1 | ////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id: freegb.lib,v 1.20 2009-04-08 11:11:20 Singular 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 via Letterplace |
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6 | AUTHOR: Viktor Levandovskyy, levandov@math.rwth-aachen.de |
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7 | |
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8 | THEORY: See chapter 'LETTERPLACE' in the Singular Manual. |
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9 | |
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10 | PROCEDURES: |
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11 | makeLetterplaceRing(d); creates a ring with d blocks of shifted original variables |
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12 | freeGBasis(L, n); compute two-sided Groebner basis of ideal, encoded via L, up to degree n |
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13 | setLetterplaceAttributes(R,d,b); supplies ring R with the letterplace structure |
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14 | |
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15 | AUXILIARY PROCEDURES: |
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16 | |
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17 | lpMult(f,g); letterplace multiplication of letterplace polynomials |
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18 | lp2lstr(K, s); convert letter-place ideal to a list of modules |
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19 | lst2str(L[, n]); convert a list (of modules) into polynomials in free algebra |
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20 | mod2str(M[, n]); convert a module into a polynomial in free algebra |
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21 | vct2str(M[, n]); convert a vector into a word in free algebra |
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22 | lieBracket(a,b[, N]); compute Lie bracket ab-ba of two letterplace polynomials |
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23 | serreRelations(A,z); compute the ideal of Serre's relations associated to a generalized Cartan matrix A |
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24 | isVar(p); check whether p is a power of a single variable |
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25 | ademRelations(i,j); compute the ideal of Adem relations for i<2j in char 0 |
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26 | |
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27 | SEE ALSO: LETTERPLACE |
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28 | " |
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29 | |
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30 | // this library computes two-sided GB of an ideal |
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31 | // in a free associative algebra |
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32 | |
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33 | // a monomial is encoded via a vector V |
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34 | // where V[1] = coefficient |
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35 | // V[1+i] = the corresponding symbol |
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36 | |
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37 | LIB "qhmoduli.lib"; // for Max |
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38 | |
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39 | proc testfreegblib() |
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40 | { |
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41 | example makeLetterplaceRing; |
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42 | example freeGBasis; |
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43 | example setLetterplaceAttributes; |
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44 | "AUXILIARY PROCEDURES: "; |
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45 | example shiftPoly; |
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46 | example lpMult; |
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47 | example lp2lstr; |
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48 | example lst2str; |
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49 | example mod2str; |
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50 | example vct2str; |
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51 | example lieBracket; |
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52 | example serreRelations; |
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53 | example isVar; |
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54 | } |
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55 | |
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56 | |
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57 | proc setLetterplaceAttributes(def R, int uptodeg, int lV) |
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58 | "USAGE: setLetterplaceAttributes(R, d, b); R a ring, b,d integers |
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59 | RETURN: ring with special attributes set |
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60 | PURPOSE: sets attributes for a letterplace ring: |
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61 | @* 'isLetterplaceRing' = true, 'uptodeg' = d, 'lV' = b, where |
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62 | @* 'uptodeg' stands for the degree bound, |
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63 | @* 'lV' for the number of variables in the block 0. |
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64 | " |
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65 | { |
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66 | if (uptodeg*lV != nvars(R)) |
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67 | { |
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68 | ERROR("uptodeg and lV do not agree on the basering!"); |
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69 | } |
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70 | |
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71 | // Set letterplace-specific attributes for the output ring! |
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72 | attrib(R, "uptodeg", uptodeg); |
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73 | attrib(R, "lV", lV); |
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74 | attrib(R, "isLetterplaceRing", 1); |
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75 | return (R); |
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76 | } |
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77 | example |
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78 | { |
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79 | "EXAMPLE:"; echo = 2; |
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80 | ring r = 0,(x(1),y(1),x(2),y(2),x(3),y(3),x(4),y(4)),dp; |
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81 | def R = setLetterplaceAttributes(r, 4, 2); setring R; |
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82 | attrib(R,"isLetterplaceRing"); |
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83 | lieBracket(x(1),y(1),2); |
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84 | } |
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85 | |
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86 | |
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87 | // obsolete? |
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88 | |
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89 | static proc lshift(module M, int s, string varing, def lpring) |
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90 | { |
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91 | // FINALLY IMPLEMENTED AS A PART OT THE CODE |
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92 | // shifts a poly from the ring @R to s positions |
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93 | // M lives in varing, the result in lpring |
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94 | // to be run from varing |
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95 | int i, j, k, sm, sv; |
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96 | vector v; |
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97 | // execute("setring "+lpring); |
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98 | setring lpring; |
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99 | poly @@p; |
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100 | ideal I; |
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101 | execute("setring "+varing); |
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102 | sm = ncols(M); |
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103 | for (i=1; i<=s; i++) |
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104 | { |
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105 | // modules, e.g. free polynomials |
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106 | for (j=1; j<=sm; j++) |
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107 | { |
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108 | //vectors, e.g. free monomials |
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109 | v = M[j]; |
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110 | sv = size(v); |
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111 | sp = "@@p = @@p + "; |
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112 | for (k=2; k<=sv; k++) |
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113 | { |
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114 | sp = sp + string(v[k])+"("+string(k-1+s)+")*"; |
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115 | } |
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116 | sp = sp + string(v[1])+";"; // coef; |
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117 | setring lpring; |
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118 | // execute("setring "+lpring); |
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119 | execute(sp); |
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120 | execute("setring "+varing); |
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121 | } |
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122 | setring lpring; |
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123 | // execute("setring "+lpring); |
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124 | I = I,@@p; |
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125 | @@p = 0; |
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126 | } |
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127 | setring lpring; |
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128 | //execute("setring "+lpring); |
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129 | export(I); |
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130 | // setring varing; |
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131 | execute("setring "+varing); |
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132 | } |
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133 | |
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134 | static proc skip0(vector v) |
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135 | { |
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136 | // skips zeros in a vector, producing another vector |
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137 | if ( (v[1]==0) || (v==0) ) { return(vector(0)); } |
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138 | int sv = nrows(v); |
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139 | int sw = size(v); |
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140 | if (sv == sw) |
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141 | { |
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142 | return(v); |
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143 | } |
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144 | int i; |
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145 | int j=1; |
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146 | vector w; |
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147 | for (i=1; i<=sv; i++) |
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148 | { |
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149 | if (v[i] != 0) |
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150 | { |
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151 | w = w + v[i]*gen(j); |
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152 | j++; |
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153 | } |
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154 | } |
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155 | return(w); |
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156 | } |
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157 | |
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158 | proc lst2str(list L, list #) |
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159 | "USAGE: lst2str(L[,n]); L a list of modules, n an optional integer |
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160 | RETURN: list (of strings) |
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161 | PURPOSE: convert a list (of modules) into polynomials in free algebra |
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162 | EXAMPLE: example lst2str; shows examples |
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163 | NOTE: if an optional integer is not 0, stars signs are used in multiplication |
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164 | " |
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165 | { |
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166 | // returns a list of strings |
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167 | // being sentences in words built from L |
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168 | // if #[1] = 1, use * between generators |
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169 | int useStar = 0; |
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170 | if ( size(#)>0 ) |
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171 | { |
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172 | if ( typeof(#[1]) != "int") |
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173 | { |
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174 | ERROR("Second argument of type int expected"); |
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175 | } |
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176 | if (#[1]) |
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177 | { |
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178 | useStar = 1; |
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179 | } |
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180 | } |
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181 | int i; |
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182 | int s = size(L); |
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183 | if (s<1) { return(list(""));} |
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184 | list N; |
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185 | for(i=1; i<=s; i++) |
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186 | { |
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187 | if ((typeof(L[i]) == "module") || (typeof(L[i]) == "matrix") ) |
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188 | { |
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189 | N[i] = mod2str(L[i],useStar); |
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190 | } |
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191 | else |
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192 | { |
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193 | "module or matrix expected in the list"; |
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194 | return(N); |
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195 | } |
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196 | } |
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197 | return(N); |
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198 | } |
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199 | example |
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200 | { |
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201 | "EXAMPLE:"; echo = 2; |
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202 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
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203 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
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204 | module N = [1,x,y,x,y],[-2,y,x,y,x],[6,x,y,y,x,y]; |
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205 | list L; L[1] = M; L[2] = N; |
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206 | lst2str(L); |
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207 | lst2str(L[1],1); |
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208 | } |
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209 | |
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210 | |
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211 | proc mod2str(module M, list #) |
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212 | "USAGE: mod2str(M[,n]); M a module, n an optional integer |
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213 | RETURN: string |
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214 | PURPOSE: convert a module into a polynomial in free algebra |
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215 | EXAMPLE: example mod2str; shows examples |
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216 | NOTE: if an optional integer is not 0, stars signs are used in multiplication |
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217 | " |
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218 | { |
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219 | if (size(M)==0) { return(""); } |
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220 | // returns a string |
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221 | // a sentence in words built from M |
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222 | // if #[1] = 1, use * between generators |
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223 | int useStar = 0; |
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224 | if ( size(#)>0 ) |
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225 | { |
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226 | if ( typeof(#[1]) != "int") |
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227 | { |
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228 | ERROR("Second argument of type int expected"); |
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229 | } |
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230 | if (#[1]) |
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231 | { |
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232 | useStar = 1; |
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233 | } |
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234 | } |
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235 | int i; |
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236 | int s = ncols(M); |
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237 | string t; |
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238 | string mp; |
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239 | for(i=1; i<=s; i++) |
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240 | { |
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241 | mp = vct2str(M[i],useStar); |
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242 | if (mp[1] == "-") |
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243 | { |
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244 | t = t + mp; |
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245 | } |
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246 | else |
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247 | { |
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248 | if (mp != "") |
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249 | { |
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250 | t = t + "+" + mp; |
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251 | } |
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252 | } |
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253 | } |
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254 | if (t[1]=="+") |
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255 | { |
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256 | t = t[2..size(t)]; // remove first "+" |
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257 | } |
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258 | return(t); |
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259 | } |
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260 | example |
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261 | { |
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262 | "EXAMPLE:"; echo = 2; |
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263 | ring r = 0,(x,y,z),(dp); |
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264 | module M = [1,x,y,x,y],[-2,y,x,y,x],[6,x,y,y,x,y]; |
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265 | mod2str(M); |
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266 | mod2str(M,1); |
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267 | } |
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268 | |
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269 | proc vct2str(vector v, list #) |
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270 | "USAGE: vct2str(v[,n]); v a vector, n an optional integer |
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271 | RETURN: string |
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272 | PURPOSE: convert a vector into a word in free algebra |
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273 | EXAMPLE: example vct2str; shows examples |
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274 | NOTE: if an optional integer is not 0, stars signs are used in multiplication |
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275 | " |
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276 | { |
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277 | if (v==0) { return(""); } |
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278 | // if #[1] = 1, use * between generators |
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279 | int useStar = 0; |
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280 | if ( size(#)>0 ) |
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281 | { |
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282 | if (#[1]) |
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283 | { |
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284 | useStar = 1; |
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285 | } |
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286 | } |
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287 | int ppl = printlevel-voice+2; |
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288 | // for a word, encoded by v |
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289 | // produces a string for it |
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290 | v = skip0(v); |
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291 | if (v==0) { return(string(""));} |
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292 | number cf = leadcoef(v[1]); |
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293 | int s = size(v); |
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294 | string vs,vv,vp,err; |
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295 | int i,j,p,q; |
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296 | for (i=1; i<=s-1; i++) |
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297 | { |
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298 | p = isVar(v[i+1]); |
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299 | if (p==0) |
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300 | { |
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301 | err = "Error: monomial expected at nonzero position " + string(i+1); |
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302 | ERROR(err+" in vct2str"); |
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303 | // dbprint(ppl,err); |
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304 | // return("_"); |
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305 | } |
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306 | if (p==1) |
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307 | { |
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308 | if (useStar && (size(vs) >0)) { vs = vs + "*"; } |
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309 | vs = vs + string(v[i+1]); |
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310 | } |
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311 | else //power |
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312 | { |
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313 | vv = string(v[i+1]); |
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314 | q = find(vv,"^"); |
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315 | if (q==0) |
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316 | { |
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317 | q = find(vv,string(p)); |
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318 | if (q==0) |
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319 | { |
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320 | err = "error in find for string "+vv; |
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321 | dbprint(ppl,err); |
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322 | return("_"); |
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323 | } |
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324 | } |
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325 | // q>0 |
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326 | vp = vv[1..q-1]; |
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327 | for(j=1;j<=p;j++) |
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328 | { |
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329 | if (useStar && (size(vs) >0)) { vs = vs + "*"; } |
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330 | vs = vs + vp; |
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331 | } |
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332 | } |
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333 | } |
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334 | string scf; |
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335 | if (cf == -1) |
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336 | { |
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337 | scf = "-"; |
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338 | } |
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339 | else |
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340 | { |
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341 | scf = string(cf); |
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342 | if ( (cf == 1) && (size(vs)>0) ) |
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343 | { |
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344 | scf = ""; |
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345 | } |
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346 | } |
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347 | if (useStar && (size(scf) >0) && (scf!="-") ) { scf = scf + "*"; } |
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348 | vs = scf + vs; |
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349 | return(vs); |
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350 | } |
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351 | example |
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352 | { |
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353 | "EXAMPLE:"; echo = 2; |
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354 | ring r = (0,a),(x,y3,z(1)),dp; |
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355 | vector v = [-7,x,y3^4,x2,z(1)^3]; |
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356 | vct2str(v); |
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357 | vct2str(v,1); |
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358 | vector w = [-7a^5+6a,x,y3,y3,x,z(1),z(1)]; |
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359 | vct2str(w); |
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360 | vct2str(w,1); |
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361 | } |
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362 | |
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363 | proc isVar(poly p) |
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364 | "USAGE: isVar(p); poly p |
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365 | RETURN: int |
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366 | PURPOSE: check, whether leading monomial of p is a power of a single variable |
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367 | @* from the basering. Returns the exponent or 0 if p is multivariate. |
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368 | EXAMPLE: example isVar; shows examples |
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369 | " |
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370 | { |
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371 | // checks whether p is a variable indeed |
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372 | // if it's a power of a variable, returns the power |
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373 | if (p==0) { return(0); } //"p=0"; |
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374 | poly q = leadmonom(p); |
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375 | if ( (p-lead(p)) !=0 ) { return(0); } // "p-lm(p)>0"; |
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376 | intvec v = leadexp(p); |
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377 | int s = size(v); |
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378 | int i=1; |
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379 | int cnt = 0; |
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380 | int pwr = 0; |
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381 | for (i=1; i<=s; i++) |
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382 | { |
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383 | if (v[i] != 0) |
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384 | { |
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385 | cnt++; |
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386 | pwr = v[i]; |
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387 | } |
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388 | } |
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389 | // "cnt:"; cnt; |
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390 | if (cnt==1) { return(pwr); } |
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391 | else { return(0); } |
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392 | } |
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393 | example |
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394 | { |
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395 | "EXAMPLE:"; echo = 2; |
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396 | ring r = 0,(x,y),dp; |
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397 | poly f = xy+1; |
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398 | isVar(f); |
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399 | poly g = y^3; |
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400 | isVar(g); |
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401 | poly h = 7*x^3; |
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402 | isVar(h); |
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403 | poly i = 1; |
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404 | isVar(i); |
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405 | } |
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406 | |
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407 | // new conversion routines |
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408 | |
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409 | static proc id2words(ideal I, int d) |
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410 | { |
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411 | // NOT FINISHED |
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412 | // input: ideal I of polys in letter-place notation |
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413 | // in the ring with d real vars |
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414 | // output: the list of strings: associative words |
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415 | // extract names of vars |
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416 | int i,m,n; string s; string place = "(1)"; |
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417 | list lv; |
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418 | for(i=1; i<=d; i++) |
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419 | { |
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420 | s = string(var(i)); |
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421 | // get rid of place |
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422 | n = find(s, place); |
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423 | if (n>0) |
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424 | { |
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425 | s = s[1..n-1]; |
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426 | } |
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427 | lv[i] = s; |
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428 | } |
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429 | poly p,q; |
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430 | for (i=1; i<=ncols(I); i++) |
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431 | { |
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432 | if (I[i] != 0) |
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433 | { |
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434 | p = I[i]; |
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435 | while (p!=0) |
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436 | { |
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437 | q = leadmonom(p); |
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438 | } |
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439 | } |
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440 | } |
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441 | |
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442 | return(lv); |
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443 | } |
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444 | example |
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445 | { |
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446 | "EXAMPLE:"; echo = 2; |
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447 | ring r = 0,(x(1),y(1),z(1),x(2),y(2),z(2)),dp; |
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448 | ideal I = x(1)*y(2) -z(1)*x(2); |
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449 | id2words(I,3); |
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450 | } |
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451 | |
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452 | static proc mono2word(poly p, int d) |
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453 | { |
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454 | } |
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455 | |
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456 | // 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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457 | // use the orig ord on (x,y,z) and expand it blockwise to (x(i),y(i),z(i)) |
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458 | |
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459 | // the correspondences: |
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460 | // monomial in K<x,y,z> <<--->> vector in R |
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461 | // polynomial in K<x,y,z> <<--->> list of vectors (matrix/module) in R |
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462 | // ideal in K<x,y,z> <<--->> list of matrices/modules in R |
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463 | |
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464 | |
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465 | // 1. form a new ring |
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466 | // 2. NOP |
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467 | // 3. compute GB -> with the kernel stuff |
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468 | // 4. skip shifted elts (check that no such exist?) |
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469 | // 5. go back to orig vars, produce strings/modules |
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470 | // 6. return the result |
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471 | |
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472 | proc freeGBasis(list LM, int d) |
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473 | "USAGE: freeGBasis(L, d); L a list of modules, d an integer |
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474 | RETURN: ring |
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475 | PURPOSE: compute the two-sided Groebner basis of an ideal, encoded by L in |
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476 | the free associative algebra, up to degree d |
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477 | EXAMPLE: example freeGBasis; shows examples |
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478 | " |
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479 | { |
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480 | // d = up to degree, will be shifted to d+1 |
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481 | if (d<1) {"bad d"; return(0);} |
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482 | |
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483 | int ppl = printlevel-voice+2; |
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484 | string err = ""; |
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485 | |
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486 | int i,j,s; |
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487 | def save = basering; |
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488 | // determine max no of places in the input |
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489 | int slm = size(LM); // numbers of polys in the ideal |
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490 | int sm; |
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491 | intvec iv; |
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492 | module M; |
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493 | for (i=1; i<=slm; i++) |
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494 | { |
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495 | // modules, e.g. free polynomials |
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496 | M = LM[i]; |
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497 | sm = ncols(M); |
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498 | for (j=1; j<=sm; j++) |
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499 | { |
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500 | //vectors, e.g. free monomials |
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501 | iv = iv, size(M[j])-1; // 1 place is reserved by the coeff |
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502 | } |
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503 | } |
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504 | int D = Max(iv); // max size of input words |
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505 | if (d<D) {"bad d"; return(LM);} |
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506 | D = D + d-1; |
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507 | // D = d; |
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508 | list LR = ringlist(save); |
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509 | list L, tmp; |
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510 | L[1] = LR[1]; // ground field |
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511 | L[4] = LR[4]; // quotient ideal |
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512 | tmp = LR[2]; // varnames |
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513 | s = size(LR[2]); |
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514 | for (i=1; i<=D; i++) |
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515 | { |
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516 | for (j=1; j<=s; j++) |
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517 | { |
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518 | tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")"; |
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519 | } |
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520 | } |
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521 | for (i=1; i<=s; i++) |
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522 | { |
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523 | tmp[i] = string(tmp[i])+"("+string(1)+")"; |
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524 | } |
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525 | L[2] = tmp; |
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526 | list OrigNames = LR[2]; |
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527 | // ordering: d blocks of the ord on r |
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528 | // try to get whether the ord on r is blockord itself |
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529 | s = size(LR[3]); |
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530 | if (s==2) |
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531 | { |
---|
532 | // not a blockord, 1 block + module ord |
---|
533 | tmp = LR[3][s]; // module ord |
---|
534 | for (i=1; i<=D; i++) |
---|
535 | { |
---|
536 | LR[3][s-1+i] = LR[3][1]; |
---|
537 | } |
---|
538 | LR[3][s+D] = tmp; |
---|
539 | } |
---|
540 | if (s>2) |
---|
541 | { |
---|
542 | // there are s-1 blocks |
---|
543 | int nb = s-1; |
---|
544 | tmp = LR[3][s]; // module ord |
---|
545 | for (i=1; i<=D; i++) |
---|
546 | { |
---|
547 | for (j=1; j<=nb; j++) |
---|
548 | { |
---|
549 | LR[3][i*nb+j] = LR[3][j]; |
---|
550 | } |
---|
551 | } |
---|
552 | // size(LR[3]); |
---|
553 | LR[3][nb*(D+1)+1] = tmp; |
---|
554 | } |
---|
555 | L[3] = LR[3]; |
---|
556 | def @R = ring(L); |
---|
557 | setring @R; |
---|
558 | ideal I; |
---|
559 | poly @p; |
---|
560 | s = size(OrigNames); |
---|
561 | // "s:";s; |
---|
562 | // convert LM to canonical vectors (no powers) |
---|
563 | setring save; |
---|
564 | kill M; // M was defined earlier |
---|
565 | module M; |
---|
566 | slm = size(LM); // numbers of polys in the ideal |
---|
567 | int sv,k,l; |
---|
568 | vector v; |
---|
569 | // poly p; |
---|
570 | string sp; |
---|
571 | setring @R; |
---|
572 | poly @@p=0; |
---|
573 | setring save; |
---|
574 | for (l=1; l<=slm; l++) |
---|
575 | { |
---|
576 | // modules, e.g. free polynomials |
---|
577 | M = LM[l]; |
---|
578 | sm = ncols(M); // in intvec iv the sizes are stored |
---|
579 | // modules, e.g. free polynomials |
---|
580 | for (j=1; j<=sm; j++) |
---|
581 | { |
---|
582 | //vectors, e.g. free monomials |
---|
583 | v = M[j]; |
---|
584 | sv = size(v); |
---|
585 | // "sv:";sv; |
---|
586 | sp = "@@p = @@p + "; |
---|
587 | for (k=2; k<=sv; k++) |
---|
588 | { |
---|
589 | sp = sp + string(v[k])+"("+string(k-1)+")*"; |
---|
590 | } |
---|
591 | sp = sp + string(v[1])+";"; // coef; |
---|
592 | setring @R; |
---|
593 | execute(sp); |
---|
594 | setring save; |
---|
595 | } |
---|
596 | setring @R; |
---|
597 | // "@@p:"; @@p; |
---|
598 | I = I,@@p; |
---|
599 | @@p = 0; |
---|
600 | setring save; |
---|
601 | } |
---|
602 | kill sp; |
---|
603 | // 3. compute GB |
---|
604 | setring @R; |
---|
605 | dbprint(ppl,"computing GB"); |
---|
606 | ideal J = system("freegb",I,d,nvars(save)); |
---|
607 | // ideal J = slimgb(I); |
---|
608 | dbprint(ppl,J); |
---|
609 | // 4. skip shifted elts |
---|
610 | ideal K = select1(J,1..s); // s = size(OrigNames) |
---|
611 | dbprint(ppl,K); |
---|
612 | dbprint(ppl, "done with GB"); |
---|
613 | // K contains vars x(1),...z(1) = images of originals |
---|
614 | // 5. go back to orig vars, produce strings/modules |
---|
615 | if (K[1] == 0) |
---|
616 | { |
---|
617 | "no reasonable output, GB gives 0"; |
---|
618 | return(0); |
---|
619 | } |
---|
620 | int sk = size(K); |
---|
621 | int sp, sx, a, b; |
---|
622 | intvec x; |
---|
623 | poly p,q; |
---|
624 | poly pn; |
---|
625 | // vars in 'save' |
---|
626 | setring save; |
---|
627 | module N; |
---|
628 | list LN; |
---|
629 | vector V; |
---|
630 | poly pn; |
---|
631 | // test and skip exponents >=2 |
---|
632 | setring @R; |
---|
633 | for(i=1; i<=sk; i++) |
---|
634 | { |
---|
635 | p = K[i]; |
---|
636 | while (p!=0) |
---|
637 | { |
---|
638 | q = lead(p); |
---|
639 | // "processing q:";q; |
---|
640 | x = leadexp(q); |
---|
641 | sx = size(x); |
---|
642 | for(k=1; k<=sx; k++) |
---|
643 | { |
---|
644 | if ( x[k] >= 2 ) |
---|
645 | { |
---|
646 | err = "skip: the value x[k] is " + string(x[k]); |
---|
647 | dbprint(ppl,err); |
---|
648 | // return(0); |
---|
649 | K[i] = 0; |
---|
650 | p = 0; |
---|
651 | q = 0; |
---|
652 | break; |
---|
653 | } |
---|
654 | } |
---|
655 | p = p - q; |
---|
656 | } |
---|
657 | } |
---|
658 | K = simplify(K,2); |
---|
659 | sk = size(K); |
---|
660 | for(i=1; i<=sk; i++) |
---|
661 | { |
---|
662 | // setring save; |
---|
663 | // V = 0; |
---|
664 | setring @R; |
---|
665 | p = K[i]; |
---|
666 | while (p!=0) |
---|
667 | { |
---|
668 | q = lead(p); |
---|
669 | err = "processing q:" + string(q); |
---|
670 | dbprint(ppl,err); |
---|
671 | x = leadexp(q); |
---|
672 | sx = size(x); |
---|
673 | pn = leadcoef(q); |
---|
674 | setring save; |
---|
675 | pn = imap(@R,pn); |
---|
676 | V = V + leadcoef(pn)*gen(1); |
---|
677 | for(k=1; k<=sx; k++) |
---|
678 | { |
---|
679 | if (x[k] ==1) |
---|
680 | { |
---|
681 | a = k / s; // block number=a+1, a!=0 |
---|
682 | b = k % s; // remainder |
---|
683 | // printf("a: %s, b: %s",a,b); |
---|
684 | if (b == 0) |
---|
685 | { |
---|
686 | // that is it's the last var in the block |
---|
687 | b = s; |
---|
688 | a = a-1; |
---|
689 | } |
---|
690 | V = V + var(b)*gen(a+2); |
---|
691 | } |
---|
692 | // else |
---|
693 | // { |
---|
694 | // printf("error: the value x[k] is %s", x[k]); |
---|
695 | // return(0); |
---|
696 | // } |
---|
697 | } |
---|
698 | err = "V: " + string(V); |
---|
699 | dbprint(ppl,err); |
---|
700 | // printf("V: %s", string(V)); |
---|
701 | N = N,V; |
---|
702 | V = 0; |
---|
703 | setring @R; |
---|
704 | p = p - q; |
---|
705 | pn = 0; |
---|
706 | } |
---|
707 | setring save; |
---|
708 | LN[i] = simplify(N,2); |
---|
709 | N = 0; |
---|
710 | } |
---|
711 | setring save; |
---|
712 | return(LN); |
---|
713 | } |
---|
714 | example |
---|
715 | { |
---|
716 | "EXAMPLE:"; echo = 2; |
---|
717 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
718 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
---|
719 | module N = [1,x,y,x],[-1,y,x,y]; |
---|
720 | list L; L[1] = M; L[2] = N; |
---|
721 | lst2str(L); |
---|
722 | def U = freeGBasis(L,5); |
---|
723 | lst2str(U); |
---|
724 | } |
---|
725 | |
---|
726 | static proc crs(list LM, int d) |
---|
727 | "USAGE: crs(L, d); L a list of modules, d an integer |
---|
728 | RETURN: ring |
---|
729 | PURPOSE: create a ring and shift the ideal |
---|
730 | EXAMPLE: example crs; shows examples |
---|
731 | " |
---|
732 | { |
---|
733 | // d = up to degree, will be shifted to d+1 |
---|
734 | if (d<1) {"bad d"; return(0);} |
---|
735 | |
---|
736 | int ppl = printlevel-voice+2; |
---|
737 | string err = ""; |
---|
738 | |
---|
739 | int i,j,s; |
---|
740 | def save = basering; |
---|
741 | // determine max no of places in the input |
---|
742 | int slm = size(LM); // numbers of polys in the ideal |
---|
743 | int sm; |
---|
744 | intvec iv; |
---|
745 | module M; |
---|
746 | for (i=1; i<=slm; i++) |
---|
747 | { |
---|
748 | // modules, e.g. free polynomials |
---|
749 | M = LM[i]; |
---|
750 | sm = ncols(M); |
---|
751 | for (j=1; j<=sm; j++) |
---|
752 | { |
---|
753 | //vectors, e.g. free monomials |
---|
754 | iv = iv, size(M[j])-1; // 1 place is reserved by the coeff |
---|
755 | } |
---|
756 | } |
---|
757 | int D = Max(iv); // max size of input words |
---|
758 | if (d<D) {"bad d"; return(LM);} |
---|
759 | D = D + d-1; |
---|
760 | // D = d; |
---|
761 | list LR = ringlist(save); |
---|
762 | list L, tmp; |
---|
763 | L[1] = LR[1]; // ground field |
---|
764 | L[4] = LR[4]; // quotient ideal |
---|
765 | tmp = LR[2]; // varnames |
---|
766 | s = size(LR[2]); |
---|
767 | for (i=1; i<=D; i++) |
---|
768 | { |
---|
769 | for (j=1; j<=s; j++) |
---|
770 | { |
---|
771 | tmp[i*s+j] = string(tmp[j])+"("+string(i)+")"; |
---|
772 | } |
---|
773 | } |
---|
774 | for (i=1; i<=s; i++) |
---|
775 | { |
---|
776 | tmp[i] = string(tmp[i])+"("+string(0)+")"; |
---|
777 | } |
---|
778 | L[2] = tmp; |
---|
779 | list OrigNames = LR[2]; |
---|
780 | // ordering: d blocks of the ord on r |
---|
781 | // try to get whether the ord on r is blockord itself |
---|
782 | s = size(LR[3]); |
---|
783 | if (s==2) |
---|
784 | { |
---|
785 | // not a blockord, 1 block + module ord |
---|
786 | tmp = LR[3][s]; // module ord |
---|
787 | for (i=1; i<=D; i++) |
---|
788 | { |
---|
789 | LR[3][s-1+i] = LR[3][1]; |
---|
790 | } |
---|
791 | LR[3][s+D] = tmp; |
---|
792 | } |
---|
793 | if (s>2) |
---|
794 | { |
---|
795 | // there are s-1 blocks |
---|
796 | int nb = s-1; |
---|
797 | tmp = LR[3][s]; // module ord |
---|
798 | for (i=1; i<=D; i++) |
---|
799 | { |
---|
800 | for (j=1; j<=nb; j++) |
---|
801 | { |
---|
802 | LR[3][i*nb+j] = LR[3][j]; |
---|
803 | } |
---|
804 | } |
---|
805 | // size(LR[3]); |
---|
806 | LR[3][nb*(D+1)+1] = tmp; |
---|
807 | } |
---|
808 | L[3] = LR[3]; |
---|
809 | def @R = ring(L); |
---|
810 | setring @R; |
---|
811 | ideal I; |
---|
812 | poly @p; |
---|
813 | s = size(OrigNames); |
---|
814 | // "s:";s; |
---|
815 | // convert LM to canonical vectors (no powers) |
---|
816 | setring save; |
---|
817 | kill M; // M was defined earlier |
---|
818 | module M; |
---|
819 | slm = size(LM); // numbers of polys in the ideal |
---|
820 | int sv,k,l; |
---|
821 | vector v; |
---|
822 | // poly p; |
---|
823 | string sp; |
---|
824 | setring @R; |
---|
825 | poly @@p=0; |
---|
826 | setring save; |
---|
827 | for (l=1; l<=slm; l++) |
---|
828 | { |
---|
829 | // modules, e.g. free polynomials |
---|
830 | M = LM[l]; |
---|
831 | sm = ncols(M); // in intvec iv the sizes are stored |
---|
832 | for (i=0; i<=d-iv[l]; i++) |
---|
833 | { |
---|
834 | // modules, e.g. free polynomials |
---|
835 | for (j=1; j<=sm; j++) |
---|
836 | { |
---|
837 | //vectors, e.g. free monomials |
---|
838 | v = M[j]; |
---|
839 | sv = size(v); |
---|
840 | // "sv:";sv; |
---|
841 | sp = "@@p = @@p + "; |
---|
842 | for (k=2; k<=sv; k++) |
---|
843 | { |
---|
844 | sp = sp + string(v[k])+"("+string(k-2+i)+")*"; |
---|
845 | } |
---|
846 | sp = sp + string(v[1])+";"; // coef; |
---|
847 | setring @R; |
---|
848 | execute(sp); |
---|
849 | setring save; |
---|
850 | } |
---|
851 | setring @R; |
---|
852 | // "@@p:"; @@p; |
---|
853 | I = I,@@p; |
---|
854 | @@p = 0; |
---|
855 | setring save; |
---|
856 | } |
---|
857 | } |
---|
858 | setring @R; |
---|
859 | export I; |
---|
860 | return(@R); |
---|
861 | } |
---|
862 | example |
---|
863 | { |
---|
864 | "EXAMPLE:"; echo = 2; |
---|
865 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
866 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
---|
867 | module N = [1,x,y,x],[-1,y,x,y]; |
---|
868 | list L; L[1] = M; L[2] = N; |
---|
869 | lst2str(L); |
---|
870 | def U = crs(L,5); |
---|
871 | setring U; U; |
---|
872 | I; |
---|
873 | } |
---|
874 | |
---|
875 | static proc polylen(ideal I) |
---|
876 | { |
---|
877 | // returns the ideal of length of polys |
---|
878 | int i; |
---|
879 | intvec J; |
---|
880 | number s = 0; |
---|
881 | for(i=1;i<=ncols(I);i++) |
---|
882 | { |
---|
883 | J[i] = size(I[i]); |
---|
884 | s = s + J[i]; |
---|
885 | } |
---|
886 | printf("the sum of length %s",s); |
---|
887 | // print(s); |
---|
888 | return(J); |
---|
889 | } |
---|
890 | |
---|
891 | //proc freegbRing(int d) |
---|
892 | proc makeLetterplaceRing(int d) |
---|
893 | "USAGE: makeLetterplaceRing(d); d an integer |
---|
894 | RETURN: ring |
---|
895 | PURPOSE: creates a ring with d blocks of shifted original variables |
---|
896 | EXAMPLE: example makeLetterplaceRing; shows examples |
---|
897 | " |
---|
898 | { |
---|
899 | // d = up to degree, will be shifted to d+1 |
---|
900 | if (d<1) {"bad d"; return(0);} |
---|
901 | |
---|
902 | int uptodeg = d; int lV = nvars(basering); |
---|
903 | |
---|
904 | int ppl = printlevel-voice+2; |
---|
905 | string err = ""; |
---|
906 | |
---|
907 | int i,j,s; |
---|
908 | def save = basering; |
---|
909 | int D = d-1; |
---|
910 | list LR = ringlist(save); |
---|
911 | list L, tmp; |
---|
912 | L[1] = LR[1]; // ground field |
---|
913 | L[4] = LR[4]; // quotient ideal |
---|
914 | tmp = LR[2]; // varnames |
---|
915 | s = size(LR[2]); |
---|
916 | for (i=1; i<=D; i++) |
---|
917 | { |
---|
918 | for (j=1; j<=s; j++) |
---|
919 | { |
---|
920 | tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")"; |
---|
921 | } |
---|
922 | } |
---|
923 | for (i=1; i<=s; i++) |
---|
924 | { |
---|
925 | tmp[i] = string(tmp[i])+"("+string(1)+")"; |
---|
926 | } |
---|
927 | L[2] = tmp; |
---|
928 | list OrigNames = LR[2]; |
---|
929 | // ordering: d blocks of the ord on r |
---|
930 | // try to get whether the ord on r is blockord itself |
---|
931 | // TODO: make L(2) ordering! exponent is maximally 2 |
---|
932 | s = size(LR[3]); |
---|
933 | if (s==2) |
---|
934 | { |
---|
935 | // not a blockord, 1 block + module ord |
---|
936 | tmp = LR[3][s]; // module ord |
---|
937 | for (i=1; i<=D; i++) |
---|
938 | { |
---|
939 | LR[3][s-1+i] = LR[3][1]; |
---|
940 | } |
---|
941 | LR[3][s+D] = tmp; |
---|
942 | } |
---|
943 | if (s>2) |
---|
944 | { |
---|
945 | // there are s-1 blocks |
---|
946 | int nb = s-1; |
---|
947 | tmp = LR[3][s]; // module ord |
---|
948 | for (i=1; i<=D; i++) |
---|
949 | { |
---|
950 | for (j=1; j<=nb; j++) |
---|
951 | { |
---|
952 | LR[3][i*nb+j] = LR[3][j]; |
---|
953 | } |
---|
954 | } |
---|
955 | // size(LR[3]); |
---|
956 | LR[3][nb*(D+1)+1] = tmp; |
---|
957 | } |
---|
958 | L[3] = LR[3]; |
---|
959 | def @R = ring(L); |
---|
960 | // setring @R; |
---|
961 | // int uptodeg = d; int lV = nvars(basering); // were defined before |
---|
962 | def @@R = setLetterplaceAttributes(@R,uptodeg,lV); |
---|
963 | return (@@R); |
---|
964 | } |
---|
965 | example |
---|
966 | { |
---|
967 | "EXAMPLE:"; echo = 2; |
---|
968 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
969 | def A = makeLetterplaceRing(2); |
---|
970 | setring A; |
---|
971 | A; |
---|
972 | attrib(A,"isLetterplaceRing"); |
---|
973 | attrib(A,"uptodeg"); // degree bound |
---|
974 | attrib(A,"lV"); // number of variables in the main block |
---|
975 | } |
---|
976 | |
---|
977 | /* EXAMPLES: |
---|
978 | |
---|
979 | //static proc ex_shift() |
---|
980 | { |
---|
981 | LIB "freegb.lib"; |
---|
982 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
983 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
---|
984 | module N = [1,x,y,x],[-1,y,x,y]; |
---|
985 | list L; L[1] = M; L[2] = N; |
---|
986 | lst2str(L); |
---|
987 | def U = crs(L,5); |
---|
988 | setring U; U; |
---|
989 | I; |
---|
990 | poly p = I[2]; // I[8]; |
---|
991 | p; |
---|
992 | system("stest",p,7,7,3); // error -> the world is ok |
---|
993 | poly q1 = system("stest",p,1,7,3); //ok |
---|
994 | poly q6 = system("stest",p,6,7,3); //ok |
---|
995 | system("btest",p,3); //ok |
---|
996 | system("btest",q1,3); //ok |
---|
997 | system("btest",q6,3); //ok |
---|
998 | } |
---|
999 | |
---|
1000 | //static proc test_shrink() |
---|
1001 | { |
---|
1002 | LIB "freegb.lib"; |
---|
1003 | ring r =0,(x,y,z),dp; |
---|
1004 | int d = 5; |
---|
1005 | def R = makeLetterplaceRing(d); |
---|
1006 | setring R; |
---|
1007 | poly p1 = x(1)*y(2)*z(3); |
---|
1008 | poly p2 = x(1)*y(4)*z(5); |
---|
1009 | poly p3 = x(1)*y(1)*z(3); |
---|
1010 | poly p4 = x(1)*y(2)*z(2); |
---|
1011 | poly p5 = x(3)*z(5); |
---|
1012 | poly p6 = x(1)*y(1)*x(3)*z(5); |
---|
1013 | poly p7 = x(1)*y(2)*x(3)*y(4)*z(5); |
---|
1014 | poly p8 = p1+p2+p3+p4+p5 + p6 + p7; |
---|
1015 | p1; system("shrinktest",p1,3); |
---|
1016 | p2; system("shrinktest",p2,3); |
---|
1017 | p3; system("shrinktest",p3,3); |
---|
1018 | p4; system("shrinktest",p4,3); |
---|
1019 | p5; system("shrinktest",p5,3); |
---|
1020 | p6; system("shrinktest",p6,3); |
---|
1021 | p7; system("shrinktest",p7,3); |
---|
1022 | p8; system("shrinktest",p8,3); |
---|
1023 | poly p9 = p1 + 2*p2 + 5*p5 + 7*p7; |
---|
1024 | p9; system("shrinktest",p9,3); |
---|
1025 | } |
---|
1026 | |
---|
1027 | //static proc ex2() |
---|
1028 | { |
---|
1029 | option(prot); |
---|
1030 | LIB "freegb.lib"; |
---|
1031 | ring r = 0,(x,y),dp; |
---|
1032 | module M = [-1,x,y],[3,x,x]; // 3x^2 - xy |
---|
1033 | def U = freegb(M,7); |
---|
1034 | lst2str(U); |
---|
1035 | } |
---|
1036 | |
---|
1037 | //static proc ex_nonhomog() |
---|
1038 | { |
---|
1039 | option(prot); |
---|
1040 | LIB "freegb.lib"; |
---|
1041 | ring r = 0,(x,y,h),dp; |
---|
1042 | list L; |
---|
1043 | module M; |
---|
1044 | M = [-1,y,y],[1,x,x,x]; // x3-y2 |
---|
1045 | L[1] = M; |
---|
1046 | M = [1,x,h],[-1,h,x]; // xh-hx |
---|
1047 | L[2] = M; |
---|
1048 | M = [1,y,h],[-1,h,y]; // yh-hy |
---|
1049 | L[3] = M; |
---|
1050 | def U = freegb(L,4); |
---|
1051 | lst2str(U); |
---|
1052 | // strange elements in the basis |
---|
1053 | } |
---|
1054 | |
---|
1055 | //static proc ex_nonhomog_comm() |
---|
1056 | { |
---|
1057 | option(prot); |
---|
1058 | LIB "freegb.lib"; |
---|
1059 | ring r = 0,(x,y),dp; |
---|
1060 | module M = [-1,y,y],[1,x,x,x]; |
---|
1061 | def U = freegb(M,5); |
---|
1062 | lst2str(U); |
---|
1063 | } |
---|
1064 | |
---|
1065 | //static proc ex_nonhomog_h() |
---|
1066 | { |
---|
1067 | option(prot); |
---|
1068 | LIB "freegb.lib"; |
---|
1069 | ring r = 0,(x,y,h),(a(1,1),dp); |
---|
1070 | module M = [-1,y,y,h],[1,x,x,x]; // x3 - y2h |
---|
1071 | def U = freegb(M,6); |
---|
1072 | lst2str(U); |
---|
1073 | } |
---|
1074 | |
---|
1075 | //static proc ex_nonhomog_h2() |
---|
1076 | { |
---|
1077 | option(prot); |
---|
1078 | LIB "freegb.lib"; |
---|
1079 | ring r = 0,(x,y,h),(dp); |
---|
1080 | list L; |
---|
1081 | module M; |
---|
1082 | M = [-1,y,y,h],[1,x,x,x]; // x3 - y2h |
---|
1083 | L[1] = M; |
---|
1084 | M = [1,x,h],[-1,h,x]; // xh - hx |
---|
1085 | L[2] = M; |
---|
1086 | M = [1,y,h],[-1,h,y]; // yh - hy |
---|
1087 | L[3] = M; |
---|
1088 | def U = freeGBasis(L,3); |
---|
1089 | lst2str(U); |
---|
1090 | // strange answer CHECK |
---|
1091 | } |
---|
1092 | |
---|
1093 | |
---|
1094 | //static proc ex_nonhomog_3() |
---|
1095 | { |
---|
1096 | option(prot); |
---|
1097 | LIB "./freegb.lib"; |
---|
1098 | ring r = 0,(x,y,z),(dp); |
---|
1099 | list L; |
---|
1100 | module M; |
---|
1101 | M = [1,z,y],[-1,x]; // zy - x |
---|
1102 | L[1] = M; |
---|
1103 | M = [1,z,x],[-1,y]; // zx - y |
---|
1104 | L[2] = M; |
---|
1105 | M = [1,y,x],[-1,z]; // yx - z |
---|
1106 | L[3] = M; |
---|
1107 | lst2str(L); |
---|
1108 | list U = freegb(L,4); |
---|
1109 | lst2str(U); |
---|
1110 | // strange answer CHECK |
---|
1111 | } |
---|
1112 | |
---|
1113 | //static proc ex_densep_2() |
---|
1114 | { |
---|
1115 | option(prot); |
---|
1116 | LIB "freegb.lib"; |
---|
1117 | ring r = (0,a,b,c),(x,y),(Dp); // deglex |
---|
1118 | module M = [1,x,x], [a,x,y], [b,y,x], [c,y,y]; |
---|
1119 | lst2str(M); |
---|
1120 | list U = freegb(M,5); |
---|
1121 | lst2str(U); |
---|
1122 | // a=b is important -> finite basis!!! |
---|
1123 | module M = [1,x,x], [a,x,y], [a,y,x], [c,y,y]; |
---|
1124 | lst2str(M); |
---|
1125 | list U = freegb(M,5); |
---|
1126 | lst2str(U); |
---|
1127 | } |
---|
1128 | |
---|
1129 | // END COMMENTED EXAMPLES |
---|
1130 | |
---|
1131 | */ |
---|
1132 | |
---|
1133 | // 1. form a new ring |
---|
1134 | // 2. produce shifted generators |
---|
1135 | // 3. compute GB |
---|
1136 | // 4. skip shifted elts |
---|
1137 | // 5. go back to orig vars, produce strings/modules |
---|
1138 | // 6. return the result |
---|
1139 | |
---|
1140 | static proc freegbold(list LM, int d) |
---|
1141 | "USAGE: freegbold(L, d); L a list of modules, d an integer |
---|
1142 | RETURN: ring |
---|
1143 | PURPOSE: compute the two-sided Groebner basis of an ideal, encoded by L in |
---|
1144 | the free associative algebra, up to degree d |
---|
1145 | EXAMPLE: example freegbold; shows examples |
---|
1146 | " |
---|
1147 | { |
---|
1148 | // d = up to degree, will be shifted to d+1 |
---|
1149 | if (d<1) {"bad d"; return(0);} |
---|
1150 | |
---|
1151 | int ppl = printlevel-voice+2; |
---|
1152 | string err = ""; |
---|
1153 | |
---|
1154 | int i,j,s; |
---|
1155 | def save = basering; |
---|
1156 | // determine max no of places in the input |
---|
1157 | int slm = size(LM); // numbers of polys in the ideal |
---|
1158 | int sm; |
---|
1159 | intvec iv; |
---|
1160 | module M; |
---|
1161 | for (i=1; i<=slm; i++) |
---|
1162 | { |
---|
1163 | // modules, e.g. free polynomials |
---|
1164 | M = LM[i]; |
---|
1165 | sm = ncols(M); |
---|
1166 | for (j=1; j<=sm; j++) |
---|
1167 | { |
---|
1168 | //vectors, e.g. free monomials |
---|
1169 | iv = iv, size(M[j])-1; // 1 place is reserved by the coeff |
---|
1170 | } |
---|
1171 | } |
---|
1172 | int D = Max(iv); // max size of input words |
---|
1173 | if (d<D) {"bad d"; return(LM);} |
---|
1174 | D = D + d-1; |
---|
1175 | // D = d; |
---|
1176 | list LR = ringlist(save); |
---|
1177 | list L, tmp; |
---|
1178 | L[1] = LR[1]; // ground field |
---|
1179 | L[4] = LR[4]; // quotient ideal |
---|
1180 | tmp = LR[2]; // varnames |
---|
1181 | s = size(LR[2]); |
---|
1182 | for (i=1; i<=D; i++) |
---|
1183 | { |
---|
1184 | for (j=1; j<=s; j++) |
---|
1185 | { |
---|
1186 | tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")"; |
---|
1187 | } |
---|
1188 | } |
---|
1189 | for (i=1; i<=s; i++) |
---|
1190 | { |
---|
1191 | tmp[i] = string(tmp[i])+"("+string(1)+")"; |
---|
1192 | } |
---|
1193 | L[2] = tmp; |
---|
1194 | list OrigNames = LR[2]; |
---|
1195 | // ordering: d blocks of the ord on r |
---|
1196 | // try to get whether the ord on r is blockord itself |
---|
1197 | // TODO: make L(2) ordering! exponent is maximally 2 |
---|
1198 | s = size(LR[3]); |
---|
1199 | if (s==2) |
---|
1200 | { |
---|
1201 | // not a blockord, 1 block + module ord |
---|
1202 | tmp = LR[3][s]; // module ord |
---|
1203 | for (i=1; i<=D; i++) |
---|
1204 | { |
---|
1205 | LR[3][s-1+i] = LR[3][1]; |
---|
1206 | } |
---|
1207 | LR[3][s+D] = tmp; |
---|
1208 | } |
---|
1209 | if (s>2) |
---|
1210 | { |
---|
1211 | // there are s-1 blocks |
---|
1212 | int nb = s-1; |
---|
1213 | tmp = LR[3][s]; // module ord |
---|
1214 | for (i=1; i<=D; i++) |
---|
1215 | { |
---|
1216 | for (j=1; j<=nb; j++) |
---|
1217 | { |
---|
1218 | LR[3][i*nb+j] = LR[3][j]; |
---|
1219 | } |
---|
1220 | } |
---|
1221 | // size(LR[3]); |
---|
1222 | LR[3][nb*(D+1)+1] = tmp; |
---|
1223 | } |
---|
1224 | L[3] = LR[3]; |
---|
1225 | def @R = ring(L); |
---|
1226 | setring @R; |
---|
1227 | ideal I; |
---|
1228 | poly @p; |
---|
1229 | s = size(OrigNames); |
---|
1230 | // "s:";s; |
---|
1231 | // convert LM to canonical vectors (no powers) |
---|
1232 | setring save; |
---|
1233 | kill M; // M was defined earlier |
---|
1234 | module M; |
---|
1235 | slm = size(LM); // numbers of polys in the ideal |
---|
1236 | int sv,k,l; |
---|
1237 | vector v; |
---|
1238 | // poly p; |
---|
1239 | string sp; |
---|
1240 | setring @R; |
---|
1241 | poly @@p=0; |
---|
1242 | setring save; |
---|
1243 | for (l=1; l<=slm; l++) |
---|
1244 | { |
---|
1245 | // modules, e.g. free polynomials |
---|
1246 | M = LM[l]; |
---|
1247 | sm = ncols(M); // in intvec iv the sizes are stored |
---|
1248 | for (i=0; i<=d-iv[l]; i++) |
---|
1249 | { |
---|
1250 | // modules, e.g. free polynomials |
---|
1251 | for (j=1; j<=sm; j++) |
---|
1252 | { |
---|
1253 | //vectors, e.g. free monomials |
---|
1254 | v = M[j]; |
---|
1255 | sv = size(v); |
---|
1256 | // "sv:";sv; |
---|
1257 | sp = "@@p = @@p + "; |
---|
1258 | for (k=2; k<=sv; k++) |
---|
1259 | { |
---|
1260 | sp = sp + string(v[k])+"("+string(k-1+i)+")*"; |
---|
1261 | } |
---|
1262 | sp = sp + string(v[1])+";"; // coef; |
---|
1263 | setring @R; |
---|
1264 | execute(sp); |
---|
1265 | setring save; |
---|
1266 | } |
---|
1267 | setring @R; |
---|
1268 | // "@@p:"; @@p; |
---|
1269 | I = I,@@p; |
---|
1270 | @@p = 0; |
---|
1271 | setring save; |
---|
1272 | } |
---|
1273 | } |
---|
1274 | kill sp; |
---|
1275 | // 3. compute GB |
---|
1276 | setring @R; |
---|
1277 | dbprint(ppl,"computing GB"); |
---|
1278 | // ideal J = groebner(I); |
---|
1279 | ideal J = slimgb(I); |
---|
1280 | dbprint(ppl,J); |
---|
1281 | // 4. skip shifted elts |
---|
1282 | ideal K = select1(J,1..s); // s = size(OrigNames) |
---|
1283 | dbprint(ppl,K); |
---|
1284 | dbprint(ppl, "done with GB"); |
---|
1285 | // K contains vars x(1),...z(1) = images of originals |
---|
1286 | // 5. go back to orig vars, produce strings/modules |
---|
1287 | if (K[1] == 0) |
---|
1288 | { |
---|
1289 | "no reasonable output, GB gives 0"; |
---|
1290 | return(0); |
---|
1291 | } |
---|
1292 | int sk = size(K); |
---|
1293 | int sp, sx, a, b; |
---|
1294 | intvec x; |
---|
1295 | poly p,q; |
---|
1296 | poly pn; |
---|
1297 | // vars in 'save' |
---|
1298 | setring save; |
---|
1299 | module N; |
---|
1300 | list LN; |
---|
1301 | vector V; |
---|
1302 | poly pn; |
---|
1303 | // test and skip exponents >=2 |
---|
1304 | setring @R; |
---|
1305 | for(i=1; i<=sk; i++) |
---|
1306 | { |
---|
1307 | p = K[i]; |
---|
1308 | while (p!=0) |
---|
1309 | { |
---|
1310 | q = lead(p); |
---|
1311 | // "processing q:";q; |
---|
1312 | x = leadexp(q); |
---|
1313 | sx = size(x); |
---|
1314 | for(k=1; k<=sx; k++) |
---|
1315 | { |
---|
1316 | if ( x[k] >= 2 ) |
---|
1317 | { |
---|
1318 | err = "skip: the value x[k] is " + string(x[k]); |
---|
1319 | dbprint(ppl,err); |
---|
1320 | // return(0); |
---|
1321 | K[i] = 0; |
---|
1322 | p = 0; |
---|
1323 | q = 0; |
---|
1324 | break; |
---|
1325 | } |
---|
1326 | } |
---|
1327 | p = p - q; |
---|
1328 | } |
---|
1329 | } |
---|
1330 | K = simplify(K,2); |
---|
1331 | sk = size(K); |
---|
1332 | for(i=1; i<=sk; i++) |
---|
1333 | { |
---|
1334 | // setring save; |
---|
1335 | // V = 0; |
---|
1336 | setring @R; |
---|
1337 | p = K[i]; |
---|
1338 | while (p!=0) |
---|
1339 | { |
---|
1340 | q = lead(p); |
---|
1341 | err = "processing q:" + string(q); |
---|
1342 | dbprint(ppl,err); |
---|
1343 | x = leadexp(q); |
---|
1344 | sx = size(x); |
---|
1345 | pn = leadcoef(q); |
---|
1346 | setring save; |
---|
1347 | pn = imap(@R,pn); |
---|
1348 | V = V + leadcoef(pn)*gen(1); |
---|
1349 | for(k=1; k<=sx; k++) |
---|
1350 | { |
---|
1351 | if (x[k] ==1) |
---|
1352 | { |
---|
1353 | a = k / s; // block number=a+1, a!=0 |
---|
1354 | b = k % s; // remainder |
---|
1355 | // printf("a: %s, b: %s",a,b); |
---|
1356 | if (b == 0) |
---|
1357 | { |
---|
1358 | // that is it's the last var in the block |
---|
1359 | b = s; |
---|
1360 | a = a-1; |
---|
1361 | } |
---|
1362 | V = V + var(b)*gen(a+2); |
---|
1363 | } |
---|
1364 | // else |
---|
1365 | // { |
---|
1366 | // printf("error: the value x[k] is %s", x[k]); |
---|
1367 | // return(0); |
---|
1368 | // } |
---|
1369 | } |
---|
1370 | err = "V: " + string(V); |
---|
1371 | dbprint(ppl,err); |
---|
1372 | // printf("V: %s", string(V)); |
---|
1373 | N = N,V; |
---|
1374 | V = 0; |
---|
1375 | setring @R; |
---|
1376 | p = p - q; |
---|
1377 | pn = 0; |
---|
1378 | } |
---|
1379 | setring save; |
---|
1380 | LN[i] = simplify(N,2); |
---|
1381 | N = 0; |
---|
1382 | } |
---|
1383 | setring save; |
---|
1384 | return(LN); |
---|
1385 | } |
---|
1386 | example |
---|
1387 | { |
---|
1388 | "EXAMPLE:"; echo = 2; |
---|
1389 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
1390 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
---|
1391 | module N = [1,x,y,x],[-1,y,x,y]; |
---|
1392 | list L; L[1] = M; L[2] = N; |
---|
1393 | lst2str(L); |
---|
1394 | def U = freegbold(L,5); |
---|
1395 | lst2str(U); |
---|
1396 | } |
---|
1397 | |
---|
1398 | /* begin older procs and tests |
---|
1399 | |
---|
1400 | static proc sgb(ideal I, int d) |
---|
1401 | { |
---|
1402 | // new code |
---|
1403 | // map x_i to x_i(1) via map() |
---|
1404 | //LIB "freegb.lib"; |
---|
1405 | def save = basering; |
---|
1406 | //int d =7;// degree |
---|
1407 | int nv = nvars(save); |
---|
1408 | def R = makeLetterplaceRing(d); |
---|
1409 | setring R; |
---|
1410 | int i; |
---|
1411 | ideal Imap; |
---|
1412 | for (i=1; i<=nv; i++) |
---|
1413 | { |
---|
1414 | Imap[i] = var(i); |
---|
1415 | } |
---|
1416 | //ideal I = x(1)*y(2), y(1)*x(2)+z(1)*z(2); |
---|
1417 | ideal I = x(1)*x(2),x(1)*y(2) + z(1)*x(2); |
---|
1418 | option(prot); |
---|
1419 | //option(teach); |
---|
1420 | ideal J = system("freegb",I,d,nv); |
---|
1421 | } |
---|
1422 | |
---|
1423 | static proc checkCeq() |
---|
1424 | { |
---|
1425 | ring r = 0,(x,y),Dp; |
---|
1426 | def A = makeLetterplaceRing(4); |
---|
1427 | setring A; |
---|
1428 | A; |
---|
1429 | // I = x2-xy |
---|
1430 | 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); |
---|
1431 | 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); |
---|
1432 | ideal K = I,C; |
---|
1433 | groebner(K); |
---|
1434 | } |
---|
1435 | |
---|
1436 | static proc exHom1() |
---|
1437 | { |
---|
1438 | // we start with |
---|
1439 | // z*y - x, z*x - y, y*x - z |
---|
1440 | LIB "freegb.lib"; |
---|
1441 | LIB "elim.lib"; |
---|
1442 | ring r = 0,(x,y,z,h),dp; |
---|
1443 | list L; |
---|
1444 | module M; |
---|
1445 | M = [1,z,y],[-1,x,h]; // zy - xh |
---|
1446 | L[1] = M; |
---|
1447 | M = [1,z,x],[-1,y,h]; // zx - yh |
---|
1448 | L[2] = M; |
---|
1449 | M = [1,y,x],[-1,z,h]; // yx - zh |
---|
1450 | L[3] = M; |
---|
1451 | lst2str(L); |
---|
1452 | def U = crs(L,4); |
---|
1453 | setring U; |
---|
1454 | I = I, |
---|
1455 | y(2)*h(3)+z(2)*x(3), y(3)*h(4)+z(3)*x(4), |
---|
1456 | y(2)*x(3)-z(2)*h(3), y(3)*x(4)-z(3)*h(4); |
---|
1457 | I = simplify(I,2); |
---|
1458 | ring r2 = 0,(x(0..4),y(0..4),z(0..4),h(0..4)),dp; |
---|
1459 | ideal J = imap(U,I); |
---|
1460 | // ideal K = homog(J,h); |
---|
1461 | option(redSB); |
---|
1462 | option(redTail); |
---|
1463 | ideal L = groebner(J); //(K); |
---|
1464 | ideal LL = sat(L,ideal(h))[1]; |
---|
1465 | ideal M = subst(LL,h,1); |
---|
1466 | M = simplify(M,2); |
---|
1467 | setring U; |
---|
1468 | ideal M = imap(r2,M); |
---|
1469 | lst2str(U); |
---|
1470 | } |
---|
1471 | |
---|
1472 | static proc test1() |
---|
1473 | { |
---|
1474 | LIB "freegb.lib"; |
---|
1475 | ring r = 0,(x,y),Dp; |
---|
1476 | int d = 10; // degree |
---|
1477 | def R = makeLetterplaceRing(d); |
---|
1478 | setring R; |
---|
1479 | ideal I = x(1)*x(2) - y(1)*y(2); |
---|
1480 | option(prot); |
---|
1481 | option(teach); |
---|
1482 | ideal J = system("freegb",I,d,2); |
---|
1483 | J; |
---|
1484 | } |
---|
1485 | |
---|
1486 | static proc test2() |
---|
1487 | { |
---|
1488 | LIB "freegb.lib"; |
---|
1489 | ring r = 0,(x,y),Dp; |
---|
1490 | int d = 10; // degree |
---|
1491 | def R = makeLetterplaceRing(d); |
---|
1492 | setring R; |
---|
1493 | ideal I = x(1)*x(2) - x(1)*y(2); |
---|
1494 | option(prot); |
---|
1495 | option(teach); |
---|
1496 | ideal J = system("freegb",I,d,2); |
---|
1497 | J; |
---|
1498 | } |
---|
1499 | |
---|
1500 | static proc test3() |
---|
1501 | { |
---|
1502 | LIB "freegb.lib"; |
---|
1503 | ring r = 0,(x,y,z),dp; |
---|
1504 | int d =5; // degree |
---|
1505 | def R = makeLetterplaceRing(d); |
---|
1506 | setring R; |
---|
1507 | ideal I = x(1)*y(2), y(1)*x(2)+z(1)*z(2); |
---|
1508 | option(prot); |
---|
1509 | option(teach); |
---|
1510 | ideal J = system("freegb",I,d,3); |
---|
1511 | } |
---|
1512 | |
---|
1513 | static proc schur2-3() |
---|
1514 | { |
---|
1515 | // nonhomog: |
---|
1516 | // h^4-10*h^2+9,f*e-e*f+h, h*2-e*h-2*e,h*f-f*h+2*f |
---|
1517 | // homogenized with t |
---|
1518 | // 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, |
---|
1519 | // t*h - h*t, t*f - f*t, t*e - e*t |
---|
1520 | } |
---|
1521 | |
---|
1522 | end older procs and tests */ |
---|
1523 | |
---|
1524 | proc ademRelations(int i, int j) |
---|
1525 | "USAGE: ademRelations(i,j); i,j int |
---|
1526 | RETURN: ring (and exports ideal) |
---|
1527 | ASSUME: there are at least i+j variables in the basering |
---|
1528 | PURPOSE: compute the ideal of Adem relations for i<2j in characteristic 0 |
---|
1529 | @* the ideal is exported under the name AdemRel in the output ring |
---|
1530 | EXAMPLE: example ademRelations; shows examples |
---|
1531 | " |
---|
1532 | { |
---|
1533 | // produces Adem relations for i<2j in char 0 |
---|
1534 | // assume: 0<i<2j |
---|
1535 | // requires presence of vars up to i+j |
---|
1536 | if ( (i<0) || (i >= 2*j) ) |
---|
1537 | { |
---|
1538 | ERROR("arguments out of range"); return(0); |
---|
1539 | } |
---|
1540 | ring @r = 0,(s(i+j..0)),lp; |
---|
1541 | poly p,q; |
---|
1542 | number n; |
---|
1543 | int ii = i div 2; int k; |
---|
1544 | // k=0 => s(0)=1 |
---|
1545 | n = binomial(j-1,i); |
---|
1546 | q = n*s(i+j)*s(0); |
---|
1547 | // printf("k=0, term=%s",q); |
---|
1548 | p = p + q; |
---|
1549 | for (k=1; k<= ii; k++) |
---|
1550 | { |
---|
1551 | n = binomial(j-k-1,i-2*k); |
---|
1552 | q = n*s(i+j-k)*s(k);; |
---|
1553 | // printf("k=%s, term=%s",k,q); |
---|
1554 | p = p + q; |
---|
1555 | } |
---|
1556 | poly AdemRel = p; |
---|
1557 | export AdemRel; |
---|
1558 | return(@r); |
---|
1559 | } |
---|
1560 | example |
---|
1561 | { |
---|
1562 | "EXAMPLE:"; echo = 2; |
---|
1563 | def A = ademRelations(2,5); |
---|
1564 | setring A; |
---|
1565 | AdemRel; |
---|
1566 | } |
---|
1567 | |
---|
1568 | /* |
---|
1569 | 1,1: 0 |
---|
1570 | 1,2: s(3)*s(0) == s(3) -> def for s(3):=s(1)s(2) |
---|
1571 | 2,1: adm |
---|
1572 | 2,2: s(3)*s(1) == s(1)s(2)s(1) |
---|
1573 | 1,3: 0 ( since 2*s(4)*s(0) = 0 mod 2) |
---|
1574 | 3,1: adm |
---|
1575 | 2,3: s(5)*s(0)+s(4)*s(1) == s(5)+s(4)*s(1) |
---|
1576 | 3,2: 0 |
---|
1577 | 3,3: s(5)*s(1) |
---|
1578 | 1,4: 3*s(5)*s(0) == s(5) -> def for s(5):=s(1)*s(4) |
---|
1579 | 4,1: adm |
---|
1580 | 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) |
---|
1581 | 4,2: adm |
---|
1582 | 4,3: s(5)*s(2) |
---|
1583 | 3,4: s(7)*s(0)+2*s(6)*s(1) == s(7) -> def for s(7):=s(3)*s(4) |
---|
1584 | 4,4: s(7)*s(1)+s(6)*s(2) |
---|
1585 | */ |
---|
1586 | |
---|
1587 | /* s1,s2: |
---|
1588 | s1*s1 =0, s2*s2 = s1*s2*s1 |
---|
1589 | */ |
---|
1590 | |
---|
1591 | /* |
---|
1592 | try char 0: |
---|
1593 | s1,s2: |
---|
1594 | 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) |
---|
1595 | hence 2==0! only in char 2 |
---|
1596 | */ |
---|
1597 | |
---|
1598 | // Adem rels modulo 2 are interesting |
---|
1599 | |
---|
1600 | static proc stringpoly2lplace(string s) |
---|
1601 | { |
---|
1602 | // decomposes sentence into terms |
---|
1603 | s = replace(s,newline,""); // get rid of newlines |
---|
1604 | s = replace(s," ",""); // get rid of empties |
---|
1605 | //arith symbols: +,- |
---|
1606 | // decompose into words with coeffs |
---|
1607 | list LS; |
---|
1608 | int i,j,ie,je,k,cnt; |
---|
1609 | // s[1]="-" situation |
---|
1610 | if (s[1]=="-") |
---|
1611 | { |
---|
1612 | LS = stringpoly2lplace(string(s[2..size(s)])); |
---|
1613 | LS[1] = string("-"+string(LS[1])); |
---|
1614 | return(LS); |
---|
1615 | } |
---|
1616 | i = find(s,"-",2); |
---|
1617 | // i==1 might happen if the 1st symbol coeff is negative |
---|
1618 | j = find(s,"+"); |
---|
1619 | list LL; |
---|
1620 | if (i==j) |
---|
1621 | { |
---|
1622 | "return a monomial"; |
---|
1623 | // that is both are 0 -> s is a monomial |
---|
1624 | LS[1] = s; |
---|
1625 | return(LS); |
---|
1626 | } |
---|
1627 | if (i==0) |
---|
1628 | { |
---|
1629 | "i==0 situation"; |
---|
1630 | // no minuses at all => pluses only |
---|
1631 | cnt++; |
---|
1632 | LS[cnt] = string(s[1..j-1]); |
---|
1633 | s = s[j+1..size(s)]; |
---|
1634 | while (s!= "") |
---|
1635 | { |
---|
1636 | j = find(s,"+"); |
---|
1637 | cnt++; |
---|
1638 | if (j==0) |
---|
1639 | { |
---|
1640 | LS[cnt] = string(s); |
---|
1641 | s = ""; |
---|
1642 | } |
---|
1643 | else |
---|
1644 | { |
---|
1645 | LS[cnt] = string(s[1..j-1]); |
---|
1646 | s = s[j+1..size(s)]; |
---|
1647 | } |
---|
1648 | } |
---|
1649 | return(LS); |
---|
1650 | } |
---|
1651 | if (j==0) |
---|
1652 | { |
---|
1653 | "j==0 situation"; |
---|
1654 | // no pluses at all except the lead coef => the rest are minuses only |
---|
1655 | cnt++; |
---|
1656 | LS[cnt] = string(s[1..i-1]); |
---|
1657 | s = s[i..size(s)]; |
---|
1658 | while (s!= "") |
---|
1659 | { |
---|
1660 | i = find(s,"-",2); |
---|
1661 | cnt++; |
---|
1662 | if (i==0) |
---|
1663 | { |
---|
1664 | LS[cnt] = string(s); |
---|
1665 | s = ""; |
---|
1666 | } |
---|
1667 | else |
---|
1668 | { |
---|
1669 | LS[cnt] = string(s[1..i-1]); |
---|
1670 | s = s[i..size(s)]; |
---|
1671 | } |
---|
1672 | } |
---|
1673 | return(LS); |
---|
1674 | } |
---|
1675 | // now i, j are nonzero |
---|
1676 | if (i>j) |
---|
1677 | { |
---|
1678 | "i>j situation"; |
---|
1679 | // + comes first, at place j |
---|
1680 | cnt++; |
---|
1681 | // "cnt:"; cnt; "j:"; j; |
---|
1682 | LS[cnt] = string(s[1..j-1]); |
---|
1683 | s = s[j+1..size(s)]; |
---|
1684 | LL = stringpoly2lplace(s); |
---|
1685 | LS = LS + LL; |
---|
1686 | kill LL; |
---|
1687 | return(LS); |
---|
1688 | } |
---|
1689 | else |
---|
1690 | { |
---|
1691 | "j>i situation"; |
---|
1692 | // - might come first, at place i |
---|
1693 | if (i>1) |
---|
1694 | { |
---|
1695 | cnt++; |
---|
1696 | LS[cnt] = string(s[1..i-1]); |
---|
1697 | s = s[i..size(s)]; |
---|
1698 | } |
---|
1699 | else |
---|
1700 | { |
---|
1701 | // i==1-> minus at leadcoef |
---|
1702 | ie = find(s,"-",i+1); |
---|
1703 | je = find(s,"+",i+1); |
---|
1704 | if (je == ie) |
---|
1705 | { |
---|
1706 | "ie=je situation"; |
---|
1707 | //monomial |
---|
1708 | cnt++; |
---|
1709 | LS[cnt] = s; |
---|
1710 | return(LS); |
---|
1711 | } |
---|
1712 | if (je < ie) |
---|
1713 | { |
---|
1714 | "je<ie situation"; |
---|
1715 | // + comes first |
---|
1716 | cnt++; |
---|
1717 | LS[cnt] = s[1..je-1]; |
---|
1718 | s = s[je+1..size(s)]; |
---|
1719 | } |
---|
1720 | else |
---|
1721 | { |
---|
1722 | // ie < je |
---|
1723 | "ie<je situation"; |
---|
1724 | cnt++; |
---|
1725 | LS[cnt] = s[1..ie-1]; |
---|
1726 | s = s[ie..size(s)]; |
---|
1727 | } |
---|
1728 | } |
---|
1729 | "going into recursion with "+s; |
---|
1730 | LL = stringpoly2lplace(s); |
---|
1731 | LS = LS + LL; |
---|
1732 | return(LS); |
---|
1733 | } |
---|
1734 | } |
---|
1735 | example |
---|
1736 | { |
---|
1737 | "EXAMPLE:"; echo = 2; |
---|
1738 | string s = "x*y+y*z+z*t"; // + only |
---|
1739 | stringpoly2lplace(s); |
---|
1740 | string s2 = "x*y - y*z-z*t*w*w"; // +1, - only |
---|
1741 | stringpoly2lplace(s2); |
---|
1742 | string s3 = "-x*y + y - 2*x +7*w*w*w"; |
---|
1743 | stringpoly2lplace(s3); |
---|
1744 | } |
---|
1745 | |
---|
1746 | static proc addplaces(list L) |
---|
1747 | { |
---|
1748 | // adds places to the list of strings |
---|
1749 | // according to their order in the list |
---|
1750 | int s = size(L); |
---|
1751 | int i; |
---|
1752 | for (i=1; i<=s; i++) |
---|
1753 | { |
---|
1754 | if (typeof(L[i]) == "string") |
---|
1755 | { |
---|
1756 | L[i] = L[i] + "(" + string(i) + ")"; |
---|
1757 | } |
---|
1758 | else |
---|
1759 | { |
---|
1760 | ERROR("entry of type string expected"); |
---|
1761 | return(0); |
---|
1762 | } |
---|
1763 | } |
---|
1764 | return(L); |
---|
1765 | } |
---|
1766 | example |
---|
1767 | { |
---|
1768 | "EXAMPLE:"; echo = 2; |
---|
1769 | string a = "f1"; string b = "f2"; |
---|
1770 | list L = a,b,a; |
---|
1771 | addplaces(L); |
---|
1772 | } |
---|
1773 | |
---|
1774 | static proc sent2lplace(string s) |
---|
1775 | { |
---|
1776 | // SENTence of words TO LetterPLACE |
---|
1777 | list L = stringpoly2lplace(s); |
---|
1778 | int i; int ss = size(L); |
---|
1779 | for(i=1; i<=ss; i++) |
---|
1780 | { |
---|
1781 | L[i] = str2lplace(L[i]); |
---|
1782 | } |
---|
1783 | return(L); |
---|
1784 | } |
---|
1785 | example |
---|
1786 | { |
---|
1787 | "EXAMPLE:"; echo = 2; |
---|
1788 | ring r = 0,(f2,f1),dp; |
---|
1789 | string s = "f2*f1*f1 - 2*f1*f2*f1+ f1*f1*f2"; |
---|
1790 | sent2lplace(s); |
---|
1791 | } |
---|
1792 | |
---|
1793 | static proc testnumber(string s) |
---|
1794 | { |
---|
1795 | string t; |
---|
1796 | if (s[1]=="-") |
---|
1797 | { |
---|
1798 | // two situations: either there's a negative number |
---|
1799 | t = s[2..size(s)]; |
---|
1800 | if (testnumber(t)) |
---|
1801 | { |
---|
1802 | //a negative number |
---|
1803 | } |
---|
1804 | else |
---|
1805 | { |
---|
1806 | // a variable times -1 |
---|
1807 | } |
---|
1808 | // or just a "-" for -1 |
---|
1809 | } |
---|
1810 | t = "ring @r=("; |
---|
1811 | t = t + charstr(basering)+"),"; |
---|
1812 | t = t + string(var(1))+",dp;"; |
---|
1813 | // write(":w tstnum.tst",t); |
---|
1814 | t = t+ "number @@Nn = " + s + ";"+"$"; |
---|
1815 | write(":w tstnum.tst",t); |
---|
1816 | string runsing = system("Singular"); |
---|
1817 | int k; |
---|
1818 | t = runsing+ " -teq <tstnum.tst >tstnum.out"; |
---|
1819 | k = system("sh",t); |
---|
1820 | if (k!=0) |
---|
1821 | { |
---|
1822 | ERROR("Problems running Singular"); |
---|
1823 | } |
---|
1824 | int i = system("sh", "grep error tstnum.out > /dev/NULL"); |
---|
1825 | if (i!=0) |
---|
1826 | { |
---|
1827 | // no error: s is a number |
---|
1828 | i = 1; |
---|
1829 | } |
---|
1830 | k = system("sh","rm tstnum.tst tstnum.out > /dev/NULL"); |
---|
1831 | return(i); |
---|
1832 | } |
---|
1833 | example |
---|
1834 | { |
---|
1835 | "EXAMPLE:"; echo = 2; |
---|
1836 | ring r = (0,a),x,dp; |
---|
1837 | string s = "a^2+7*a-2"; |
---|
1838 | testnumber(s); |
---|
1839 | s = "b+a"; |
---|
1840 | testnumber(s); |
---|
1841 | } |
---|
1842 | |
---|
1843 | static proc str2lplace(string s) |
---|
1844 | { |
---|
1845 | // converts a word (monomial) with coeff into letter-place |
---|
1846 | // string: coef*var1^exp1*var2^exp2*...varN^expN |
---|
1847 | s = strpower2rep(s); // expand powers |
---|
1848 | if (size(s)==0) { return(0); } |
---|
1849 | int i,j,k,insC; |
---|
1850 | string a,b,c,d,t; |
---|
1851 | // 1. get coeff |
---|
1852 | i = find(s,"*"); |
---|
1853 | if (i==0) { return(s); } |
---|
1854 | list VN; |
---|
1855 | c = s[1..i-1]; // incl. the case like (-a^2+1) |
---|
1856 | int tn = testnumber(c); |
---|
1857 | if (tn == 0) |
---|
1858 | { |
---|
1859 | // failed test |
---|
1860 | if (c[1]=="-") |
---|
1861 | { |
---|
1862 | // two situations: either there's a negative number |
---|
1863 | t = c[2..size(c)]; |
---|
1864 | if (testnumber(t)) |
---|
1865 | { |
---|
1866 | //a negative number |
---|
1867 | // nop here |
---|
1868 | } |
---|
1869 | else |
---|
1870 | { |
---|
1871 | // a variable times -1 |
---|
1872 | c = "-1"; |
---|
1873 | j++; VN[j] = t; //string(c[2..size(c)]); |
---|
1874 | insC = 1; |
---|
1875 | } |
---|
1876 | } |
---|
1877 | else |
---|
1878 | { |
---|
1879 | // just a variable with coeff 1 |
---|
1880 | j++; VN[j] = string(c); |
---|
1881 | c = "1"; |
---|
1882 | insC = 1; |
---|
1883 | } |
---|
1884 | } |
---|
1885 | // get vars |
---|
1886 | t = s; |
---|
1887 | // t = s[i+1..size(s)]; |
---|
1888 | k = size(t); //j = 0; |
---|
1889 | while (k>0) |
---|
1890 | { |
---|
1891 | t = t[i+1..size(t)]; //next part |
---|
1892 | i = find(t,"*"); // next * |
---|
1893 | if (i==0) |
---|
1894 | { |
---|
1895 | // last monomial |
---|
1896 | j++; |
---|
1897 | VN[j] = t; |
---|
1898 | k = size(t); |
---|
1899 | break; |
---|
1900 | } |
---|
1901 | b = t[1..i-1]; |
---|
1902 | // print(b); |
---|
1903 | j++; |
---|
1904 | VN[j] = b; |
---|
1905 | k = size(t); |
---|
1906 | } |
---|
1907 | VN = addplaces(VN); |
---|
1908 | VN[size(VN)+1] = string(c); |
---|
1909 | return(VN); |
---|
1910 | } |
---|
1911 | example |
---|
1912 | { |
---|
1913 | "EXAMPLE:"; echo = 2; |
---|
1914 | ring r = (0,a),(f2,f1),dp; |
---|
1915 | str2lplace("-2*f2^2*f1^2*f2"); |
---|
1916 | str2lplace("-f1*f2"); |
---|
1917 | str2lplace("(-a^2+7a)*f1*f2"); |
---|
1918 | } |
---|
1919 | |
---|
1920 | static proc strpower2rep(string s) |
---|
1921 | { |
---|
1922 | // makes x*x*x*x out of x^4 ., rep statys for repetitions |
---|
1923 | // looks for "-" problem |
---|
1924 | // exception: "-" as coeff |
---|
1925 | string ex,t; |
---|
1926 | int i,j,k; |
---|
1927 | |
---|
1928 | i = find(s,"^"); // first ^ |
---|
1929 | if (i==0) { return(s); } // no ^ signs |
---|
1930 | |
---|
1931 | if (s[1] == "-") |
---|
1932 | { |
---|
1933 | // either -coef or -1 |
---|
1934 | // got the coeff: |
---|
1935 | i = find(s,"*"); |
---|
1936 | if (i==0) |
---|
1937 | { |
---|
1938 | // no *'s => coef == -1 or s == -23 |
---|
1939 | i = size(s)+1; |
---|
1940 | } |
---|
1941 | t = string(s[2..i-1]); // without "-" |
---|
1942 | if ( testnumber(t) ) |
---|
1943 | { |
---|
1944 | // a good number |
---|
1945 | t = strpower2rep(string(s[2..size(s)])); |
---|
1946 | t = "-"+t; |
---|
1947 | return(t); |
---|
1948 | } |
---|
1949 | else |
---|
1950 | { |
---|
1951 | // a variable |
---|
1952 | t = strpower2rep(string(s[2..size(s)])); |
---|
1953 | t = "-1*"+ t; |
---|
1954 | return(t); |
---|
1955 | } |
---|
1956 | } |
---|
1957 | // the case when leadcoef is a number in () |
---|
1958 | if (s[1] == "(") |
---|
1959 | { |
---|
1960 | i = find(s,")",2); // must be nonzero |
---|
1961 | t = s[2..i-1]; |
---|
1962 | if ( testnumber(t) ) |
---|
1963 | { |
---|
1964 | // a good number |
---|
1965 | } |
---|
1966 | else {"strpower2rep: bad number as coef";} |
---|
1967 | ex = string(s[i+2..size(s)]); // 2 because of * |
---|
1968 | ex = strpower2rep(ex); |
---|
1969 | t = "("+t+")*"+ex; |
---|
1970 | return(t); |
---|
1971 | } |
---|
1972 | |
---|
1973 | i = find(s,"^"); // first ^ |
---|
1974 | j = find(s,"*",i+1); // next * == end of ^ |
---|
1975 | if (j==0) |
---|
1976 | { |
---|
1977 | ex = s[i+1..size(s)]; |
---|
1978 | } |
---|
1979 | else |
---|
1980 | { |
---|
1981 | ex = s[i+1..j-1]; |
---|
1982 | } |
---|
1983 | execute("int @exp = " + ex + ";"); //@exp = exponent |
---|
1984 | // got varname |
---|
1985 | for (k=i-1; k>0; k--) |
---|
1986 | { |
---|
1987 | if (s[k] == "*") break; |
---|
1988 | } |
---|
1989 | string varn = s[k+1..i-1]; |
---|
1990 | // "varn:"; varn; |
---|
1991 | string pref; |
---|
1992 | if (k>0) |
---|
1993 | { |
---|
1994 | pref = s[1..k]; // with * on the k-th place |
---|
1995 | } |
---|
1996 | // "pref:"; pref; |
---|
1997 | string suf; |
---|
1998 | if ( (j>0) && (j+1 <= size(s)) ) |
---|
1999 | { |
---|
2000 | suf = s[j+1..size(s)]; // without * on the 1st place |
---|
2001 | } |
---|
2002 | // "suf:"; suf; |
---|
2003 | string toins; |
---|
2004 | for (k=1; k<=@exp; k++) |
---|
2005 | { |
---|
2006 | toins = toins + varn+"*"; |
---|
2007 | } |
---|
2008 | // "toins: "; toins; |
---|
2009 | if (size(suf) == 0) |
---|
2010 | { |
---|
2011 | toins = toins[1..size(toins)-1]; // get rid of trailing * |
---|
2012 | } |
---|
2013 | else |
---|
2014 | { |
---|
2015 | suf = strpower2rep(suf); |
---|
2016 | } |
---|
2017 | ex = pref + toins + suf; |
---|
2018 | return(ex); |
---|
2019 | // return(strpower2rep(ex)); |
---|
2020 | } |
---|
2021 | example |
---|
2022 | { |
---|
2023 | "EXAMPLE:"; echo = 2; |
---|
2024 | ring r = (0,a),(x,y,z,t),dp; |
---|
2025 | strpower2rep("-x^4"); |
---|
2026 | strpower2rep("-2*x^4*y^3*z*t^2"); |
---|
2027 | strpower2rep("-a^2*x^4"); |
---|
2028 | } |
---|
2029 | |
---|
2030 | proc lieBracket(poly a, poly b, list #) |
---|
2031 | "USAGE: lieBracket(a,b[,N]); a,b letterplace polynomials, N an optional integer |
---|
2032 | RETURN: poly |
---|
2033 | ASSUME: basering has a letterplace ring structure |
---|
2034 | PURPOSE: compute the Lie bracket [a,b] = ab - ba between letterplace polynomials |
---|
2035 | NOTE: if N>1 is specified, then the left normed bracket [a,[...[a,b]]]] is computed. |
---|
2036 | EXAMPLE: example lieBracket; shows examples |
---|
2037 | " |
---|
2038 | { |
---|
2039 | if (lpAssumeViolation()) |
---|
2040 | { |
---|
2041 | // ERROR("Either 'uptodeg' or 'lV' global variables are not set!"); |
---|
2042 | ERROR("Incomplete Letterplace structure on the basering!"); |
---|
2043 | } |
---|
2044 | // alias ppLiebr; |
---|
2045 | //if int N is given compute [a,[...[a,b]]]] left normed bracket |
---|
2046 | poly q; |
---|
2047 | int N=1; |
---|
2048 | if (size(#)>0) |
---|
2049 | { |
---|
2050 | if (typeof(#[1])=="int") |
---|
2051 | { |
---|
2052 | N = int(#[1]); |
---|
2053 | } |
---|
2054 | } |
---|
2055 | if (N<=0) { return(q); } |
---|
2056 | while (b!=0) |
---|
2057 | { |
---|
2058 | q = q + pmLiebr(a,lead(b)); |
---|
2059 | b = b - lead(b); |
---|
2060 | } |
---|
2061 | int i; |
---|
2062 | if (N >1) |
---|
2063 | { |
---|
2064 | for(i=1; i<=N; i++) |
---|
2065 | { |
---|
2066 | q = lieBracket(a,q); |
---|
2067 | } |
---|
2068 | } |
---|
2069 | return(q); |
---|
2070 | } |
---|
2071 | example |
---|
2072 | { |
---|
2073 | "EXAMPLE:"; echo = 2; |
---|
2074 | ring r = 0,(x(1),y(1),x(2),y(2),x(3),y(3),x(4),y(4)),dp; |
---|
2075 | def R = setLetterplaceAttributes(r,4,2); // supply R with letterplace structure |
---|
2076 | setring R; |
---|
2077 | poly a = x(1)*y(2); poly b = y(1); |
---|
2078 | lieBracket(a,b); |
---|
2079 | lieBracket(x(1),y(1),2); |
---|
2080 | } |
---|
2081 | |
---|
2082 | static proc pmLiebr(poly a, poly b) |
---|
2083 | { |
---|
2084 | // a poly, b mono |
---|
2085 | poly s; |
---|
2086 | while (a!=0) |
---|
2087 | { |
---|
2088 | s = s + mmLiebr(lead(a),lead(b)); |
---|
2089 | a = a - lead(a); |
---|
2090 | } |
---|
2091 | return(s); |
---|
2092 | } |
---|
2093 | |
---|
2094 | proc shiftPoly(poly a, int i) |
---|
2095 | "USAGE: shiftPoly(p,i); p letterplace poly, i int |
---|
2096 | RETURN: poly |
---|
2097 | ASSUME: basering has letterplace ring structure |
---|
2098 | PURPOSE: compute the i-th shift of letterplace polynomial p |
---|
2099 | EXAMPLE: example shiftPoly; shows examples |
---|
2100 | " |
---|
2101 | { |
---|
2102 | // shifts a monomial a by i |
---|
2103 | // calls pLPshift(p,sh,uptodeg,lVblock); |
---|
2104 | if (lpAssumeViolation()) |
---|
2105 | { |
---|
2106 | ERROR("Incomplete Letterplace structure on the basering!"); |
---|
2107 | } |
---|
2108 | int uptodeg = attrib(basering,"uptodeg"); |
---|
2109 | int lV = attrib(basering,"lV"); |
---|
2110 | if (deg(a) + i > uptodeg) |
---|
2111 | { |
---|
2112 | ERROR("degree bound violated by the shift!"); |
---|
2113 | } |
---|
2114 | return(system("stest",a,i,uptodeg,lV)); |
---|
2115 | } |
---|
2116 | example |
---|
2117 | { |
---|
2118 | "EXAMPLE:"; echo = 2; |
---|
2119 | ring r = 0,(x,y,z),dp; |
---|
2120 | int uptodeg = 5; int lV = 3; |
---|
2121 | def R = makeLetterplaceRing(uptodeg); |
---|
2122 | setring R; |
---|
2123 | poly f = x(1)*z(2)*y(3) - 2*z(1)*y(2) + 3*x(1); |
---|
2124 | shiftPoly(f,1); |
---|
2125 | shiftPoly(f,2); |
---|
2126 | } |
---|
2127 | |
---|
2128 | |
---|
2129 | static proc mmLiebr(poly a, poly b) |
---|
2130 | { |
---|
2131 | // a,b, monomials |
---|
2132 | a = lead(a); |
---|
2133 | b = lead(b); |
---|
2134 | int sa = deg(a); |
---|
2135 | int sb = deg(b); |
---|
2136 | poly v = a*shiftPoly(b,sa) - b*shiftPoly(a,sb); |
---|
2137 | return(v); |
---|
2138 | } |
---|
2139 | |
---|
2140 | static proc test_shift() |
---|
2141 | { |
---|
2142 | LIB "freegb.lib"; |
---|
2143 | ring r = 0,(a,b),dp; |
---|
2144 | int d =5; |
---|
2145 | def R = makeLetterplaceRing(d); |
---|
2146 | setring R; |
---|
2147 | int uptodeg = d; |
---|
2148 | int lV = 2; |
---|
2149 | def R = setLetterplaceAttributes(r,uptodeg,2); // supply R with letterplace structure |
---|
2150 | setring R; |
---|
2151 | poly p = mmLiebr(a(1),b(1)); |
---|
2152 | poly p = lieBracket(a(1),b(1)); |
---|
2153 | } |
---|
2154 | |
---|
2155 | proc serreRelations(intmat A, int zu) |
---|
2156 | "USAGE: serreRelations(A,z); A an intmat, z an int |
---|
2157 | RETURN: ideal |
---|
2158 | ASSUME: basering has a letterplace ring structure and |
---|
2159 | @* A is a generalized Cartan matrix with integer entries |
---|
2160 | PURPOSE: compute the ideal of Serre's relations associated to A |
---|
2161 | EXAMPLE: example serreRelations; shows examples |
---|
2162 | " |
---|
2163 | { |
---|
2164 | // zu = 1 -> with commutators [f_i,f_j]; zu == 0 without them |
---|
2165 | // suppose that A is cartan matrix |
---|
2166 | // then Serre's relations are |
---|
2167 | // (ad f_j)^{1-A_{ij}} ( f_i) |
---|
2168 | int ppl = printlevel-voice+2; |
---|
2169 | int n = ncols(A); // hence n variables |
---|
2170 | int i,j,k,el; |
---|
2171 | poly p,q; |
---|
2172 | ideal I; |
---|
2173 | for (i=1; i<=n; i++) |
---|
2174 | { |
---|
2175 | for (j=1; j<=n; j++) |
---|
2176 | { |
---|
2177 | el = 1 - A[i,j]; |
---|
2178 | // printf("i:%s, j: %s, l: %s",i,j,l); |
---|
2179 | dbprint(ppl,"i, j, l: ",i,j,el); |
---|
2180 | // if ((i!=j) && (l >0)) |
---|
2181 | // if ( (i!=j) && ( ((zu ==0) && (l >=2)) || ((zu ==1) && (l >=1)) ) ) |
---|
2182 | if ((i!=j) && (el >0)) |
---|
2183 | { |
---|
2184 | q = lieBracket(var(j),var(i)); |
---|
2185 | dbprint(ppl,"first bracket: ",q); |
---|
2186 | // if (l >=2) |
---|
2187 | // { |
---|
2188 | for (k=1; k<=el-1; k++) |
---|
2189 | { |
---|
2190 | q = lieBracket(var(j),q); |
---|
2191 | dbprint(ppl,"further bracket:",q); |
---|
2192 | } |
---|
2193 | // } |
---|
2194 | } |
---|
2195 | if (q!=0) { I = I,q; q=0;} |
---|
2196 | } |
---|
2197 | } |
---|
2198 | I = simplify(I,2); |
---|
2199 | return(I); |
---|
2200 | } |
---|
2201 | example |
---|
2202 | { |
---|
2203 | "EXAMPLE:"; echo = 2; |
---|
2204 | intmat A[3][3] = |
---|
2205 | 2, -1, 0, |
---|
2206 | -1, 2, -3, |
---|
2207 | 0, -1, 2; // G^1_2 Cartan matrix |
---|
2208 | ring r = 0,(f1,f2,f3),dp; |
---|
2209 | int uptodeg = 5; |
---|
2210 | def R = makeLetterplaceRing(uptodeg); |
---|
2211 | setring R; |
---|
2212 | ideal I = serreRelations(A,1); I = simplify(I,1+2+8); |
---|
2213 | I; |
---|
2214 | } |
---|
2215 | |
---|
2216 | /* setup for older example: |
---|
2217 | intmat A[2][2] = 2, -1, -1, 2; // sl_3 == A_2 |
---|
2218 | ring r = 0,(f1,f2),dp; |
---|
2219 | int uptodeg = 5; int lV = 2; |
---|
2220 | */ |
---|
2221 | |
---|
2222 | proc lp2lstr(ideal K, def save) |
---|
2223 | "USAGE: lp2lstr(K,s); K an ideal, s a ring |
---|
2224 | RETURN: nothing (exports object @LN into s) |
---|
2225 | ASSUME: basering has a letterplace ring structure |
---|
2226 | PURPOSE: converts letterplace ideal to list of modules |
---|
2227 | NOTE: useful as preprocessing to 'lst2str' |
---|
2228 | EXAMPLE: example lp2lstr; shows examples |
---|
2229 | " |
---|
2230 | { |
---|
2231 | def @R = basering; |
---|
2232 | string err; |
---|
2233 | int s = nvars(save); |
---|
2234 | int i,j,k; |
---|
2235 | // K contains vars x(1),...z(1) = images of originals |
---|
2236 | // 5. go back to orig vars, produce strings/modules |
---|
2237 | int sk = size(K); |
---|
2238 | int sp, sx, a, b; |
---|
2239 | intvec x; |
---|
2240 | poly p,q; |
---|
2241 | poly pn; |
---|
2242 | // vars in 'save' |
---|
2243 | setring save; |
---|
2244 | module N; |
---|
2245 | list LN; |
---|
2246 | vector V; |
---|
2247 | poly pn; |
---|
2248 | // test and skip exponents >=2 |
---|
2249 | setring @R; |
---|
2250 | for(i=1; i<=sk; i++) |
---|
2251 | { |
---|
2252 | p = K[i]; |
---|
2253 | while (p!=0) |
---|
2254 | { |
---|
2255 | q = lead(p); |
---|
2256 | // "processing q:";q; |
---|
2257 | x = leadexp(q); |
---|
2258 | sx = size(x); |
---|
2259 | for(k=1; k<=sx; k++) |
---|
2260 | { |
---|
2261 | if ( x[k] >= 2 ) |
---|
2262 | { |
---|
2263 | err = "skip: the value x[k] is " + string(x[k]); |
---|
2264 | dbprint(ppl,err); |
---|
2265 | // return(0); |
---|
2266 | K[i] = 0; |
---|
2267 | p = 0; |
---|
2268 | q = 0; |
---|
2269 | break; |
---|
2270 | } |
---|
2271 | } |
---|
2272 | p = p - q; |
---|
2273 | } |
---|
2274 | } |
---|
2275 | K = simplify(K,2); |
---|
2276 | sk = size(K); |
---|
2277 | for(i=1; i<=sk; i++) |
---|
2278 | { |
---|
2279 | // setring save; |
---|
2280 | // V = 0; |
---|
2281 | setring @R; |
---|
2282 | p = K[i]; |
---|
2283 | while (p!=0) |
---|
2284 | { |
---|
2285 | q = lead(p); |
---|
2286 | err = "processing q:" + string(q); |
---|
2287 | dbprint(ppl,err); |
---|
2288 | x = leadexp(q); |
---|
2289 | sx = size(x); |
---|
2290 | pn = leadcoef(q); |
---|
2291 | setring save; |
---|
2292 | pn = imap(@R,pn); |
---|
2293 | V = V + leadcoef(pn)*gen(1); |
---|
2294 | for(k=1; k<=sx; k++) |
---|
2295 | { |
---|
2296 | if (x[k] ==1) |
---|
2297 | { |
---|
2298 | a = k / s; // block number=a+1, a!=0 |
---|
2299 | b = k % s; // remainder |
---|
2300 | // printf("a: %s, b: %s",a,b); |
---|
2301 | if (b == 0) |
---|
2302 | { |
---|
2303 | // that is it's the last var in the block |
---|
2304 | b = s; |
---|
2305 | a = a-1; |
---|
2306 | } |
---|
2307 | V = V + var(b)*gen(a+2); |
---|
2308 | } |
---|
2309 | } |
---|
2310 | err = "V: " + string(V); |
---|
2311 | dbprint(ppl,err); |
---|
2312 | // printf("V: %s", string(V)); |
---|
2313 | N = N,V; |
---|
2314 | V = 0; |
---|
2315 | setring @R; |
---|
2316 | p = p - q; |
---|
2317 | pn = 0; |
---|
2318 | } |
---|
2319 | setring save; |
---|
2320 | LN[i] = simplify(N,2); |
---|
2321 | N = 0; |
---|
2322 | } |
---|
2323 | setring save; |
---|
2324 | list @LN = LN; |
---|
2325 | export @LN; |
---|
2326 | // return(LN); |
---|
2327 | } |
---|
2328 | example |
---|
2329 | { |
---|
2330 | "EXAMPLE:"; echo = 2; |
---|
2331 | intmat A[2][2] = 2, -1, -1, 2; // sl_3 == A_2 |
---|
2332 | ring r = 0,(f1,f2),dp; |
---|
2333 | def R = makeLetterplaceRing(3); |
---|
2334 | setring R; |
---|
2335 | ideal I = serreRelations(A,1); |
---|
2336 | lp2lstr(I,r); |
---|
2337 | setring r; |
---|
2338 | lst2str(@LN,1); |
---|
2339 | } |
---|
2340 | |
---|
2341 | static proc strList2poly(list L) |
---|
2342 | { |
---|
2343 | // list L comes from sent2lplace (which takes a poly on the input) |
---|
2344 | // each entry of L is a sublist with the coef on the last place |
---|
2345 | int s = size(L); int t; |
---|
2346 | int i,j; |
---|
2347 | list M; |
---|
2348 | poly p,q; |
---|
2349 | string Q; |
---|
2350 | for(i=1; i<=s; i++) |
---|
2351 | { |
---|
2352 | M = L[i]; |
---|
2353 | t = size(M); |
---|
2354 | // q = M[t]; // a constant |
---|
2355 | Q = string(M[t]); |
---|
2356 | for(j=1; j<t; j++) |
---|
2357 | { |
---|
2358 | // q = q*M[j]; |
---|
2359 | Q = Q+"*"+string(M[j]); |
---|
2360 | } |
---|
2361 | execute("q="+Q+";"); |
---|
2362 | // q; |
---|
2363 | p = p + q; |
---|
2364 | } |
---|
2365 | kill Q; |
---|
2366 | return(p); |
---|
2367 | } |
---|
2368 | example |
---|
2369 | { |
---|
2370 | "EXAMPLE:"; echo = 2; |
---|
2371 | ring r =0,(x,y,z,t),Dp; |
---|
2372 | def A = makeLetterplaceRing(4); |
---|
2373 | setring A; |
---|
2374 | string t = "-2*y*z*y*z + y*t*z*z - z*x*x*y + 2*z*y*z*y"; |
---|
2375 | list L = sent2lplace(t); |
---|
2376 | L; |
---|
2377 | poly p = strList2poly(L); |
---|
2378 | p; |
---|
2379 | } |
---|
2380 | |
---|
2381 | static proc file2lplace(string fname) |
---|
2382 | "USAGE: file2lplace(fnm); fnm a string |
---|
2383 | RETURN: ideal |
---|
2384 | PURPOSE: convert the contents of the file fnm into ideal of polynomials in free algebra |
---|
2385 | EXAMPLE: example file2lplace; shows examples |
---|
2386 | " |
---|
2387 | { |
---|
2388 | // format: from the usual string to letterplace |
---|
2389 | string s = read(fname); |
---|
2390 | // assume: file is a comma-sep list of polys |
---|
2391 | // the vars are declared before |
---|
2392 | // the file ends with ";" |
---|
2393 | string t; int i; |
---|
2394 | ideal I; |
---|
2395 | list tst; |
---|
2396 | while (s!="") |
---|
2397 | { |
---|
2398 | i = find(s,","); |
---|
2399 | "i"; i; |
---|
2400 | if (i==0) |
---|
2401 | { |
---|
2402 | i = find(s,";"); |
---|
2403 | if (i==0) |
---|
2404 | { |
---|
2405 | // no ; ?? |
---|
2406 | "no colon or semicolon found anymore"; |
---|
2407 | return(I); |
---|
2408 | } |
---|
2409 | // no "," but ";" on the i-th place |
---|
2410 | t = s[1..i-1]; |
---|
2411 | s = ""; |
---|
2412 | "processing: "; t; |
---|
2413 | tst = sent2lplace(t); |
---|
2414 | tst; |
---|
2415 | I = I, strList2poly(tst); |
---|
2416 | return(I); |
---|
2417 | } |
---|
2418 | // here i !=0 |
---|
2419 | t = s[1..i-1]; |
---|
2420 | s = s[i+1..size(s)]; |
---|
2421 | "processing: "; t; |
---|
2422 | tst = sent2lplace(t); |
---|
2423 | tst; |
---|
2424 | I = I, strList2poly(tst); |
---|
2425 | } |
---|
2426 | return(I); |
---|
2427 | } |
---|
2428 | example |
---|
2429 | { |
---|
2430 | "EXAMPLE:"; echo = 2; |
---|
2431 | ring r =0,(x,y,z,t),dp; |
---|
2432 | def A = makeLetterplaceRing(4); |
---|
2433 | setring A; |
---|
2434 | string fn = "myfile"; |
---|
2435 | string s1 = "z*y*y*y - 3*y*z*x*y + 3*y*y*z*y - y*x*y*z,"; |
---|
2436 | string s2 = "-2*y*x*y*z + y*y*z*z - z*z*y*y + 2*z*y*z*y,"; |
---|
2437 | 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;"; |
---|
2438 | write(":w "+fn,s1); write(":a "+fn,s2); write(":a "+fn,s3); |
---|
2439 | read(fn); |
---|
2440 | ideal I = file2lplace(fn); |
---|
2441 | I; |
---|
2442 | } |
---|
2443 | |
---|
2444 | /* EXAMPLES AGAIN: |
---|
2445 | //static proc get_ls3nilp() |
---|
2446 | { |
---|
2447 | //first app of file2lplace |
---|
2448 | ring r =0,(x,y,z,t),dp; |
---|
2449 | int d = 10; |
---|
2450 | def A = makeLetterplaceRing(d); |
---|
2451 | setring A; |
---|
2452 | ideal I = file2lplace("./ls3nilp.bg"); |
---|
2453 | // and now test the correctness: go back from lplace to strings |
---|
2454 | lp2lstr(I,r); |
---|
2455 | setring r; |
---|
2456 | lst2str(@LN,1); // agree! |
---|
2457 | } |
---|
2458 | |
---|
2459 | //static proc doc_example() |
---|
2460 | { |
---|
2461 | LIB "freegb.lib"; |
---|
2462 | ring r = 0,(x,y,z),dp; |
---|
2463 | int d =4; // degree bound |
---|
2464 | def R = makeLetterplaceRing(d); |
---|
2465 | setring R; |
---|
2466 | 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); |
---|
2467 | option(redSB);option(redTail); |
---|
2468 | ideal J = system("freegb",I,d,nvars(r)); |
---|
2469 | J; |
---|
2470 | // visualization: |
---|
2471 | lp2lstr(J,r); // export an object called @LN to the ring r |
---|
2472 | setring r; // change to the ring r |
---|
2473 | lst2str(@LN,1); // output the strings |
---|
2474 | } |
---|
2475 | |
---|
2476 | */ |
---|
2477 | |
---|
2478 | // TODO: |
---|
2479 | // multiply two letterplace polynomials, lpMult |
---|
2480 | // reduction/ Normalform? needs kernel stuff |
---|
2481 | |
---|
2482 | proc lpMult(poly f, poly g) |
---|
2483 | "USAGE: lpMult(f,g); f,g letterplace polynomials |
---|
2484 | RETURN: poly |
---|
2485 | ASSUME: basering has a letterplace ring structure |
---|
2486 | PURPOSE: compute the letterplace form of f*g |
---|
2487 | EXAMPLE: example lpMult; shows examples |
---|
2488 | " |
---|
2489 | { |
---|
2490 | if (lpAssumeViolation()) |
---|
2491 | { |
---|
2492 | ERROR("Incomplete Letterplace structure on the basering!"); |
---|
2493 | } |
---|
2494 | int sf = deg(f); |
---|
2495 | int sg = deg(g); |
---|
2496 | int uptodeg = attrib(basering, "uptodeg"); |
---|
2497 | if (sf+sg > uptodeg) |
---|
2498 | { |
---|
2499 | ERROR("degree bound violated by the product!"); |
---|
2500 | } |
---|
2501 | // if (sf>1) { sf = sf -1; } |
---|
2502 | poly v = f*shiftPoly(g,sf); |
---|
2503 | return(v); |
---|
2504 | } |
---|
2505 | example |
---|
2506 | { |
---|
2507 | "EXAMPLE:"; echo = 2; |
---|
2508 | // define a ring in letterplace form as follows: |
---|
2509 | ring r = 0,(x(1),y(1),x(2),y(2),x(3),y(3),x(4),y(4)),dp; |
---|
2510 | poly a = x(1)*y(2); poly b = y(1); |
---|
2511 | def R = setLetterplaceAttributes(r,4,2); // supply R with letterplace structure |
---|
2512 | setring R; |
---|
2513 | lpMult(b,a); |
---|
2514 | lpMult(a,b); |
---|
2515 | } |
---|
2516 | |
---|
2517 | static proc lpAssumeViolation() |
---|
2518 | { |
---|
2519 | // checks whether the global vars |
---|
2520 | // uptodeg and lV are defined |
---|
2521 | // returns Boolean : yes/no [for assume violation] |
---|
2522 | def lpring = attrib(basering,"isLetterplaceRing"); |
---|
2523 | if ( typeof(lpring)!="int" ) |
---|
2524 | { |
---|
2525 | // if ( typeof(lpring)=="string" ) ?? |
---|
2526 | // basering is NOT lp Ring |
---|
2527 | |
---|
2528 | return(1); |
---|
2529 | } |
---|
2530 | def uptodeg = attrib(basering,"uptodeg"); |
---|
2531 | if ( typeof(uptodeg)!="int" ) |
---|
2532 | { |
---|
2533 | return(1); |
---|
2534 | } |
---|
2535 | def lV = attrib(basering,"lV"); |
---|
2536 | if ( typeof(lV)!="int" ) |
---|
2537 | { |
---|
2538 | return(1); |
---|
2539 | } |
---|
2540 | // int i = ( defined(uptodeg) && (defined(lV)) ); |
---|
2541 | // return ( !i ); |
---|
2542 | return(0); |
---|
2543 | } |
---|
2544 | |
---|
2545 | // alias libs for compatibility with older examples |
---|
2546 | proc freegbRing(int d) |
---|
2547 | { |
---|
2548 | return(makeLetterplaceRing(d)); |
---|
2549 | } |
---|
2550 | |
---|
2551 | proc freegbasis( list L, int n) |
---|
2552 | { |
---|
2553 | return(freeGBasis(L, n)); |
---|
2554 | } |
---|