1 | version="$Id: involut.lib,v 1.2 2005-02-23 18:10:45 levandov Exp $"; |
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2 | category="Noncommutative"; |
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3 | info=" |
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4 | LIBRARY: involution.lib Procedures for Computations and Operations with Involutions |
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5 | AUTHORS: Oleksandr Iena, yena@mathematik.uni-kl.de, |
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6 | @* Markus Becker, mbecker@mathematik.uni-kl.de, |
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7 | @* Viktor Levandovskyy, levandov@mathematik.uni-kl.de |
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8 | |
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9 | SUPPORT: Forschungsschwerpunkt 'Mathematik und Praxis' (Project of Dr. E. Zerz |
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10 | and V. Levandovskyy), Uni Kaiserslautern |
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11 | |
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12 | NOTE: This library provides algebraic tools for computations and operations |
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13 | with algebraic involutions and linear automorphisms of noncommutative algebras |
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14 | |
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15 | PROCEDURES: |
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16 | find_invo(); describes a variety of linear involutions on a basering; |
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17 | find_invo_diag(); describes a variety of homothetic (diagonal) involutions on a basering; |
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18 | find_auto(); describes a variety of linear automorphisms of a basering; |
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19 | ncdetection(ring r); computes an ideal, presenting an involution map on some classical noncommutative algebras; |
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20 | involution(m, map theta); applies the involution, presented by theta, to the object m = |
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21 | "; |
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22 | |
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23 | LIB "ncalg.lib"; |
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24 | LIB "poly.lib"; |
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25 | LIB "primdec.lib"; |
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26 | /////////////////////////////////////////////////////////////////////////////// |
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27 | proc ncdetection(def r) |
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28 | "USAGE: ncdetection(r), r a ring |
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29 | RETURN: ideal, presenting an involution map on a noncommutative algebra r |
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30 | NOTE: returns optimized involutions for some classical noncomm algebras, |
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31 | arising in the Control Theory, namely algebras with |
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32 | differential, shift or advance operators |
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33 | EXAMPLE: example ncdetection; shows an example |
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34 | " |
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35 | { |
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36 | // in this procedure an involution map is generated from the NCRelations |
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37 | // that will be used in the function involution |
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38 | // in dieser proc. wird eine matrix erzeugt, die in der i-ten zeile die indices |
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39 | // der differential-, shift- oder advance-operatoren enthaelt mit denen die i-te |
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40 | // variable nicht kommutiert. |
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41 | int i,j,k,LExp; |
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42 | int NVars = nvars(r); |
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43 | matrix rel = NCRelations(r)[2]; |
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44 | intmat M[NVars][3]; |
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45 | int NRows = nrows(rel); |
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46 | intvec v,w; |
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47 | poly d,d_lead; |
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48 | ideal I; |
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49 | map theta; |
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50 | for( j=NRows; j>=2; j-- ) |
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51 | { |
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52 | if( rel[j] == w ) //the whole column is zero |
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53 | { |
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54 | j--; |
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55 | continue; |
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56 | } |
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57 | for( i=1; i<j; i++ ) |
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58 | { |
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59 | if( rel[i,j]==1 ) //relation of type var(j)*var(i) = var(i)*var(j) +1 |
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60 | { |
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61 | M[i,1]=j; |
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62 | } |
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63 | if( rel[i,j] == -1 ) //relation of type var(i)*var(j) = var(j)*var(i) -1 |
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64 | { |
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65 | M[j,1]=i; |
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66 | } |
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67 | d = rel[i,j]; |
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68 | d_lead = lead(d); |
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69 | v = leadexp(d_lead); //in the next lines we check wether we have a relation of differential or shift type |
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70 | LExp=0; |
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71 | for(k=1; k<=NVars; k++) |
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72 | { |
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73 | LExp = LExp + v[k]; |
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74 | } |
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75 | // if( (d-d_lead != 0) || (LExp > 1) ) |
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76 | if ( ( (d-d_lead) != 0) || (LExp > 1) || ( (LExp==0) && ((d_lead>1) || (d_lead<-1)) ) ) |
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77 | { |
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78 | return(theta); |
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79 | } |
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80 | |
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81 | if( v[j] == 1) //relation of type var(j)*var(i) = var(i)*var(j) -lambda*var(j) |
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82 | { |
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83 | if (leadcoef(d) < 0) |
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84 | { |
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85 | M[i,2] = j; |
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86 | } |
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87 | else |
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88 | { |
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89 | M[i,3] = j; |
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90 | } |
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91 | } |
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92 | if( v[i]==1 ) //relation of type var(j)*var(i) = var(i)*var(j) -lambda*var(i) |
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93 | { |
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94 | if (leadcoef(d) > 0) |
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95 | { |
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96 | M[j,2] = i; |
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97 | } |
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98 | else |
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99 | { |
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100 | M[j,3] = i; |
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101 | } |
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102 | } |
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103 | } |
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104 | } |
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105 | // from here on, the map is computed |
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106 | for(i=1;i<=NVars;i++) |
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107 | { |
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108 | I=I+var(i); |
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109 | } |
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110 | |
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111 | for(i=1;i<=NVars;i++) |
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112 | { |
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113 | if( M[i,1..3]==(0,0,0) ) |
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114 | { |
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115 | i++; |
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116 | continue; |
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117 | } |
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118 | if( M[i,1]!=0 ) |
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119 | { |
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120 | if( (M[i,2]!=0) && (M[i,3]!=0) ) |
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121 | { |
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122 | I[M[i,1]] = -var(M[i,1]); |
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123 | I[M[i,2]] = var(M[i,3]); |
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124 | I[M[i,3]] = var(M[i,2]); |
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125 | } |
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126 | if( (M[i,2]==0) && (M[i,3]==0) ) |
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127 | { |
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128 | I[M[i,1]] = -var(M[i,1]); |
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129 | } |
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130 | if( ( (M[i,2]!=0) && (M[i,3]==0) )|| ( (M[i,2]!=0) && (M[i,3]==0) ) |
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131 | ) |
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132 | { |
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133 | I[i] = -var(i); |
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134 | } |
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135 | } |
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136 | else |
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137 | { |
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138 | if( (M[i,2]!=0) && (M[i,3]!=0) ) |
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139 | { |
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140 | I[i] = -var(i); |
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141 | I[M[i,2]] = var(M[i,3]); |
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142 | I[M[i,3]] = var(M[i,2]); |
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143 | } |
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144 | else |
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145 | { |
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146 | I[i] = -var(i); |
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147 | } |
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148 | } |
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149 | } |
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150 | return(I); |
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151 | } |
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152 | example |
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153 | { |
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154 | "EXAMPLE:"; echo = 2; |
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155 | ring r=0,(x,y,z,D(1..3)),dp; |
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156 | matrix D[6][6]; |
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157 | D[1,4]=1; D[2,5]=1; D[3,6]=1; |
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158 | ncalgebra(1,D); |
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159 | ncdetection(r); |
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160 | kill r; |
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161 | //---------------------------------------- |
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162 | ring r=0,(x,S),dp; |
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163 | ncalgebra(1,-S); |
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164 | ncdetection(r); |
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165 | kill r; |
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166 | //---------------------------------------- |
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167 | ring r=0,(x,D(1),S),dp; |
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168 | matrix D[3][3]; |
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169 | D[1,2]=1; D[1,3]=-S; |
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170 | ncalgebra(1,D); |
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171 | ncdetection(r); |
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172 | } |
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173 | |
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174 | static proc In_Poly(poly mm, list l, int NVars) |
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175 | // applies the involution to the poly mm |
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176 | // entries of a list l are images of variables under invo |
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177 | // more general than invo_poly; used in many rings setting |
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178 | { |
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179 | int i,j; |
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180 | intvec v; |
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181 | poly pp, zz; |
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182 | poly nn = 0; |
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183 | i = 1; |
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184 | while(mm[i]!=0) |
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185 | { |
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186 | v = leadexp(mm[i]); |
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187 | zz = 1; |
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188 | for( j=NVars; j>=1; j--) |
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189 | { |
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190 | if (v[j]!=0) |
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191 | { |
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192 | pp = l[j]; |
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193 | zz = zz*(pp^v[j]); |
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194 | } |
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195 | } |
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196 | nn = nn + (leadcoef(mm[i])*zz); |
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197 | i++; |
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198 | } |
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199 | return(nn); |
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200 | } |
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201 | |
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202 | static proc Hom_Poly(poly mm, list l, int NVars) |
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203 | // applies the endomorphism to the poly mm |
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204 | // entries of a list l are images of variables under endo |
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205 | // should not be replaced by map-based stuff! used in |
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206 | // many rings setting |
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207 | { |
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208 | int i,j; |
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209 | intvec v; |
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210 | poly pp, zz; |
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211 | poly nn = 0; |
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212 | i = 1; |
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213 | while(mm[i]!=0) |
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214 | { |
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215 | v = leadexp(mm[i]); |
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216 | zz = 1; |
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217 | for( j=NVars; j>=1; j--) |
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218 | { |
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219 | if (v[j]!=0) |
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220 | { |
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221 | pp = l[j]; |
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222 | zz = (pp^v[j])*zz; |
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223 | } |
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224 | } |
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225 | nn = nn + (leadcoef(mm[i])*zz); |
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226 | i++; |
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227 | } |
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228 | return(nn); |
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229 | } |
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230 | |
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231 | static proc invo_poly(poly m, map theta) |
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232 | // applies the involution map theta to m, where m=polynomial |
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233 | { |
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234 | // compatibility: |
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235 | ideal l = ideal(theta); |
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236 | int i; |
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237 | list L; |
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238 | for (i=1; i<=size(l); i++) |
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239 | { |
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240 | L[i] = l[i]; |
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241 | } |
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242 | int nv = nvars(basering); |
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243 | return (In_Poly(m,L,nv)); |
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244 | // if (m==0) { return(m); } |
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245 | // int i,j; |
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246 | // intvec v; |
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247 | // poly p,z; |
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248 | // poly n = 0; |
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249 | // i = 1; |
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250 | // while(m[i]!=0) |
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251 | // { |
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252 | // v = leadexp(m[i]); |
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253 | // z =1; |
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254 | // for(j=nvars(basering); j>=1; j--) |
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255 | // { |
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256 | // if (v[j]!=0) |
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257 | // { |
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258 | // p = var(j); |
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259 | // p = theta(p); |
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260 | // z = z*(p^v[j]); |
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261 | // } |
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262 | // } |
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263 | // n = n + (leadcoef(m[i])*z); |
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264 | // i++; |
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265 | // } |
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266 | // return(n); |
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267 | } |
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268 | /////////////////////////////////////////////////////////////////////////////////// |
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269 | proc involution(m, map theta) |
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270 | "USAGE: involution(m, theta); m is a poly/vector/ideal/matrix/module, theta is a map |
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271 | PURPOSE: applies the involution, presented by theta to the input m |
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272 | RETURN: object of the same type as m |
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273 | EXAMPLE: example involution; shows an example |
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274 | " |
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275 | { |
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276 | // applies the involution map theta to m, |
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277 | // where m= vector, polynomial, module, matrix, ideal |
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278 | int i,j; |
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279 | intvec v; |
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280 | poly p,z; |
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281 | if (typeof(m)=="poly") |
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282 | { |
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283 | return (invo_poly(m,theta)); |
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284 | } |
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285 | if ( typeof(m)=="ideal" ) |
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286 | { |
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287 | ideal n; |
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288 | for (i=1; i<=size(m); i++) |
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289 | { |
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290 | n[i] = invo_poly(m[i], theta); |
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291 | } |
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292 | return(n); |
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293 | } |
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294 | if (typeof(m)=="vector") |
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295 | { |
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296 | for(i=1; i<=size(m); i++) |
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297 | { |
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298 | m[i] = invo_poly(m[i], theta); |
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299 | } |
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300 | return (m); |
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301 | } |
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302 | if ( (typeof(m)=="matrix") || (typeof(m)=="module")) |
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303 | { |
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304 | matrix n = matrix(m); |
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305 | int @R=nrows(n); |
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306 | int @C=ncols(n); |
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307 | for(i=1; i<=@R; i++) |
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308 | { |
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309 | for(j=1; j<=@C; j++) |
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310 | { |
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311 | if (m[i,j]!=0) |
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312 | { |
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313 | n[i,j] = invo_poly( m[i,j], theta); |
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314 | } |
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315 | } |
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316 | } |
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317 | if (typeof(m)=="module") |
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318 | { |
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319 | return (module(n)); |
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320 | } |
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321 | else // matrix |
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322 | { |
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323 | return(n); |
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324 | } |
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325 | } |
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326 | // if m is not of the supported type: |
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327 | "Error: unsupported argument type!"; |
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328 | return(); |
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329 | } |
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330 | example |
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331 | { |
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332 | "EXAMPLE:";echo = 2; |
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333 | ring r = 0,(x,d),dp; |
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334 | ncalgebra(1,1); // Weyl-Algebra |
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335 | map F = r,x,-d; |
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336 | poly f = x*d^2+d; |
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337 | poly If = involution(f,F); |
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338 | f-If; |
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339 | poly g = x^2*d+2*x*d+3*x+7*d; |
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340 | poly tg = -d*x^2-2*d*x+3*x-7*d; |
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341 | poly Ig = involution(g,F); |
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342 | tg-Ig; |
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343 | ideal I = f,g; |
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344 | ideal II = involution(I,F); |
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345 | II; |
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346 | I - involution(II,F); |
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347 | module M = [f,g,0],[g,0,x^2*d]; |
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348 | module IM = involution(M,F); |
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349 | print(IM); |
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350 | print(M - involution(IM,F)); |
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351 | } |
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352 | /////////////////////////////////////////////////////////////////////////////////// |
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353 | static proc new_var() |
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354 | //generates a string of new variables |
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355 | { |
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356 | |
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357 | int NVars=nvars(basering); |
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358 | int i,j; |
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359 | string s="@_1_1"; |
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360 | for(i=1; i<=NVars; i++) |
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361 | { |
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362 | for(j=1; j<=NVars; j++) |
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363 | { |
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364 | if(i*j!=1) |
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365 | { |
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366 | s = s+ ","+NVAR(i,j); |
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367 | }; |
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368 | }; |
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369 | }; |
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370 | return(s); |
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371 | }; |
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372 | |
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373 | static proc NVAR(int i, int j) |
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374 | { |
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375 | return("@_"+string(i)+"_"+string(j)); |
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376 | }; |
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377 | /////////////////////////////////////////////////////////////////////////////////// |
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378 | static proc new_var_special() |
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379 | //generates a string of new variables |
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380 | { |
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381 | int NVars=nvars(basering); |
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382 | int i; |
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383 | string s="@_1_1"; |
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384 | for(i=2; i<=NVars; i++) |
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385 | { |
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386 | s = s+ ","+NVAR(i,i); |
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387 | }; |
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388 | return(s); |
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389 | }; |
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390 | /////////////////////////////////////////////////////////////////////////////////// |
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391 | static proc RelMatr() |
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392 | // returns the matrix of relations |
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393 | // only Lie-type relations x_j x_i= x_i x_j + .. are taken into account |
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394 | { |
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395 | int i,j; |
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396 | int NVars = nvars(basering); |
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397 | matrix Rel[NVars][NVars]; |
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398 | for(i=1; i<NVars; i++) |
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399 | { |
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400 | for(j=i+1; j<=NVars; j++) |
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401 | { |
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402 | Rel[i,j]=var(j)*var(i)-var(i)*var(j); |
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403 | }; |
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404 | }; |
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405 | return(Rel); |
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406 | }; |
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407 | ///////////////////////////////////////////////////////////////// |
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408 | proc find_invo() |
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409 | "USAGE: find_invo(); |
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410 | PURPOSE: describes a variety of linear involutions on a basering |
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411 | RETURN: a ring together with a list of pairs L, where |
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412 | @* L[i][1] = Groebner Basis of an i-th associated prime, |
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413 | @* L[i][2] = multiplication matrix, reduced wrt L[i][1] |
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414 | NOTE: for convenience, the full ideal of relations 'idJ' |
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415 | and the matrix with indeterminates 'matD' are exported in the output ring. |
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416 | " |
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417 | { |
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418 | def @B = basering; //save the name of basering |
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419 | int NVars = nvars(@B); //number of variables in basering |
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420 | int i, j; |
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421 | |
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422 | matrix Rel = RelMatr(); //the matrix of relations |
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423 | |
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424 | string s = new_var(); //string of new variables |
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425 | string Par = parstr(@B); //string of parameters in old ring |
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426 | |
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427 | if (Par=="") // if there are no parameters |
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428 | { |
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429 | execute("ring @@K=0,("+varstr(@B)+","+s+"), dp;"); //new ring with new variables |
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430 | } |
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431 | else //if there exist parameters |
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432 | { |
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433 | execute("ring @@K=(0,"+Par+") ,("+varstr(@B)+","+s+"), dp;");//new ring with new variables |
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434 | }; |
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435 | |
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436 | matrix Rel = imap(@B, Rel); //consider the matrix of relations in new ring |
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437 | |
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438 | int Sz = NVars*NVars+NVars; // number of variables in new ring |
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439 | |
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440 | matrix M[Sz][Sz]; //to be the matrix of relations in new ring |
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441 | |
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442 | for(i=1; i<NVars; i++) //initialize that matrix of relations |
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443 | { |
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444 | for(j=i+1; j<=NVars; j++) |
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445 | { |
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446 | M[i,j] = Rel[i,j]; |
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447 | }; |
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448 | }; |
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449 | |
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450 | ncalgebra(1, M); //now new ring @@K become a noncommutative ring |
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451 | |
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452 | list l; //list to define an involution |
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453 | poly @@F; |
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454 | for(i=1; i<=NVars; i++) //initializing list for involution |
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455 | { |
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456 | @@F=0; |
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457 | for(j=1; j<=NVars; j++) |
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458 | { |
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459 | execute( "@@F = @@F+"+NVAR(i,j)+"*"+string( var(j) )+";" ); |
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460 | }; |
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461 | l=l+list(@@F); |
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462 | }; |
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463 | |
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464 | matrix N = Rel; //imap(@B,Rel); |
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465 | |
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466 | for(i=1; i<NVars; i++)//get matrix by applying the involution to relations |
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467 | { |
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468 | for(j=i+1; j<=NVars; j++) |
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469 | { |
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470 | N[i,j]= l[j]*l[i] - l[i]*l[j] + In_Poly( N[i,j], l, NVars); |
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471 | }; |
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472 | }; |
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473 | kill l; |
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474 | //--------------------------------------------- |
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475 | //get the ideal of coefficients of N |
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476 | ideal J; |
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477 | ideal idN = simplify(ideal(N),2); |
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478 | J = ideal(coeffs( idN, var(1) ) ); |
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479 | for(i=2; i<=NVars; i++) |
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480 | { |
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481 | J = ideal( coeffs( J, var(i) ) ); |
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482 | }; |
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483 | J = simplify(J,2); |
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484 | //------------------------------------------------- |
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485 | if ( Par=="" ) //initializes the ring of relations |
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486 | { |
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487 | execute("ring @@KK=0,("+s+"), dp;"); |
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488 | } |
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489 | else |
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490 | { |
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491 | execute("ring @@KK=(0,"+Par+"),("+s+"), dp;"); |
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492 | }; |
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493 | ideal J = imap(@@K,J); // ideal, considered in @@KK now |
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494 | string snv = "["+string(NVars)+"]"; |
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495 | execute("matrix @@D"+snv+snv+"="+s+";"); // matrix with entries=new variables |
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496 | |
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497 | J = J, ideal( @@D*@@D-matrix( freemodule(NVars) ) ); // add the condition that involution to square is just identity |
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498 | J = simplify(J,2); // without extra zeros |
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499 | list mL = minAssGTZ(J); // components not in GB |
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500 | int sL = size(mL); |
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501 | option(redSB); // important for reduced GBs |
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502 | option(redTail); |
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503 | matrix IM = @@D; // involution map |
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504 | list L = list(); // the answer |
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505 | list TL; |
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506 | for (i=1; i<=sL; i++) // compute GBs of components |
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507 | { |
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508 | TL = list(); |
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509 | TL[1] = std(mL[i]); |
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510 | TL[2] = NF( ideal(IM), TL[1] ); |
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511 | L[i] = TL; |
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512 | } |
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513 | export(L); // main export |
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514 | ideal idJ = J; // debug-comfortable exports |
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515 | matrix matD = @@D; |
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516 | export(idJ); |
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517 | export(matD); |
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518 | return(@@KK); |
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519 | } |
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520 | example |
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521 | { "EXAMPLE:"; echo = 2; |
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522 | def a = CreateWeyl(1); |
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523 | setring a; |
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524 | def X = find_invo(); |
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525 | setring X; |
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526 | L; |
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527 | } |
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528 | /////////////////////////////////////////////////////////////////////////// |
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529 | proc find_invo_diag() |
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530 | "USAGE: find_invo_diag(); |
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531 | PURPOSE: describes a variety of homothetic (diagonal) involutions on a basering |
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532 | RETURN: a ring together with a list of pairs L, where |
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533 | @* L[i][1] = Groebner Basis of an i-th associated prime, |
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534 | @* L[i][2] = multiplication matrix, reduced wrt L[i][1] |
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535 | NOTE: for convenience, the full ideal of relations 'idJ' |
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536 | and the matrix with indeterminates 'matD' are exported in the output ring. |
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537 | " |
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538 | { |
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539 | def @B = basering; //save the name of basering |
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540 | int NVars = nvars(@B); //number of variables in basering |
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541 | int i, j; |
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542 | |
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543 | matrix Rel = RelMatr(); //the matrix of relations |
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544 | |
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545 | string s = new_var_special(); //string of new variables |
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546 | string Par = parstr(@B); //string of parameters in old ring |
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547 | |
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548 | if (Par=="") // if there are no parameters |
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549 | { |
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550 | execute("ring @@K=0,("+varstr(@B)+","+s+"), dp;"); //new ring with new variables |
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551 | } |
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552 | else //if there exist parameters |
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553 | { |
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554 | execute("ring @@K=(0,"+Par+") ,("+varstr(@B)+","+s+"), dp;");//new ring with new variables |
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555 | }; |
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556 | |
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557 | matrix Rel = imap(@B, Rel); //consider the matrix of relations in new ring |
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558 | |
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559 | int Sz = 2*NVars; // number of variables in new ring |
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560 | |
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561 | matrix M[Sz][Sz]; //to be the matrix of relations in new ring |
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562 | for(i=1; i<NVars; i++) //initialize that matrix of relations |
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563 | { |
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564 | for(j=i+1; j<=NVars; j++) |
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565 | { |
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566 | M[i,j] = Rel[i,j]; |
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567 | }; |
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568 | }; |
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569 | |
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570 | ncalgebra(1, M); //now new ring @@K become a noncommutative ring |
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571 | |
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572 | list l; //list to define an involution |
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573 | |
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574 | for(i=1; i<=NVars; i++) //initializing list for involution |
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575 | { |
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576 | execute( "l["+string(i)+"]="+NVAR(i,i)+"*"+string( var(i) )+";" ); |
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577 | |
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578 | }; |
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579 | matrix N = Rel; //imap(@B,Rel); |
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580 | |
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581 | for(i=1; i<NVars; i++)//get matrix by applying the involution to relations |
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582 | { |
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583 | for(j=i+1; j<=NVars; j++) |
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584 | { |
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585 | N[i,j]= l[j]*l[i] - l[i]*l[j] + In_Poly( N[i,j], l, NVars); |
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586 | }; |
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587 | }; |
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588 | kill l; |
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589 | //--------------------------------------------- |
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590 | //get the ideal of coefficients of N |
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591 | |
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592 | ideal J; |
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593 | ideal idN = simplify(ideal(N),2); |
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594 | J = ideal(coeffs( idN, var(1) ) ); |
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595 | for(i=2; i<=NVars; i++) |
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596 | { |
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597 | J = ideal( coeffs( J, var(i) ) ); |
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598 | }; |
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599 | J = simplify(J,2); |
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600 | //------------------------------------------------- |
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601 | |
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602 | if ( Par=="" ) //initializes the ring of relations |
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603 | { |
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604 | execute("ring @@KK=0,("+s+"), dp;"); |
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605 | } |
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606 | else |
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607 | { |
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608 | execute("ring @@KK=(0,"+Par+"),("+s+"), dp;"); |
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609 | }; |
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610 | |
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611 | ideal J = imap(@@K,J); // ideal, considered in @@KK now |
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612 | |
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613 | matrix @@D[NVars][NVars]; // matrix with entries=new variables to square i.e. @@D=@@D^2 |
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614 | for(i=1;i<=NVars;i++) |
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615 | { |
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616 | execute("@@D["+string(i)+","+string(i)+"]="+NVAR(i,i)+";"); |
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617 | }; |
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618 | J = J, ideal( @@D*@@D - matrix( freemodule(NVars) ) ); // add the condition that involution to square is just identity |
---|
619 | J = simplify(J,2); // without extra zeros |
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620 | |
---|
621 | list mL = minAssGTZ(J); // components not in GB |
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622 | int sL = size(mL); |
---|
623 | option(redSB); // important for reduced GBs |
---|
624 | option(redTail); |
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625 | matrix IM = @@D; // involution map |
---|
626 | list L = list(); // the answer |
---|
627 | list TL; |
---|
628 | for (i=1; i<=sL; i++)// compute GBs of components |
---|
629 | { |
---|
630 | TL = list(); |
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631 | TL[1] = std(mL[i]); |
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632 | TL[2] = NF( ideal(IM), TL[1] ); |
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633 | L[i] = TL; |
---|
634 | } |
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635 | export(L); |
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636 | ideal idJ = J; // debug-comfortable exports |
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637 | matrix matD = @@D; |
---|
638 | export(idJ); |
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639 | export(matD); |
---|
640 | return(@@KK); |
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641 | } |
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642 | example |
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643 | { "EXAMPLE:"; echo = 2; |
---|
644 | def a = CreateWeyl(1); |
---|
645 | setring a; |
---|
646 | def X = find_invo_diag(); |
---|
647 | setring X; |
---|
648 | L; |
---|
649 | } |
---|
650 | ///////////////////////////////////////////////////////////////////// |
---|
651 | proc find_auto() |
---|
652 | "USAGE: find_auto(); |
---|
653 | PURPOSE: describes a variety of linear automorphisms of a basering |
---|
654 | RETURN: a ring together with a list of pairs L, where |
---|
655 | @* L[i][1] = Groebner Basis of an i-th associated prime, |
---|
656 | @* L[i][2] = multiplication matrix, reduced wrt L[i][1] |
---|
657 | NOTE: for convenience, the full ideal of relations 'idJ' |
---|
658 | and the matrix with indeterminates 'matD' are exported in the output ring. |
---|
659 | " |
---|
660 | { |
---|
661 | def @B = basering; //save the name of basering |
---|
662 | int NVars = nvars(@B); //number of variables in basering |
---|
663 | int i, j; |
---|
664 | |
---|
665 | matrix Rel = RelMatr(); //the matrix of relations |
---|
666 | |
---|
667 | string s = new_var(); //string of new variables |
---|
668 | string Par = parstr(@B); //string of parameters in old ring |
---|
669 | |
---|
670 | if (Par=="") // if there are no parameters |
---|
671 | { |
---|
672 | execute("ring @@K=0,("+varstr(@B)+","+s+"), dp;"); //new ring with new variables |
---|
673 | } |
---|
674 | else //if there exist parameters |
---|
675 | { |
---|
676 | execute("ring @@K=(0,"+Par+") ,("+varstr(@B)+","+s+"), dp;");//new ring with new variables |
---|
677 | }; |
---|
678 | |
---|
679 | matrix Rel = imap(@B, Rel); //consider the matrix of relations in new ring |
---|
680 | |
---|
681 | int Sz = NVars*NVars+NVars; // number of variables in new ring |
---|
682 | |
---|
683 | matrix M[Sz][Sz]; //to be the matrix of relations in new ring |
---|
684 | |
---|
685 | for(i=1; i<NVars; i++) //initialize that matrix of relations |
---|
686 | { |
---|
687 | for(j=i+1; j<=NVars; j++) |
---|
688 | { |
---|
689 | M[i,j] = Rel[i,j]; |
---|
690 | }; |
---|
691 | }; |
---|
692 | |
---|
693 | ncalgebra(1, M); //now new ring @@K become a noncommutative ring |
---|
694 | |
---|
695 | list l; //list to define a homomorphism(isomorphism) |
---|
696 | poly @@F; |
---|
697 | for(i=1; i<=NVars; i++) //initializing list for involution |
---|
698 | { |
---|
699 | @@F=0; |
---|
700 | for(j=1; j<=NVars; j++) |
---|
701 | { |
---|
702 | execute( "@@F = @@F+"+NVAR(i,j)+"*"+string( var(j) )+";" ); |
---|
703 | }; |
---|
704 | l=l+list(@@F); |
---|
705 | }; |
---|
706 | |
---|
707 | matrix N = Rel; //imap(@B,Rel); |
---|
708 | |
---|
709 | for(i=1; i<NVars; i++)//get matrix by applying the homomorphism to relations |
---|
710 | { |
---|
711 | for(j=i+1; j<=NVars; j++) |
---|
712 | { |
---|
713 | N[i,j]= l[j]*l[i] - l[i]*l[j] - Hom_Poly( N[i,j], l, NVars); |
---|
714 | }; |
---|
715 | }; |
---|
716 | kill l; |
---|
717 | //--------------------------------------------- |
---|
718 | //get the ideal of coefficients of N |
---|
719 | ideal J; |
---|
720 | ideal idN = simplify(ideal(N),2); |
---|
721 | J = ideal(coeffs( idN, var(1) ) ); |
---|
722 | for(i=2; i<=NVars; i++) |
---|
723 | { |
---|
724 | J = ideal( coeffs( J, var(i) ) ); |
---|
725 | }; |
---|
726 | J = simplify(J,2); |
---|
727 | //------------------------------------------------- |
---|
728 | if ( Par=="" ) //initializes the ring of relations |
---|
729 | { |
---|
730 | execute("ring @@KK=0,("+s+"), dp;"); |
---|
731 | } |
---|
732 | else |
---|
733 | { |
---|
734 | execute("ring @@KK=(0,"+Par+"),("+s+"), dp;"); |
---|
735 | }; |
---|
736 | ideal J = imap(@@K,J); // ideal, considered in @@KK now |
---|
737 | string snv = "["+string(NVars)+"]"; |
---|
738 | execute("matrix @@D"+snv+snv+"="+s+";"); // matrix with entries=new variables |
---|
739 | |
---|
740 | J = J, ideal( @@D*@@D-matrix( freemodule(NVars) ) ); // add the condition that homomorphism to square is just identity |
---|
741 | J = simplify(J,2); // without extra zeros |
---|
742 | list mL = minAssGTZ(J); // components not in GB |
---|
743 | int sL = size(mL); |
---|
744 | option(redSB); // important for reduced GBs |
---|
745 | option(redTail); |
---|
746 | matrix IM = @@D; // map |
---|
747 | list L = list(); // the answer |
---|
748 | list TL; |
---|
749 | for (i=1; i<=sL; i++)// compute GBs of components |
---|
750 | { |
---|
751 | TL = list(); |
---|
752 | TL[1] = std(mL[i]); |
---|
753 | TL[2] = NF( ideal(IM), TL[1] ); |
---|
754 | L[i] = TL; |
---|
755 | } |
---|
756 | export(L); |
---|
757 | ideal idJ = J; // debug-comfortable exports |
---|
758 | matrix matD = @@D; |
---|
759 | export(idJ); |
---|
760 | export(matD); |
---|
761 | return(@@KK); |
---|
762 | } |
---|
763 | example |
---|
764 | { "EXAMPLE:"; echo = 2; |
---|
765 | def a = CreateWeyl(1); |
---|
766 | setring a; |
---|
767 | def X = find_auto(); |
---|
768 | setring X; |
---|
769 | L; |
---|
770 | } |
---|