(0);} |
---|
3962 | } |
---|
3963 | |
---|
3964 | // dimP0: Auxiliary routine |
---|
3965 | // if the dimension in @P of an ideal in the parameters has dimension 0 then it returns 0 |
---|
3966 | // else it retuns 1 |
---|
3967 | static proc dimP0(ideal N) |
---|
3968 | { |
---|
3969 | def R=basering; |
---|
3970 | setring(@P); |
---|
3971 | int te=1; |
---|
3972 | def NP=imap(R,N); |
---|
3973 | attrib(NP,"IsSB",1); |
---|
3974 | int d=dim(std(NP)); |
---|
3975 | if(d==0){te=0;} |
---|
3976 | setring(R); |
---|
3977 | return(te); |
---|
3978 | } |
---|
3979 | |
---|
3980 | // Takes a list of intvec and sorts it and eliminates repeated elements. |
---|
3981 | // Auxiliary routine |
---|
3982 | static proc sortpairs(L) |
---|
3983 | { |
---|
3984 | def L1=sort(L); |
---|
3985 | def L2=elimrepeated(L1[1]); |
---|
3986 | return(L2); |
---|
3987 | } |
---|
3988 | |
---|
3989 | // Eliminates the pairs of L1 that are also in L2. |
---|
3990 | // Auxiliary routine |
---|
3991 | static proc minuselements(list L1,list L2) |
---|
3992 | { |
---|
3993 | int i; |
---|
3994 | list L3; |
---|
3995 | for (i=1;i<=size(L1);i++) |
---|
3996 | { |
---|
3997 | if(not(memberpos(L1[i],L2)[1])){L3[size(L3)+1]=L1[i];} |
---|
3998 | } |
---|
3999 | return(L3); |
---|
4000 | } |
---|
4001 | |
---|
4002 | // NorSing |
---|
4003 | // Input: |
---|
4004 | // ideal B: the basis of a component of the grobcov |
---|
4005 | // ideal E: the top of the component (assumed to be of dimension > 0 (single equation) |
---|
4006 | // ideal N: the holes of the component |
---|
4007 | // Output: |
---|
4008 | // int d: the dimension of B on the variables reduced by the holes. |
---|
4009 | // if d>0 then the component is 'Normal' |
---|
4010 | // if d==0 then the component is 'Singular' |
---|
4011 | static proc NorSing(ideal B, ideal E, ideal N, list #) |
---|
4012 | { |
---|
4013 | int i; int j; int Fenv=0; int env; int dd; |
---|
4014 | list DD=#; |
---|
4015 | def RR=basering; |
---|
4016 | int moverdim=2; |
---|
4017 | int version=0; |
---|
4018 | int nv=nvars(RR); |
---|
4019 | if(nv<4){version=1;} |
---|
4020 | int d; |
---|
4021 | poly F; |
---|
4022 | for(i=1;i<=(size(DD) div 2);i++) |
---|
4023 | { |
---|
4024 | if(DD[2*i-1]=="movdim"){moverdim=DD[2*i];} |
---|
4025 | if(DD[2*i-1]=="version"){version=DD[2*i];} |
---|
4026 | if(DD[2*i-1]=="family"){F=DD[2*i];} |
---|
4027 | } |
---|
4028 | if(F!=0){Fenv=1;} |
---|
4029 | //"T_B="; B; "E="; E; "N="; N; |
---|
4030 | list L0; |
---|
4031 | if(dimP0(E)==0){L0=2,"Normal";} // 2 es fals pero ha de ser >0 encara que sigui 0 |
---|
4032 | else |
---|
4033 | { |
---|
4034 | if(version==0) |
---|
4035 | { |
---|
4036 | //"T_B="; B; // Computing std(B+E,plex(x,y,x1,..xn)) one can detect if there is a first part |
---|
4037 | // independent of parameters giving the variables with dimension 0 |
---|
4038 | dd=indepparameters(B); |
---|
4039 | if (dd==1){d=0; L0=d,string(B),determineF(B,F,E);} |
---|
4040 | else{d=1; L0=2,"Normal";} |
---|
4041 | } |
---|
4042 | else |
---|
4043 | { |
---|
4044 | def RH=ringlist(RR); |
---|
4045 | //"T_RH="; RH; |
---|
4046 | def H=RH; |
---|
4047 | H[1]=0; |
---|
4048 | H[2]=RH[1][2]+RH[2]; |
---|
4049 | int n=size(H[2]); |
---|
4050 | intvec ll; |
---|
4051 | for(i=1;i<=n;i++) |
---|
4052 | { |
---|
4053 | ll[i]=1; |
---|
4054 | } |
---|
4055 | H[3][1][1]="lp"; |
---|
4056 | H[3][1][2]=ll; |
---|
4057 | def RRH=ring(H); |
---|
4058 | setring(RRH); |
---|
4059 | ideal BH=imap(RR,B); |
---|
4060 | ideal EH=imap(RR,E); |
---|
4061 | ideal NH=imap(RR,N); |
---|
4062 | if(Fenv==1){poly FH=imap(RR,F);} |
---|
4063 | for(i=1;i<=size(EH);i++){BH[size(BH)+1]=EH[i];} |
---|
4064 | BH=std(BH); // MOLT COSTOS!!! |
---|
4065 | ideal G; |
---|
4066 | ideal r; poly q; |
---|
4067 | for(i=1;i<=size(BH);i++) |
---|
4068 | { |
---|
4069 | r=factorize(BH[i],1); |
---|
4070 | q=1; |
---|
4071 | for(j=1;j<=size(r);j++) |
---|
4072 | { |
---|
4073 | if((pdivi(r[j],NH)[1] != 0) or (equalideals(ideal(NH),ideal(1)))) |
---|
4074 | { |
---|
4075 | q=q*r[j]; |
---|
4076 | } |
---|
4077 | } |
---|
4078 | if(q!=1){G[size(G)+1]=q;} |
---|
4079 | } |
---|
4080 | setring RR; |
---|
4081 | def GG=imap(RRH,G); |
---|
4082 | ideal GGG; |
---|
4083 | if(defined(L0)){kill L0; list L0;} |
---|
4084 | for(i=1;i<=size(GG);i++) |
---|
4085 | { |
---|
4086 | if(indepparameters(GG[i])){GGG[size(GGG)+1]=GG[i];} |
---|
4087 | } |
---|
4088 | GGG=std(GGG); |
---|
4089 | ideal GLM; |
---|
4090 | for(i=1;i<=size(GGG);i++) |
---|
4091 | { |
---|
4092 | GLM[i]=leadmonom(GGG[i]); |
---|
4093 | } |
---|
4094 | attrib(GLM,"IsSB",1); |
---|
4095 | d=dim(std(GLM)); |
---|
4096 | string antiim=string(GGG); |
---|
4097 | L0=d,antiim; |
---|
4098 | if(d==0) |
---|
4099 | { |
---|
4100 | //" ";string("Antiimage of Special component = ", GGG); |
---|
4101 | if(Fenv==1) |
---|
4102 | { |
---|
4103 | L0[3]=determineF(GGG,F,E); |
---|
4104 | } |
---|
4105 | } |
---|
4106 | else |
---|
4107 | { |
---|
4108 | L0[2]="Normal"; |
---|
4109 | } |
---|
4110 | } |
---|
4111 | } |
---|
4112 | return(L0); |
---|
4113 | } |
---|
4114 | |
---|
4115 | static proc determineF(ideal A,poly F,ideal E) |
---|
4116 | { |
---|
4117 | int env; int i; |
---|
4118 | def RR=basering; |
---|
4119 | def RH=ringlist(RR); |
---|
4120 | def H=RH; |
---|
4121 | H[1]=0; |
---|
4122 | H[2]=RH[1][2]+RH[2]; |
---|
4123 | int n=size(H[2]); |
---|
4124 | intvec ll; |
---|
4125 | for(i=1;i<=n;i++) |
---|
4126 | { |
---|
4127 | ll[i]=1; |
---|
4128 | } |
---|
4129 | H[3][1][1]="lp"; |
---|
4130 | H[3][1][2]=ll; |
---|
4131 | def RRH=ring(H); |
---|
4132 | |
---|
4133 | //" ";string("Antiimage of Special component = ", GGG); |
---|
4134 | |
---|
4135 | setring(RRH); |
---|
4136 | list LL; |
---|
4137 | def AA=imap(RR,A); |
---|
4138 | def FH=imap(RR,F); |
---|
4139 | def EH=imap(RR,E); |
---|
4140 | ideal M=std(AA+FH); |
---|
4141 | def rh=reduce(EH,M); |
---|
4142 | if(rh==0){env=1;} else{env=0;} |
---|
4143 | setring RR; |
---|
4144 | //L0[3]=env; |
---|
4145 | return(env); |
---|
4146 | } |
---|
4147 | |
---|
4148 | // DimPar |
---|
4149 | // Auxilliary routine to NorSing determining the dimension of a parametric ideal |
---|
4150 | // Does not use @P and define it directly because is assumes that |
---|
4151 | // variables and parameters have been inverted |
---|
4152 | static proc DimPar(ideal E) |
---|
4153 | { |
---|
4154 | def RRH=basering; |
---|
4155 | def RHx=ringlist(RRH); |
---|
4156 | def Prin=ring(RHx[1]); |
---|
4157 | setring(Prin); |
---|
4158 | def E2=std(imap(RRH,E)); |
---|
4159 | def d=dim(E2); |
---|
4160 | setring RRH; |
---|
4161 | return(d); |
---|
4162 | } |
---|
4163 | |
---|
4164 | // locus0(G): Private routine used by locus (the public routine), that |
---|
4165 | // builds the diferent components. |
---|
4166 | // input: The output G of the grobcov (in generic representation, which is the default option) |
---|
4167 | // Options: The algorithm allows the following options ar pair of arguments: |
---|
4168 | // "movdim", d : by default movdim is 2 but it can be set to other values |
---|
4169 | // "version", v : There are two versions of the algorithm. ('version',1) is |
---|
4170 | // a full algorithm that always distinguishes correctly between 'Normal' |
---|
4171 | // and 'Special' components, whereas ('version',0) can decalre a component |
---|
4172 | // as 'Normal' being really 'Special', but is more effective. By default ('version',1) |
---|
4173 | // is used when the number of variables is less than 4 and 0 if not. |
---|
4174 | // The user can force to use one or other version, but it is not recommended. |
---|
4175 | // "system", ideal F: if the initial systrem is passed as an argument. This is actually not used. |
---|
4176 | // "comments", c: by default it is 0, but it can be set to 1. |
---|
4177 | // Usually locus problems have mover coordinates, variables and tracer coordinates. |
---|
4178 | // The mover coordinates are to be placed as the last variables, and by default, |
---|
4179 | // its number is 2. If one consider locus problems in higer dimensions, the number of |
---|
4180 | // mover coordinates (placed as the last variables) is to be given as an option. |
---|
4181 | // output: |
---|
4182 | // list, the canonical P-representation of the Normal and Non-Normal locus: |
---|
4183 | // The Normal locus has two kind of components: Normal and Special. |
---|
4184 | // The Non-normal locus has two kind of components: Accumulation and Degenerate. |
---|
4185 | // This routine is compemented by locus that calls it in order to eliminate problems |
---|
4186 | // with degenerate points of the mover. |
---|
4187 | // The output components are given as |
---|
4188 | // ((p1,(p11,..p1s_1),type_1,level_1),..,(pk,(pk1,..pks_k),type_k,level_k) |
---|
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 | static proc locus0(list GG, list #) |
---|
4193 | { |
---|
4194 | int te=0; |
---|
4195 | int t1=1; int t2=1; int i; |
---|
4196 | def R=basering; |
---|
4197 | //if(defined(@P)==1){te=1; kill @P; kill @R; kill @RP; } |
---|
4198 | //setglobalrings(); |
---|
4199 | // Options |
---|
4200 | list DD=#; |
---|
4201 | int moverdim=nvars(R); |
---|
4202 | int version=0; |
---|
4203 | int nv=nvars(R); |
---|
4204 | if(nv<4){version=1;} |
---|
4205 | int comment=0; |
---|
4206 | ideal Fm; |
---|
4207 | poly F; |
---|
4208 | for(i=1;i<=(size(DD) div 2);i++) |
---|
4209 | { |
---|
4210 | if(DD[2*i-1]=="movdim"){moverdim=DD[2*i];} |
---|
4211 | if(DD[2*i-1]=="version"){version=DD[2*i];} |
---|
4212 | if(DD[2*i-1]=="system"){Fm=DD[2*i];} |
---|
4213 | if(DD[2*i-1]=="comment"){comment=DD[2*i];} |
---|
4214 | if(DD[2*i-1]=="family"){F=DD[2*i];} |
---|
4215 | } |
---|
4216 | list HHH; |
---|
4217 | 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);} |
---|
4218 | list G1; list G2; |
---|
4219 | def G=GG; |
---|
4220 | list Q1; list Q2; |
---|
4221 | int d; int j; int k; |
---|
4222 | t1=1; |
---|
4223 | for(i=1;i<=size(G);i++) |
---|
4224 | { |
---|
4225 | attrib(G[i][1],"IsSB",1); |
---|
4226 | d=locusdim(G[i][2],moverdim); |
---|
4227 | if(d==0){G1[size(G1)+1]=G[i];} |
---|
4228 | else |
---|
4229 | { |
---|
4230 | if(d>0){G2[size(G2)+1]=G[i];} |
---|
4231 | } |
---|
4232 | } |
---|
4233 | if(size(G1)==0){t1=0;} |
---|
4234 | if(size(G2)==0){t2=0;} |
---|
4235 | setring(@RP); |
---|
4236 | if(t1) |
---|
4237 | { |
---|
4238 | list G1RP=imap(R,G1); |
---|
4239 | } |
---|
4240 | else {list G1RP;} |
---|
4241 | list P1RP; |
---|
4242 | ideal B; |
---|
4243 | for(i=1;i<=size(G1RP);i++) |
---|
4244 | { |
---|
4245 | kill B; |
---|
4246 | ideal B; |
---|
4247 | for(k=1;k<=size(G1RP[i][3]);k++) |
---|
4248 | { |
---|
4249 | attrib(G1RP[i][3][k][1],"IsSB",1); |
---|
4250 | G1RP[i][3][k][1]=std(G1RP[i][3][k][1]); |
---|
4251 | for(j=1;j<=size(G1RP[i][2]);j++) |
---|
4252 | { |
---|
4253 | B[j]=reduce(G1RP[i][2][j],G1RP[i][3][k][1]); |
---|
4254 | } |
---|
4255 | P1RP[size(P1RP)+1]=list(G1RP[i][3][k][1],G1RP[i][3][k][2],B); |
---|
4256 | } |
---|
4257 | } |
---|
4258 | setring(R); |
---|
4259 | ideal h; |
---|
4260 | if(t1) |
---|
4261 | { |
---|
4262 | def P1=imap(@RP,P1RP); |
---|
4263 | for(i=1;i<=size(P1);i++) |
---|
4264 | { |
---|
4265 | for(j=1;j<=size(P1[i][3]);j++) |
---|
4266 | { |
---|
4267 | h=factorize(P1[i][3][j],1); |
---|
4268 | P1[i][3][j]=h[1]; |
---|
4269 | for(k=2;k<=size(h);k++) |
---|
4270 | { |
---|
4271 | P1[i][3][j]=P1[i][3][j]*h[k]; |
---|
4272 | } |
---|
4273 | } |
---|
4274 | } |
---|
4275 | } |
---|
4276 | else{list P1;} |
---|
4277 | ideal BB; int dd; list NS; |
---|
4278 | for(i=1;i<=size(P1);i++) |
---|
4279 | { |
---|
4280 | NS=NorSing(P1[i][3],P1[i][1],P1[i][2][1],DD); |
---|
4281 | dd=NS[1]; |
---|
4282 | if(dd==0){P1[i][3]=NS;} //"Special"; |
---|
4283 | else{P1[i][3]="Normal";} |
---|
4284 | } |
---|
4285 | list P2; |
---|
4286 | for(i=1;i<=size(G2);i++) |
---|
4287 | { |
---|
4288 | for(k=1;k<=size(G2[i][3]);k++) |
---|
4289 | { |
---|
4290 | P2[size(P2)+1]=list(G2[i][3][k][1],G2[i][3][k][2]); |
---|
4291 | } |
---|
4292 | } |
---|
4293 | list l; |
---|
4294 | for(i=1;i<=size(P1);i++){Q1[i]=l; Q1[i][1]=P1[i];} P1=Q1; |
---|
4295 | for(i=1;i<=size(P2);i++){Q2[i]=l; Q2[i][1]=P2[i];} P2=Q2; |
---|
4296 | |
---|
4297 | setring(@P); |
---|
4298 | ideal J; |
---|
4299 | if(t1==1) |
---|
4300 | { |
---|
4301 | def C1=imap(R,P1); |
---|
4302 | def L1=AddLocus(C1); |
---|
4303 | } |
---|
4304 | else{list C1; list L1; kill P1; list P1;} |
---|
4305 | if(t2==1) |
---|
4306 | { |
---|
4307 | def C2=imap(R,P2); |
---|
4308 | def L2=AddLocus(C2); |
---|
4309 | } |
---|
4310 | else{list L2; list C2; kill P2; list P2;} |
---|
4311 | for(i=1;i<=size(L2);i++) |
---|
4312 | { |
---|
4313 | J=std(L2[i][2]); |
---|
4314 | d=dim(J); // AQUI |
---|
4315 | if(d==0) |
---|
4316 | { |
---|
4317 | L2[i][4]=string("Accumulation",L2[i][4]); |
---|
4318 | } |
---|
4319 | else{L2[i][4]=string("Degenerate",L2[i][4]);} |
---|
4320 | } |
---|
4321 | list LN; |
---|
4322 | if(t1==1) |
---|
4323 | { |
---|
4324 | for(i=1;i<=size(L1);i++){LN[size(LN)+1]=L1[i];} |
---|
4325 | } |
---|
4326 | if(t2==1) |
---|
4327 | { |
---|
4328 | for(i=1;i<=size(L2);i++){LN[size(LN)+1]=L2[i];} |
---|
4329 | } |
---|
4330 | setring(R); |
---|
4331 | def L=imap(@P,LN); |
---|
4332 | for(i=1;i<=size(L);i++){if(size(L[i][2])==0){L[i][2]=ideal(1);}} |
---|
4333 | //if(te==0){kill @R; kill @RP; kill @P;} |
---|
4334 | list LL; |
---|
4335 | for(i=1;i<=size(L);i++) |
---|
4336 | { |
---|
4337 | if(typeof(L[i][4])=="list") {L[i][4][1]="Special";} |
---|
4338 | l[1]=L[i][2]; |
---|
4339 | l[2]=L[i][3]; |
---|
4340 | l[3]=L[i][4]; |
---|
4341 | l[4]=L[i][5]; |
---|
4342 | L[i]=l; |
---|
4343 | } |
---|
4344 | return(L); |
---|
4345 | } |
---|
4346 | |
---|
4347 | // locus(G): Special routine for determining the locus of points |
---|
4348 | // of geometrical constructions. |
---|
4349 | // input: The output G of the grobcov (in generic representation, which is the default option for grobcov) |
---|
4350 | // output: |
---|
4351 | // list, the canonical P-representation of the Normal and Non-Normal locus: |
---|
4352 | // The Normal locus has two kind of components: Normal and Special. |
---|
4353 | // The Non-normal locus has two kind of components: Accumulation and Degenerate. |
---|
4354 | // Normal components: for each point in the component, |
---|
4355 | // the number of solutions in the variables is finite, and |
---|
4356 | // the solutions depend on the point in the component if the component is not 0-dimensional. |
---|
4357 | // Special components: for each point in the component, |
---|
4358 | // the number of solutions in the variables is finite, |
---|
4359 | // the component is not 0-dimensional, but the solutions do not depend on the |
---|
4360 | // values of the parameters in the component. |
---|
4361 | // Accumlation points: are 0-dimensional components for which it exist |
---|
4362 | // an infinite number of solutions. |
---|
4363 | // Degenerate components: are components of dimension greater than 0 for which |
---|
4364 | // for every point in the component there exist infinite solutions. |
---|
4365 | // The output components are given as |
---|
4366 | // ((p1,(p11,..p1s_1),type_1,level_1),..,(pk,(pk1,..pks_k),type_k,level_k) |
---|
4367 | // The components are given in canonical P-representation of the subset. |
---|
4368 | // If all levels of a class of locus are 1, then the set is locally closed. Otherwise the level |
---|
4369 | // gives the depth of the component of the constructible set. |
---|
4370 | proc locus(list GG, list #) |
---|
4371 | "USAGE: locus(G) |
---|
4372 | The input must be the grobcov of a parametrical ideal in Q[a][x], |
---|
4373 | (a=parameters, x=variables). In fact a must be the tracer coordinates |
---|
4374 | and x the mover coordinates and other auxiliary ones. |
---|
4375 | Special routine for determining the locus of points |
---|
4376 | of geometrical constructions. Given a parametric ideal J |
---|
4377 | representing the system determining the locus of points (a) |
---|
4378 | who verify certain properties, the call to locus on the output of grobcov( J ) |
---|
4379 | determines the different classes of locus components, following |
---|
4380 | the taxonomy defined in |
---|
4381 | Abanades, Botana, Montes, Recio: |
---|
4382 | \"An Algebraic Taxonomy for Locus Computation in Dynamic Geometry\". |
---|
4383 | Computer-Aided Design 56 (2014) 22-33. |
---|
4384 | The components can be Normal, Special, Accumulation, Degenerate. |
---|
4385 | OPTIONS: The algorithm allows the following options as pair of arguments: |
---|
4386 | \"movdim\", d : by default movdim is 2 but it can be set to other values, |
---|
4387 | and represents the number of mever variables. they should be given as |
---|
4388 | the last variables of the ring. |
---|
4389 | \"version\", v : There are two versions of the algorithm. (\"version\",1) is |
---|
4390 | a full algorithm that always distinguishes correctly between 'Normal' |
---|
4391 | and 'Special' components, whereas \("version\",0) can decalre a component |
---|
4392 | as 'Normal' being really 'Special', but is more effective. By default (\"version\",1) |
---|
4393 | is used when the number of variables is less than 4 and 0 if not. |
---|
4394 | The user can force to use one or other version, but it is not recommended. |
---|
4395 | \"system\", ideal F: if the initial system is passed as an argument. This is actually not used. |
---|
4396 | \"comments\", c: by default it is 0, but it can be set to 1. |
---|
4397 | Usually locus problems have mover coordinates, variables and tracer coordinates. |
---|
4398 | The mover coordinates are to be placed as the last variables, and by default, |
---|
4399 | its number is 2. If one consider locus problems in higer dimensions, the number of |
---|
4400 | mover coordinates (placed as the last variables) is to be given as an option. |
---|
4401 | RETURN: The locus. The output is a list of the components ( C_1,.. C_n ) where |
---|
4402 | C_i = (p_i,(p_i1,..p_is_i), type_i,l evel_i ) and type_i can be |
---|
4403 | 'Normal', 'Special', Accumulation', 'Degenerate'. The 'Special' components |
---|
4404 | return more information, namely the antiimage of the component, that |
---|
4405 | is 0-dimensional for these kind of components. |
---|
4406 | Normal components: for each point in the component, |
---|
4407 | the number of solutions in the variables is finite, and |
---|
4408 | the solutions depend on the point in the component. |
---|
4409 | Special components: for each point in the component, |
---|
4410 | the number of solutions in the variables is finite. The |
---|
4411 | antiimage of the component is 0-dimensional. |
---|
4412 | Accumlation points: are 0-dimensional components whose |
---|
4413 | antiimage is not zero-dimansional. |
---|
4414 | Degenerate components: are components of dimension greater than 0 |
---|
4415 | whose antiimage is not-zero-diemansional. |
---|
4416 | The components are given in canonical P-representation. |
---|
4417 | The levels of a class of locus are 1, |
---|
4418 | because they represent locally closed. sets. |
---|
4419 | NOTE: It can only be called after computing the grobcov of the |
---|
4420 | parametrical ideal in generic representation ('ext',0), |
---|
4421 | which is the default. |
---|
4422 | The basering R, must be of the form Q[a_1,..,a_m][x_1,..,x_n]. |
---|
4423 | KEYWORDS: geometrical locus, locus, loci. |
---|
4424 | EXAMPLE: locus; shows an example" |
---|
4425 | { |
---|
4426 | int tes=0; int i; |
---|
4427 | def R=basering; |
---|
4428 | if(defined(@P)==1){tes=1; kill @P; kill @R; kill @RP;} |
---|
4429 | setglobalrings(); |
---|
4430 | // Options |
---|
4431 | list DD=#; |
---|
4432 | int moverdim=nvars(R); |
---|
4433 | int version=0; |
---|
4434 | int nv=nvars(R); |
---|
4435 | if(nv<4){version=1;} |
---|
4436 | int comment=0; |
---|
4437 | ideal Fm; |
---|
4438 | for(i=1;i<=(size(DD) div 2);i++) |
---|
4439 | { |
---|
4440 | if(DD[2*i-1]=="movdim"){moverdim=DD[2*i];} |
---|
4441 | if(DD[2*i-1]=="version"){version=DD[2*i];} |
---|
4442 | if(DD[2*i-1]=="system"){Fm=DD[2*i];} |
---|
4443 | if(DD[2*i-1]=="comment"){comment=DD[2*i];} |
---|
4444 | if(DD[2*i-1]=="family"){poly F=DD[2*i];} |
---|
4445 | } |
---|
4446 | int j; int k; |
---|
4447 | def B0=GG[1][2]; |
---|
4448 | def H0=GG[1][3][1][1]; |
---|
4449 | if (equalideals(B0,ideal(1)) or (equalideals(H0,ideal(0))==0)) |
---|
4450 | { |
---|
4451 | return(locus0(GG,DD)); |
---|
4452 | } |
---|
4453 | else |
---|
4454 | { |
---|
4455 | int n=nvars(R); |
---|
4456 | ideal vmov=var(n-1),var(n); |
---|
4457 | ideal N; |
---|
4458 | intvec xw; intvec yw; |
---|
4459 | for(i=1;i<=n-1;i++){xw[i]=0;} |
---|
4460 | xw[n]=1; |
---|
4461 | for(i=1;i<=n;i++){yw[i]=0;} |
---|
4462 | yw[n-1]=1; |
---|
4463 | poly px; poly py; |
---|
4464 | int te=1; |
---|
4465 | i=1; |
---|
4466 | while( te and i<=size(B0)) |
---|
4467 | { |
---|
4468 | if((deg(B0[i],xw)==1) and (deg(B0[i])==1)){px=B0[i]; te=0;} |
---|
4469 | i++; |
---|
4470 | } |
---|
4471 | i=1; te=1; |
---|
4472 | while( te and i<=size(B0)) |
---|
4473 | { |
---|
4474 | if((deg(B0[i],yw)==1) and (deg(B0[i])==1)){py=B0[i]; te=0;} |
---|
4475 | i++; |
---|
4476 | } |
---|
4477 | N=px,py; |
---|
4478 | te=indepparameters(N); |
---|
4479 | if(te) |
---|
4480 | { |
---|
4481 | string("locus detected that the mover must avoid point (",N,") in order to obtain the correct locus"); |
---|
4482 | // eliminates segments of GG where N is contained in the basis |
---|
4483 | list nGP; |
---|
4484 | def GP=GG; |
---|
4485 | ideal BP; |
---|
4486 | for(j=1;j<=size(GP);j++) |
---|
4487 | { |
---|
4488 | te=1; k=1; |
---|
4489 | BP=GP[j][2]; |
---|
4490 | while((te==1) and (k<=size(N))) |
---|
4491 | { |
---|
4492 | if(pdivi(N[k],BP)[1]!=0){te=0;} |
---|
4493 | k++; |
---|
4494 | } |
---|
4495 | if(te==0){nGP[size(nGP)+1]=GP[j];} |
---|
4496 | } |
---|
4497 | } |
---|
4498 | else{"Warning! Problem with more than one mover. Not able to solve it."; list L; return(L);} |
---|
4499 | } |
---|
4500 | def LL=locus0(nGP,DD); |
---|
4501 | kill @RP; kill @P; kill @R; |
---|
4502 | return(LL); |
---|
4503 | } |
---|
4504 | example |
---|
4505 | { |
---|
4506 | "EXAMPLE:"; echo = 2; |
---|
4507 | ring R=(0,a,b),(x,y),dp; |
---|
4508 | short=0; |
---|
4509 | |
---|
4510 | // Concoid |
---|
4511 | ideal S96=x^2+y^2-4,(b-2)*x-a*y+2*a,(a-x)^2+(b-y)^2-1; |
---|
4512 | // System S96= |
---|
4513 | S96; |
---|
4514 | locus(grobcov(S96)); |
---|
4515 | kill R; |
---|
4516 | // ******************************************** |
---|
4517 | ring R=(0,a,b),(x4,x3,x2,x1),dp; |
---|
4518 | short=0; |
---|
4519 | ideal S=(x1-3)^2+(x2-1)^2-9, |
---|
4520 | (4-x2)*(x3-3)+(x1-3)*(x4-1), |
---|
4521 | (3-x1)*(x3-x1)+(4-x2)*(x4-x2), |
---|
4522 | (4-x4)*a+(x3-3)*b+3*x4-4*x3, |
---|
4523 | (a-x1)^2+(b-x2)^2-(x1-x3)^2-(x2-x4)^2; |
---|
4524 | // System S= |
---|
4525 | S; |
---|
4526 | locus(grobcov(S)); |
---|
4527 | kill R; |
---|
4528 | "********************************************"; |
---|
4529 | |
---|
4530 | ring R=(0,x,y),(x1,x2),dp; |
---|
4531 | short=0; |
---|
4532 | ideal S=-(x - 5)*(x1 - 1) - (x2 - 2)*(y - 2), |
---|
4533 | (x1 - 5)^2 + (x2 - 2)^2 - 4, |
---|
4534 | -2*(x - 5)*(x2 - 2) + 2*(x1 - 5)*(y - 2); |
---|
4535 | // System S= |
---|
4536 | S; |
---|
4537 | locus(grobcov(S)); |
---|
4538 | } |
---|
4539 | |
---|
4540 | // locusdg(G): Special routine for determining the locus of points |
---|
4541 | // of geometrical constructions. Given a parametric ideal J with |
---|
4542 | // parameters (a_1,..a_m) and variables (x_1,..,xn), |
---|
4543 | // representing the system determining |
---|
4544 | // the locus of points (a_1,..,a_m) who verify certain |
---|
4545 | // properties, computing the grobcov G of |
---|
4546 | // J and applying to it locus, determines the different |
---|
4547 | // classes of locus components. The components can be |
---|
4548 | // Normal, Special, Accumulation point, Degenerate. |
---|
4549 | // The output are the components given in P-canonical form |
---|
4550 | // of at most 4 constructible sets: Normal, Special, Accumulation, |
---|
4551 | // Degenerate. |
---|
4552 | // The description of the algorithm and definitions is |
---|
4553 | // given in a forthcoming paper by Abanades, Botana, Montes, Recio. |
---|
4554 | // Usually locus problems have mover coordinates, variables and tracer coordinates. |
---|
4555 | // The mover coordinates are to be placed as the last variables, and by default, |
---|
4556 | // its number is 2. If onw consider locus problems in higer dimensions, the number of |
---|
4557 | // mover coordinates (placed as the last variables) is to be given as an option. |
---|
4558 | // |
---|
4559 | // input: The output of locus(G); |
---|
4560 | // output: |
---|
4561 | // list, the canonical P-representation of the Normal and Non-Normal locus: |
---|
4562 | // The Normal locus has two kind of components: Normal and Special. |
---|
4563 | // The Non-normal locus has two kind of components: Accumulation and Degenerate. |
---|
4564 | // Normal components: for each point in the component, |
---|
4565 | // the number of solutions in the variables is finite, and |
---|
4566 | // the solutions depend on the point in the component if the component is not 0-dimensional. |
---|
4567 | // Special components: for each point in the component, |
---|
4568 | // the number of solutions in the variables is finite, |
---|
4569 | // the component is not 0-dimensional, but the solutions do not depend on the |
---|
4570 | // values of the parameters in the component. |
---|
4571 | // Accumlation points: are 0-dimensional components for which it exist |
---|
4572 | // an infinite number of solutions. |
---|
4573 | // Degenerate components: are components of dimension greater than 0 for which |
---|
4574 | // for every point in the component there exist infinite solutions. |
---|
4575 | // The output components are given as |
---|
4576 | // ((p1,(p11,..p1s_1),type_1,level_1),..,(pk,(pk1,..pks_k),type_k,level_k) |
---|
4577 | // The components are given in canonical P-representation of the subset. |
---|
4578 | // If all levels of a class of locus are 1, then the set is locally closed. Otherwise the level |
---|
4579 | // gives the depth of the component of the constructible set. |
---|
4580 | proc locusdg(list L) |
---|
4581 | "USAGE: locusdg(L) The call must be locusdg(locus(grobcov(S))). |
---|
4582 | RETURN: The output is the list of the 'Relevant' components of the locus |
---|
4583 | in Dynamic Geometry:(C1,..,C:m), where |
---|
4584 | C_i= ( p_i,(p_i1,..p_is_i), 'Relevant', level_i ) |
---|
4585 | The 'Relevant' components are the 'Normal' and 'Accumulation' components |
---|
4586 | of the locus. (See help for locus). |
---|
4587 | |
---|
4588 | NOTE: It can only be called after computing the locus. |
---|
4589 | Calling sequence: locusdg(locus(grobcov(S))); |
---|
4590 | KEYWORDS: geometrical locus, locus, loci, dynamic geometry |
---|
4591 | EXAMPLE: locusdg; shows an example" |
---|
4592 | { |
---|
4593 | list LL; |
---|
4594 | int i; |
---|
4595 | for(i=1;i<=size(L);i++) |
---|
4596 | { |
---|
4597 | if(typeof(L[i][3])=="string") |
---|
4598 | { |
---|
4599 | if((L[i][3]=="Normal") or (L[i][3]=="Accumulation")){L[i][3]="Relevant"; LL[size(LL)+1]=L[i];} |
---|
4600 | } |
---|
4601 | } |
---|
4602 | return(LL); |
---|
4603 | } |
---|
4604 | example |
---|
4605 | { |
---|
4606 | "EXAMPLE:"; echo = 2; |
---|
4607 | ring R=(0,a,b),(x,y),dp; |
---|
4608 | short=0; |
---|
4609 | |
---|
4610 | // Concoid |
---|
4611 | ideal S96=x^2+y^2-4,(b-2)*x-a*y+2*a,(a-x)^2+(b-y)^2-1; |
---|
4612 | // System S96= |
---|
4613 | S96; |
---|
4614 | locus(grobcov(S96)); |
---|
4615 | locusdg(locus(grobcov(S96))); |
---|
4616 | kill R; |
---|
4617 | "********************************************"; |
---|
4618 | ring R=(0,a,b),(x4,x3,x2,x1),dp; |
---|
4619 | ideal S=(x1-3)^2+(x2-1)^2-9, |
---|
4620 | (4-x2)*(x3-3)+(x1-3)*(x4-1), |
---|
4621 | (3-x1)*(x3-x1)+(4-x2)*(x4-x2), |
---|
4622 | (4-x4)*a+(x3-3)*b+3*x4-4*x3, |
---|
4623 | (a-x1)^2+(b-x2)^2-(x1-x3)^2-(x2-x4)^2; |
---|
4624 | short=0; |
---|
4625 | locus(grobcov(S)); |
---|
4626 | locusdg(locus(grobcov(S))); |
---|
4627 | kill R; |
---|
4628 | "********************************************"; |
---|
4629 | |
---|
4630 | ring R=(0,x,y),(x1,x2),dp; |
---|
4631 | short=0; |
---|
4632 | ideal S=-(x - 5)*(x1 - 1) - (x2 - 2)*(y - 2), |
---|
4633 | (x1 - 5)^2 + (x2 - 2)^2 - 4, |
---|
4634 | -2*(x - 5)*(x2 - 2) + 2*(x1 - 5)*(y - 2); |
---|
4635 | locus(grobcov(S)); |
---|
4636 | locusdg(locus(grobcov(S))); |
---|
4637 | } |
---|
4638 | |
---|
4639 | // locusto: Transforms the output of locus to a string that |
---|
4640 | // can be read by different computational systems. |
---|
4641 | // input: |
---|
4642 | // list L: The output of locus |
---|
4643 | // output: |
---|
4644 | // string s: The output of locus converted to a string readable by other programs |
---|
4645 | proc locusto(list L) |
---|
4646 | "USAGE: locusto(L); |
---|
4647 | The argument must be the output of locus or locusdg or |
---|
4648 | envelop or envelopdg. |
---|
4649 | It transforms the output into a string in standard form |
---|
4650 | readable in many languages (Geogebra). |
---|
4651 | RETURN: The locus in string standard form |
---|
4652 | NOTE: It can only be called after computing either |
---|
4653 | - locus(grobcov(F)) -> locusto( locus(grobcov(F)) ) |
---|
4654 | - locusdg(locus(grobcov(F))) -> locusto( locusdg(locus(grobcov(F))) ) |
---|
4655 | - envelop(F,C) -> locusto( envelop(F,C) ) |
---|
4656 | -envelopdg(envelop(F,C)) -> locusto( envelopdg(envelop(F,C)) ) |
---|
4657 | KEYWORDS: geometrical locus, locus, loci. |
---|
4658 | EXAMPLE: locusto; shows an example" |
---|
4659 | { |
---|
4660 | int i; int j; int k; |
---|
4661 | string s="["; string sf="]"; string st=s+sf; |
---|
4662 | if(size(L)==0){return(st);} |
---|
4663 | ideal p; |
---|
4664 | ideal q; |
---|
4665 | for(i=1;i<=size(L);i++) |
---|
4666 | { |
---|
4667 | s=string(s,"[["); |
---|
4668 | for (j=1;j<=size(L[i][1]);j++) |
---|
4669 | { |
---|
4670 | s=string(s,L[i][1][j],","); |
---|
4671 | } |
---|
4672 | s[size(s)]="]"; |
---|
4673 | s=string(s,",["); |
---|
4674 | for(j=1;j<=size(L[i][2]);j++) |
---|
4675 | { |
---|
4676 | s=string(s,"["); |
---|
4677 | for(k=1;k<=size(L[i][2][j]);k++) |
---|
4678 | { |
---|
4679 | s=string(s,L[i][2][j][k],","); |
---|
4680 | } |
---|
4681 | s[size(s)]="]"; |
---|
4682 | s=string(s,","); |
---|
4683 | } |
---|
4684 | s[size(s)]="]"; |
---|
4685 | s=string(s,"]"); |
---|
4686 | if(size(L[i])>=3) |
---|
4687 | { |
---|
4688 | s=string(s,",["); |
---|
4689 | if(typeof(L[i][3])=="string") |
---|
4690 | { |
---|
4691 | s=string(s,string(L[i][3]),"]]"); |
---|
4692 | } |
---|
4693 | else |
---|
4694 | { |
---|
4695 | for(k=1;k<=size(L[i][3]);k++) |
---|
4696 | { |
---|
4697 | s=string(s,"[",L[i][3][k],"],"); |
---|
4698 | } |
---|
4699 | s[size(s)]="]"; |
---|
4700 | s=string(s,"]"); |
---|
4701 | } |
---|
4702 | } |
---|
4703 | if(size(L[i])>=4) |
---|
4704 | { |
---|
4705 | s[size(s)]=","; |
---|
4706 | s=string(s,string(L[i][4]),"],"); |
---|
4707 | } |
---|
4708 | s[size(s)]="]"; |
---|
4709 | s=string(s,","); |
---|
4710 | } |
---|
4711 | s[size(s)]="]"; |
---|
4712 | return(s); |
---|
4713 | } |
---|
4714 | example |
---|
4715 | { |
---|
4716 | "EXAMPLE:"; echo = 2; |
---|
4717 | ring R=(0,x,y),(x1,y1),dp; |
---|
4718 | short=0; |
---|
4719 | ideal S=x1^2+y1^2-4,(y-2)*x1-x*y1+2*x,(x-x1)^2+(y-y1)^2-1; |
---|
4720 | locusto(locus(grobcov(S))); |
---|
4721 | locusto(locusdg(locus(grobcov(S)))); |
---|
4722 | kill R; |
---|
4723 | "********************************************"; |
---|
4724 | |
---|
4725 | // 1. Take a fixed line l: x1-y1=0 and consider |
---|
4726 | // the family F of a lines parallel to l passing through the mover point M |
---|
4727 | // 2. Consider a circle x1^2+x2^2-25, and a mover point M(x1,x2) on it. |
---|
4728 | // 3. Compute the envelop of the family of lines. |
---|
4729 | |
---|
4730 | ring R=(0,x,y),(x1,y1),lp; |
---|
4731 | poly F=(y-y1)-(x-x1); |
---|
4732 | ideal C=x1^2+y1^2-25; |
---|
4733 | short=0; |
---|
4734 | |
---|
4735 | // Curves Family F= |
---|
4736 | F; |
---|
4737 | // Conditions C= |
---|
4738 | C; |
---|
4739 | |
---|
4740 | locusto(envelop(F,C)); |
---|
4741 | locusto(envelopdg(envelop(F,C))); |
---|
4742 | kill R; |
---|
4743 | "********************************************"; |
---|
4744 | |
---|
4745 | // Steiner Deltoid |
---|
4746 | // 1. Consider the circle x1^2+y1^2-1=0, and a mover point M(x1,y1) on it. |
---|
4747 | // 2. Consider the triangle A(0,1), B(-1,0), C(1,0). |
---|
4748 | // 3. Consider lines passing through M perpendicular to two sides of ABC triangle. |
---|
4749 | // 4. Obtain the envelop of the lines above. |
---|
4750 | |
---|
4751 | ring R=(0,x,y),(x1,y1,x2,y2),lp; |
---|
4752 | ideal C=(x1)^2+(y1)^2-1, |
---|
4753 | x2+y2-1, |
---|
4754 | x2-y2-x1+y1; |
---|
4755 | matrix M[3][3]=x,y,1,x2,y2,1,x1,0,1; |
---|
4756 | poly F=det(M); |
---|
4757 | |
---|
4758 | short=0; |
---|
4759 | |
---|
4760 | // Curves Family F= |
---|
4761 | F; |
---|
4762 | // Conditions C= |
---|
4763 | C; |
---|
4764 | |
---|
4765 | locusto(envelop(F,C)); |
---|
4766 | locusto(envelopdg(envelop(F,C))); |
---|
4767 | } |
---|
4768 | |
---|
4769 | // Auxiliary routine |
---|
4770 | // locusdim |
---|
4771 | // input: |
---|
4772 | // B: ideal, a basis of a segment of the grobcov |
---|
4773 | // dgdim: int, the dimension of the mover (for locus) |
---|
4774 | // by default dgdim is equal to the number of variables |
---|
4775 | static proc locusdim(ideal B, list #) |
---|
4776 | { |
---|
4777 | def R=basering; |
---|
4778 | int dgdim; |
---|
4779 | int nv=nvars(R); |
---|
4780 | if (size(#)>0){dgdim=#[1];} |
---|
4781 | else {dgdim=nv;} |
---|
4782 | int d; |
---|
4783 | list v; |
---|
4784 | ideal vi; |
---|
4785 | int i; |
---|
4786 | for(i=1;i<=dgdim;i++) |
---|
4787 | { |
---|
4788 | v[size(v)+1]=varstr(nv-dgdim+i); |
---|
4789 | vi[size(v)+1]=var(nv-dgdim+i); |
---|
4790 | } |
---|
4791 | ideal B0; |
---|
4792 | for(i=1;i<=size(B);i++) |
---|
4793 | { |
---|
4794 | if(subset(variables(B[i]),vi)) |
---|
4795 | { |
---|
4796 | B0[size(B0)+1]=B[i]; |
---|
4797 | } |
---|
4798 | } |
---|
4799 | def RR=ringlist(R); |
---|
4800 | def RR0=RR; |
---|
4801 | RR0[2]=v; |
---|
4802 | def R0=ring(RR0); |
---|
4803 | setring(R0); |
---|
4804 | def B0r=imap(R,B0); |
---|
4805 | B0r=std(B0r); |
---|
4806 | d=dim(B0r); |
---|
4807 | setring R; |
---|
4808 | return(d); |
---|
4809 | } |
---|
4810 | |
---|
4811 | // // locusdgto: Transforms the output of locusdg to a string that |
---|
4812 | // // can be read by different computational systems. |
---|
4813 | // // input: |
---|
4814 | // // list L: The output of locus |
---|
4815 | // // output: |
---|
4816 | // // string s: The output of locus converted to a string readable by other programs |
---|
4817 | // // Outputs only the relevant dynamical geometry components. |
---|
4818 | // // Without unnecessary parenthesis |
---|
4819 | // proc locusdgto(list LL) |
---|
4820 | // "USAGE: locusdgto(L); |
---|
4821 | // The argument must be the output of locusdg of a parametrical ideal |
---|
4822 | // It transforms the output into a string in standard form |
---|
4823 | // readable in many languages (Geogebra). |
---|
4824 | // RETURN: The locusdg in string standard form |
---|
4825 | // NOTE: It can only be called after computing the locusdg(grobcov(F)) of the |
---|
4826 | // parametrical ideal. |
---|
4827 | // The basering R, must be of the form Q[a,b,..][x,y,..]. |
---|
4828 | // KEYWORDS: geometrical locus, locus, loci. |
---|
4829 | // EXAMPLE: locusdgto; shows an example" |
---|
4830 | // { |
---|
4831 | // int i; int j; int k; int short0=short; int ojo; |
---|
4832 | // int te=0; |
---|
4833 | // short=0; |
---|
4834 | // if(size(LL)==0){ojo=1; list L;} |
---|
4835 | // else |
---|
4836 | // { |
---|
4837 | // def L=LL; |
---|
4838 | // } |
---|
4839 | // string s="["; string sf="]"; string st=s+sf; |
---|
4840 | // if(size(L)==0){return(st);} |
---|
4841 | // ideal p; |
---|
4842 | // ideal q; |
---|
4843 | // for(i=1;i<=size(L);i++) |
---|
4844 | // { |
---|
4845 | // if(L[i][3]=="Relevant") |
---|
4846 | // { |
---|
4847 | // s=string(s,"[["); |
---|
4848 | // for (j=1;j<=size(L[i][1]);j++) |
---|
4849 | // { |
---|
4850 | // s=string(s,L[i][1][j],","); |
---|
4851 | // } |
---|
4852 | // s[size(s)]="]"; |
---|
4853 | // s=string(s,",["); |
---|
4854 | // for(j=1;j<=size(L[i][2]);j++) |
---|
4855 | // { |
---|
4856 | // s=string(s,"["); |
---|
4857 | // for(k=1;k<=size(L[i][2][j]);k++) |
---|
4858 | // { |
---|
4859 | // s=string(s,L[i][2][j][k],","); |
---|
4860 | // } |
---|
4861 | // s[size(s)]="]"; |
---|
4862 | // s=string(s,","); |
---|
4863 | // } |
---|
4864 | // s[size(s)]="]"; |
---|
4865 | // s=string(s,"]"); |
---|
4866 | // s[size(s)]="]"; |
---|
4867 | // s=string(s,","); |
---|
4868 | // } |
---|
4869 | // } |
---|
4870 | // if(s=="["){s="[]";} |
---|
4871 | // else{s[size(s)]="]";} |
---|
4872 | // short=short0; |
---|
4873 | // return(s); |
---|
4874 | // } |
---|
4875 | // example |
---|
4876 | // {"EXAMPLE:"; echo = 2; |
---|
4877 | // ring R=(0,a,b),(x,y),dp; |
---|
4878 | // short=0; |
---|
4879 | // ideal S96=x^2+y^2-4,(b-2)*x-a*y+2*a,(a-x)^2+(b-y)^2-1; |
---|
4880 | // "System="; S96; " "; |
---|
4881 | // "locusdgto(locusdg(grobcov(S96)))="; |
---|
4882 | // locusdgto(locusdg(grobcov(S96))); |
---|
4883 | // } |
---|
4884 | |
---|
4885 | static proc norspec(ideal F) |
---|
4886 | { |
---|
4887 | def RR=basering; |
---|
4888 | def Rx=ringlist(RR); |
---|
4889 | |
---|
4890 | def Rx=ringlist(RR); |
---|
4891 | def @P=ring(Rx[1]); |
---|
4892 | list Lx; |
---|
4893 | Lx[1]=0; |
---|
4894 | Lx[2]=Rx[2]+Rx[1][2]; |
---|
4895 | Lx[3]=Rx[1][3]; |
---|
4896 | Lx[4]=Rx[1][4]; |
---|
4897 | Rx[1]=0; |
---|
4898 | def D=ring(Rx); |
---|
4899 | def @RP=D+@P; |
---|
4900 | exportto(Top,@R); // global ring Q[a][x] |
---|
4901 | exportto(Top,@P); // global ring Q[a] |
---|
4902 | exportto(Top,@RP); // global ring K[x,a] with product order |
---|
4903 | setring(RR); |
---|
4904 | |
---|
4905 | } |
---|
4906 | |
---|
4907 | // envelop |
---|
4908 | // Input: n arguments |
---|
4909 | // poly F: the polynomial defining the family of curves in ring R=0,(x,y),(x1,..,xn),lp; |
---|
4910 | // ideal C=g1,..,g_{n-1}: the set of constraints |
---|
4911 | // Output: the components of the envolvent; |
---|
4912 | proc envelop(poly F, ideal C, list #) |
---|
4913 | "USAGE: envelop(F,C); |
---|
4914 | The first argument F must be the family of curves of which |
---|
4915 | on want to compute the envelop. |
---|
4916 | The second argument C must be the ideal of conditions |
---|
4917 | over the variables, and should contain as polynomials |
---|
4918 | as the number of variables -1. |
---|
4919 | RETURN: The components of the envelop with its taxonomy: |
---|
4920 | The taxonomy distinguishes 'Normal', |
---|
4921 | 'Special', 'Accumulation', 'Degenerate' components. |
---|
4922 | In the case of 'Special' components, it also |
---|
4923 | outputs the antiimage of the component |
---|
4924 | and an integer (0-1). If the integer is 0 |
---|
4925 | the component is not a curve of the family and is |
---|
4926 | not considered as 'Relevant' by the envelopdg routine |
---|
4927 | applied to it, but is considered as 'Relevant' if the integer is 1. |
---|
4928 | NOTE: grobcov is called internally. |
---|
4929 | The basering R, must be of the form Q[a][x] (a=parameters, x=variables). |
---|
4930 | KEYWORDS: geometrical locus, locus, loci, envelop |
---|
4931 | EXAMPLE: envelop; shows an example" |
---|
4932 | { |
---|
4933 | int tes=0; int i; int j; |
---|
4934 | def R=basering; |
---|
4935 | if(defined(@P)==1){tes=1; kill @P; kill @R; kill @RP;} |
---|
4936 | setglobalrings(); |
---|
4937 | // Options |
---|
4938 | list DD=#; |
---|
4939 | int moverdim=nvars(R); |
---|
4940 | int version=0; |
---|
4941 | int nv=nvars(R); |
---|
4942 | if(nv<4){version=1;} |
---|
4943 | int comment=0; |
---|
4944 | ideal Fm; |
---|
4945 | for(i=1;i<=(size(DD) div 2);i++) |
---|
4946 | { |
---|
4947 | if(DD[2*i-1]=="movdim"){moverdim=DD[2*i];} |
---|
4948 | if(DD[2*i-1]=="version"){version=DD[2*i];} |
---|
4949 | if(DD[2*i-1]=="system"){Fm=DD[2*i];} |
---|
4950 | if(DD[2*i-1]=="comment"){comment=DD[2*i];} |
---|
4951 | } |
---|
4952 | int n=nvars(R); |
---|
4953 | list v; |
---|
4954 | for(i=1;i<=n;i++){v[size(v)+1]=var(i);} |
---|
4955 | def MF=jacob(F); |
---|
4956 | def TMF=transpose(MF); |
---|
4957 | def Mg=MF; |
---|
4958 | def TMg=TMF; |
---|
4959 | for(i=1;i<=n-1;i++) |
---|
4960 | { |
---|
4961 | Mg=jacob(C[i]); |
---|
4962 | TMg=transpose(Mg); |
---|
4963 | TMF=concat(TMF,TMg); |
---|
4964 | } |
---|
4965 | poly J=det(TMF); |
---|
4966 | ideal S=ideal(F)+C+ideal(J); |
---|
4967 | DD[size(DD)+1]="family"; |
---|
4968 | DD[size(DD)+1]=F; |
---|
4969 | def G=grobcov(S,DD); |
---|
4970 | def L=locus(G, DD); |
---|
4971 | return(L); |
---|
4972 | } |
---|
4973 | example |
---|
4974 | { |
---|
4975 | "EXAMPLE:"; echo = 2; |
---|
4976 | // Steiner Deltoid |
---|
4977 | // 1. Consider the circle x1^2+y1^2-1=0, and a mover point M(x1,y1) on it. |
---|
4978 | // 2. Consider the triangle A(0,1), B(-1,0), C(1,0). |
---|
4979 | // 3. Consider lines passing through M perpendicular to two sides of ABC triangle. |
---|
4980 | // 4. Obtain the envelop of the lines above. |
---|
4981 | |
---|
4982 | ring R=(0,x,y),(x1,y1,x2,y2),lp; |
---|
4983 | ideal C=(x1)^2+(y1)^2-1, |
---|
4984 | x2+y2-1, |
---|
4985 | x2-y2-x1+y1; |
---|
4986 | matrix M[3][3]=x,y,1,x2,y2,1,x1,0,1; |
---|
4987 | poly F=det(M); |
---|
4988 | |
---|
4989 | short=0; |
---|
4990 | |
---|
4991 | // Curves Family F= |
---|
4992 | F; |
---|
4993 | // Conditions C= |
---|
4994 | C; |
---|
4995 | |
---|
4996 | def Env=envelop(F,C); |
---|
4997 | Env; |
---|
4998 | } |
---|
4999 | |
---|
5000 | // envelopdg |
---|
5001 | // Input: list L: the output of envelop(poly F, ideal C, list #) |
---|
5002 | // Output: the relevant components of the envolvent in dynamic geometry; |
---|
5003 | proc envelopdg(list L) |
---|
5004 | "USAGE: envelopdg(L); |
---|
5005 | The input list L must be the output of the call to |
---|
5006 | the routine 'envolop' of the family of curves |
---|
5007 | RETURN: The relevant components of the envelop in Dynamic Geometry. |
---|
5008 | 'Normal' and 'Accumulation' components are always considered |
---|
5009 | 'Relevant'. 'Special' components of the envelop outputs |
---|
5010 | three objects in its characterization: 'Special', the antiimage ideal, |
---|
5011 | and the integer 0 or 1, that indicates that the given component is |
---|
5012 | formed (1) or is not formed (0) by curves of the family. Only if yes, |
---|
5013 | 'envelopdg' considers the component as 'Relevant' . |
---|
5014 | NOTE: It must be called to the output of the 'envelop' routine. |
---|
5015 | The basering R, must be of the form Q[a,b,..][x,y,..]. |
---|
5016 | KEYWORDS: geometrical locus, locus, loci, envelop. |
---|
5017 | EXAMPLE: envelop; shows an example" |
---|
5018 | { |
---|
5019 | list LL; |
---|
5020 | list Li; |
---|
5021 | int i; |
---|
5022 | for(i=1;i<=size(L);i++) |
---|
5023 | { |
---|
5024 | if(typeof(L[i][3])=="string") |
---|
5025 | { |
---|
5026 | if((L[i][3]=="Normal") or (L[i][3]=="Accumulation")){Li=L[i]; Li[3]="Relevant"; LL[size(LL)+1]=Li;} |
---|
5027 | } |
---|
5028 | else |
---|
5029 | { |
---|
5030 | if(typeof(L[i][3])=="list") |
---|
5031 | { |
---|
5032 | if(L[i][3][3]==1) |
---|
5033 | { |
---|
5034 | Li=L[i]; Li[3]="Relevant"; LL[size(LL)+1]=Li; |
---|
5035 | } |
---|
5036 | } |
---|
5037 | } |
---|
5038 | } |
---|
5039 | return(LL); |
---|
5040 | } |
---|
5041 | example |
---|
5042 | { |
---|
5043 | "EXAMPLE:"; echo = 2; |
---|
5044 | |
---|
5045 | // 1. Take a fixed line l: x1-y1=0 and consider |
---|
5046 | // the family F of a lines parallel to l passing through the mover point M |
---|
5047 | // 2. Consider a circle x1^2+x2^2-25, and a mover point M(x1,x2) on it. |
---|
5048 | // 3. Compute the envelop of the family of lines. |
---|
5049 | |
---|
5050 | ring R=(0,x,y),(x1,y1),lp; |
---|
5051 | short=0; |
---|
5052 | poly F=(y-y1)-(x-x1); |
---|
5053 | ideal C=x1^2+y1^2-25; |
---|
5054 | short=0; |
---|
5055 | |
---|
5056 | // Curves Family F= |
---|
5057 | F; |
---|
5058 | // Conditions C= |
---|
5059 | C; |
---|
5060 | |
---|
5061 | envelop(F,C); |
---|
5062 | envelopdg(envelop(F,C)); |
---|
5063 | } |
---|
5064 | |
---|
5065 | //********************* End locus **************************** |
---|
5066 | |
---|
5067 | //********************* Begin AddCons ********************** |
---|
5068 | |
---|
5069 | // Input: L1,L2: lists of components with common top |
---|
5070 | // Output L: list of the union of L1 and L2. |
---|
5071 | static proc Add2ComWithCommonTop(list L1, list L2) |
---|
5072 | { |
---|
5073 | int i; int j; ideal pij; list L; list Lp; list PR; int k; |
---|
5074 | for(i=1;i<=size(L1[2]);i++) |
---|
5075 | { |
---|
5076 | for(j=1;j<=size(L2[2]);j++) |
---|
5077 | { |
---|
5078 | pij=std(L1[2][i]+L2[2][j]); |
---|
5079 | PR=minGTZ(pij); |
---|
5080 | for(k=1;k<=size(PR);k++) |
---|
5081 | { |
---|
5082 | Lp[size(Lp)+1]=PR[k]; |
---|
5083 | } |
---|
5084 | } |
---|
5085 | } |
---|
5086 | for(i=1; i<=size(Lp);i++) |
---|
5087 | { |
---|
5088 | for(j=i+1;j<=size(Lp);j++) |
---|
5089 | { |
---|
5090 | if(idcontains(Lp[i],Lp[j])) {Lp=delete(Lp,j);} |
---|
5091 | } |
---|
5092 | for(j=1;j<i;j++) |
---|
5093 | { |
---|
5094 | if(idcontains(Lp[i],Lp[j])){Lp=delete(Lp,j); j=j-1; i=i-1;} |
---|
5095 | } |
---|
5096 | } |
---|
5097 | L[1]=L1[1]; |
---|
5098 | L[2]=Lp; |
---|
5099 | return(L); |
---|
5100 | } |
---|
5101 | |
---|
5102 | // Input: L list od P-rep of components to be added. L[i]=[p_i,[p_{i1},...p_{ir_i}]] |
---|
5103 | // Output: lists A,B,L |
---|
5104 | // where no top in the lists are repeated |
---|
5105 | // A: list of components with higher tops |
---|
5106 | // B: list of components with lower tops |
---|
5107 | // L1: A,B |
---|
5108 | static proc SepareAB(list L) |
---|
5109 | { |
---|
5110 | int i; int j; list Ln=L; |
---|
5111 | for(i=1;i<=size(Ln);i++) |
---|
5112 | { |
---|
5113 | for(j=i+1;j<=size(Ln);j++) |
---|
5114 | { |
---|
5115 | if (equalideals(Ln[j][1],Ln[i][1])) |
---|
5116 | { |
---|
5117 | Ln[i]=Add2ComWithCommonTop(Ln[i],Ln[j]); |
---|
5118 | Ln=delete(Ln,j); |
---|
5119 | j=j-1; |
---|
5120 | } |
---|
5121 | } |
---|
5122 | } |
---|
5123 | list A; list B; int clas; list T; list L1; |
---|
5124 | for(i=1;i<=size(Ln);i++) |
---|
5125 | { |
---|
5126 | j=1; |
---|
5127 | clas=0; |
---|
5128 | while((clas==0) and (j<=size(Ln))) |
---|
5129 | { |
---|
5130 | if(j!=i) |
---|
5131 | { |
---|
5132 | if(idcontains(Ln[i][1],Ln[j][1]) ) {B[size(B)+1]=Ln[i]; clas=1;} |
---|
5133 | } |
---|
5134 | j++; |
---|
5135 | } |
---|
5136 | if(clas==0) {A[size(A)+1]=Ln[i];} |
---|
5137 | } |
---|
5138 | L1=A; for(j=1;j<=size(B);j++){L1[size(L1)+1]=B[j];} |
---|
5139 | T[1]=A; T[2]=B; T[3]=L1; |
---|
5140 | return(T); |
---|
5141 | } |
---|
5142 | |
---|
5143 | // Input: |
---|
5144 | // A1: list of high set of P-reps to be completed by the remaining P-reps |
---|
5145 | // L1: the list A1, completed with the list of lower P-reps. |
---|
5146 | // Output: |
---|
5147 | // A: list A1 completed with all possible parts of the remaining parts of L1 with the |
---|
5148 | // condition of building locally closed sets. |
---|
5149 | static proc CompleteA(list A1,list L1) |
---|
5150 | { |
---|
5151 | int modif; int i; int j; int k; int l; |
---|
5152 | ideal pij; ideal pk; ideal pijkl; ideal pkl; |
---|
5153 | int n; list nl; int te; int clas; list vvv; list AAA; |
---|
5154 | list Lp; int m; ideal Pij; |
---|
5155 | list A0; |
---|
5156 | modif=1; |
---|
5157 | list A=A1; |
---|
5158 | while(modif==1) |
---|
5159 | { |
---|
5160 | modif=0; |
---|
5161 | A0=A; |
---|
5162 | for(i=1;i<=size(A);i++) |
---|
5163 | { |
---|
5164 | for(j=1;j<=size(A[i][2]); j++) |
---|
5165 | { |
---|
5166 | pij=A[i][2][j]; |
---|
5167 | for(k=1;k<=size(L1);k++) |
---|
5168 | { |
---|
5169 | if(k!=i) |
---|
5170 | { |
---|
5171 | pk=L1[k][1]; |
---|
5172 | if(idcontains(pij,pk)==1) |
---|
5173 | { |
---|
5174 | te=0; |
---|
5175 | kill nl; |
---|
5176 | list nl; l=1; |
---|
5177 | while((te==0) and (l<=size(L1[k][2]))) |
---|
5178 | { |
---|
5179 | pkl=L1[k][2][l]; |
---|
5180 | if((equalideals(pij,pkl)==1) or (idcontains(pij,pkl)==1)) {te=1;} |
---|
5181 | l++; |
---|
5182 | } // end while ((te=0) and (l>... |
---|
5183 | //"T_te="; te; pause(); |
---|
5184 | if(te==0) |
---|
5185 | { |
---|
5186 | modif=1; |
---|
5187 | kill Pij; ideal Pij=1; |
---|
5188 | for(l=1; l<=size(L1[k][2]);l++) |
---|
5189 | { |
---|
5190 | pkl=L1[k][2][l]; |
---|
5191 | pijkl=std(pij+pkl); |
---|
5192 | Pij=intersect(Pij,pijkl); |
---|
5193 | } |
---|
5194 | kill Lp; list Lp; |
---|
5195 | Lp=minGTZ(Pij); |
---|
5196 | for(m=1;m<=size(Lp);m++) |
---|
5197 | { |
---|
5198 | nl[size(nl)+1]=Lp[m]; |
---|
5199 | } |
---|
5200 | A[i][2]=delete(A[i][2], j); |
---|
5201 | for(n=1;n<=size(nl);n++) |
---|
5202 | { |
---|
5203 | A[i][2][size(A[i][2])+1]=nl[n]; |
---|
5204 | } |
---|
5205 | } // end if(te==0) |
---|
5206 | } // end if(idcontains(pij,pk)==1) |
---|
5207 | } // end if (k!=i) |
---|
5208 | } // end for k |
---|
5209 | } // end for j |
---|
5210 | kill vvv; list vvv; |
---|
5211 | if(modif==1) |
---|
5212 | // Select the maximal ideals of the set A[I][2][j] |
---|
5213 | { |
---|
5214 | kill nl; list nl; |
---|
5215 | nl=selectminideals(A[i][2]); |
---|
5216 | kill AAA; list AAA; |
---|
5217 | for(j=1;j<=size(nl);j++) |
---|
5218 | { |
---|
5219 | AAA[size(AAA)+1]=A[i][2][nl[j]]; |
---|
5220 | } |
---|
5221 | A[i][2]=AAA; |
---|
5222 | } // end if(modif=1) |
---|
5223 | } // end for i |
---|
5224 | modif=1-equallistsAall(A,A0); |
---|
5225 | } // end while(modif==1) |
---|
5226 | return(A); |
---|
5227 | } |
---|
5228 | |
---|
5229 | // Input: |
---|
5230 | // A: list of the top P-reps of one level |
---|
5231 | // B: list of remaining lower P-reps that have not yeen be possible to add |
---|
5232 | // Output: |
---|
5233 | // Bn: list B where the elements that are completely included in A are removed and the parts that are |
---|
5234 | // included in A also. |
---|
5235 | static proc ReduceB(list A,list B) |
---|
5236 | { |
---|
5237 | int i; int j; list nl; list Bn; int te; int k; int elim; |
---|
5238 | ideal pC; ideal pD; ideal pCD; ideal pBC; list nB; int j0; |
---|
5239 | for(i=1;i<=size(B);i++) |
---|
5240 | { |
---|
5241 | j=1; te=0; |
---|
5242 | while((te==0) and (j<=size(A))) |
---|
5243 | { |
---|
5244 | if(idcontains(B[i][1],A[j][1])){te=1; j0=j;} |
---|
5245 | else{j++;} |
---|
5246 | } |
---|
5247 | pD=B[i][2][1]; |
---|
5248 | for(k=2;k<=size(B[i][2]);k++){pD=intersect(pD,B[i][2][k]);} |
---|
5249 | pC=A[j0][2][1]; |
---|
5250 | for(k=2;k<=size(A[j0][2]);k++) {pC=intersect(pC,A[j0][2][k]);} |
---|
5251 | pCD=std(pD+pC); |
---|
5252 | pBC=std(B[i][1]+pC); |
---|
5253 | elim=0; |
---|
5254 | if(idcontains(pBC,pCD)){elim=1;} // B=delfromlist(B,i);i=i-1; |
---|
5255 | else |
---|
5256 | { |
---|
5257 | nB=Prep(pBC,pCD); |
---|
5258 | if(equalideals(nB[1][1],ideal(1))==0) |
---|
5259 | { |
---|
5260 | for(k=1;k<=size(nB);k++){Bn[size(Bn)+1]=nB[k];} |
---|
5261 | } |
---|
5262 | } |
---|
5263 | } // end for i |
---|
5264 | return(Bn); |
---|
5265 | } |
---|
5266 | |
---|
5267 | // AddConsP: given a set of components of locally closed sets in P-representation, it builds the |
---|
5268 | // canonical P-representation of the corresponding constructible set, of its union, |
---|
5269 | // including levels it they are. |
---|
5270 | proc AddConsP(list L) |
---|
5271 | "USAGE: AddConsP(L) |
---|
5272 | Input L: list of components in P-rep to be added |
---|
5273 | [ [[p_1,[p_11,..,p_1,r1]],..[p_k,[p_k1,..,p_kr_k]] ] |
---|
5274 | RETURN: |
---|
5275 | list of lists of levels of the different locally closed sets of |
---|
5276 | the canonical P-rep of the constructible. |
---|
5277 | [ [level_1,[ [Comp_11,..Comp_1r_1] ] ], .. , |
---|
5278 | [level_s,[ [Comp_s1,..Comp_sr_1] ] |
---|
5279 | ] |
---|
5280 | where level_i=i, Comp_ij=[ p_i,[p_i1,..,p_it_i] ] is a prime component. |
---|
5281 | NOTE: Operates in a ring R=Q[u_1,..,u_m] |
---|
5282 | KEYWORDS: Constructible set, Canoncial form |
---|
5283 | EXAMPLE: AddConsP; shows an example" |
---|
5284 | { |
---|
5285 | list LL; list A; list B; list L1; list T; int level=0; list h; |
---|
5286 | LL=L; int i; |
---|
5287 | while(size(LL)!=0) |
---|
5288 | { |
---|
5289 | level++; |
---|
5290 | L1=SepareAB(LL); |
---|
5291 | A=L1[1]; B=L1[2]; LL=L1[3]; |
---|
5292 | A=CompleteA(A,LL); |
---|
5293 | for(i=1;i<=size(A);i++) |
---|
5294 | { |
---|
5295 | LL[i]=A[i]; |
---|
5296 | } |
---|
5297 | h[1]=level; h[2]=A; |
---|
5298 | T[size(T)+1]=h; |
---|
5299 | LL=ReduceB(A,B); |
---|
5300 | } |
---|
5301 | return(T); |
---|
5302 | } |
---|
5303 | example |
---|
5304 | { |
---|
5305 | "EXAMPLE:"; echo = 2; |
---|
5306 | if (defined(Grobcov::@P)){kill Grobcov::@P; kill Grobcov::@R; kill Grobcov::@RP;} |
---|
5307 | ring R=0,(x,y,z),lp; |
---|
5308 | short=0; |
---|
5309 | |
---|
5310 | ideal P1=x; |
---|
5311 | ideal Q1=x,y; |
---|
5312 | ideal P2=y; |
---|
5313 | ideal Q2=y,z; |
---|
5314 | |
---|
5315 | list L=list(Prep(P1,Q1)[1],Prep(P2,Q2)[1]); |
---|
5316 | L; |
---|
5317 | AddConsP(L); |
---|
5318 | } |
---|
5319 | |
---|
5320 | // AddCons: given a set of locally closed sets by pairs of ideal, it builds the |
---|
5321 | // canonical P-representation of the corresponding constructible set, of its union, |
---|
5322 | // including levels it they are. |
---|
5323 | // It makes a call to AddConsP after transforming the input. |
---|
5324 | // Input list of lists of pairs of ideals representing locally colsed sets: |
---|
5325 | // L= [ [E1,N1], .. , [E_s,N_s] ] |
---|
5326 | // Output: The canonical frepresentation of the constructible set union of the V(E_i) \ V(N_i) |
---|
5327 | // T=[ [level_1,[ [p_1,[p_11,..,p_1s_1]],.., [p_k,[p_k1,..,p_ks_k]] ]],, .. , [level_r,[.. ]] ] |
---|
5328 | proc AddCons(list L) |
---|
5329 | "USAGE: AddCons(L) |
---|
5330 | Calls internally AddConsP and allows a different input. |
---|
5331 | Input L: list of pairs of of ideals [ [P_1,Q_1], .., [Pr,Qr] ] |
---|
5332 | representing a set of locally closed setsV(P_i) \ V(Q_i) |
---|
5333 | to be added. |
---|
5334 | RETURN: |
---|
5335 | list of lists of levels of the different locally closed sets of |
---|
5336 | the canonical P-rep of the constructible. |
---|
5337 | [ [level_1,[ [Comp_11,..Comp_1r_1] ] ], .. , |
---|
5338 | [level_s,[ [Comp_s1,..Comp_sr_1] ] |
---|
5339 | ] |
---|
5340 | where level_i=i, Comp_ij=[ p_i,[p_i1,..,p_it_i] ] is a prime component. |
---|
5341 | NOTE: Operates in a ring R=Q[u_1,..,u_m] |
---|
5342 | KEYWORDS: Constructible set, Canoncial form |
---|
5343 | EXAMPLE: AddCons; shows an example" |
---|
5344 | { |
---|
5345 | int i; list H; list P; int j; |
---|
5346 | for(i=1;i<=size(L);i++) |
---|
5347 | { |
---|
5348 | P=Prep(L[i][1],L[i][2]); |
---|
5349 | for(j=1;j<=size(P);j++) |
---|
5350 | { |
---|
5351 | H[size(H)+1]=P[j]; |
---|
5352 | } |
---|
5353 | } |
---|
5354 | return(AddConsP(H)); |
---|
5355 | } |
---|
5356 | example |
---|
5357 | { |
---|
5358 | "EXAMPLE:"; echo = 2; |
---|
5359 | if (defined(Grobcov::@P)){kill Grobcov::@P; kill Grobcov::@R; kill Grobcov::@RP;} |
---|
5360 | ring R=0,(x,y,z),lp; |
---|
5361 | short=0; |
---|
5362 | |
---|
5363 | ideal P1=x; |
---|
5364 | ideal Q1=x,y; |
---|
5365 | ideal P2=y; |
---|
5366 | ideal Q2=y,z; |
---|
5367 | |
---|
5368 | list L=list(list(P1,Q1), list(P2,Q2) ); |
---|
5369 | L; |
---|
5370 | |
---|
5371 | AddCons(L); |
---|
5372 | } |
---|
5373 | |
---|
5374 | // AddLocus: auxilliary routine for locus0 that computes the components of the constructible: |
---|
5375 | // Input: the list of locally closed sets to be added, each with its type as third argument |
---|
5376 | // L=[ [LC[11],,,LC[1k_1], .., [LC[r1],,,LC[rk_r] ] where |
---|
5377 | // LC[1]=[p1,[p11,..,p1k],typ] |
---|
5378 | // Output: the list of components of the constructible union of the L, with the type of the corresponding top |
---|
5379 | // and the level of the constructible |
---|
5380 | // L4= [[v1,p1,[p11,..,p1l],typ_1,level]_1 ,.. [vs,ps,[ps1,..,psl],typ_s,level_s] |
---|
5381 | static proc AddLocus(list L) |
---|
5382 | { |
---|
5383 | // int te0=0; |
---|
5384 | // def RR=basering; |
---|
5385 | // if(defined(@P)){te0=1; def Rx=@R; kill @P; setring RR;} |
---|
5386 | list L1; int i; int j; list L2; list L3; |
---|
5387 | list l1; list l2; |
---|
5388 | intvec v; |
---|
5389 | for(i=1; i<=size(L); i++) |
---|
5390 | { |
---|
5391 | for(j=1;j<=size(L[i]);j++) |
---|
5392 | { |
---|
5393 | l1[1]=L[i][j][1]; |
---|
5394 | l1[2]=L[i][j][2]; |
---|
5395 | l2[1]=l1[1]; |
---|
5396 | if(size(L[i][j])>2){l2[3]=L[i][j][3];} |
---|
5397 | v[1]=i; v[2]=j; |
---|
5398 | l2[2]=v; |
---|
5399 | L1[size(L1)+1]=l1; |
---|
5400 | L2[size(L2)+1]=l2; |
---|
5401 | } |
---|
5402 | } |
---|
5403 | L3=AddConsP(L1); |
---|
5404 | list L4; int level; |
---|
5405 | ideal p1; ideal pp1; int t; int k; int k0; string typ; list l4; |
---|
5406 | for(i=1;i<=size(L3);i++) |
---|
5407 | { |
---|
5408 | level=L3[i][1]; |
---|
5409 | for(j=1;j<=size(L3[i][2]);j++) |
---|
5410 | { |
---|
5411 | p1=L3[i][2][j][1]; |
---|
5412 | t=1; k=1; |
---|
5413 | while((t==1) and (k<=size(L2))) |
---|
5414 | { |
---|
5415 | pp1=L2[k][1]; |
---|
5416 | if(equalideals(p1,pp1)){t=0; k0=k;} |
---|
5417 | k++; |
---|
5418 | } |
---|
5419 | if(t==0) |
---|
5420 | { |
---|
5421 | v=L2[k0][2]; |
---|
5422 | } |
---|
5423 | else{"ERROR p1 NOT FOUND";} |
---|
5424 | l4[1]=v; l4[2]=p1; l4[3]=L3[i][2][j][2]; l4[5]=level; |
---|
5425 | if(size(L2[k0])>2){l4[4]=L2[k0][3];} |
---|
5426 | L4[size(L4)+1]=l4; |
---|
5427 | } |
---|
5428 | } |
---|
5429 | return(L4); |
---|
5430 | } |
---|
5431 | |
---|
5432 | //********************* End AddCons ********************** |
---|
5433 | ; |
---|