1 | /////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id: compregb.lib,v 1.3 2009-01-14 16:07:03 Singular Exp $"; |
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3 | category="General purpose"; |
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4 | info=" |
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5 | LIBRARY: compregb.lib experimental implementation for comprehensive Groebner systems |
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6 | AUTHOR: Akira Suzuki (http://kurt.scitec.kobe-u.ac.jp/~sakira/CGBusingGB/) |
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7 | (<sakira@kobe-u.ac.jp>) |
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8 | OVERVIEW: |
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9 | see \"A Simple Algorithm to compute Comprehensive Groebner Bases using Groebner |
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10 | Bases\" by Akira Suzuki and Yosuke Sato for details. |
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11 | |
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12 | PROCEDURES: |
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13 | cgs(polys,vars,pars,R1,R2); comprehensive Groebner systems |
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14 | base2str(G); pretty print of the result G |
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15 | |
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16 | KEYWORDS: comprehensive Groebner system |
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17 | "; |
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18 | /////////////////////////////////////////////////////////////////////////////// |
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19 | // global variables are the followings: |
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20 | // Parameters, Variables, VMinDPoly, |
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21 | // RingAll, RingVar; |
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22 | |
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23 | /////////////////////////////////////////////////////////////////////////////// |
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24 | static proc setup_special_dpolys() |
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25 | { |
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26 | poly VMinDPoly = Variables[size(Variables)]; |
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27 | export(VMinDPoly); |
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28 | } |
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29 | |
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30 | static proc newcasebasis(poly Case, ideal Basis) |
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31 | { |
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32 | list CB = Case, Basis; |
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33 | return(CB); |
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34 | } |
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35 | |
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36 | static proc contains(poly Vari, list Varis) |
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37 | { |
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38 | int l = size(Varis); |
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39 | for (int i = 1; i <= l; i ++) |
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40 | { |
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41 | if (Vari == Varis[i]) |
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42 | { |
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43 | return(1); |
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44 | } |
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45 | } |
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46 | |
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47 | return(0); |
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48 | } |
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49 | |
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50 | static proc polys_heads(list PolyL) |
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51 | { |
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52 | int i, j; |
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53 | int length = size(PolyL); |
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54 | if (!length) |
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55 | { |
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56 | return(PolyL); |
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57 | } |
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58 | |
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59 | setring(RingVar); |
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60 | list PolyL = imap(RingAll, PolyL); |
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61 | list HCoefs = list(); |
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62 | length = size(PolyL); |
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63 | for (i = 1; i <= length; i ++) |
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64 | { |
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65 | HCoefs = insert(HCoefs, leadcoef(PolyL[i])); |
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66 | } |
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67 | setring(RingAll); |
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68 | list HCoefs = imap(RingVar, HCoefs); |
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69 | |
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70 | list CoefL; |
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71 | ideal CoefLsub; |
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72 | poly Coef; |
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73 | for (i = 1; i <= length; i ++) |
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74 | { |
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75 | Coef = HCoefs[i]; |
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76 | if (typeof(Coef) == "poly") |
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77 | { |
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78 | CoefLsub = factorize(Coef, 1); |
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79 | for (j = 1; j <= size(CoefLsub); j ++) |
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80 | { |
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81 | if (CoefLsub[j] > 1) |
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82 | { |
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83 | CoefL = insert(CoefL, CoefLsub[j]); |
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84 | } |
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85 | } |
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86 | } |
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87 | } |
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88 | |
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89 | for (i = 1; i <= size(CoefL); i ++) |
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90 | { |
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91 | Coef = CoefL[i]; |
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92 | for (j = i + 1; j <= size(CoefL); j ++) |
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93 | { |
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94 | if (Coef == CoefL[j]) |
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95 | { |
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96 | CoefL = delete(CoefL, j); |
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97 | j --; |
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98 | } |
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99 | } |
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100 | } |
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101 | |
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102 | return(CoefL); |
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103 | } |
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104 | |
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105 | static proc polys_restrict_v(ideal Polys) |
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106 | { |
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107 | return(polys_separate_v_p(Polys)[0]); |
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108 | } |
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109 | |
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110 | static proc polys_restrict_p(ideal Polys) |
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111 | { |
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112 | return(polys_separate_v_p(Polys)[1]); |
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113 | } |
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114 | |
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115 | static proc polys_separate_v_p(ideal Polys) |
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116 | { |
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117 | list R_v, R_p; |
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118 | poly P; |
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119 | int length = size(Polys); |
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120 | |
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121 | for (int i = 1; i <= length; i ++) |
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122 | { |
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123 | P = Polys[i]; |
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124 | if (P < VMinDPoly) |
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125 | { |
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126 | R_p = insert(R_p, P); |
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127 | } |
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128 | else |
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129 | { |
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130 | R_v = insert(R_v, P); |
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131 | } |
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132 | } |
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133 | |
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134 | return(R_v, R_p); |
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135 | } |
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136 | |
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137 | static proc cgs_main(ideal Polys) |
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138 | { |
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139 | ideal F; |
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140 | list FP, FV, HFact, Bases; |
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141 | poly H; |
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142 | int i; |
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143 | |
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144 | // F = groebner(Polys); |
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145 | F = slimgb(Polys); |
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146 | |
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147 | if (F[1] == 1) |
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148 | { |
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149 | return(list()); |
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150 | } |
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151 | |
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152 | FV, FP = polys_separate_v_p(F); |
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153 | |
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154 | HFact = polys_heads(FV); |
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155 | int HFL = size(HFact); |
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156 | |
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157 | H = 1; |
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158 | for (i = 1; i <= HFL; i ++) |
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159 | { |
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160 | H = H * HFact[i]; |
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161 | } |
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162 | |
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163 | Bases = insert(Bases, list(H, F)); |
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164 | |
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165 | for (i = 1; i <= HFL; i ++) |
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166 | { |
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167 | // print("paras:" + string(FP)); |
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168 | // print("ideal:" + string(HFact[i])); |
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169 | Bases = cgs_main(F + ideal(HFact[i])) + Bases; |
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170 | } |
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171 | |
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172 | return(Bases); |
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173 | } |
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174 | |
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175 | proc cgs(ideal Polys, list Vars, list Paras, RingVar, RingAll) |
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176 | "USAGE: cgs(Polys,Vars,Paras,RingVar,RingAll); Polys an ideal, Vars, the list |
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177 | of variables, Paras the list of parameters, RingVar the ring with |
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178 | Paras as parameters, RingAll the ring with Paras as variables |
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179 | (RingAll should be the current ring) |
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180 | RETURN: a list L of lists L[i] of a polynomial and an ideal: |
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181 | L[i][1] the polynomial giving the condition on the parameters |
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182 | L[i][2] the Groebner basis for this case |
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183 | EXAMPLE: example cgs; shows an example |
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184 | " |
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185 | { |
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186 | option(redSB); |
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187 | list Parameters = Paras; |
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188 | list Variables = Vars; |
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189 | poly VMinDPoly = Vars[size(Vars)]; |
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190 | export(Parameters, Variables, VMinDPoly); |
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191 | |
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192 | export(RingVar, RingAll); |
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193 | setring(RingAll); |
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194 | |
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195 | list G = cgs_main(Polys); |
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196 | |
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197 | keepring(RingAll); |
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198 | return(G); |
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199 | } |
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200 | example |
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201 | { "EXAMPLE:";echo=2; |
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202 | ring RingVar=(0,a,b),(x,y,t),lp; |
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203 | ring RingAll=0,(x,y,t,a,b),(lp(3),dp); |
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204 | ideal polys=x^3-a,y^4-b,x+y-t; |
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205 | list vars=x,y,t; |
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206 | list paras=a,b; |
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207 | list G = cgs(polys,vars,paras,RingVar,RingAll); |
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208 | G; |
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209 | } |
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210 | proc basis2str(list B) |
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211 | { |
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212 | string Str; |
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213 | ideal Factors; |
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214 | int i; |
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215 | |
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216 | Str = "("; |
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217 | Factors = factorize(B[1], 1); |
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218 | for (i = 1; i <= size(Factors); i ++) |
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219 | { |
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220 | Str = Str + "(" + string(Factors[i]) + ")"; |
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221 | } |
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222 | Str = Str + "!=0,"; |
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223 | |
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224 | list FV, FP; |
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225 | FV, FP = polys_separate_v_p(B[2]); |
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226 | for (i = 1; i <= size(FP); i ++) |
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227 | { |
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228 | Str = Str + string(FP[i]) + "=0,"; |
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229 | } |
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230 | |
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231 | if (size(Str) > 1) |
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232 | { |
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233 | Str = Str[1, size(Str) - 1] + ")["; |
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234 | } |
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235 | else |
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236 | { |
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237 | Str = "()["; |
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238 | } |
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239 | |
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240 | if (size(FV)) |
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241 | { |
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242 | for (i = 1; i <= size(FV); i ++) |
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243 | { |
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244 | Str = Str + string(FV[i]) + ","; |
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245 | } |
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246 | |
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247 | Str = Str[1, size(Str) - 1] + "]"; |
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248 | } |
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249 | else |
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250 | { |
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251 | Str += "]"; |
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252 | } |
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253 | |
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254 | return(Str); |
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255 | } |
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256 | |
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257 | proc bases2str(list Bases) |
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258 | { |
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259 | string Str; |
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260 | int i; |
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261 | |
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262 | Str = ""; |
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263 | for (i = 1; i <= size(Bases); i ++) |
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264 | { |
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265 | Str = Str + basis2str(Bases[i]) + newline; |
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266 | } |
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267 | |
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268 | return(Str); |
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269 | } |
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270 | |
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271 | /* |
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272 | ring RingVar=(0,a,b),(x,y,t),lp; ring RingAll=0,(x,y,t,a,b),(lp(3),dp); |
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273 | ideal polys=x^3-a,y^4-b,x+y-t; list vars=x,y,t; list paras=a,b; |
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274 | list G = cgs(polys,vars,paras,RingVar,RingAll); |
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275 | bases2str(G); |
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276 | */ |
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