1 | ////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id: freegb.lib,v 1.10 2008-08-07 18:08:37 levandov Exp $"; |
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3 | category="Noncommutative"; |
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4 | info=" |
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5 | LIBRARY: freegb.lib Twosided Noncommutative Groebner bases in Free Algebras |
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6 | AUTHOR: Viktor Levandovskyy, levandov@math.rwth-aachen.de |
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7 | |
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8 | PROCEDURES: |
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9 | freegbRing(int d); creates a ring with d blocks of shifted original variables |
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10 | freegbasis(list L, int n); compute two-sided Groebner basis of ideal, encoded via L, up to degree n |
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11 | |
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12 | CONVERSION ROUTINES: |
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13 | |
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14 | lp2lstr(ideal K, def save): converts letter-place ideal to a list of modules |
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15 | lst2str(list L[,int n]); convert a list (of modules) into polynomials in free algebra |
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16 | mod2str(module M[,int n]); convert a module into a polynomial in free algebra |
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17 | vct2str(module M[,int n]); convert a vector into a word in free algebra |
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18 | " |
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19 | |
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20 | // this library computes two-sided GB of an ideal |
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21 | // in a free associative algebra |
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22 | |
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23 | // a monomial is encoded via a vector V |
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24 | // where V[1] = coefficient |
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25 | // V[1+i] = the corresponding symbol |
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26 | |
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27 | LIB "discretize.lib"; // for replace |
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28 | LIB "qhmoduli.lib"; // for Max |
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29 | |
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30 | |
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31 | // obsolete? |
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32 | |
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33 | proc lshift(module M, int s, string varing, def lpring) |
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34 | { |
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35 | // FINALLY IMPLEMENTED AS A PART OT THE CODE |
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36 | // shifts a poly from the ring @R to s positions |
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37 | // M lives in varing, the result in lpring |
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38 | // to be run from varing |
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39 | int i, j, k, sm, sv; |
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40 | vector v; |
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41 | // execute("setring "+lpring); |
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42 | setring lpring; |
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43 | poly @@p; |
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44 | ideal I; |
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45 | execute("setring "+varing); |
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46 | sm = ncols(M); |
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47 | for (i=1; i<=s; i++) |
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48 | { |
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49 | // modules, e.g. free polynomials |
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50 | for (j=1; j<=sm; j++) |
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51 | { |
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52 | //vectors, e.g. free monomials |
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53 | v = M[j]; |
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54 | sv = size(v); |
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55 | sp = "@@p = @@p + "; |
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56 | for (k=2; k<=sv; k++) |
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57 | { |
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58 | sp = sp + string(v[k])+"("+string(k-1+s)+")*"; |
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59 | } |
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60 | sp = sp + string(v[1])+";"; // coef; |
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61 | setring lpring; |
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62 | // execute("setring "+lpring); |
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63 | execute(sp); |
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64 | execute("setring "+varing); |
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65 | } |
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66 | setring lpring; |
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67 | // execute("setring "+lpring); |
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68 | I = I,@@p; |
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69 | @@p = 0; |
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70 | } |
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71 | setring lpring; |
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72 | //execute("setring "+lpring); |
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73 | export(I); |
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74 | // setring varing; |
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75 | execute("setring "+varing); |
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76 | } |
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77 | |
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78 | proc skip0(vector v) |
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79 | { |
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80 | // skips zeros in a vector, producing another vector |
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81 | int sv = nrows(v); |
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82 | int sw = size(v); |
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83 | if (sv == sw) |
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84 | { |
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85 | return(v); |
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86 | } |
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87 | int i; |
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88 | int j=1; |
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89 | vector w; |
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90 | for (i=1; i<=sv; i++) |
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91 | { |
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92 | if (v[i] != 0) |
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93 | { |
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94 | w = w + v[i]*gen(j); |
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95 | j++; |
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96 | } |
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97 | } |
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98 | return(w); |
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99 | } |
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100 | |
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101 | proc lst2str(list L, list #) |
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102 | "USAGE: lst2str(L[,n]); L a list of modules, n an optional integer |
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103 | RETURN: list (of strings) |
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104 | PURPOSE: convert a list (of modules) into polynomials in free algebra |
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105 | EXAMPLE: example lst2str; shows examples |
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106 | NOTE: if an optional integer is not 0, stars signs are used in multiplication |
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107 | " |
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108 | { |
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109 | // returns a list of strings |
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110 | // being sentences in words built from L |
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111 | // if #[1] = 1, use * between generators |
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112 | int useStar = 0; |
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113 | if ( size(#)>0 ) |
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114 | { |
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115 | if (#[1]) |
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116 | { |
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117 | useStar = 1; |
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118 | } |
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119 | } |
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120 | int i; |
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121 | int s = size(L); |
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122 | list N; |
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123 | for(i=1; i<=s; i++) |
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124 | { |
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125 | if ((typeof(L[i]) == "module") || (typeof(L[i]) == "matrix") ) |
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126 | { |
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127 | N[i] = mod2str(L[i],useStar); |
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128 | } |
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129 | else |
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130 | { |
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131 | "module or matrix expected in the list"; |
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132 | return(N); |
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133 | } |
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134 | } |
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135 | return(N); |
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136 | } |
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137 | example |
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138 | { |
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139 | "EXAMPLE:"; echo = 2; |
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140 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
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141 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
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142 | module N = [1,x,y,x,y],[-2,y,x,y,x],[6,x,y,y,x,y]; |
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143 | list L; L[1] = M; L[2] = N; |
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144 | lst2str(L); |
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145 | lst2str(L[1],1); |
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146 | } |
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147 | |
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148 | |
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149 | proc mod2str(module M, list #) |
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150 | "USAGE: mod2str(M[,n]); M a module, n an optional integer |
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151 | RETURN: string |
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152 | PURPOSE: convert a module into a polynomial in free algebra |
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153 | EXAMPLE: example mod2str; shows examples |
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154 | NOTE: if an optional integer is not 0, stars signs are used in multiplication |
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155 | " |
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156 | { |
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157 | // returns a string |
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158 | // a sentence in words built from M |
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159 | // if #[1] = 1, use * between generators |
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160 | int useStar = 0; |
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161 | if ( size(#)>0 ) |
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162 | { |
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163 | if (#[1]) |
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164 | { |
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165 | useStar = 1; |
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166 | } |
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167 | } |
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168 | int i; |
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169 | int s = ncols(M); |
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170 | string t; |
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171 | string mp; |
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172 | for(i=1; i<=s; i++) |
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173 | { |
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174 | mp = vct2str(M[i],useStar); |
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175 | if (mp[1] == "-") |
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176 | { |
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177 | t = t + mp; |
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178 | } |
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179 | else |
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180 | { |
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181 | t = t + "+" + mp; |
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182 | } |
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183 | } |
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184 | if (t[1]=="+") |
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185 | { |
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186 | t = t[2..size(t)]; // remove first "+" |
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187 | } |
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188 | return(t); |
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189 | } |
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190 | example |
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191 | { |
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192 | "EXAMPLE:"; echo = 2; |
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193 | ring r = 0,(x,y,z),(dp); |
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194 | module M = [1,x,y,x,y],[-2,y,x,y,x],[6,x,y,y,x,y]; |
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195 | mod2str(M); |
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196 | mod2str(M,1); |
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197 | } |
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198 | |
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199 | proc vct2str(vector v, list #) |
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200 | "USAGE: vct2str(v[,n]); v a vector, n an optional integer |
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201 | RETURN: string |
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202 | PURPOSE: convert a vector into a word in free algebra |
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203 | EXAMPLE: example vct2str; shows examples |
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204 | NOTE: if an optional integer is not 0, stars signs are used in multiplication |
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205 | " |
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206 | { |
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207 | // if #[1] = 1, use * between generators |
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208 | int useStar = 0; |
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209 | if ( size(#)>0 ) |
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210 | { |
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211 | if (#[1]) |
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212 | { |
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213 | useStar = 1; |
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214 | } |
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215 | } |
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216 | int ppl = printlevel-voice+2; |
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217 | // for a word, encoded by v |
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218 | // produces a string for it |
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219 | v = skip0(v); |
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220 | number cf = leadcoef(v[1]); |
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221 | int s = size(v); |
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222 | string vs,vv,vp,err; |
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223 | int i,j,p,q; |
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224 | for (i=1; i<=s-1; i++) |
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225 | { |
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226 | p = IsVar(v[i+1]); |
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227 | if (p==0) |
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228 | { |
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229 | err = "Error: monomial expected at" + string(i+1); |
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230 | dbprint(ppl,err); |
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231 | return("_"); |
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232 | } |
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233 | if (p==1) |
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234 | { |
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235 | if (useStar && (size(vs) >0)) { vs = vs + "*"; } |
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236 | vs = vs + string(v[i+1]); |
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237 | } |
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238 | else //power |
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239 | { |
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240 | vv = string(v[i+1]); |
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241 | q = find(vv,"^"); |
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242 | if (q==0) |
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243 | { |
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244 | q = find(vv,string(p)); |
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245 | if (q==0) |
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246 | { |
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247 | err = "error in find for string "+vv; |
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248 | dbprint(ppl,err); |
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249 | return("_"); |
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250 | } |
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251 | } |
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252 | // q>0 |
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253 | vp = vv[1..q-1]; |
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254 | for(j=1;j<=p;j++) |
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255 | { |
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256 | if (useStar && (size(vs) >0)) { vs = vs + "*"; } |
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257 | vs = vs + vp; |
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258 | } |
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259 | } |
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260 | } |
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261 | string scf; |
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262 | if (cf == -1) |
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263 | { |
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264 | scf = "-"; |
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265 | } |
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266 | else |
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267 | { |
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268 | scf = string(cf); |
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269 | if (cf == 1) |
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270 | { |
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271 | scf = ""; |
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272 | } |
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273 | } |
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274 | if (useStar && (size(scf) >0) && (scf!="-") ) { scf = scf + "*"; } |
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275 | vs = scf + vs; |
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276 | return(vs); |
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277 | } |
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278 | example |
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279 | { |
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280 | "EXAMPLE:"; echo = 2; |
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281 | ring r = (0,a),(x,y3,z(1)),dp; |
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282 | vector v = [-7,x,y3^4,x2,z(1)^3]; |
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283 | vct2str(v); |
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284 | vct2str(v,1); |
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285 | vector w = [-7a^5+6a,x,y3,y3,x,z(1),z(1)]; |
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286 | vct2str(w); |
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287 | vct2str(w,1); |
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288 | } |
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289 | |
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290 | proc IsVar(poly p) |
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291 | { |
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292 | // checks whether p is a variable indeed |
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293 | // if it's a power of a variable, returns the power |
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294 | if (p==0) { return(0); } //"p=0"; |
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295 | poly q = leadmonom(p); |
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296 | if ( (p-lead(p)) !=0 ) { return(0); } // "p-lm(p)>0"; |
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297 | intvec v = leadexp(p); |
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298 | int s = size(v); |
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299 | int i=1; |
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300 | int cnt = 0; |
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301 | int pwr = 0; |
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302 | for (i=1; i<=s; i++) |
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303 | { |
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304 | if (v[i] != 0) |
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305 | { |
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306 | cnt++; |
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307 | pwr = v[i]; |
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308 | } |
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309 | } |
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310 | // "cnt:"; cnt; |
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311 | if (cnt==1) { return(pwr); } |
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312 | else { return(0); } |
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313 | } |
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314 | example |
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315 | { |
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316 | "EXAMPLE:"; echo = 2; |
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317 | ring r = 0,(x,y),dp; |
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318 | poly f = xy+1; |
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319 | IsVar(f); |
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320 | poly g = xy; |
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321 | IsVar(g); |
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322 | poly h = y^3; |
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323 | IsVar(h); |
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324 | poly i = 1; |
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325 | IsVar(i); |
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326 | } |
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327 | |
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328 | // new conversion routines |
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329 | |
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330 | proc id2words(ideal I, int d) |
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331 | { |
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332 | // NOT FINISHED |
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333 | // input: ideal I of polys in letter-place notation |
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334 | // in the ring with d real vars |
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335 | // output: the list of strings: associative words |
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336 | // extract names of vars |
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337 | int i,m,n; string s; string place = "(1)"; |
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338 | list lv; |
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339 | for(i=1; i<=d; i++) |
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340 | { |
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341 | s = string(var(i)); |
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342 | // get rid of place |
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343 | n = find(s, place); |
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344 | if (n>0) |
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345 | { |
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346 | s = s[1..n-1]; |
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347 | } |
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348 | lv[i] = s; |
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349 | } |
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350 | poly p,q; |
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351 | for (i=1; i<=ncols(I); i++) |
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352 | { |
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353 | if (I[i] != 0) |
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354 | { |
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355 | p = I[i]; |
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356 | while (p!=0) |
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357 | { |
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358 | q = leadmonom(p); |
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359 | |
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360 | } |
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361 | } |
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362 | } |
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363 | |
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364 | return(lv); |
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365 | } |
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366 | example |
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367 | { |
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368 | "EXAMPLE:"; echo = 2; |
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369 | ring r = 0,(x(1),y(1),z(1),x(2),y(2),z(2)),dp; |
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370 | ideal I = x(1)*y(2) -z(1)*x(2); |
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371 | id2words(I,3); |
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372 | } |
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373 | |
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374 | proc mono2word(poly p, int d) |
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375 | { |
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376 | } |
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377 | |
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378 | // given the element -7xy^2x, it is represented as [-7,x,y^2,x] or as [-7,x,y,y,x] |
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379 | // use the orig ord on (x,y,z) and expand it blockwise to (x(i),y(i),z(i)) |
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380 | |
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381 | // the correspondences: |
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382 | // monomial in K<x,y,z> <<--->> vector in R |
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383 | // polynomial in K<x,y,z> <<--->> list of vectors (matrix/module) in R |
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384 | // ideal in K<x,y,z> <<--->> list of matrices/modules in R |
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385 | |
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386 | |
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387 | // 1. form a new ring |
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388 | // 2. NOP |
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389 | // 3. compute GB -> with the kernel stuff |
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390 | // 4. skip shifted elts (check that no such exist?) |
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391 | // 5. go back to orig vars, produce strings/modules |
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392 | // 6. return the result |
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393 | |
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394 | proc freegbasis(list LM, int d) |
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395 | "USAGE: freegbasis(L, d); L a list of modules, d an integer |
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396 | RETURN: ring |
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397 | PURPOSE: compute the two-sided Groebner basis of an ideal, encoded by L in |
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398 | the free associative algebra, up to degree d |
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399 | EXAMPLE: example freegbasis; shows examples |
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400 | " |
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401 | { |
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402 | // d = up to degree, will be shifted to d+1 |
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403 | if (d<1) {"bad d"; return(0);} |
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404 | |
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405 | int ppl = printlevel-voice+2; |
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406 | string err = ""; |
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407 | |
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408 | int i,j,s; |
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409 | def save = basering; |
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410 | // determine max no of places in the input |
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411 | int slm = size(LM); // numbers of polys in the ideal |
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412 | int sm; |
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413 | intvec iv; |
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414 | module M; |
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415 | for (i=1; i<=slm; i++) |
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416 | { |
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417 | // modules, e.g. free polynomials |
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418 | M = LM[i]; |
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419 | sm = ncols(M); |
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420 | for (j=1; j<=sm; j++) |
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421 | { |
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422 | //vectors, e.g. free monomials |
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423 | iv = iv, size(M[j])-1; // 1 place is reserved by the coeff |
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424 | } |
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425 | } |
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426 | int D = Max(iv); // max size of input words |
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427 | if (d<D) {"bad d"; return(LM);} |
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428 | D = D + d-1; |
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429 | // D = d; |
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430 | list LR = ringlist(save); |
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431 | list L, tmp; |
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432 | L[1] = LR[1]; // ground field |
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433 | L[4] = LR[4]; // quotient ideal |
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434 | tmp = LR[2]; // varnames |
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435 | s = size(LR[2]); |
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436 | for (i=1; i<=D; i++) |
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437 | { |
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438 | for (j=1; j<=s; j++) |
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439 | { |
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440 | tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")"; |
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441 | } |
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442 | } |
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443 | for (i=1; i<=s; i++) |
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444 | { |
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445 | tmp[i] = string(tmp[i])+"("+string(1)+")"; |
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446 | } |
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447 | L[2] = tmp; |
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448 | list OrigNames = LR[2]; |
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449 | // ordering: d blocks of the ord on r |
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450 | // try to get whether the ord on r is blockord itself |
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451 | s = size(LR[3]); |
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452 | if (s==2) |
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453 | { |
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454 | // not a blockord, 1 block + module ord |
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455 | tmp = LR[3][s]; // module ord |
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456 | for (i=1; i<=D; i++) |
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457 | { |
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458 | LR[3][s-1+i] = LR[3][1]; |
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459 | } |
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460 | LR[3][s+D] = tmp; |
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461 | } |
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462 | if (s>2) |
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463 | { |
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464 | // there are s-1 blocks |
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465 | int nb = s-1; |
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466 | tmp = LR[3][s]; // module ord |
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467 | for (i=1; i<=D; i++) |
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468 | { |
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469 | for (j=1; j<=nb; j++) |
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470 | { |
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471 | LR[3][i*nb+j] = LR[3][j]; |
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472 | } |
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473 | } |
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474 | // size(LR[3]); |
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475 | LR[3][nb*(D+1)+1] = tmp; |
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476 | } |
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477 | L[3] = LR[3]; |
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478 | def @R = ring(L); |
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479 | setring @R; |
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480 | ideal I; |
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481 | poly @p; |
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482 | s = size(OrigNames); |
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483 | // "s:";s; |
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484 | // convert LM to canonical vectors (no powers) |
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485 | setring save; |
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486 | kill M; // M was defined earlier |
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487 | module M; |
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488 | slm = size(LM); // numbers of polys in the ideal |
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489 | int sv,k,l; |
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490 | vector v; |
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491 | // poly p; |
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492 | string sp; |
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493 | setring @R; |
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494 | poly @@p=0; |
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495 | setring save; |
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496 | for (l=1; l<=slm; l++) |
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497 | { |
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498 | // modules, e.g. free polynomials |
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499 | M = LM[l]; |
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500 | sm = ncols(M); // in intvec iv the sizes are stored |
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501 | // modules, e.g. free polynomials |
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502 | for (j=1; j<=sm; j++) |
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503 | { |
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504 | //vectors, e.g. free monomials |
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505 | v = M[j]; |
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506 | sv = size(v); |
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507 | // "sv:";sv; |
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508 | sp = "@@p = @@p + "; |
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509 | for (k=2; k<=sv; k++) |
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510 | { |
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511 | sp = sp + string(v[k])+"("+string(k-1)+")*"; |
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512 | } |
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513 | sp = sp + string(v[1])+";"; // coef; |
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514 | setring @R; |
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515 | execute(sp); |
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516 | setring save; |
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517 | } |
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518 | setring @R; |
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519 | // "@@p:"; @@p; |
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520 | I = I,@@p; |
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521 | @@p = 0; |
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522 | setring save; |
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523 | } |
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524 | kill sp; |
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525 | // 3. compute GB |
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526 | setring @R; |
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527 | dbprint(ppl,"computing GB"); |
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528 | ideal J = system("freegb",I,d,nvars(save)); |
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529 | // ideal J = slimgb(I); |
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530 | dbprint(ppl,J); |
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531 | // 4. skip shifted elts |
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532 | ideal K = select1(J,1,s); // s = size(OrigNames) |
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533 | dbprint(ppl,K); |
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534 | dbprint(ppl, "done with GB"); |
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535 | // K contains vars x(1),...z(1) = images of originals |
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536 | // 5. go back to orig vars, produce strings/modules |
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537 | if (K[1] == 0) |
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538 | { |
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539 | "no reasonable output, GB gives 0"; |
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540 | return(0); |
---|
541 | } |
---|
542 | int sk = size(K); |
---|
543 | int sp, sx, a, b; |
---|
544 | intvec x; |
---|
545 | poly p,q; |
---|
546 | poly pn; |
---|
547 | // vars in 'save' |
---|
548 | setring save; |
---|
549 | module N; |
---|
550 | list LN; |
---|
551 | vector V; |
---|
552 | poly pn; |
---|
553 | // test and skip exponents >=2 |
---|
554 | setring @R; |
---|
555 | for(i=1; i<=sk; i++) |
---|
556 | { |
---|
557 | p = K[i]; |
---|
558 | while (p!=0) |
---|
559 | { |
---|
560 | q = lead(p); |
---|
561 | // "processing q:";q; |
---|
562 | x = leadexp(q); |
---|
563 | sx = size(x); |
---|
564 | for(k=1; k<=sx; k++) |
---|
565 | { |
---|
566 | if ( x[k] >= 2 ) |
---|
567 | { |
---|
568 | err = "skip: the value x[k] is " + string(x[k]); |
---|
569 | dbprint(ppl,err); |
---|
570 | // return(0); |
---|
571 | K[i] = 0; |
---|
572 | p = 0; |
---|
573 | q = 0; |
---|
574 | break; |
---|
575 | } |
---|
576 | } |
---|
577 | p = p - q; |
---|
578 | } |
---|
579 | } |
---|
580 | K = simplify(K,2); |
---|
581 | sk = size(K); |
---|
582 | for(i=1; i<=sk; i++) |
---|
583 | { |
---|
584 | // setring save; |
---|
585 | // V = 0; |
---|
586 | setring @R; |
---|
587 | p = K[i]; |
---|
588 | while (p!=0) |
---|
589 | { |
---|
590 | q = lead(p); |
---|
591 | err = "processing q:" + string(q); |
---|
592 | dbprint(ppl,err); |
---|
593 | x = leadexp(q); |
---|
594 | sx = size(x); |
---|
595 | pn = leadcoef(q); |
---|
596 | setring save; |
---|
597 | pn = imap(@R,pn); |
---|
598 | V = V + leadcoef(pn)*gen(1); |
---|
599 | for(k=1; k<=sx; k++) |
---|
600 | { |
---|
601 | if (x[k] ==1) |
---|
602 | { |
---|
603 | a = k / s; // block number=a+1, a!=0 |
---|
604 | b = k % s; // remainder |
---|
605 | // printf("a: %s, b: %s",a,b); |
---|
606 | if (b == 0) |
---|
607 | { |
---|
608 | // that is it's the last var in the block |
---|
609 | b = s; |
---|
610 | a = a-1; |
---|
611 | } |
---|
612 | V = V + var(b)*gen(a+2); |
---|
613 | } |
---|
614 | // else |
---|
615 | // { |
---|
616 | // printf("error: the value x[k] is %s", x[k]); |
---|
617 | // return(0); |
---|
618 | // } |
---|
619 | } |
---|
620 | err = "V: " + string(V); |
---|
621 | dbprint(ppl,err); |
---|
622 | // printf("V: %s", string(V)); |
---|
623 | N = N,V; |
---|
624 | V = 0; |
---|
625 | setring @R; |
---|
626 | p = p - q; |
---|
627 | pn = 0; |
---|
628 | } |
---|
629 | setring save; |
---|
630 | LN[i] = simplify(N,2); |
---|
631 | N = 0; |
---|
632 | } |
---|
633 | setring save; |
---|
634 | return(LN); |
---|
635 | } |
---|
636 | example |
---|
637 | { |
---|
638 | "EXAMPLE:"; echo = 2; |
---|
639 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
640 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
---|
641 | module N = [1,x,y,x],[-1,y,x,y]; |
---|
642 | list L; L[1] = M; L[2] = N; |
---|
643 | lst2str(L); |
---|
644 | def U = freegbasis(L,5); |
---|
645 | lst2str(U); |
---|
646 | } |
---|
647 | |
---|
648 | proc crs(list LM, int d) |
---|
649 | "USAGE: crs(L, d); L a list of modules, d an integer |
---|
650 | RETURN: ring |
---|
651 | PURPOSE: create a ring and shift the ideal |
---|
652 | EXAMPLE: example crs; shows examples |
---|
653 | " |
---|
654 | { |
---|
655 | // d = up to degree, will be shifted to d+1 |
---|
656 | if (d<1) {"bad d"; return(0);} |
---|
657 | |
---|
658 | int ppl = printlevel-voice+2; |
---|
659 | string err = ""; |
---|
660 | |
---|
661 | int i,j,s; |
---|
662 | def save = basering; |
---|
663 | // determine max no of places in the input |
---|
664 | int slm = size(LM); // numbers of polys in the ideal |
---|
665 | int sm; |
---|
666 | intvec iv; |
---|
667 | module M; |
---|
668 | for (i=1; i<=slm; i++) |
---|
669 | { |
---|
670 | // modules, e.g. free polynomials |
---|
671 | M = LM[i]; |
---|
672 | sm = ncols(M); |
---|
673 | for (j=1; j<=sm; j++) |
---|
674 | { |
---|
675 | //vectors, e.g. free monomials |
---|
676 | iv = iv, size(M[j])-1; // 1 place is reserved by the coeff |
---|
677 | } |
---|
678 | } |
---|
679 | int D = Max(iv); // max size of input words |
---|
680 | if (d<D) {"bad d"; return(LM);} |
---|
681 | D = D + d-1; |
---|
682 | // D = d; |
---|
683 | list LR = ringlist(save); |
---|
684 | list L, tmp; |
---|
685 | L[1] = LR[1]; // ground field |
---|
686 | L[4] = LR[4]; // quotient ideal |
---|
687 | tmp = LR[2]; // varnames |
---|
688 | s = size(LR[2]); |
---|
689 | for (i=1; i<=D; i++) |
---|
690 | { |
---|
691 | for (j=1; j<=s; j++) |
---|
692 | { |
---|
693 | tmp[i*s+j] = string(tmp[j])+"("+string(i)+")"; |
---|
694 | } |
---|
695 | } |
---|
696 | for (i=1; i<=s; i++) |
---|
697 | { |
---|
698 | tmp[i] = string(tmp[i])+"("+string(0)+")"; |
---|
699 | } |
---|
700 | L[2] = tmp; |
---|
701 | list OrigNames = LR[2]; |
---|
702 | // ordering: d blocks of the ord on r |
---|
703 | // try to get whether the ord on r is blockord itself |
---|
704 | s = size(LR[3]); |
---|
705 | if (s==2) |
---|
706 | { |
---|
707 | // not a blockord, 1 block + module ord |
---|
708 | tmp = LR[3][s]; // module ord |
---|
709 | for (i=1; i<=D; i++) |
---|
710 | { |
---|
711 | LR[3][s-1+i] = LR[3][1]; |
---|
712 | } |
---|
713 | LR[3][s+D] = tmp; |
---|
714 | } |
---|
715 | if (s>2) |
---|
716 | { |
---|
717 | // there are s-1 blocks |
---|
718 | int nb = s-1; |
---|
719 | tmp = LR[3][s]; // module ord |
---|
720 | for (i=1; i<=D; i++) |
---|
721 | { |
---|
722 | for (j=1; j<=nb; j++) |
---|
723 | { |
---|
724 | LR[3][i*nb+j] = LR[3][j]; |
---|
725 | } |
---|
726 | } |
---|
727 | // size(LR[3]); |
---|
728 | LR[3][nb*(D+1)+1] = tmp; |
---|
729 | } |
---|
730 | L[3] = LR[3]; |
---|
731 | def @R = ring(L); |
---|
732 | setring @R; |
---|
733 | ideal I; |
---|
734 | poly @p; |
---|
735 | s = size(OrigNames); |
---|
736 | // "s:";s; |
---|
737 | // convert LM to canonical vectors (no powers) |
---|
738 | setring save; |
---|
739 | kill M; // M was defined earlier |
---|
740 | module M; |
---|
741 | slm = size(LM); // numbers of polys in the ideal |
---|
742 | int sv,k,l; |
---|
743 | vector v; |
---|
744 | // poly p; |
---|
745 | string sp; |
---|
746 | setring @R; |
---|
747 | poly @@p=0; |
---|
748 | setring save; |
---|
749 | for (l=1; l<=slm; l++) |
---|
750 | { |
---|
751 | // modules, e.g. free polynomials |
---|
752 | M = LM[l]; |
---|
753 | sm = ncols(M); // in intvec iv the sizes are stored |
---|
754 | for (i=0; i<=d-iv[l]; i++) |
---|
755 | { |
---|
756 | // modules, e.g. free polynomials |
---|
757 | for (j=1; j<=sm; j++) |
---|
758 | { |
---|
759 | //vectors, e.g. free monomials |
---|
760 | v = M[j]; |
---|
761 | sv = size(v); |
---|
762 | // "sv:";sv; |
---|
763 | sp = "@@p = @@p + "; |
---|
764 | for (k=2; k<=sv; k++) |
---|
765 | { |
---|
766 | sp = sp + string(v[k])+"("+string(k-2+i)+")*"; |
---|
767 | } |
---|
768 | sp = sp + string(v[1])+";"; // coef; |
---|
769 | setring @R; |
---|
770 | execute(sp); |
---|
771 | setring save; |
---|
772 | } |
---|
773 | setring @R; |
---|
774 | // "@@p:"; @@p; |
---|
775 | I = I,@@p; |
---|
776 | @@p = 0; |
---|
777 | setring save; |
---|
778 | } |
---|
779 | } |
---|
780 | setring @R; |
---|
781 | export I; |
---|
782 | return(@R); |
---|
783 | } |
---|
784 | example |
---|
785 | { |
---|
786 | "EXAMPLE:"; echo = 2; |
---|
787 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
788 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
---|
789 | module N = [1,x,y,x],[-1,y,x,y]; |
---|
790 | list L; L[1] = M; L[2] = N; |
---|
791 | lst2str(L); |
---|
792 | def U = crs(L,5); |
---|
793 | setring U; U; |
---|
794 | I; |
---|
795 | } |
---|
796 | |
---|
797 | proc polylen(ideal I) |
---|
798 | { |
---|
799 | // returns the ideal of length of polys |
---|
800 | int i; |
---|
801 | intvec J; |
---|
802 | number s = 0; |
---|
803 | for(i=1;i<=ncols(I);i++) |
---|
804 | { |
---|
805 | J[i] = size(I[i]); |
---|
806 | s = s + J[i]; |
---|
807 | } |
---|
808 | printf("the sum of length %s",s); |
---|
809 | // print(s); |
---|
810 | return(J); |
---|
811 | } |
---|
812 | |
---|
813 | proc freegbRing(int d) |
---|
814 | "USAGE: freegbRing(d); d an integer |
---|
815 | RETURN: ring |
---|
816 | PURPOSE: creates a ring with d blocks of shifted original variables |
---|
817 | EXAMPLE: example freegbRing; shows examples |
---|
818 | " |
---|
819 | { |
---|
820 | // d = up to degree, will be shifted to d+1 |
---|
821 | if (d<1) {"bad d"; return(0);} |
---|
822 | |
---|
823 | int ppl = printlevel-voice+2; |
---|
824 | string err = ""; |
---|
825 | |
---|
826 | int i,j,s; |
---|
827 | def save = basering; |
---|
828 | int D = d-1; |
---|
829 | list LR = ringlist(save); |
---|
830 | list L, tmp; |
---|
831 | L[1] = LR[1]; // ground field |
---|
832 | L[4] = LR[4]; // quotient ideal |
---|
833 | tmp = LR[2]; // varnames |
---|
834 | s = size(LR[2]); |
---|
835 | for (i=1; i<=D; i++) |
---|
836 | { |
---|
837 | for (j=1; j<=s; j++) |
---|
838 | { |
---|
839 | tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")"; |
---|
840 | } |
---|
841 | } |
---|
842 | for (i=1; i<=s; i++) |
---|
843 | { |
---|
844 | tmp[i] = string(tmp[i])+"("+string(1)+")"; |
---|
845 | } |
---|
846 | L[2] = tmp; |
---|
847 | list OrigNames = LR[2]; |
---|
848 | // ordering: d blocks of the ord on r |
---|
849 | // try to get whether the ord on r is blockord itself |
---|
850 | // TODO: make L(2) ordering! exponent is maximally 2 |
---|
851 | s = size(LR[3]); |
---|
852 | if (s==2) |
---|
853 | { |
---|
854 | // not a blockord, 1 block + module ord |
---|
855 | tmp = LR[3][s]; // module ord |
---|
856 | for (i=1; i<=D; i++) |
---|
857 | { |
---|
858 | LR[3][s-1+i] = LR[3][1]; |
---|
859 | } |
---|
860 | LR[3][s+D] = tmp; |
---|
861 | } |
---|
862 | if (s>2) |
---|
863 | { |
---|
864 | // there are s-1 blocks |
---|
865 | int nb = s-1; |
---|
866 | tmp = LR[3][s]; // module ord |
---|
867 | for (i=1; i<=D; i++) |
---|
868 | { |
---|
869 | for (j=1; j<=nb; j++) |
---|
870 | { |
---|
871 | LR[3][i*nb+j] = LR[3][j]; |
---|
872 | } |
---|
873 | } |
---|
874 | // size(LR[3]); |
---|
875 | LR[3][nb*(D+1)+1] = tmp; |
---|
876 | } |
---|
877 | L[3] = LR[3]; |
---|
878 | def @R = ring(L); |
---|
879 | // setring @R; |
---|
880 | return (@R); |
---|
881 | } |
---|
882 | example |
---|
883 | { |
---|
884 | "EXAMPLE:"; echo = 2; |
---|
885 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
886 | def A = freegbRing(2); |
---|
887 | setring A; |
---|
888 | A; |
---|
889 | } |
---|
890 | |
---|
891 | proc ex_shift() |
---|
892 | { |
---|
893 | LIB "freegb.lib"; |
---|
894 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
895 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
---|
896 | module N = [1,x,y,x],[-1,y,x,y]; |
---|
897 | list L; L[1] = M; L[2] = N; |
---|
898 | lst2str(L); |
---|
899 | def U = crs(L,5); |
---|
900 | setring U; U; |
---|
901 | I; |
---|
902 | poly p = I[2]; // I[8]; |
---|
903 | p; |
---|
904 | system("stest",p,7,7,3); // error -> the world is ok |
---|
905 | poly q1 = system("stest",p,1,7,3); //ok |
---|
906 | poly q6 = system("stest",p,6,7,3); //ok |
---|
907 | system("btest",p,3); //ok |
---|
908 | system("btest",q1,3); //ok |
---|
909 | system("btest",q6,3); //ok |
---|
910 | } |
---|
911 | |
---|
912 | proc test_shrink() |
---|
913 | { |
---|
914 | LIB "freegb.lib"; |
---|
915 | ring r =0,(x,y,z),dp; |
---|
916 | int d = 5; |
---|
917 | def R = freegbRing(d); |
---|
918 | setring R; |
---|
919 | poly p1 = x(1)*y(2)*z(3); |
---|
920 | poly p2 = x(1)*y(4)*z(5); |
---|
921 | poly p3 = x(1)*y(1)*z(3); |
---|
922 | poly p4 = x(1)*y(2)*z(2); |
---|
923 | poly p5 = x(3)*z(5); |
---|
924 | poly p6 = x(1)*y(1)*x(3)*z(5); |
---|
925 | poly p7 = x(1)*y(2)*x(3)*y(4)*z(5); |
---|
926 | poly p8 = p1+p2+p3+p4+p5 + p6 + p7; |
---|
927 | p1; system("shrinktest",p1,3); |
---|
928 | p2; system("shrinktest",p2,3); |
---|
929 | p3; system("shrinktest",p3,3); |
---|
930 | p4; system("shrinktest",p4,3); |
---|
931 | p5; system("shrinktest",p5,3); |
---|
932 | p6; system("shrinktest",p6,3); |
---|
933 | p7; system("shrinktest",p7,3); |
---|
934 | p8; system("shrinktest",p8,3); |
---|
935 | poly p9 = p1 + 2*p2 + 5*p5 + 7*p7; |
---|
936 | p9; system("shrinktest",p9,3); |
---|
937 | } |
---|
938 | |
---|
939 | proc ex2() |
---|
940 | { |
---|
941 | option(prot); |
---|
942 | LIB "freegb.lib"; |
---|
943 | ring r = 0,(x,y),dp; |
---|
944 | module M = [-1,x,y],[3,x,x]; // 3x^2 - xy |
---|
945 | def U = freegb(M,7); |
---|
946 | lst2str(U); |
---|
947 | } |
---|
948 | |
---|
949 | proc ex_nonhomog() |
---|
950 | { |
---|
951 | option(prot); |
---|
952 | LIB "freegb.lib"; |
---|
953 | ring r = 0,(x,y,h),dp; |
---|
954 | list L; |
---|
955 | module M; |
---|
956 | M = [-1,y,y],[1,x,x,x]; // x3-y2 |
---|
957 | L[1] = M; |
---|
958 | M = [1,x,h],[-1,h,x]; // xh-hx |
---|
959 | L[2] = M; |
---|
960 | M = [1,y,h],[-1,h,y]; // yh-hy |
---|
961 | L[3] = M; |
---|
962 | def U = freegb(L,4); |
---|
963 | lst2str(U); |
---|
964 | // strange elements in the basis |
---|
965 | } |
---|
966 | |
---|
967 | proc ex_nonhomog_comm() |
---|
968 | { |
---|
969 | option(prot); |
---|
970 | LIB "freegb.lib"; |
---|
971 | ring r = 0,(x,y),dp; |
---|
972 | module M = [-1,y,y],[1,x,x,x]; |
---|
973 | def U = freegb(M,5); |
---|
974 | lst2str(U); |
---|
975 | } |
---|
976 | |
---|
977 | proc ex_nonhomog_h() |
---|
978 | { |
---|
979 | option(prot); |
---|
980 | LIB "freegb.lib"; |
---|
981 | ring r = 0,(x,y,h),(a(1,1),dp); |
---|
982 | module M = [-1,y,y,h],[1,x,x,x]; // x3 - y2h |
---|
983 | def U = freegb(M,6); |
---|
984 | lst2str(U); |
---|
985 | } |
---|
986 | |
---|
987 | proc ex_nonhomog_h2() |
---|
988 | { |
---|
989 | option(prot); |
---|
990 | LIB "freegb.lib"; |
---|
991 | ring r = 0,(x,y,h),(dp); |
---|
992 | list L; |
---|
993 | module M; |
---|
994 | M = [-1,y,y,h],[1,x,x,x]; // x3 - y2h |
---|
995 | L[1] = M; |
---|
996 | M = [1,x,h],[-1,h,x]; // xh - hx |
---|
997 | L[2] = M; |
---|
998 | M = [1,y,h],[-1,h,y]; // yh - hy |
---|
999 | L[3] = M; |
---|
1000 | def U = freegbasis(L,3); |
---|
1001 | lst2str(U); |
---|
1002 | // strange answer CHECK |
---|
1003 | } |
---|
1004 | |
---|
1005 | |
---|
1006 | proc ex_nonhomog_3() |
---|
1007 | { |
---|
1008 | option(prot); |
---|
1009 | LIB "./freegb.lib"; |
---|
1010 | ring r = 0,(x,y,z),(dp); |
---|
1011 | list L; |
---|
1012 | module M; |
---|
1013 | M = [1,z,y],[-1,x]; // zy - x |
---|
1014 | L[1] = M; |
---|
1015 | M = [1,z,x],[-1,y]; // zx - y |
---|
1016 | L[2] = M; |
---|
1017 | M = [1,y,x],[-1,z]; // yx - z |
---|
1018 | L[3] = M; |
---|
1019 | lst2str(L); |
---|
1020 | list U = freegb(L,4); |
---|
1021 | lst2str(U); |
---|
1022 | // strange answer CHECK |
---|
1023 | } |
---|
1024 | |
---|
1025 | proc ex_densep_2() |
---|
1026 | { |
---|
1027 | option(prot); |
---|
1028 | LIB "freegb.lib"; |
---|
1029 | ring r = (0,a,b,c),(x,y),(Dp); // deglex |
---|
1030 | module M = [1,x,x], [a,x,y], [b,y,x], [c,y,y]; |
---|
1031 | lst2str(M); |
---|
1032 | list U = freegb(M,5); |
---|
1033 | lst2str(U); |
---|
1034 | // a=b is important -> finite basis!!! |
---|
1035 | module M = [1,x,x], [a,x,y], [a,y,x], [c,y,y]; |
---|
1036 | lst2str(M); |
---|
1037 | list U = freegb(M,5); |
---|
1038 | lst2str(U); |
---|
1039 | } |
---|
1040 | |
---|
1041 | |
---|
1042 | // 1. form a new ring |
---|
1043 | // 2. produce shifted generators |
---|
1044 | // 3. compute GB |
---|
1045 | // 4. skip shifted elts |
---|
1046 | // 5. go back to orig vars, produce strings/modules |
---|
1047 | // 6. return the result |
---|
1048 | |
---|
1049 | proc freegbold(list LM, int d) |
---|
1050 | "USAGE: freegbold(L, d); L a list of modules, d an integer |
---|
1051 | RETURN: ring |
---|
1052 | PURPOSE: compute the two-sided Groebner basis of an ideal, encoded by L in |
---|
1053 | the free associative algebra, up to degree d |
---|
1054 | EXAMPLE: example freegbold; shows examples |
---|
1055 | " |
---|
1056 | { |
---|
1057 | // d = up to degree, will be shifted to d+1 |
---|
1058 | if (d<1) {"bad d"; return(0);} |
---|
1059 | |
---|
1060 | int ppl = printlevel-voice+2; |
---|
1061 | string err = ""; |
---|
1062 | |
---|
1063 | int i,j,s; |
---|
1064 | def save = basering; |
---|
1065 | // determine max no of places in the input |
---|
1066 | int slm = size(LM); // numbers of polys in the ideal |
---|
1067 | int sm; |
---|
1068 | intvec iv; |
---|
1069 | module M; |
---|
1070 | for (i=1; i<=slm; i++) |
---|
1071 | { |
---|
1072 | // modules, e.g. free polynomials |
---|
1073 | M = LM[i]; |
---|
1074 | sm = ncols(M); |
---|
1075 | for (j=1; j<=sm; j++) |
---|
1076 | { |
---|
1077 | //vectors, e.g. free monomials |
---|
1078 | iv = iv, size(M[j])-1; // 1 place is reserved by the coeff |
---|
1079 | } |
---|
1080 | } |
---|
1081 | int D = Max(iv); // max size of input words |
---|
1082 | if (d<D) {"bad d"; return(LM);} |
---|
1083 | D = D + d-1; |
---|
1084 | // D = d; |
---|
1085 | list LR = ringlist(save); |
---|
1086 | list L, tmp; |
---|
1087 | L[1] = LR[1]; // ground field |
---|
1088 | L[4] = LR[4]; // quotient ideal |
---|
1089 | tmp = LR[2]; // varnames |
---|
1090 | s = size(LR[2]); |
---|
1091 | for (i=1; i<=D; i++) |
---|
1092 | { |
---|
1093 | for (j=1; j<=s; j++) |
---|
1094 | { |
---|
1095 | tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")"; |
---|
1096 | } |
---|
1097 | } |
---|
1098 | for (i=1; i<=s; i++) |
---|
1099 | { |
---|
1100 | tmp[i] = string(tmp[i])+"("+string(1)+")"; |
---|
1101 | } |
---|
1102 | L[2] = tmp; |
---|
1103 | list OrigNames = LR[2]; |
---|
1104 | // ordering: d blocks of the ord on r |
---|
1105 | // try to get whether the ord on r is blockord itself |
---|
1106 | // TODO: make L(2) ordering! exponent is maximally 2 |
---|
1107 | s = size(LR[3]); |
---|
1108 | if (s==2) |
---|
1109 | { |
---|
1110 | // not a blockord, 1 block + module ord |
---|
1111 | tmp = LR[3][s]; // module ord |
---|
1112 | for (i=1; i<=D; i++) |
---|
1113 | { |
---|
1114 | LR[3][s-1+i] = LR[3][1]; |
---|
1115 | } |
---|
1116 | LR[3][s+D] = tmp; |
---|
1117 | } |
---|
1118 | if (s>2) |
---|
1119 | { |
---|
1120 | // there are s-1 blocks |
---|
1121 | int nb = s-1; |
---|
1122 | tmp = LR[3][s]; // module ord |
---|
1123 | for (i=1; i<=D; i++) |
---|
1124 | { |
---|
1125 | for (j=1; j<=nb; j++) |
---|
1126 | { |
---|
1127 | LR[3][i*nb+j] = LR[3][j]; |
---|
1128 | } |
---|
1129 | } |
---|
1130 | // size(LR[3]); |
---|
1131 | LR[3][nb*(D+1)+1] = tmp; |
---|
1132 | } |
---|
1133 | L[3] = LR[3]; |
---|
1134 | def @R = ring(L); |
---|
1135 | setring @R; |
---|
1136 | ideal I; |
---|
1137 | poly @p; |
---|
1138 | s = size(OrigNames); |
---|
1139 | // "s:";s; |
---|
1140 | // convert LM to canonical vectors (no powers) |
---|
1141 | setring save; |
---|
1142 | kill M; // M was defined earlier |
---|
1143 | module M; |
---|
1144 | slm = size(LM); // numbers of polys in the ideal |
---|
1145 | int sv,k,l; |
---|
1146 | vector v; |
---|
1147 | // poly p; |
---|
1148 | string sp; |
---|
1149 | setring @R; |
---|
1150 | poly @@p=0; |
---|
1151 | setring save; |
---|
1152 | for (l=1; l<=slm; l++) |
---|
1153 | { |
---|
1154 | // modules, e.g. free polynomials |
---|
1155 | M = LM[l]; |
---|
1156 | sm = ncols(M); // in intvec iv the sizes are stored |
---|
1157 | for (i=0; i<=d-iv[l]; i++) |
---|
1158 | { |
---|
1159 | // modules, e.g. free polynomials |
---|
1160 | for (j=1; j<=sm; j++) |
---|
1161 | { |
---|
1162 | //vectors, e.g. free monomials |
---|
1163 | v = M[j]; |
---|
1164 | sv = size(v); |
---|
1165 | // "sv:";sv; |
---|
1166 | sp = "@@p = @@p + "; |
---|
1167 | for (k=2; k<=sv; k++) |
---|
1168 | { |
---|
1169 | sp = sp + string(v[k])+"("+string(k-1+i)+")*"; |
---|
1170 | } |
---|
1171 | sp = sp + string(v[1])+";"; // coef; |
---|
1172 | setring @R; |
---|
1173 | execute(sp); |
---|
1174 | setring save; |
---|
1175 | } |
---|
1176 | setring @R; |
---|
1177 | // "@@p:"; @@p; |
---|
1178 | I = I,@@p; |
---|
1179 | @@p = 0; |
---|
1180 | setring save; |
---|
1181 | } |
---|
1182 | } |
---|
1183 | kill sp; |
---|
1184 | // 3. compute GB |
---|
1185 | setring @R; |
---|
1186 | dbprint(ppl,"computing GB"); |
---|
1187 | // ideal J = groebner(I); |
---|
1188 | ideal J = slimgb(I); |
---|
1189 | dbprint(ppl,J); |
---|
1190 | // 4. skip shifted elts |
---|
1191 | ideal K = select1(J,1,s); // s = size(OrigNames) |
---|
1192 | dbprint(ppl,K); |
---|
1193 | dbprint(ppl, "done with GB"); |
---|
1194 | // K contains vars x(1),...z(1) = images of originals |
---|
1195 | // 5. go back to orig vars, produce strings/modules |
---|
1196 | if (K[1] == 0) |
---|
1197 | { |
---|
1198 | "no reasonable output, GB gives 0"; |
---|
1199 | return(0); |
---|
1200 | } |
---|
1201 | int sk = size(K); |
---|
1202 | int sp, sx, a, b; |
---|
1203 | intvec x; |
---|
1204 | poly p,q; |
---|
1205 | poly pn; |
---|
1206 | // vars in 'save' |
---|
1207 | setring save; |
---|
1208 | module N; |
---|
1209 | list LN; |
---|
1210 | vector V; |
---|
1211 | poly pn; |
---|
1212 | // test and skip exponents >=2 |
---|
1213 | setring @R; |
---|
1214 | for(i=1; i<=sk; i++) |
---|
1215 | { |
---|
1216 | p = K[i]; |
---|
1217 | while (p!=0) |
---|
1218 | { |
---|
1219 | q = lead(p); |
---|
1220 | // "processing q:";q; |
---|
1221 | x = leadexp(q); |
---|
1222 | sx = size(x); |
---|
1223 | for(k=1; k<=sx; k++) |
---|
1224 | { |
---|
1225 | if ( x[k] >= 2 ) |
---|
1226 | { |
---|
1227 | err = "skip: the value x[k] is " + string(x[k]); |
---|
1228 | dbprint(ppl,err); |
---|
1229 | // return(0); |
---|
1230 | K[i] = 0; |
---|
1231 | p = 0; |
---|
1232 | q = 0; |
---|
1233 | break; |
---|
1234 | } |
---|
1235 | } |
---|
1236 | p = p - q; |
---|
1237 | } |
---|
1238 | } |
---|
1239 | K = simplify(K,2); |
---|
1240 | sk = size(K); |
---|
1241 | for(i=1; i<=sk; i++) |
---|
1242 | { |
---|
1243 | // setring save; |
---|
1244 | // V = 0; |
---|
1245 | setring @R; |
---|
1246 | p = K[i]; |
---|
1247 | while (p!=0) |
---|
1248 | { |
---|
1249 | q = lead(p); |
---|
1250 | err = "processing q:" + string(q); |
---|
1251 | dbprint(ppl,err); |
---|
1252 | x = leadexp(q); |
---|
1253 | sx = size(x); |
---|
1254 | pn = leadcoef(q); |
---|
1255 | setring save; |
---|
1256 | pn = imap(@R,pn); |
---|
1257 | V = V + leadcoef(pn)*gen(1); |
---|
1258 | for(k=1; k<=sx; k++) |
---|
1259 | { |
---|
1260 | if (x[k] ==1) |
---|
1261 | { |
---|
1262 | a = k / s; // block number=a+1, a!=0 |
---|
1263 | b = k % s; // remainder |
---|
1264 | // printf("a: %s, b: %s",a,b); |
---|
1265 | if (b == 0) |
---|
1266 | { |
---|
1267 | // that is it's the last var in the block |
---|
1268 | b = s; |
---|
1269 | a = a-1; |
---|
1270 | } |
---|
1271 | V = V + var(b)*gen(a+2); |
---|
1272 | } |
---|
1273 | // else |
---|
1274 | // { |
---|
1275 | // printf("error: the value x[k] is %s", x[k]); |
---|
1276 | // return(0); |
---|
1277 | // } |
---|
1278 | } |
---|
1279 | err = "V: " + string(V); |
---|
1280 | dbprint(ppl,err); |
---|
1281 | // printf("V: %s", string(V)); |
---|
1282 | N = N,V; |
---|
1283 | V = 0; |
---|
1284 | setring @R; |
---|
1285 | p = p - q; |
---|
1286 | pn = 0; |
---|
1287 | } |
---|
1288 | setring save; |
---|
1289 | LN[i] = simplify(N,2); |
---|
1290 | N = 0; |
---|
1291 | } |
---|
1292 | setring save; |
---|
1293 | return(LN); |
---|
1294 | } |
---|
1295 | example |
---|
1296 | { |
---|
1297 | "EXAMPLE:"; echo = 2; |
---|
1298 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
1299 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
---|
1300 | module N = [1,x,y,x],[-1,y,x,y]; |
---|
1301 | list L; L[1] = M; L[2] = N; |
---|
1302 | lst2str(L); |
---|
1303 | def U = freegbold(L,5); |
---|
1304 | lst2str(U); |
---|
1305 | } |
---|
1306 | |
---|
1307 | proc sgb(ideal I, int d) |
---|
1308 | { |
---|
1309 | // new code |
---|
1310 | // map x_i to x_i(1) via map() |
---|
1311 | //LIB "freegb.lib"; |
---|
1312 | def save = basering; |
---|
1313 | //int d =7;// degree |
---|
1314 | int nv = nvars(save); |
---|
1315 | def R = freegbRing(d); |
---|
1316 | setring R; |
---|
1317 | int i; |
---|
1318 | ideal Imap; |
---|
1319 | for (i=1; i<=nv; i++) |
---|
1320 | { |
---|
1321 | Imap[i] = var(i); |
---|
1322 | } |
---|
1323 | //ideal I = x(1)*y(2), y(1)*x(2)+z(1)*z(2); |
---|
1324 | ideal I = x(1)*x(2),x(1)*y(2) + z(1)*x(2); |
---|
1325 | option(prot); |
---|
1326 | //option(teach); |
---|
1327 | ideal J = system("freegb",I,d,nv); |
---|
1328 | } |
---|
1329 | |
---|
1330 | |
---|
1331 | |
---|
1332 | static proc checkCeq() |
---|
1333 | { |
---|
1334 | ring r = 0,(x,y),Dp; |
---|
1335 | def A = freegbRing(4); |
---|
1336 | setring A; |
---|
1337 | A; |
---|
1338 | // I = x2-xy |
---|
1339 | ideal I = x(1)*x(2) - x(1)*y(2), x(2)*x(3) - x(2)*y(3), x(3)*x(4) - x(3)*y(4); |
---|
1340 | ideal C = x(2)-x(1),x(3)-x(2),x(4)-x(3),y(2)-y(1),y(3)-y(2),y(4)-y(3); |
---|
1341 | ideal K = I,C; |
---|
1342 | groebner(K); |
---|
1343 | } |
---|
1344 | |
---|
1345 | |
---|
1346 | proc exHom1() |
---|
1347 | { |
---|
1348 | // we start with |
---|
1349 | // z*y - x, z*x - y, y*x - z |
---|
1350 | LIB "freegb.lib"; |
---|
1351 | LIB "elim.lib"; |
---|
1352 | ring r = 0,(x,y,z,h),dp; |
---|
1353 | list L; |
---|
1354 | module M; |
---|
1355 | M = [1,z,y],[-1,x,h]; // zy - xh |
---|
1356 | L[1] = M; |
---|
1357 | M = [1,z,x],[-1,y,h]; // zx - yh |
---|
1358 | L[2] = M; |
---|
1359 | M = [1,y,x],[-1,z,h]; // yx - zh |
---|
1360 | L[3] = M; |
---|
1361 | lst2str(L); |
---|
1362 | def U = crs(L,4); |
---|
1363 | setring U; |
---|
1364 | I = I, |
---|
1365 | y(2)*h(3)+z(2)*x(3), y(3)*h(4)+z(3)*x(4), |
---|
1366 | y(2)*x(3)-z(2)*h(3), y(3)*x(4)-z(3)*h(4); |
---|
1367 | I = simplify(I,2); |
---|
1368 | ring r2 = 0,(x(0..4),y(0..4),z(0..4),h(0..4)),dp; |
---|
1369 | ideal J = imap(U,I); |
---|
1370 | // ideal K = homog(J,h); |
---|
1371 | option(redSB); |
---|
1372 | option(redTail); |
---|
1373 | ideal L = groebner(J); //(K); |
---|
1374 | ideal LL = sat(L,ideal(h))[1]; |
---|
1375 | ideal M = subst(LL,h,1); |
---|
1376 | M = simplify(M,2); |
---|
1377 | setring U; |
---|
1378 | ideal M = imap(r2,M); |
---|
1379 | lst2str(U); |
---|
1380 | } |
---|
1381 | |
---|
1382 | static proc test1() |
---|
1383 | { |
---|
1384 | LIB "freegb.lib"; |
---|
1385 | ring r = 0,(x,y),Dp; |
---|
1386 | int d = 10; // degree |
---|
1387 | def R = freegbRing(d); |
---|
1388 | setring R; |
---|
1389 | ideal I = x(1)*x(2) - y(1)*y(2); |
---|
1390 | option(prot); |
---|
1391 | option(teach); |
---|
1392 | ideal J = system("freegb",I,d,2); |
---|
1393 | J; |
---|
1394 | } |
---|
1395 | |
---|
1396 | static proc test2() |
---|
1397 | { |
---|
1398 | LIB "freegb.lib"; |
---|
1399 | ring r = 0,(x,y),Dp; |
---|
1400 | int d = 10; // degree |
---|
1401 | def R = freegbRing(d); |
---|
1402 | setring R; |
---|
1403 | ideal I = x(1)*x(2) - x(1)*y(2); |
---|
1404 | option(prot); |
---|
1405 | option(teach); |
---|
1406 | ideal J = system("freegb",I,d,2); |
---|
1407 | J; |
---|
1408 | } |
---|
1409 | |
---|
1410 | static proc test3() |
---|
1411 | { |
---|
1412 | LIB "freegb.lib"; |
---|
1413 | ring r = 0,(x,y,z),dp; |
---|
1414 | int d =5; // degree |
---|
1415 | def R = freegbRing(d); |
---|
1416 | setring R; |
---|
1417 | ideal I = x(1)*y(2), y(1)*x(2)+z(1)*z(2); |
---|
1418 | option(prot); |
---|
1419 | option(teach); |
---|
1420 | ideal J = system("freegb",I,d,3); |
---|
1421 | } |
---|
1422 | |
---|
1423 | proc schur2-3() |
---|
1424 | { |
---|
1425 | // nonhomog: |
---|
1426 | // h^4-10*h^2+9,f*e-e*f+h, h*2-e*h-2*e,h*f-f*h+2*f |
---|
1427 | // homogenized with t |
---|
1428 | // h^4-10*h^2*t^2+9*t^4,f*e-e*f+h*t, h*2-e*h-2*e*t,h*f-f*h+2*f*t, |
---|
1429 | // t*h - h*t, t*f - f*t, t*e - e*t |
---|
1430 | } |
---|
1431 | |
---|
1432 | proc adem(int i, int j) |
---|
1433 | { |
---|
1434 | // produces Adem relations for i<2j in char 0 |
---|
1435 | // assume: 0<i<2j |
---|
1436 | // requires presence of vars up to i+j |
---|
1437 | if ( (i<0) || (i >= 2*j) ) |
---|
1438 | { |
---|
1439 | ERROR("arguments out of range"); return(0); |
---|
1440 | } |
---|
1441 | ring @r = 0,(s(i+j..0)),lp; |
---|
1442 | poly p,q; |
---|
1443 | number n; |
---|
1444 | int ii = i div 2; int k; |
---|
1445 | // k=0 => s(0)=1 |
---|
1446 | n = binomial(j-1,i); |
---|
1447 | q = n*s(i+j)*s(0); |
---|
1448 | printf("k=0, term=%s",q); |
---|
1449 | p = p + q; |
---|
1450 | for (k=1; k<= ii; k++) |
---|
1451 | { |
---|
1452 | n = binomial(j-k-1,i-2*k); |
---|
1453 | q = n*s(i+j-k)*s(k);; |
---|
1454 | printf("k=%s, term=%s",k,q); |
---|
1455 | p = p + q; |
---|
1456 | } |
---|
1457 | poly AdemRel = p; |
---|
1458 | export AdemRel; |
---|
1459 | return(@r); |
---|
1460 | } |
---|
1461 | example |
---|
1462 | { |
---|
1463 | "EXAMPLE:"; echo = 2; |
---|
1464 | def A = adem(2,5); |
---|
1465 | setring A; |
---|
1466 | AdemRel; |
---|
1467 | } |
---|
1468 | |
---|
1469 | /* |
---|
1470 | 1,1: 0 |
---|
1471 | 1,2: s(3)*s(0) == s(3) -> def for s(3):=s(1)s(2) |
---|
1472 | 2,1: adm |
---|
1473 | 2,2: s(3)*s(1) == s(1)s(2)s(1) |
---|
1474 | 1,3: 0 ( since 2*s(4)*s(0) = 0 mod 2) |
---|
1475 | 3,1: adm |
---|
1476 | 2,3: s(5)*s(0)+s(4)*s(1) == s(5)+s(4)*s(1) |
---|
1477 | 3,2: 0 |
---|
1478 | 3,3: s(5)*s(1) |
---|
1479 | 1,4: 3*s(5)*s(0) == s(5) -> def for s(5):=s(1)*s(4) |
---|
1480 | 4,1: adm |
---|
1481 | 2,4: 3*s(6)*s(0)+s(5)*s(1) == s(6) + s(5)*s(1) == s(6) + s(1)*s(4)*s(1) |
---|
1482 | 4,2: adm |
---|
1483 | 4,3: s(5)*s(2) |
---|
1484 | 3,4: s(7)*s(0)+2*s(6)*s(1) == s(7) -> def for s(7):=s(3)*s(4) |
---|
1485 | 4,4: s(7)*s(1)+s(6)*s(2) |
---|
1486 | */ |
---|
1487 | |
---|
1488 | /* s1,s2: |
---|
1489 | s1*s1 =0, s2*s2 = s1*s2*s1 |
---|
1490 | */ |
---|
1491 | |
---|
1492 | /* |
---|
1493 | try char 0: |
---|
1494 | s1,s2: |
---|
1495 | s1*s1 =0, s2*s2 = s1*s2*s1, s(1)*s(3)== s(1)*s(1)*s(3) == 0 = 2*s(4) ->def for s(4) |
---|
1496 | hence 2==0! only in char 2 |
---|
1497 | */ |
---|
1498 | |
---|
1499 | proc adem2mod(int n) |
---|
1500 | { |
---|
1501 | // Adem rels modulo 2 |
---|
1502 | } |
---|
1503 | |
---|
1504 | proc stringpoly2lplace(string s) |
---|
1505 | { |
---|
1506 | // decomposes sentence into terms |
---|
1507 | s = replace(s,newline,""); // get rid of newlines |
---|
1508 | s = replace(s," ",""); // get rid of empties |
---|
1509 | //arith symbols: +,- |
---|
1510 | // decompose into words with coeffs |
---|
1511 | list LS; |
---|
1512 | int i,j,ie,je,k,cnt; |
---|
1513 | // s[1]="-" situation |
---|
1514 | if (s[1]=="-") |
---|
1515 | { |
---|
1516 | LS = stringpoly2lplace(string(s[2..size(s)])); |
---|
1517 | LS[1] = string("-"+string(LS[1])); |
---|
1518 | return(LS); |
---|
1519 | } |
---|
1520 | i = find(s,"-",2); |
---|
1521 | // i==1 might happen if the 1st symbol coeff is negative |
---|
1522 | j = find(s,"+"); |
---|
1523 | list LL; |
---|
1524 | if (i==j) |
---|
1525 | { |
---|
1526 | "return a monomial"; |
---|
1527 | // that is both are 0 -> s is a monomial |
---|
1528 | LS[1] = s; |
---|
1529 | return(LS); |
---|
1530 | } |
---|
1531 | if (i==0) |
---|
1532 | { |
---|
1533 | "i==0 situation"; |
---|
1534 | // no minuses at all => pluses only |
---|
1535 | cnt++; |
---|
1536 | LS[cnt] = string(s[1..j-1]); |
---|
1537 | s = s[j+1..size(s)]; |
---|
1538 | while (s!= "") |
---|
1539 | { |
---|
1540 | j = find(s,"+"); |
---|
1541 | cnt++; |
---|
1542 | if (j==0) |
---|
1543 | { |
---|
1544 | LS[cnt] = string(s); |
---|
1545 | s = ""; |
---|
1546 | } |
---|
1547 | else |
---|
1548 | { |
---|
1549 | LS[cnt] = string(s[1..j-1]); |
---|
1550 | s = s[j+1..size(s)]; |
---|
1551 | } |
---|
1552 | } |
---|
1553 | return(LS); |
---|
1554 | } |
---|
1555 | if (j==0) |
---|
1556 | { |
---|
1557 | "j==0 situation"; |
---|
1558 | // no pluses at all except the lead coef => the rest are minuses only |
---|
1559 | cnt++; |
---|
1560 | LS[cnt] = string(s[1..i-1]); |
---|
1561 | s = s[i..size(s)]; |
---|
1562 | while (s!= "") |
---|
1563 | { |
---|
1564 | i = find(s,"-",2); |
---|
1565 | cnt++; |
---|
1566 | if (i==0) |
---|
1567 | { |
---|
1568 | LS[cnt] = string(s); |
---|
1569 | s = ""; |
---|
1570 | } |
---|
1571 | else |
---|
1572 | { |
---|
1573 | LS[cnt] = string(s[1..i-1]); |
---|
1574 | s = s[i..size(s)]; |
---|
1575 | } |
---|
1576 | } |
---|
1577 | return(LS); |
---|
1578 | } |
---|
1579 | // now i, j are nonzero |
---|
1580 | if (i>j) |
---|
1581 | { |
---|
1582 | "i>j situation"; |
---|
1583 | // + comes first, at place j |
---|
1584 | cnt++; |
---|
1585 | // "cnt:"; cnt; "j:"; j; |
---|
1586 | LS[cnt] = string(s[1..j-1]); |
---|
1587 | s = s[j+1..size(s)]; |
---|
1588 | LL = stringpoly2lplace(s); |
---|
1589 | LS = LS + LL; |
---|
1590 | kill LL; |
---|
1591 | return(LS); |
---|
1592 | } |
---|
1593 | else |
---|
1594 | { |
---|
1595 | "j>i situation"; |
---|
1596 | // - might come first, at place i |
---|
1597 | if (i>1) |
---|
1598 | { |
---|
1599 | cnt++; |
---|
1600 | LS[cnt] = string(s[1..i-1]); |
---|
1601 | s = s[i..size(s)]; |
---|
1602 | } |
---|
1603 | else |
---|
1604 | { |
---|
1605 | // i==1-> minus at leadcoef |
---|
1606 | ie = find(s,"-",i+1); |
---|
1607 | je = find(s,"+",i+1); |
---|
1608 | if (je == ie) |
---|
1609 | { |
---|
1610 | "ie=je situation"; |
---|
1611 | //monomial |
---|
1612 | cnt++; |
---|
1613 | LS[cnt] = s; |
---|
1614 | return(LS); |
---|
1615 | } |
---|
1616 | if (je < ie) |
---|
1617 | { |
---|
1618 | "je<ie situation"; |
---|
1619 | // + comes first |
---|
1620 | cnt++; |
---|
1621 | LS[cnt] = s[1..je-1]; |
---|
1622 | s = s[je+1..size(s)]; |
---|
1623 | } |
---|
1624 | else |
---|
1625 | { |
---|
1626 | // ie < je |
---|
1627 | "ie<je situation"; |
---|
1628 | cnt++; |
---|
1629 | LS[cnt] = s[1..ie-1]; |
---|
1630 | s = s[ie..size(s)]; |
---|
1631 | } |
---|
1632 | } |
---|
1633 | "going into recursion with "+s; |
---|
1634 | LL = stringpoly2lplace(s); |
---|
1635 | LS = LS + LL; |
---|
1636 | return(LS); |
---|
1637 | } |
---|
1638 | } |
---|
1639 | example |
---|
1640 | { |
---|
1641 | "EXAMPLE:"; echo = 2; |
---|
1642 | string s = "x*y+y*z+z*t"; // + only |
---|
1643 | stringpoly2lplace(s); |
---|
1644 | string s2 = "x*y - y*z-z*t*w*w"; // +1, - only |
---|
1645 | stringpoly2lplace(s2); |
---|
1646 | string s3 = "-x*y + y - 2*x +7*w*w*w"; |
---|
1647 | stringpoly2lplace(s3); |
---|
1648 | } |
---|
1649 | |
---|
1650 | proc addplaces(list L) |
---|
1651 | { |
---|
1652 | // adds places to the list of strings |
---|
1653 | // according to their order in the list |
---|
1654 | int s = size(L); |
---|
1655 | int i; |
---|
1656 | for (i=1; i<=s; i++) |
---|
1657 | { |
---|
1658 | if (typeof(L[i]) == "string") |
---|
1659 | { |
---|
1660 | L[i] = L[i] + "(" + string(i) + ")"; |
---|
1661 | } |
---|
1662 | else |
---|
1663 | { |
---|
1664 | ERROR("entry of type string expected"); |
---|
1665 | return(0); |
---|
1666 | } |
---|
1667 | } |
---|
1668 | return(L); |
---|
1669 | } |
---|
1670 | example |
---|
1671 | { |
---|
1672 | "EXAMPLE:"; echo = 2; |
---|
1673 | string a = "f1"; string b = "f2"; |
---|
1674 | list L = a,b,a; |
---|
1675 | addplaces(L); |
---|
1676 | } |
---|
1677 | |
---|
1678 | proc sent2lplace(string s) |
---|
1679 | { |
---|
1680 | // SENTence of words TO LetterPLACE |
---|
1681 | list L = stringpoly2lplace(s); |
---|
1682 | int i; int ss = size(L); |
---|
1683 | for(i=1; i<=ss; i++) |
---|
1684 | { |
---|
1685 | L[i] = str2lplace(L[i]); |
---|
1686 | } |
---|
1687 | return(L); |
---|
1688 | } |
---|
1689 | example |
---|
1690 | { |
---|
1691 | "EXAMPLE:"; echo = 2; |
---|
1692 | ring r = 0,(f2,f1),dp; |
---|
1693 | string s = "f2*f1*f1 - 2*f1*f2*f1+ f1*f1*f2"; |
---|
1694 | sent2lplace(s); |
---|
1695 | } |
---|
1696 | |
---|
1697 | proc testnumber(string s) |
---|
1698 | { |
---|
1699 | string t; |
---|
1700 | if (s[1]=="-") |
---|
1701 | { |
---|
1702 | // two situations: either there's a negative number |
---|
1703 | t = s[2..size(s)]; |
---|
1704 | if (testnumber(t)) |
---|
1705 | { |
---|
1706 | //a negative number |
---|
1707 | } |
---|
1708 | else |
---|
1709 | { |
---|
1710 | // a variable times -1 |
---|
1711 | } |
---|
1712 | // or just a "-" for -1 |
---|
1713 | } |
---|
1714 | t = "ring @r=("; |
---|
1715 | t = t + charstr(basering)+"),"; |
---|
1716 | t = t + string(var(1))+",dp;"; |
---|
1717 | // write(":w tstnum.tst",t); |
---|
1718 | t = t+ "number @@Nn = " + s + ";"+"$"; |
---|
1719 | write(":w tstnum.tst",t); |
---|
1720 | string runsing = system("Singular"); |
---|
1721 | int k; |
---|
1722 | t = runsing+ " -teq <tstnum.tst >tstnum.out"; |
---|
1723 | k = system("sh",t); |
---|
1724 | if (k!=0) |
---|
1725 | { |
---|
1726 | ERROR("Problems running Singular"); |
---|
1727 | } |
---|
1728 | int i = system("sh", "grep error tstnum.out > /dev/NULL"); |
---|
1729 | if (i!=0) |
---|
1730 | { |
---|
1731 | // no error: s is a number |
---|
1732 | i = 1; |
---|
1733 | } |
---|
1734 | k = system("sh","rm tstnum.tst tstnum.out > /dev/NULL"); |
---|
1735 | return(i); |
---|
1736 | } |
---|
1737 | example |
---|
1738 | { |
---|
1739 | "EXAMPLE:"; echo = 2; |
---|
1740 | ring r = (0,a),x,dp; |
---|
1741 | string s = "a^2+7*a-2"; |
---|
1742 | testnumber(s); |
---|
1743 | s = "b+a"; |
---|
1744 | testnumber(s); |
---|
1745 | } |
---|
1746 | |
---|
1747 | proc str2lplace(string s) |
---|
1748 | { |
---|
1749 | // converts a word (monomial) with coeff into letter-place |
---|
1750 | // string: coef*var1^exp1*var2^exp2*...varN^expN |
---|
1751 | s = strpower2rep(s); // expand powers |
---|
1752 | if (size(s)==0) { return(0); } |
---|
1753 | int i,j,k,insC; |
---|
1754 | string a,b,c,d,t; |
---|
1755 | // 1. get coeff |
---|
1756 | i = find(s,"*"); |
---|
1757 | if (i==0) { return(s); } |
---|
1758 | list VN; |
---|
1759 | c = s[1..i-1]; // incl. the case like (-a^2+1) |
---|
1760 | int tn = testnumber(c); |
---|
1761 | if (tn == 0) |
---|
1762 | { |
---|
1763 | // failed test |
---|
1764 | if (c[1]=="-") |
---|
1765 | { |
---|
1766 | // two situations: either there's a negative number |
---|
1767 | t = c[2..size(c)]; |
---|
1768 | if (testnumber(t)) |
---|
1769 | { |
---|
1770 | //a negative number |
---|
1771 | // nop here |
---|
1772 | } |
---|
1773 | else |
---|
1774 | { |
---|
1775 | // a variable times -1 |
---|
1776 | c = "-1"; |
---|
1777 | j++; VN[j] = t; //string(c[2..size(c)]); |
---|
1778 | insC = 1; |
---|
1779 | } |
---|
1780 | } |
---|
1781 | else |
---|
1782 | { |
---|
1783 | // just a variable with coeff 1 |
---|
1784 | j++; VN[j] = string(c); |
---|
1785 | c = "1"; |
---|
1786 | insC = 1; |
---|
1787 | } |
---|
1788 | } |
---|
1789 | // get vars |
---|
1790 | t = s; |
---|
1791 | // t = s[i+1..size(s)]; |
---|
1792 | k = size(t); //j = 0; |
---|
1793 | while (k>0) |
---|
1794 | { |
---|
1795 | t = t[i+1..size(t)]; //next part |
---|
1796 | i = find(t,"*"); // next * |
---|
1797 | if (i==0) |
---|
1798 | { |
---|
1799 | // last monomial |
---|
1800 | j++; |
---|
1801 | VN[j] = t; |
---|
1802 | k = size(t); |
---|
1803 | break; |
---|
1804 | } |
---|
1805 | b = t[1..i-1]; |
---|
1806 | // print(b); |
---|
1807 | j++; |
---|
1808 | VN[j] = b; |
---|
1809 | k = size(t); |
---|
1810 | } |
---|
1811 | VN = addplaces(VN); |
---|
1812 | VN[size(VN)+1] = string(c); |
---|
1813 | return(VN); |
---|
1814 | } |
---|
1815 | example |
---|
1816 | { |
---|
1817 | "EXAMPLE:"; echo = 2; |
---|
1818 | ring r = (0,a),(f2,f1),dp; |
---|
1819 | str2lplace("-2*f2^2*f1^2*f2"); |
---|
1820 | str2lplace("-f1*f2"); |
---|
1821 | str2lplace("(-a^2+7a)*f1*f2"); |
---|
1822 | } |
---|
1823 | |
---|
1824 | proc strpower2rep(string s) |
---|
1825 | { |
---|
1826 | // makes x*x*x*x out of x^4 ., rep statys for repetitions |
---|
1827 | // looks for "-" problem |
---|
1828 | // exception: "-" as coeff |
---|
1829 | string ex,t; |
---|
1830 | int i,j,k; |
---|
1831 | |
---|
1832 | i = find(s,"^"); // first ^ |
---|
1833 | if (i==0) { return(s); } // no ^ signs |
---|
1834 | |
---|
1835 | if (s[1] == "-") |
---|
1836 | { |
---|
1837 | // either -coef or -1 |
---|
1838 | // got the coeff: |
---|
1839 | i = find(s,"*"); |
---|
1840 | if (i==0) |
---|
1841 | { |
---|
1842 | // no *'s => coef == -1 or s == -23 |
---|
1843 | i = size(s)+1; |
---|
1844 | } |
---|
1845 | t = string(s[2..i-1]); // without "-" |
---|
1846 | if ( testnumber(t) ) |
---|
1847 | { |
---|
1848 | // a good number |
---|
1849 | t = strpower2rep(string(s[2..size(s)])); |
---|
1850 | t = "-"+t; |
---|
1851 | return(t); |
---|
1852 | } |
---|
1853 | else |
---|
1854 | { |
---|
1855 | // a variable |
---|
1856 | t = strpower2rep(string(s[2..size(s)])); |
---|
1857 | t = "-1*"+ t; |
---|
1858 | return(t); |
---|
1859 | } |
---|
1860 | } |
---|
1861 | // the case when leadcoef is a number in () |
---|
1862 | if (s[1] == "(") |
---|
1863 | { |
---|
1864 | i = find(s,")",2); // must be nonzero |
---|
1865 | t = s[2..i-1]; |
---|
1866 | if ( testnumber(t) ) |
---|
1867 | { |
---|
1868 | // a good number |
---|
1869 | } |
---|
1870 | else {"strpower2rep: bad number as coef";} |
---|
1871 | ex = string(s[i+2..size(s)]); // 2 because of * |
---|
1872 | ex = strpower2rep(ex); |
---|
1873 | t = "("+t+")*"+ex; |
---|
1874 | return(t); |
---|
1875 | } |
---|
1876 | |
---|
1877 | i = find(s,"^"); // first ^ |
---|
1878 | j = find(s,"*",i+1); // next * == end of ^ |
---|
1879 | if (j==0) |
---|
1880 | { |
---|
1881 | ex = s[i+1..size(s)]; |
---|
1882 | } |
---|
1883 | else |
---|
1884 | { |
---|
1885 | ex = s[i+1..j-1]; |
---|
1886 | } |
---|
1887 | execute("int @exp = " + ex + ";"); //@exp = exponent |
---|
1888 | // got varname |
---|
1889 | for (k=i-1; k>0; k--) |
---|
1890 | { |
---|
1891 | if (s[k] == "*") break; |
---|
1892 | } |
---|
1893 | string varn = s[k+1..i-1]; |
---|
1894 | // "varn:"; varn; |
---|
1895 | string pref; |
---|
1896 | if (k>0) |
---|
1897 | { |
---|
1898 | pref = s[1..k]; // with * on the k-th place |
---|
1899 | } |
---|
1900 | // "pref:"; pref; |
---|
1901 | string suf; |
---|
1902 | if ( (j>0) && (j+1 <= size(s)) ) |
---|
1903 | { |
---|
1904 | suf = s[j+1..size(s)]; // without * on the 1st place |
---|
1905 | } |
---|
1906 | // "suf:"; suf; |
---|
1907 | string toins; |
---|
1908 | for (k=1; k<=@exp; k++) |
---|
1909 | { |
---|
1910 | toins = toins + varn+"*"; |
---|
1911 | } |
---|
1912 | // "toins: "; toins; |
---|
1913 | if (size(suf) == 0) |
---|
1914 | { |
---|
1915 | toins = toins[1..size(toins)-1]; // get rid of trailing * |
---|
1916 | } |
---|
1917 | else |
---|
1918 | { |
---|
1919 | suf = strpower2rep(suf); |
---|
1920 | } |
---|
1921 | ex = pref + toins + suf; |
---|
1922 | return(ex); |
---|
1923 | // return(strpower2rep(ex)); |
---|
1924 | } |
---|
1925 | example |
---|
1926 | { |
---|
1927 | "EXAMPLE:"; echo = 2; |
---|
1928 | ring r = (0,a),(x,y,z,t),dp; |
---|
1929 | strpower2rep("-x^4"); |
---|
1930 | strpower2rep("-2*x^4*y^3*z*t^2"); |
---|
1931 | strpower2rep("-a^2*x^4"); |
---|
1932 | } |
---|
1933 | |
---|
1934 | proc Liebr(poly a, poly b, list #) |
---|
1935 | { |
---|
1936 | // alias ppLiebr; |
---|
1937 | //if int N is given compute [a,[...[a,b]]]] left normed bracket |
---|
1938 | poly q; |
---|
1939 | int N=1; |
---|
1940 | if (size(#)>0) |
---|
1941 | { |
---|
1942 | if (typeof(#[1])=="int") |
---|
1943 | { |
---|
1944 | N = int(#[1]); |
---|
1945 | } |
---|
1946 | } |
---|
1947 | if (N<=0) { return(q); } |
---|
1948 | while (b!=0) |
---|
1949 | { |
---|
1950 | q = q + pmLiebr(a,lead(b)); |
---|
1951 | b = b - lead(b); |
---|
1952 | } |
---|
1953 | int i; |
---|
1954 | if (N >1) |
---|
1955 | { |
---|
1956 | for(i=1; i<=N; i++) |
---|
1957 | { |
---|
1958 | q = Liebr(a,q); |
---|
1959 | } |
---|
1960 | } |
---|
1961 | return(q); |
---|
1962 | } |
---|
1963 | example |
---|
1964 | { |
---|
1965 | "EXAMPLE:"; echo = 2; |
---|
1966 | ring r = 0,(x(1),y(1),x(2),y(2),x(3),y(3),x(4),y(4)),dp; |
---|
1967 | poly a = x(1)*y(2); poly b = y(1); |
---|
1968 | int uptodeg=4; int lV=2; |
---|
1969 | export uptodeg; export lV; |
---|
1970 | Liebr(a,b); |
---|
1971 | Liebr(x(1),y(1),2); |
---|
1972 | } |
---|
1973 | |
---|
1974 | proc pmLiebr(poly a, poly b) |
---|
1975 | { |
---|
1976 | // a poly, b mono |
---|
1977 | poly s; |
---|
1978 | while (a!=0) |
---|
1979 | { |
---|
1980 | s = s + mmLiebr(lead(a),lead(b)); |
---|
1981 | a = a - lead(a); |
---|
1982 | } |
---|
1983 | return(s); |
---|
1984 | } |
---|
1985 | |
---|
1986 | //proc pshift(poly a, int i, int uptodeg, int lV) |
---|
1987 | proc pshift(poly a, int i) |
---|
1988 | { |
---|
1989 | // shifts a monomial a by i |
---|
1990 | // calls pLPshift(p,sh,uptodeg,lVblock); |
---|
1991 | return(system("stest",a,i,uptodeg,lV)); |
---|
1992 | } |
---|
1993 | |
---|
1994 | proc mmLiebr(poly a, poly b) |
---|
1995 | { |
---|
1996 | // a,b, monomials |
---|
1997 | a = lead(a); |
---|
1998 | b = lead(b); |
---|
1999 | int sa = deg(a); |
---|
2000 | int sb = deg(b); |
---|
2001 | poly v = a*pshift(b,sa) - b*pshift(a,sb); |
---|
2002 | return(v); |
---|
2003 | } |
---|
2004 | |
---|
2005 | static proc test_shift() |
---|
2006 | { |
---|
2007 | LIB "freegb.lib"; |
---|
2008 | ring r = 0,(a,b),dp; |
---|
2009 | int d =5; |
---|
2010 | def R = freegbRing(d); |
---|
2011 | setring R; |
---|
2012 | int uptodeg = d; export uptodeg; |
---|
2013 | int lV = 2; export lV; |
---|
2014 | poly p = mmLiebr(a(1),b(1)); |
---|
2015 | poly p = Liebr(a(1),b(1)); |
---|
2016 | } |
---|
2017 | |
---|
2018 | proc Serre(intmat A, int zu) |
---|
2019 | { |
---|
2020 | // zu = 1 -> with commutators [f_i,f_j]; zu == 0 without them |
---|
2021 | // suppose that A is cartan matrix |
---|
2022 | // then Serre's relations are |
---|
2023 | // (ad f_j)^{1-A_{ij}} ( f_i) |
---|
2024 | int ppl = printlevel-voice+2; |
---|
2025 | int n = ncols(A); // hence n variables |
---|
2026 | int i,j,k,l; |
---|
2027 | poly p,q; |
---|
2028 | ideal I; |
---|
2029 | for (i=1; i<=n; i++) |
---|
2030 | { |
---|
2031 | for (j=1; j<=n; j++) |
---|
2032 | { |
---|
2033 | l = 1 - A[i,j]; |
---|
2034 | // printf("i:%s, j: %s, l: %s",i,j,l); |
---|
2035 | dbprint(ppl,"i, j, l: ",i,j,l); |
---|
2036 | // if ((i!=j) && (l >0)) |
---|
2037 | // if ( (i!=j) && ( ((zu ==0) && (l >=2)) || ((zu ==1) && (l >=1)) ) ) |
---|
2038 | if ((i!=j) && (l >0)) |
---|
2039 | { |
---|
2040 | q = Liebr(var(j),var(i)); |
---|
2041 | // printf("first bracket: %s",q); |
---|
2042 | dbprint(ppl,"first bracket: ",q); |
---|
2043 | // if (l >=2) |
---|
2044 | // { |
---|
2045 | for (k=1; k<=l-1; k++) |
---|
2046 | { |
---|
2047 | q = Liebr(var(j),q); |
---|
2048 | // printf("further bracket: %s",q); |
---|
2049 | dbprint(ppl,"further bracket:",q); |
---|
2050 | } |
---|
2051 | // } |
---|
2052 | } |
---|
2053 | if (q!=0) { I = I,q; q=0;} |
---|
2054 | } |
---|
2055 | } |
---|
2056 | I = simplify(I,2); |
---|
2057 | return(I); |
---|
2058 | } |
---|
2059 | example |
---|
2060 | { |
---|
2061 | "EXAMPLE:"; echo = 2; |
---|
2062 | intmat A[2][2] = 2, -1, -1, 2; // sl_3 == A_2 |
---|
2063 | ring r = 0,(f1,f2),dp; |
---|
2064 | int uptodeg = 3; int lV = 2; |
---|
2065 | export uptodeg; export lV; |
---|
2066 | def R = freegbRing(uptodeg); |
---|
2067 | setring R; |
---|
2068 | ideal I = Serre(A,1); |
---|
2069 | I; |
---|
2070 | Serre(A,0); |
---|
2071 | } |
---|
2072 | |
---|
2073 | proc lp2lstr(ideal K, def save) |
---|
2074 | "USAGE: lp2lstr(K,save); K an ideal, save a ring |
---|
2075 | RETURN: nothing (exports object LN into save) |
---|
2076 | PURPOSE: converts letter-place ideal to list of modules |
---|
2077 | EXAMPLE: example lp2lstr; shows examples |
---|
2078 | " |
---|
2079 | { |
---|
2080 | def @R = basering; |
---|
2081 | string err; |
---|
2082 | int s = nvars(save); |
---|
2083 | int i,j,k; |
---|
2084 | // K contains vars x(1),...z(1) = images of originals |
---|
2085 | // 5. go back to orig vars, produce strings/modules |
---|
2086 | int sk = size(K); |
---|
2087 | int sp, sx, a, b; |
---|
2088 | intvec x; |
---|
2089 | poly p,q; |
---|
2090 | poly pn; |
---|
2091 | // vars in 'save' |
---|
2092 | setring save; |
---|
2093 | module N; |
---|
2094 | list LN; |
---|
2095 | vector V; |
---|
2096 | poly pn; |
---|
2097 | // test and skip exponents >=2 |
---|
2098 | setring @R; |
---|
2099 | for(i=1; i<=sk; i++) |
---|
2100 | { |
---|
2101 | p = K[i]; |
---|
2102 | while (p!=0) |
---|
2103 | { |
---|
2104 | q = lead(p); |
---|
2105 | // "processing q:";q; |
---|
2106 | x = leadexp(q); |
---|
2107 | sx = size(x); |
---|
2108 | for(k=1; k<=sx; k++) |
---|
2109 | { |
---|
2110 | if ( x[k] >= 2 ) |
---|
2111 | { |
---|
2112 | err = "skip: the value x[k] is " + string(x[k]); |
---|
2113 | dbprint(ppl,err); |
---|
2114 | // return(0); |
---|
2115 | K[i] = 0; |
---|
2116 | p = 0; |
---|
2117 | q = 0; |
---|
2118 | break; |
---|
2119 | } |
---|
2120 | } |
---|
2121 | p = p - q; |
---|
2122 | } |
---|
2123 | } |
---|
2124 | K = simplify(K,2); |
---|
2125 | sk = size(K); |
---|
2126 | for(i=1; i<=sk; i++) |
---|
2127 | { |
---|
2128 | // setring save; |
---|
2129 | // V = 0; |
---|
2130 | setring @R; |
---|
2131 | p = K[i]; |
---|
2132 | while (p!=0) |
---|
2133 | { |
---|
2134 | q = lead(p); |
---|
2135 | err = "processing q:" + string(q); |
---|
2136 | dbprint(ppl,err); |
---|
2137 | x = leadexp(q); |
---|
2138 | sx = size(x); |
---|
2139 | pn = leadcoef(q); |
---|
2140 | setring save; |
---|
2141 | pn = imap(@R,pn); |
---|
2142 | V = V + leadcoef(pn)*gen(1); |
---|
2143 | for(k=1; k<=sx; k++) |
---|
2144 | { |
---|
2145 | if (x[k] ==1) |
---|
2146 | { |
---|
2147 | a = k / s; // block number=a+1, a!=0 |
---|
2148 | b = k % s; // remainder |
---|
2149 | // printf("a: %s, b: %s",a,b); |
---|
2150 | if (b == 0) |
---|
2151 | { |
---|
2152 | // that is it's the last var in the block |
---|
2153 | b = s; |
---|
2154 | a = a-1; |
---|
2155 | } |
---|
2156 | V = V + var(b)*gen(a+2); |
---|
2157 | } |
---|
2158 | } |
---|
2159 | err = "V: " + string(V); |
---|
2160 | dbprint(ppl,err); |
---|
2161 | // printf("V: %s", string(V)); |
---|
2162 | N = N,V; |
---|
2163 | V = 0; |
---|
2164 | setring @R; |
---|
2165 | p = p - q; |
---|
2166 | pn = 0; |
---|
2167 | } |
---|
2168 | setring save; |
---|
2169 | LN[i] = simplify(N,2); |
---|
2170 | N = 0; |
---|
2171 | } |
---|
2172 | setring save; |
---|
2173 | export LN; |
---|
2174 | // return(LN); |
---|
2175 | } |
---|
2176 | example |
---|
2177 | { |
---|
2178 | "EXAMPLE:"; echo = 2; |
---|
2179 | intmat A[2][2] = 2, -1, -1, 2; // sl_3 == A_2 |
---|
2180 | ring r = 0,(f1,f2),dp; |
---|
2181 | int uptodeg = 3; int lV = 2; |
---|
2182 | export uptodeg; export lV; |
---|
2183 | def R = freegbRing(uptodeg); |
---|
2184 | setring R; |
---|
2185 | ideal I = Serre(A,1); |
---|
2186 | lp2lstr(I,r); |
---|
2187 | setring r; |
---|
2188 | lst2str(LN,1); |
---|
2189 | kill uptodeg; kill lV; |
---|
2190 | } |
---|
2191 | |
---|
2192 | proc strList2poly(list L) |
---|
2193 | { |
---|
2194 | // list L comes from sent2lplace (which takes a poly on the input) |
---|
2195 | // each entry of L is a sublist with the coef on the last place |
---|
2196 | int s = size(L); int t; |
---|
2197 | int i,j; |
---|
2198 | list M; |
---|
2199 | poly p,q; |
---|
2200 | string Q; |
---|
2201 | for(i=1; i<=s; i++) |
---|
2202 | { |
---|
2203 | M = L[i]; |
---|
2204 | t = size(M); |
---|
2205 | // q = M[t]; // a constant |
---|
2206 | Q = string(M[t]); |
---|
2207 | for(j=1; j<t; j++) |
---|
2208 | { |
---|
2209 | // q = q*M[j]; |
---|
2210 | Q = Q+"*"+string(M[j]); |
---|
2211 | } |
---|
2212 | execute("q="+Q+";"); |
---|
2213 | // q; |
---|
2214 | p = p + q; |
---|
2215 | } |
---|
2216 | kill Q; |
---|
2217 | return(p); |
---|
2218 | } |
---|
2219 | example |
---|
2220 | { |
---|
2221 | "EXAMPLE:"; echo = 2; |
---|
2222 | ring r =0,(x,y,z,t),Dp; |
---|
2223 | def A = freegbRing(4); |
---|
2224 | setring A; |
---|
2225 | string t = "-2*y*z*y*z + y*t*z*z - z*x*x*y + 2*z*y*z*y"; |
---|
2226 | list L = sent2lplace(t); |
---|
2227 | L; |
---|
2228 | poly p = strList2poly(L); |
---|
2229 | p; |
---|
2230 | } |
---|
2231 | |
---|
2232 | proc file2lplace(string fname) |
---|
2233 | { |
---|
2234 | // format: from the usual string to letterplace |
---|
2235 | string s = read(fname); |
---|
2236 | // assume: file is a comma-sep list of polys |
---|
2237 | // the vars are declared before |
---|
2238 | // the file ends with ";" |
---|
2239 | string t; int i; |
---|
2240 | ideal I; |
---|
2241 | list tst; |
---|
2242 | while (s!="") |
---|
2243 | { |
---|
2244 | i = find(s,","); |
---|
2245 | "i"; i; |
---|
2246 | if (i==0) |
---|
2247 | { |
---|
2248 | i = find(s,";"); |
---|
2249 | if (i==0) |
---|
2250 | { |
---|
2251 | // no ; ?? |
---|
2252 | "no colon or semicolon found anymore"; |
---|
2253 | return(I); |
---|
2254 | } |
---|
2255 | // no "," but ";" on the i-th place |
---|
2256 | t = s[1..i-1]; |
---|
2257 | s = ""; |
---|
2258 | "processing: "; t; |
---|
2259 | tst = sent2lplace(t); |
---|
2260 | tst; |
---|
2261 | I = I, strList2poly(tst); |
---|
2262 | return(I); |
---|
2263 | } |
---|
2264 | // here i !=0 |
---|
2265 | t = s[1..i-1]; |
---|
2266 | s = s[i+1..size(s)]; |
---|
2267 | "processing: "; t; |
---|
2268 | tst = sent2lplace(t); |
---|
2269 | tst; |
---|
2270 | I = I, strList2poly(tst); |
---|
2271 | } |
---|
2272 | return(I); |
---|
2273 | } |
---|
2274 | example |
---|
2275 | { |
---|
2276 | "EXAMPLE:"; echo = 2; |
---|
2277 | ring r =0,(x,y,z,t),dp; |
---|
2278 | def A = freegbRing(4); |
---|
2279 | setring A; |
---|
2280 | string fn = "myfile"; |
---|
2281 | string s1 = "z*y*y*y - 3*y*z*x*y + 3*y*y*z*y - y*x*y*z,"; |
---|
2282 | string s2 = "-2*y*x*y*z + y*y*z*z - z*z*y*y + 2*z*y*z*y,"; |
---|
2283 | string s3 = "z*y*x*t - 2*y*z*y*t + y*y*z*t - t*z*y*y + 2*t*y*z*y - t*x*y*z;"; |
---|
2284 | write(":w "+fn,s1); write(":a "+fn,s2); write(":a "+fn,s3); |
---|
2285 | read(fn); |
---|
2286 | ideal I = file2lplace(fn); |
---|
2287 | I; |
---|
2288 | } |
---|
2289 | |
---|
2290 | static proc get_ls3nilp() |
---|
2291 | { |
---|
2292 | //first app of file2lplace |
---|
2293 | ring r =0,(x,y,z,t),dp; |
---|
2294 | int d = 10; |
---|
2295 | def A = freegbRing(d); |
---|
2296 | setring A; |
---|
2297 | ideal I = file2lplace("./ls3nilp.bg"); |
---|
2298 | // and now test the correctness: go back from lplace to strings |
---|
2299 | lp2lstr(I,r); |
---|
2300 | setring r; |
---|
2301 | lst2str(LN,1); // agree! |
---|
2302 | } |
---|
2303 | |
---|
2304 | static proc doc_example |
---|
2305 | { |
---|
2306 | LIB "freegb.lib"; |
---|
2307 | ring r = 0,(x,y,z),dp; |
---|
2308 | int d =4; // degree bound |
---|
2309 | def R = freegbRing(d); |
---|
2310 | setring R; |
---|
2311 | ideal I = x(1)*y(2) + y(1)*z(2), x(1)*x(2) + x(1)*y(2) - y(1)*x(2) - y(1)*y(2); |
---|
2312 | option(redSB);option(redTail); |
---|
2313 | ideal J = system("freegb",I,d,nvars(r)); |
---|
2314 | J; |
---|
2315 | // visualization: |
---|
2316 | lp2lstr(J,r); // export an object called LN to the ring r |
---|
2317 | setring r; // change to the ring r |
---|
2318 | lst2str(LN,1); // output the strings |
---|
2319 | } |
---|
2320 | |
---|
2321 | |
---|
2322 | |
---|
2323 | // TODO: |
---|
2324 | // multiply two letterplace polynomials, lpMult |
---|
2325 | // reduction/ Normalform? needs kernel stuff |
---|
2326 | |
---|