1 | ////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id$"; |
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3 | category="Noncommutative"; |
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4 | info=" |
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5 | LIBRARY: freegb.lib Compute two-sided 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 @sc{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 | shiftPoly(p,i); compute the i-th shift of letterplace polynomial p |
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19 | lpPower(f,n); natural power of a letterplace polynomial |
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20 | lp2lstr(K, s); convert letter-place ideal to a list of modules |
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21 | lst2str(L[, n]); convert a list (of modules) into polynomials in free algebra |
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22 | mod2str(M[, n]); convert a module into a polynomial in free algebra |
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23 | vct2str(M[, n]); convert a vector into a word in free algebra |
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24 | lieBracket(a,b[, N]); compute Lie bracket ab-ba of two letterplace polynomials |
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25 | serreRelations(A,z); compute the homogeneous part of Serre's relations associated to a generalized Cartan matrix A |
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26 | fullSerreRelations(A,N,C,P,d); compute the ideal of all Serre's relations associated to a generalized Cartan matrix A |
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27 | isVar(p); check whether p is a power of a single variable |
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28 | ademRelations(i,j); compute the ideal of Adem relations for i<2j in char 0 |
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29 | |
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30 | SEE ALSO: LETTERPLACE |
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31 | " |
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32 | |
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33 | // this library computes two-sided GB of an ideal |
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34 | // in a free associative algebra |
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35 | |
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36 | // a monomial is encoded via a vector V |
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37 | // where V[1] = coefficient |
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38 | // V[1+i] = the corresponding symbol |
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39 | |
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40 | LIB "qhmoduli.lib"; // for Max |
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41 | |
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42 | proc tstfreegb() |
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43 | { |
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44 | /* tests all procs for consistency */ |
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45 | /* adding the new proc, add it here */ |
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46 | example makeLetterplaceRing; |
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47 | example freeGBasis; |
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48 | example setLetterplaceAttributes; |
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49 | /* secondary */ |
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50 | example lpMult; |
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51 | example shiftPoly; |
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52 | example lpPower; |
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53 | example lp2lstr; |
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54 | example lst2str; |
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55 | example mod2str; |
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56 | example vct2str; |
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57 | example lieBracket; |
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58 | example serreRelations; |
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59 | example fullSerreRelations; |
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60 | example isVar; |
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61 | example ademRelations; |
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62 | } |
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63 | |
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64 | proc setLetterplaceAttributes(def R, int uptodeg, int lV) |
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65 | "USAGE: setLetterplaceAttributes(R, d, b); R a ring, b,d integers |
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66 | RETURN: ring with special attributes set |
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67 | PURPOSE: sets attributes for a letterplace ring: |
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68 | @* 'isLetterplaceRing' = true, 'uptodeg' = d, 'lV' = b, where |
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69 | @* 'uptodeg' stands for the degree bound, |
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70 | @* 'lV' for the number of variables in the block 0. |
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71 | NOTE: Activate the resulting ring by using @code{setring} |
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72 | " |
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73 | { |
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74 | if (uptodeg*lV != nvars(R)) |
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75 | { |
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76 | ERROR("uptodeg and lV do not agree on the basering!"); |
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77 | } |
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78 | |
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79 | // Set letterplace-specific attributes for the output ring! |
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80 | attrib(R, "uptodeg", uptodeg); |
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81 | attrib(R, "lV", lV); |
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82 | attrib(R, "isLetterplaceRing", 1); |
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83 | return (R); |
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84 | } |
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85 | example |
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86 | { |
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87 | "EXAMPLE:"; echo = 2; |
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88 | ring r = 0,(x(1),y(1),x(2),y(2),x(3),y(3),x(4),y(4)),dp; |
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89 | def R = setLetterplaceAttributes(r, 4, 2); setring R; |
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90 | attrib(R,"isLetterplaceRing"); |
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91 | lieBracket(x(1),y(1),2); |
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92 | } |
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93 | |
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94 | |
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95 | // obsolete? |
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96 | |
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97 | static proc lshift(module M, int s, string varing, def lpring) |
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98 | { |
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99 | // FINALLY IMPLEMENTED AS A PART OT THE CODE |
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100 | // shifts a polynomial from the ring R to s positions |
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101 | // M lives in varing, the result in lpring |
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102 | // to be run from varing |
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103 | int i, j, k, sm, sv; |
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104 | vector v; |
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105 | // execute("setring "+lpring); |
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106 | setring lpring; |
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107 | poly @@p; |
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108 | ideal I; |
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109 | execute("setring "+varing); |
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110 | sm = ncols(M); |
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111 | for (i=1; i<=s; i++) |
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112 | { |
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113 | // modules, e.g. free polynomials |
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114 | for (j=1; j<=sm; j++) |
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115 | { |
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116 | //vectors, e.g. free monomials |
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117 | v = M[j]; |
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118 | sv = size(v); |
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119 | sp = "@@p = @@p + "; |
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120 | for (k=2; k<=sv; k++) |
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121 | { |
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122 | sp = sp + string(v[k])+"("+string(k-1+s)+")*"; |
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123 | } |
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124 | sp = sp + string(v[1])+";"; // coef; |
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125 | setring lpring; |
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126 | // execute("setring "+lpring); |
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127 | execute(sp); |
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128 | execute("setring "+varing); |
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129 | } |
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130 | setring lpring; |
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131 | // execute("setring "+lpring); |
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132 | I = I,@@p; |
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133 | @@p = 0; |
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134 | } |
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135 | setring lpring; |
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136 | //execute("setring "+lpring); |
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137 | export(I); |
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138 | // setring varing; |
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139 | execute("setring "+varing); |
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140 | } |
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141 | |
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142 | static proc skip0(vector v) |
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143 | { |
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144 | // skips zeros in a vector, producing another vector |
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145 | if ( (v[1]==0) || (v==0) ) { return(vector(0)); } |
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146 | int sv = nrows(v); |
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147 | int sw = size(v); |
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148 | if (sv == sw) |
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149 | { |
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150 | return(v); |
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151 | } |
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152 | int i; |
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153 | int j=1; |
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154 | vector w; |
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155 | for (i=1; i<=sv; i++) |
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156 | { |
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157 | if (v[i] != 0) |
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158 | { |
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159 | w = w + v[i]*gen(j); |
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160 | j++; |
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161 | } |
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162 | } |
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163 | return(w); |
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164 | } |
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165 | |
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166 | proc lst2str(list L, list #) |
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167 | "USAGE: lst2str(L[,n]); L a list of modules, n an optional integer |
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168 | RETURN: list (of strings) |
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169 | PURPOSE: convert a list (of modules) into polynomials in free algebra |
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170 | EXAMPLE: example lst2str; shows examples |
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171 | NOTE: if an optional integer is not 0, stars signs are used in multiplication |
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172 | " |
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173 | { |
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174 | // returns a list of strings |
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175 | // being sentences in words built from L |
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176 | // if #[1] = 1, use * between generators |
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177 | int useStar = 0; |
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178 | if ( size(#)>0 ) |
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179 | { |
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180 | if ( typeof(#[1]) != "int") |
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181 | { |
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182 | ERROR("Second argument of type int expected"); |
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183 | } |
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184 | if (#[1]) |
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185 | { |
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186 | useStar = 1; |
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187 | } |
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188 | } |
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189 | int i; |
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190 | int s = size(L); |
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191 | if (s<1) { return(list(""));} |
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192 | list N; |
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193 | for(i=1; i<=s; i++) |
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194 | { |
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195 | if ((typeof(L[i]) == "module") || (typeof(L[i]) == "matrix") ) |
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196 | { |
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197 | N[i] = mod2str(L[i],useStar); |
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198 | } |
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199 | else |
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200 | { |
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201 | "module or matrix expected in the list"; |
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202 | return(N); |
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203 | } |
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204 | } |
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205 | return(N); |
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206 | } |
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207 | example |
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208 | { |
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209 | "EXAMPLE:"; echo = 2; |
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210 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
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211 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
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212 | module N = [1,x,y,x,y],[-2,y,x,y,x],[6,x,y,y,x,y]; |
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213 | list L; L[1] = M; L[2] = N; |
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214 | lst2str(L); |
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215 | lst2str(L[1],1); |
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216 | } |
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217 | |
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218 | |
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219 | proc mod2str(module M, list #) |
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220 | "USAGE: mod2str(M[,n]); M a module, n an optional integer |
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221 | RETURN: string |
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222 | PURPOSE: convert a module into a polynomial in free algebra |
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223 | EXAMPLE: example mod2str; shows examples |
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224 | NOTE: if an optional integer is not 0, stars signs are used in multiplication |
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225 | " |
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226 | { |
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227 | if (size(M)==0) { return(""); } |
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228 | // returns a string |
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229 | // a sentence in words built from M |
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230 | // if #[1] = 1, use * between generators |
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231 | int useStar = 0; |
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232 | if ( size(#)>0 ) |
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233 | { |
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234 | if ( typeof(#[1]) != "int") |
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235 | { |
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236 | ERROR("Second argument of type int expected"); |
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237 | } |
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238 | if (#[1]) |
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239 | { |
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240 | useStar = 1; |
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241 | } |
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242 | } |
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243 | int i; |
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244 | int s = ncols(M); |
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245 | string t; |
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246 | string mp; |
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247 | for(i=1; i<=s; i++) |
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248 | { |
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249 | mp = vct2str(M[i],useStar); |
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250 | if (mp[1] == "-") |
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251 | { |
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252 | t = t + mp; |
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253 | } |
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254 | else |
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255 | { |
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256 | if (mp != "") |
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257 | { |
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258 | t = t + "+" + mp; |
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259 | } |
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260 | } |
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261 | } |
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262 | if (t[1]=="+") |
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263 | { |
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264 | t = t[2..size(t)]; // remove first "+" |
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265 | } |
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266 | return(t); |
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267 | } |
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268 | example |
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269 | { |
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270 | "EXAMPLE:"; echo = 2; |
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271 | ring r = 0,(x,y,z),(dp); |
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272 | module M = [1,x,y,x,y],[-2,y,x,y,x],[6,x,y,y,x,y]; |
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273 | mod2str(M); |
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274 | mod2str(M,1); |
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275 | } |
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276 | |
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277 | proc vct2str(vector v, list #) |
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278 | "USAGE: vct2str(v[,n]); v a vector, n an optional integer |
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279 | RETURN: string |
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280 | PURPOSE: convert a vector into a word in free algebra |
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281 | EXAMPLE: example vct2str; shows examples |
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282 | NOTE: if an optional integer is not 0, stars signs are used in multiplication |
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283 | " |
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284 | { |
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285 | if (v==0) { return(""); } |
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286 | // if #[1] = 1, use * between generators |
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287 | int useStar = 0; |
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288 | if ( size(#)>0 ) |
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289 | { |
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290 | if (#[1]) |
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291 | { |
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292 | useStar = 1; |
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293 | } |
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294 | } |
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295 | int ppl = printlevel-voice+2; |
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296 | // for a word, encoded by v |
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297 | // produces a string for it |
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298 | v = skip0(v); |
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299 | if (v==0) { return(string(""));} |
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300 | number cf = leadcoef(v[1]); |
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301 | int s = size(v); |
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302 | string vs,vv,vp,err; |
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303 | int i,j,p,q; |
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304 | for (i=1; i<=s-1; i++) |
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305 | { |
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306 | p = isVar(v[i+1]); |
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307 | if (p==0) |
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308 | { |
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309 | err = "Error: monomial expected at nonzero position " + string(i+1); |
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310 | ERROR(err+" in vct2str"); |
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311 | // dbprint(ppl,err); |
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312 | // return("_"); |
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313 | } |
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314 | if (p==1) |
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315 | { |
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316 | if (useStar && (size(vs) >0)) { vs = vs + "*"; } |
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317 | vs = vs + string(v[i+1]); |
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318 | } |
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319 | else //power |
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320 | { |
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321 | vv = string(v[i+1]); |
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322 | q = find(vv,"^"); |
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323 | if (q==0) |
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324 | { |
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325 | q = find(vv,string(p)); |
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326 | if (q==0) |
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327 | { |
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328 | err = "error in find for string "+vv; |
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329 | dbprint(ppl,err); |
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330 | return("_"); |
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331 | } |
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332 | } |
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333 | // q>0 |
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334 | vp = vv[1..q-1]; |
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335 | for(j=1;j<=p;j++) |
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336 | { |
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337 | if (useStar && (size(vs) >0)) { vs = vs + "*"; } |
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338 | vs = vs + vp; |
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339 | } |
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340 | } |
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341 | } |
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342 | string scf; |
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343 | if (cf == -1) |
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344 | { |
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345 | scf = "-"; |
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346 | } |
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347 | else |
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348 | { |
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349 | scf = string(cf); |
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350 | if ( (cf == 1) && (size(vs)>0) ) |
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351 | { |
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352 | scf = ""; |
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353 | } |
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354 | } |
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355 | if (useStar && (size(scf) >0) && (scf!="-") ) { scf = scf + "*"; } |
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356 | vs = scf + vs; |
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357 | return(vs); |
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358 | } |
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359 | example |
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360 | { |
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361 | "EXAMPLE:"; echo = 2; |
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362 | ring r = (0,a),(x,y3,z(1)),dp; |
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363 | vector v = [-7,x,y3^4,x2,z(1)^3]; |
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364 | vct2str(v); |
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365 | vct2str(v,1); |
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366 | vector w = [-7a^5+6a,x,y3,y3,x,z(1),z(1)]; |
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367 | vct2str(w); |
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368 | vct2str(w,1); |
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369 | } |
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370 | |
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371 | proc isVar(poly p) |
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372 | "USAGE: isVar(p); poly p |
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373 | RETURN: int |
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374 | PURPOSE: check, whether leading monomial of p is a power of a single variable |
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375 | @* from the basering. Returns the exponent or 0 if p is multivariate. |
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376 | EXAMPLE: example isVar; shows examples |
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377 | " |
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378 | { |
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379 | // checks whether p is a variable indeed |
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380 | // if it's a power of a variable, returns the power |
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381 | if (p==0) { return(0); } //"p=0"; |
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382 | poly q = leadmonom(p); |
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383 | if ( (p-lead(p)) !=0 ) { return(0); } // "p-lm(p)>0"; |
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384 | intvec v = leadexp(p); |
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385 | int s = size(v); |
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386 | int i=1; |
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387 | int cnt = 0; |
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388 | int pwr = 0; |
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389 | for (i=1; i<=s; i++) |
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390 | { |
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391 | if (v[i] != 0) |
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392 | { |
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393 | cnt++; |
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394 | pwr = v[i]; |
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395 | } |
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396 | } |
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397 | // "cnt:"; cnt; |
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398 | if (cnt==1) { return(pwr); } |
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399 | else { return(0); } |
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400 | } |
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401 | example |
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402 | { |
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403 | "EXAMPLE:"; echo = 2; |
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404 | ring r = 0,(x,y),dp; |
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405 | poly f = xy+1; |
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406 | isVar(f); |
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407 | poly g = y^3; |
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408 | isVar(g); |
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409 | poly h = 7*x^3; |
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410 | isVar(h); |
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411 | poly i = 1; |
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412 | isVar(i); |
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413 | } |
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414 | |
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415 | // new conversion routines |
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416 | |
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417 | static proc id2words(ideal I, int d) |
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418 | { |
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419 | // NOT FINISHED |
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420 | // input: ideal I of polys in letter-place notation |
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421 | // in the ring with d real vars |
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422 | // output: the list of strings: associative words |
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423 | // extract names of vars |
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424 | int i,m,n; string s; string place = "(1)"; |
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425 | list lv; |
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426 | for(i=1; i<=d; i++) |
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427 | { |
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428 | s = string(var(i)); |
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429 | // get rid of place |
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430 | n = find(s, place); |
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431 | if (n>0) |
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432 | { |
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433 | s = s[1..n-1]; |
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434 | } |
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435 | lv[i] = s; |
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436 | } |
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437 | poly p,q; |
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438 | for (i=1; i<=ncols(I); i++) |
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439 | { |
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440 | if (I[i] != 0) |
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441 | { |
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442 | p = I[i]; |
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443 | while (p!=0) |
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444 | { |
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445 | q = leadmonom(p); |
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446 | } |
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447 | } |
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448 | } |
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449 | |
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450 | return(lv); |
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451 | } |
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452 | example |
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453 | { |
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454 | "EXAMPLE:"; echo = 2; |
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455 | ring r = 0,(x(1),y(1),z(1),x(2),y(2),z(2)),dp; |
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456 | ideal I = x(1)*y(2) -z(1)*x(2); |
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457 | id2words(I,3); |
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458 | } |
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459 | |
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460 | static proc mono2word(poly p, int d) |
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461 | { |
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462 | } |
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463 | |
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464 | // 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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465 | // use the orig ord on (x,y,z) and expand it blockwise to (x(i),y(i),z(i)) |
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466 | |
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467 | // the correspondences: |
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468 | // monomial in K<x,y,z> <<--->> vector in R |
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469 | // polynomial in K<x,y,z> <<--->> list of vectors (matrix/module) in R |
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470 | // ideal in K<x,y,z> <<--->> list of matrices/modules in R |
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471 | |
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472 | |
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473 | // 1. form a new ring |
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474 | // 2. NOP |
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475 | // 3. compute GB -> with the kernel stuff |
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476 | // 4. skip shifted elts (check that no such exist?) |
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477 | // 5. go back to orig vars, produce strings/modules |
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478 | // 6. return the result |
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479 | |
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480 | proc freeGBasis(list LM, int d) |
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481 | "USAGE: freeGBasis(L, d); L a list of modules, d an integer |
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482 | RETURN: ring |
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483 | ASSUME: L has a special form. Namely, it is a list of modules, where |
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484 | @* each generator of every module stands for a monomial times coefficient in free algebra. |
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485 | @* In such a vector generator, the 1st entry is a nonzero coefficient from the ground field |
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486 | @* and each next entry hosts a variable from the basering. |
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487 | PURPOSE: compute the two-sided Groebner basis of an ideal, encoded by L |
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488 | @* in the free associative algebra, up to degree d |
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489 | NOTE: Apply @code{lst2str} to the output in order to obtain a better readable presentation |
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490 | EXAMPLE: example freeGBasis; shows examples |
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491 | " |
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492 | { |
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493 | // d = up to degree, will be shifted to d+1 |
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494 | if (d<1) {"bad d"; return(0);} |
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495 | |
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496 | int ppl = printlevel-voice+2; |
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497 | string err = ""; |
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498 | |
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499 | int i,j,s; |
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500 | def save = basering; |
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501 | // determine max no of places in the input |
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502 | int slm = size(LM); // numbers of polys in the ideal |
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503 | int sm; |
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504 | intvec iv; |
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505 | module M; |
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506 | for (i=1; i<=slm; i++) |
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507 | { |
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508 | // modules, e.g. free polynomials |
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509 | M = LM[i]; |
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510 | sm = ncols(M); |
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511 | for (j=1; j<=sm; j++) |
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512 | { |
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513 | //vectors, e.g. free monomials |
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514 | iv = iv, size(M[j])-1; // 1 place is reserved by the coeff |
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515 | } |
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516 | } |
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517 | int D = Max(iv); // max size of input words |
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518 | if (d<D) {"bad d"; return(LM);} |
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519 | D = D + d-1; |
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520 | // D = d; |
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521 | list LR = ringlist(save); |
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522 | list L, tmp; |
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523 | L[1] = LR[1]; // ground field |
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524 | L[4] = LR[4]; // quotient ideal |
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525 | tmp = LR[2]; // varnames |
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526 | s = size(LR[2]); |
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527 | for (i=1; i<=D; i++) |
---|
528 | { |
---|
529 | for (j=1; j<=s; j++) |
---|
530 | { |
---|
531 | tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")"; |
---|
532 | } |
---|
533 | } |
---|
534 | for (i=1; i<=s; i++) |
---|
535 | { |
---|
536 | tmp[i] = string(tmp[i])+"("+string(1)+")"; |
---|
537 | } |
---|
538 | L[2] = tmp; |
---|
539 | list OrigNames = LR[2]; |
---|
540 | // ordering: d blocks of the ord on r |
---|
541 | // try to get whether the ord on r is blockord itself |
---|
542 | s = size(LR[3]); |
---|
543 | if (s==2) |
---|
544 | { |
---|
545 | // not a blockord, 1 block + module ord |
---|
546 | tmp = LR[3][s]; // module ord |
---|
547 | for (i=1; i<=D; i++) |
---|
548 | { |
---|
549 | LR[3][s-1+i] = LR[3][1]; |
---|
550 | } |
---|
551 | LR[3][s+D] = tmp; |
---|
552 | } |
---|
553 | if (s>2) |
---|
554 | { |
---|
555 | // there are s-1 blocks |
---|
556 | int nb = s-1; |
---|
557 | tmp = LR[3][s]; // module ord |
---|
558 | for (i=1; i<=D; i++) |
---|
559 | { |
---|
560 | for (j=1; j<=nb; j++) |
---|
561 | { |
---|
562 | LR[3][i*nb+j] = LR[3][j]; |
---|
563 | } |
---|
564 | } |
---|
565 | // size(LR[3]); |
---|
566 | LR[3][nb*(D+1)+1] = tmp; |
---|
567 | } |
---|
568 | L[3] = LR[3]; |
---|
569 | def @R = ring(L); |
---|
570 | setring @R; |
---|
571 | ideal I; |
---|
572 | poly @p; |
---|
573 | s = size(OrigNames); |
---|
574 | // "s:";s; |
---|
575 | // convert LM to canonical vectors (no powers) |
---|
576 | setring save; |
---|
577 | kill M; // M was defined earlier |
---|
578 | module M; |
---|
579 | slm = size(LM); // numbers of polys in the ideal |
---|
580 | int sv,k,l; |
---|
581 | vector v; |
---|
582 | // poly p; |
---|
583 | string sp; |
---|
584 | setring @R; |
---|
585 | poly @@p=0; |
---|
586 | setring save; |
---|
587 | for (l=1; l<=slm; l++) |
---|
588 | { |
---|
589 | // modules, e.g. free polynomials |
---|
590 | M = LM[l]; |
---|
591 | sm = ncols(M); // in intvec iv the sizes are stored |
---|
592 | // modules, e.g. free polynomials |
---|
593 | for (j=1; j<=sm; j++) |
---|
594 | { |
---|
595 | //vectors, e.g. free monomials |
---|
596 | v = M[j]; |
---|
597 | sv = size(v); |
---|
598 | // "sv:";sv; |
---|
599 | sp = "@@p = @@p + "; |
---|
600 | for (k=2; k<=sv; k++) |
---|
601 | { |
---|
602 | sp = sp + string(v[k])+"("+string(k-1)+")*"; |
---|
603 | } |
---|
604 | sp = sp + string(v[1])+";"; // coef; |
---|
605 | setring @R; |
---|
606 | execute(sp); |
---|
607 | setring save; |
---|
608 | } |
---|
609 | setring @R; |
---|
610 | // "@@p:"; @@p; |
---|
611 | I = I,@@p; |
---|
612 | @@p = 0; |
---|
613 | setring save; |
---|
614 | } |
---|
615 | kill sp; |
---|
616 | // 3. compute GB |
---|
617 | setring @R; |
---|
618 | dbprint(ppl,"computing GB"); |
---|
619 | ideal J = system("freegb",I,d,nvars(save)); |
---|
620 | // ideal J = slimgb(I); |
---|
621 | dbprint(ppl,J); |
---|
622 | // 4. skip shifted elts |
---|
623 | ideal K = select1(J,1..s); // s = size(OrigNames) |
---|
624 | dbprint(ppl,K); |
---|
625 | dbprint(ppl, "done with GB"); |
---|
626 | // K contains vars x(1),...z(1) = images of originals |
---|
627 | // 5. go back to orig vars, produce strings/modules |
---|
628 | if (K[1] == 0) |
---|
629 | { |
---|
630 | "no reasonable output, GB gives 0"; |
---|
631 | return(0); |
---|
632 | } |
---|
633 | int sk = size(K); |
---|
634 | int sp, sx, a, b; |
---|
635 | intvec x; |
---|
636 | poly p,q; |
---|
637 | poly pn; |
---|
638 | // vars in 'save' |
---|
639 | setring save; |
---|
640 | module N; |
---|
641 | list LN; |
---|
642 | vector V; |
---|
643 | poly pn; |
---|
644 | // test and skip exponents >=2 |
---|
645 | setring @R; |
---|
646 | for(i=1; i<=sk; i++) |
---|
647 | { |
---|
648 | p = K[i]; |
---|
649 | while (p!=0) |
---|
650 | { |
---|
651 | q = lead(p); |
---|
652 | // "processing q:";q; |
---|
653 | x = leadexp(q); |
---|
654 | sx = size(x); |
---|
655 | for(k=1; k<=sx; k++) |
---|
656 | { |
---|
657 | if ( x[k] >= 2 ) |
---|
658 | { |
---|
659 | err = "skip: the value x[k] is " + string(x[k]); |
---|
660 | dbprint(ppl,err); |
---|
661 | // return(0); |
---|
662 | K[i] = 0; |
---|
663 | p = 0; |
---|
664 | q = 0; |
---|
665 | break; |
---|
666 | } |
---|
667 | } |
---|
668 | p = p - q; |
---|
669 | } |
---|
670 | } |
---|
671 | K = simplify(K,2); |
---|
672 | sk = size(K); |
---|
673 | for(i=1; i<=sk; i++) |
---|
674 | { |
---|
675 | // setring save; |
---|
676 | // V = 0; |
---|
677 | setring @R; |
---|
678 | p = K[i]; |
---|
679 | while (p!=0) |
---|
680 | { |
---|
681 | q = lead(p); |
---|
682 | err = "processing q:" + string(q); |
---|
683 | dbprint(ppl,err); |
---|
684 | x = leadexp(q); |
---|
685 | sx = size(x); |
---|
686 | pn = leadcoef(q); |
---|
687 | setring save; |
---|
688 | pn = imap(@R,pn); |
---|
689 | V = V + leadcoef(pn)*gen(1); |
---|
690 | for(k=1; k<=sx; k++) |
---|
691 | { |
---|
692 | if (x[k] ==1) |
---|
693 | { |
---|
694 | a = k / s; // block number=a+1, a!=0 |
---|
695 | b = k % s; // remainder |
---|
696 | // printf("a: %s, b: %s",a,b); |
---|
697 | if (b == 0) |
---|
698 | { |
---|
699 | // that is it's the last var in the block |
---|
700 | b = s; |
---|
701 | a = a-1; |
---|
702 | } |
---|
703 | V = V + var(b)*gen(a+2); |
---|
704 | } |
---|
705 | // else |
---|
706 | // { |
---|
707 | // printf("error: the value x[k] is %s", x[k]); |
---|
708 | // return(0); |
---|
709 | // } |
---|
710 | } |
---|
711 | err = "V: " + string(V); |
---|
712 | dbprint(ppl,err); |
---|
713 | // printf("V: %s", string(V)); |
---|
714 | N = N,V; |
---|
715 | V = 0; |
---|
716 | setring @R; |
---|
717 | p = p - q; |
---|
718 | pn = 0; |
---|
719 | } |
---|
720 | setring save; |
---|
721 | LN[i] = simplify(N,2); |
---|
722 | N = 0; |
---|
723 | } |
---|
724 | setring save; |
---|
725 | return(LN); |
---|
726 | } |
---|
727 | example |
---|
728 | { |
---|
729 | "EXAMPLE:"; echo = 2; |
---|
730 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
731 | module M = [-1,x,y],[-7,y,y],[3,x,x]; // stands for free poly -xy - 7yy - 3xx |
---|
732 | module N = [1,x,y,x],[-1,y,x,y]; // stands for free poly xyx - yxy |
---|
733 | list L; L[1] = M; L[2] = N; // list of modules stands for an ideal in free algebra |
---|
734 | lst2str(L); // list to string conversion of input polynomials |
---|
735 | def U = freeGBasis(L,5); // 5 is the degree bound |
---|
736 | lst2str(U); |
---|
737 | } |
---|
738 | |
---|
739 | static proc crs(list LM, int d) |
---|
740 | "USAGE: crs(L, d); L a list of modules, d an integer |
---|
741 | RETURN: ring |
---|
742 | PURPOSE: create a ring and shift the ideal |
---|
743 | EXAMPLE: example crs; shows examples |
---|
744 | " |
---|
745 | { |
---|
746 | // d = up to degree, will be shifted to d+1 |
---|
747 | if (d<1) {"bad d"; return(0);} |
---|
748 | |
---|
749 | int ppl = printlevel-voice+2; |
---|
750 | string err = ""; |
---|
751 | |
---|
752 | int i,j,s; |
---|
753 | def save = basering; |
---|
754 | // determine max no of places in the input |
---|
755 | int slm = size(LM); // numbers of polys in the ideal |
---|
756 | int sm; |
---|
757 | intvec iv; |
---|
758 | module M; |
---|
759 | for (i=1; i<=slm; i++) |
---|
760 | { |
---|
761 | // modules, e.g. free polynomials |
---|
762 | M = LM[i]; |
---|
763 | sm = ncols(M); |
---|
764 | for (j=1; j<=sm; j++) |
---|
765 | { |
---|
766 | //vectors, e.g. free monomials |
---|
767 | iv = iv, size(M[j])-1; // 1 place is reserved by the coeff |
---|
768 | } |
---|
769 | } |
---|
770 | int D = Max(iv); // max size of input words |
---|
771 | if (d<D) {"bad d"; return(LM);} |
---|
772 | D = D + d-1; |
---|
773 | // D = d; |
---|
774 | list LR = ringlist(save); |
---|
775 | list L, tmp; |
---|
776 | L[1] = LR[1]; // ground field |
---|
777 | L[4] = LR[4]; // quotient ideal |
---|
778 | tmp = LR[2]; // varnames |
---|
779 | s = size(LR[2]); |
---|
780 | for (i=1; i<=D; i++) |
---|
781 | { |
---|
782 | for (j=1; j<=s; j++) |
---|
783 | { |
---|
784 | tmp[i*s+j] = string(tmp[j])+"("+string(i)+")"; |
---|
785 | } |
---|
786 | } |
---|
787 | for (i=1; i<=s; i++) |
---|
788 | { |
---|
789 | tmp[i] = string(tmp[i])+"("+string(0)+")"; |
---|
790 | } |
---|
791 | L[2] = tmp; |
---|
792 | list OrigNames = LR[2]; |
---|
793 | // ordering: d blocks of the ord on r |
---|
794 | // try to get whether the ord on r is blockord itself |
---|
795 | s = size(LR[3]); |
---|
796 | if (s==2) |
---|
797 | { |
---|
798 | // not a blockord, 1 block + module ord |
---|
799 | tmp = LR[3][s]; // module ord |
---|
800 | for (i=1; i<=D; i++) |
---|
801 | { |
---|
802 | LR[3][s-1+i] = LR[3][1]; |
---|
803 | } |
---|
804 | LR[3][s+D] = tmp; |
---|
805 | } |
---|
806 | if (s>2) |
---|
807 | { |
---|
808 | // there are s-1 blocks |
---|
809 | int nb = s-1; |
---|
810 | tmp = LR[3][s]; // module ord |
---|
811 | for (i=1; i<=D; i++) |
---|
812 | { |
---|
813 | for (j=1; j<=nb; j++) |
---|
814 | { |
---|
815 | LR[3][i*nb+j] = LR[3][j]; |
---|
816 | } |
---|
817 | } |
---|
818 | // size(LR[3]); |
---|
819 | LR[3][nb*(D+1)+1] = tmp; |
---|
820 | } |
---|
821 | L[3] = LR[3]; |
---|
822 | def @R = ring(L); |
---|
823 | setring @R; |
---|
824 | ideal I; |
---|
825 | poly @p; |
---|
826 | s = size(OrigNames); |
---|
827 | // "s:";s; |
---|
828 | // convert LM to canonical vectors (no powers) |
---|
829 | setring save; |
---|
830 | kill M; // M was defined earlier |
---|
831 | module M; |
---|
832 | slm = size(LM); // numbers of polys in the ideal |
---|
833 | int sv,k,l; |
---|
834 | vector v; |
---|
835 | // poly p; |
---|
836 | string sp; |
---|
837 | setring @R; |
---|
838 | poly @@p=0; |
---|
839 | setring save; |
---|
840 | for (l=1; l<=slm; l++) |
---|
841 | { |
---|
842 | // modules, e.g. free polynomials |
---|
843 | M = LM[l]; |
---|
844 | sm = ncols(M); // in intvec iv the sizes are stored |
---|
845 | for (i=0; i<=d-iv[l]; i++) |
---|
846 | { |
---|
847 | // modules, e.g. free polynomials |
---|
848 | for (j=1; j<=sm; j++) |
---|
849 | { |
---|
850 | //vectors, e.g. free monomials |
---|
851 | v = M[j]; |
---|
852 | sv = size(v); |
---|
853 | // "sv:";sv; |
---|
854 | sp = "@@p = @@p + "; |
---|
855 | for (k=2; k<=sv; k++) |
---|
856 | { |
---|
857 | sp = sp + string(v[k])+"("+string(k-2+i)+")*"; |
---|
858 | } |
---|
859 | sp = sp + string(v[1])+";"; // coef; |
---|
860 | setring @R; |
---|
861 | execute(sp); |
---|
862 | setring save; |
---|
863 | } |
---|
864 | setring @R; |
---|
865 | // "@@p:"; @@p; |
---|
866 | I = I,@@p; |
---|
867 | @@p = 0; |
---|
868 | setring save; |
---|
869 | } |
---|
870 | } |
---|
871 | setring @R; |
---|
872 | export I; |
---|
873 | return(@R); |
---|
874 | } |
---|
875 | example |
---|
876 | { |
---|
877 | "EXAMPLE:"; echo = 2; |
---|
878 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
879 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
---|
880 | module N = [1,x,y,x],[-1,y,x,y]; |
---|
881 | list L; L[1] = M; L[2] = N; |
---|
882 | lst2str(L); |
---|
883 | def U = crs(L,5); |
---|
884 | setring U; U; |
---|
885 | I; |
---|
886 | } |
---|
887 | |
---|
888 | static proc polylen(ideal I) |
---|
889 | { |
---|
890 | // returns the ideal of length of polys |
---|
891 | int i; |
---|
892 | intvec J; |
---|
893 | number s = 0; |
---|
894 | for(i=1;i<=ncols(I);i++) |
---|
895 | { |
---|
896 | J[i] = size(I[i]); |
---|
897 | s = s + J[i]; |
---|
898 | } |
---|
899 | printf("the sum of length %s",s); |
---|
900 | // print(s); |
---|
901 | return(J); |
---|
902 | } |
---|
903 | |
---|
904 | // new: uniting both mLR1 (homog) and mLR2 (nonhomog) |
---|
905 | proc makeLetterplaceRing(int d, list #) |
---|
906 | "USAGE: makeLetterplaceRing(d [,h]); d an integer, h an optional integer |
---|
907 | RETURN: ring |
---|
908 | PURPOSE: creates a ring with the ordering, used in letterplace computations |
---|
909 | NOTE: if h is given an nonzero, the pure homogeneous letterplace block ordering will be used. |
---|
910 | EXAMPLE: example makeLetterplaceRing; shows examples |
---|
911 | " |
---|
912 | { |
---|
913 | int use_old_mlr = 0; |
---|
914 | if ( size(#)>0 ) |
---|
915 | { |
---|
916 | if (( typeof(#[1]) == "int" ) || ( typeof(#[1]) == "poly" ) ) |
---|
917 | { |
---|
918 | poly x = poly(#[1]); |
---|
919 | if (x!=0) |
---|
920 | { |
---|
921 | use_old_mlr = 1; |
---|
922 | } |
---|
923 | } |
---|
924 | } |
---|
925 | if (use_old_mlr) |
---|
926 | { |
---|
927 | def @A = makeLetterplaceRing1(d); |
---|
928 | } |
---|
929 | else |
---|
930 | { |
---|
931 | def @A = makeLetterplaceRing2(d); |
---|
932 | } |
---|
933 | return(@A); |
---|
934 | } |
---|
935 | example |
---|
936 | { |
---|
937 | "EXAMPLE:"; echo = 2; |
---|
938 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
939 | def A = makeLetterplaceRing(2); |
---|
940 | setring A; A; |
---|
941 | attrib(A,"isLetterplaceRing"); |
---|
942 | attrib(A,"uptodeg"); // degree bound |
---|
943 | attrib(A,"lV"); // number of variables in the main block |
---|
944 | setring r; def B = makeLetterplaceRing(2,1); // to compare: |
---|
945 | setring B; B; |
---|
946 | } |
---|
947 | |
---|
948 | //proc freegbRing(int d) |
---|
949 | proc makeLetterplaceRing1(int d) |
---|
950 | "USAGE: makeLetterplaceRing1(d); d an integer |
---|
951 | RETURN: ring |
---|
952 | PURPOSE: creates a ring with a special ordering, suitable for |
---|
953 | @* the use of homogeneous letterplace (d blocks of shifted original variables) |
---|
954 | EXAMPLE: example makeLetterplaceRing1; shows examples |
---|
955 | " |
---|
956 | { |
---|
957 | // d = up to degree, will be shifted to d+1 |
---|
958 | if (d<1) {"bad d"; return(0);} |
---|
959 | |
---|
960 | int uptodeg = d; int lV = nvars(basering); |
---|
961 | |
---|
962 | int ppl = printlevel-voice+2; |
---|
963 | string err = ""; |
---|
964 | |
---|
965 | int i,j,s; |
---|
966 | def save = basering; |
---|
967 | int D = d-1; |
---|
968 | list LR = ringlist(save); |
---|
969 | list L, tmp; |
---|
970 | L[1] = LR[1]; // ground field |
---|
971 | L[4] = LR[4]; // quotient ideal |
---|
972 | tmp = LR[2]; // varnames |
---|
973 | s = size(LR[2]); |
---|
974 | for (i=1; i<=D; i++) |
---|
975 | { |
---|
976 | for (j=1; j<=s; j++) |
---|
977 | { |
---|
978 | tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")"; |
---|
979 | } |
---|
980 | } |
---|
981 | for (i=1; i<=s; i++) |
---|
982 | { |
---|
983 | tmp[i] = string(tmp[i])+"("+string(1)+")"; |
---|
984 | } |
---|
985 | L[2] = tmp; |
---|
986 | list OrigNames = LR[2]; |
---|
987 | // ordering: d blocks of the ord on r |
---|
988 | // try to get whether the ord on r is blockord itself |
---|
989 | // TODO: make L(2) ordering! exponent is maximally 2 |
---|
990 | s = size(LR[3]); |
---|
991 | if (s==2) |
---|
992 | { |
---|
993 | // not a blockord, 1 block + module ord |
---|
994 | tmp = LR[3][s]; // module ord |
---|
995 | for (i=1; i<=D; i++) |
---|
996 | { |
---|
997 | LR[3][s-1+i] = LR[3][1]; |
---|
998 | } |
---|
999 | LR[3][s+D] = tmp; |
---|
1000 | } |
---|
1001 | if (s>2) |
---|
1002 | { |
---|
1003 | // there are s-1 blocks |
---|
1004 | int nb = s-1; |
---|
1005 | tmp = LR[3][s]; // module ord |
---|
1006 | for (i=1; i<=D; i++) |
---|
1007 | { |
---|
1008 | for (j=1; j<=nb; j++) |
---|
1009 | { |
---|
1010 | LR[3][i*nb+j] = LR[3][j]; |
---|
1011 | } |
---|
1012 | } |
---|
1013 | // size(LR[3]); |
---|
1014 | LR[3][nb*(D+1)+1] = tmp; |
---|
1015 | } |
---|
1016 | L[3] = LR[3]; |
---|
1017 | def @R = ring(L); |
---|
1018 | // setring @R; |
---|
1019 | // int uptodeg = d; int lV = nvars(basering); // were defined before |
---|
1020 | def @@R = setLetterplaceAttributes(@R,uptodeg,lV); |
---|
1021 | return (@@R); |
---|
1022 | } |
---|
1023 | example |
---|
1024 | { |
---|
1025 | "EXAMPLE:"; echo = 2; |
---|
1026 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
1027 | def A = makeLetterplaceRing1(2); |
---|
1028 | setring A; |
---|
1029 | A; |
---|
1030 | attrib(A,"isLetterplaceRing"); |
---|
1031 | attrib(A,"uptodeg"); // degree bound |
---|
1032 | attrib(A,"lV"); // number of variables in the main block |
---|
1033 | } |
---|
1034 | |
---|
1035 | proc makeLetterplaceRing2(int d) |
---|
1036 | "USAGE: makeLetterplaceRing2(d); d an integer |
---|
1037 | RETURN: ring |
---|
1038 | PURPOSE: creates a ring with a special ordering, suitable for |
---|
1039 | @* the use of non-homogeneous letterplace |
---|
1040 | NOTE: the matrix for the ordering looks as follows: first row is 1,1,...,1 |
---|
1041 | @* then there come 'd' blocks of shifted original variables |
---|
1042 | EXAMPLE: example makeLetterplaceRing2; shows examples |
---|
1043 | " |
---|
1044 | { |
---|
1045 | // d = up to degree, will be shifted to d+1 |
---|
1046 | if (d<1) {"bad d"; return(0);} |
---|
1047 | |
---|
1048 | int uptodeg = d; int lV = nvars(basering); |
---|
1049 | |
---|
1050 | int ppl = printlevel-voice+2; |
---|
1051 | string err = ""; |
---|
1052 | |
---|
1053 | int i,j,s; |
---|
1054 | def save = basering; |
---|
1055 | int D = d-1; |
---|
1056 | list LR = ringlist(save); |
---|
1057 | list L, tmp, tmp2, tmp3; |
---|
1058 | L[1] = LR[1]; // ground field |
---|
1059 | L[4] = LR[4]; // quotient ideal |
---|
1060 | tmp = LR[2]; // varnames |
---|
1061 | s = size(LR[2]); |
---|
1062 | for (i=1; i<=D; i++) |
---|
1063 | { |
---|
1064 | for (j=1; j<=s; j++) |
---|
1065 | { |
---|
1066 | tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")"; |
---|
1067 | } |
---|
1068 | } |
---|
1069 | for (i=1; i<=s; i++) |
---|
1070 | { |
---|
1071 | tmp[i] = string(tmp[i])+"("+string(1)+")"; |
---|
1072 | } |
---|
1073 | L[2] = tmp; |
---|
1074 | list OrigNames = LR[2]; |
---|
1075 | // ordering: one 1..1 a above |
---|
1076 | // ordering: d blocks of the ord on r |
---|
1077 | // try to get whether the ord on r is blockord itself |
---|
1078 | // TODO: make L(2) ordering! exponent is maximally 2 |
---|
1079 | s = size(LR[3]); |
---|
1080 | if (s==2) |
---|
1081 | { |
---|
1082 | // not a blockord, 1 block + module ord |
---|
1083 | tmp = LR[3][s]; // module ord |
---|
1084 | for (i=1; i<=d; i++) |
---|
1085 | { |
---|
1086 | LR[3][s-1+i] = LR[3][1]; |
---|
1087 | } |
---|
1088 | // LR[3][s+D] = tmp; |
---|
1089 | LR[3][s+1+D] = tmp; |
---|
1090 | LR[3][1] = list("a",intvec(1: int(d*lV))); // deg-ord |
---|
1091 | } |
---|
1092 | if (s>2) |
---|
1093 | { |
---|
1094 | // there are s-1 blocks |
---|
1095 | int nb = s-1; |
---|
1096 | tmp = LR[3][s]; // module ord to place at the very end |
---|
1097 | tmp2 = LR[3]; tmp2 = tmp2[1..nb]; |
---|
1098 | tmp3[1] = list("a",intvec(1: int(d*lV))); // deg-ord, insert as the 1st |
---|
1099 | for (i=1; i<=d; i++) |
---|
1100 | { |
---|
1101 | tmp3 = tmp3 + tmp2; |
---|
1102 | } |
---|
1103 | tmp3 = tmp3 + list(tmp); |
---|
1104 | LR[3] = tmp3; |
---|
1105 | // for (i=1; i<=d; i++) |
---|
1106 | // { |
---|
1107 | // for (j=1; j<=nb; j++) |
---|
1108 | // { |
---|
1109 | // // LR[3][i*nb+j+1]= LR[3][j]; |
---|
1110 | // LR[3][i*nb+j+1]= tmp2[j]; |
---|
1111 | // } |
---|
1112 | // } |
---|
1113 | // // size(LR[3]); |
---|
1114 | // LR[3][(s-1)*d+2] = tmp; |
---|
1115 | // LR[3] = list("a",intvec(1: int(d*lV))) + LR[3]; // deg-ord, insert as the 1st |
---|
1116 | // remove everything behind nb*(D+1)+1 ? |
---|
1117 | // tmp = LR[3]; |
---|
1118 | // LR[3] = tmp[1..size(tmp)-1]; |
---|
1119 | } |
---|
1120 | L[3] = LR[3]; |
---|
1121 | def @R = ring(L); |
---|
1122 | // setring @R; |
---|
1123 | // int uptodeg = d; int lV = nvars(basering); // were defined before |
---|
1124 | def @@R = setLetterplaceAttributes(@R,uptodeg,lV); |
---|
1125 | return (@@R); |
---|
1126 | } |
---|
1127 | example |
---|
1128 | { |
---|
1129 | "EXAMPLE:"; echo = 2; |
---|
1130 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
1131 | def A = makeLetterplaceRing2(2); |
---|
1132 | setring A; |
---|
1133 | A; |
---|
1134 | attrib(A,"isLetterplaceRing"); |
---|
1135 | attrib(A,"uptodeg"); // degree bound |
---|
1136 | attrib(A,"lV"); // number of variables in the main block |
---|
1137 | } |
---|
1138 | |
---|
1139 | /* EXAMPLES: |
---|
1140 | |
---|
1141 | //static proc ex_shift() |
---|
1142 | { |
---|
1143 | LIB "freegb.lib"; |
---|
1144 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
1145 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
---|
1146 | module N = [1,x,y,x],[-1,y,x,y]; |
---|
1147 | list L; L[1] = M; L[2] = N; |
---|
1148 | lst2str(L); |
---|
1149 | def U = crs(L,5); |
---|
1150 | setring U; U; |
---|
1151 | I; |
---|
1152 | poly p = I[2]; // I[8]; |
---|
1153 | p; |
---|
1154 | system("stest",p,7,7,3); // error -> the world is ok |
---|
1155 | poly q1 = system("stest",p,1,7,3); //ok |
---|
1156 | poly q6 = system("stest",p,6,7,3); //ok |
---|
1157 | system("btest",p,3); //ok |
---|
1158 | system("btest",q1,3); //ok |
---|
1159 | system("btest",q6,3); //ok |
---|
1160 | } |
---|
1161 | |
---|
1162 | //static proc test_shrink() |
---|
1163 | { |
---|
1164 | LIB "freegb.lib"; |
---|
1165 | ring r =0,(x,y,z),dp; |
---|
1166 | int d = 5; |
---|
1167 | def R = makeLetterplaceRing(d); |
---|
1168 | setring R; |
---|
1169 | poly p1 = x(1)*y(2)*z(3); |
---|
1170 | poly p2 = x(1)*y(4)*z(5); |
---|
1171 | poly p3 = x(1)*y(1)*z(3); |
---|
1172 | poly p4 = x(1)*y(2)*z(2); |
---|
1173 | poly p5 = x(3)*z(5); |
---|
1174 | poly p6 = x(1)*y(1)*x(3)*z(5); |
---|
1175 | poly p7 = x(1)*y(2)*x(3)*y(4)*z(5); |
---|
1176 | poly p8 = p1+p2+p3+p4+p5 + p6 + p7; |
---|
1177 | p1; system("shrinktest",p1,3); |
---|
1178 | p2; system("shrinktest",p2,3); |
---|
1179 | p3; system("shrinktest",p3,3); |
---|
1180 | p4; system("shrinktest",p4,3); |
---|
1181 | p5; system("shrinktest",p5,3); |
---|
1182 | p6; system("shrinktest",p6,3); |
---|
1183 | p7; system("shrinktest",p7,3); |
---|
1184 | p8; system("shrinktest",p8,3); |
---|
1185 | poly p9 = p1 + 2*p2 + 5*p5 + 7*p7; |
---|
1186 | p9; system("shrinktest",p9,3); |
---|
1187 | } |
---|
1188 | |
---|
1189 | //static proc ex2() |
---|
1190 | { |
---|
1191 | option(prot); |
---|
1192 | LIB "freegb.lib"; |
---|
1193 | ring r = 0,(x,y),dp; |
---|
1194 | module M = [-1,x,y],[3,x,x]; // 3x^2 - xy |
---|
1195 | def U = freegb(M,7); |
---|
1196 | lst2str(U); |
---|
1197 | } |
---|
1198 | |
---|
1199 | //static proc ex_nonhomog() |
---|
1200 | { |
---|
1201 | option(prot); |
---|
1202 | LIB "freegb.lib"; |
---|
1203 | ring r = 0,(x,y,h),dp; |
---|
1204 | list L; |
---|
1205 | module M; |
---|
1206 | M = [-1,y,y],[1,x,x,x]; // x3-y2 |
---|
1207 | L[1] = M; |
---|
1208 | M = [1,x,h],[-1,h,x]; // xh-hx |
---|
1209 | L[2] = M; |
---|
1210 | M = [1,y,h],[-1,h,y]; // yh-hy |
---|
1211 | L[3] = M; |
---|
1212 | def U = freegb(L,4); |
---|
1213 | lst2str(U); |
---|
1214 | // strange elements in the basis |
---|
1215 | } |
---|
1216 | |
---|
1217 | //static proc ex_nonhomog_comm() |
---|
1218 | { |
---|
1219 | option(prot); |
---|
1220 | LIB "freegb.lib"; |
---|
1221 | ring r = 0,(x,y),dp; |
---|
1222 | module M = [-1,y,y],[1,x,x,x]; |
---|
1223 | def U = freegb(M,5); |
---|
1224 | lst2str(U); |
---|
1225 | } |
---|
1226 | |
---|
1227 | //static proc ex_nonhomog_h() |
---|
1228 | { |
---|
1229 | option(prot); |
---|
1230 | LIB "freegb.lib"; |
---|
1231 | ring r = 0,(x,y,h),(a(1,1),dp); |
---|
1232 | module M = [-1,y,y,h],[1,x,x,x]; // x3 - y2h |
---|
1233 | def U = freegb(M,6); |
---|
1234 | lst2str(U); |
---|
1235 | } |
---|
1236 | |
---|
1237 | //static proc ex_nonhomog_h2() |
---|
1238 | { |
---|
1239 | option(prot); |
---|
1240 | LIB "freegb.lib"; |
---|
1241 | ring r = 0,(x,y,h),(dp); |
---|
1242 | list L; |
---|
1243 | module M; |
---|
1244 | M = [-1,y,y,h],[1,x,x,x]; // x3 - y2h |
---|
1245 | L[1] = M; |
---|
1246 | M = [1,x,h],[-1,h,x]; // xh - hx |
---|
1247 | L[2] = M; |
---|
1248 | M = [1,y,h],[-1,h,y]; // yh - hy |
---|
1249 | L[3] = M; |
---|
1250 | def U = freeGBasis(L,3); |
---|
1251 | lst2str(U); |
---|
1252 | // strange answer CHECK |
---|
1253 | } |
---|
1254 | |
---|
1255 | |
---|
1256 | //static proc ex_nonhomog_3() |
---|
1257 | { |
---|
1258 | option(prot); |
---|
1259 | LIB "./freegb.lib"; |
---|
1260 | ring r = 0,(x,y,z),(dp); |
---|
1261 | list L; |
---|
1262 | module M; |
---|
1263 | M = [1,z,y],[-1,x]; // zy - x |
---|
1264 | L[1] = M; |
---|
1265 | M = [1,z,x],[-1,y]; // zx - y |
---|
1266 | L[2] = M; |
---|
1267 | M = [1,y,x],[-1,z]; // yx - z |
---|
1268 | L[3] = M; |
---|
1269 | lst2str(L); |
---|
1270 | list U = freegb(L,4); |
---|
1271 | lst2str(U); |
---|
1272 | // strange answer CHECK |
---|
1273 | } |
---|
1274 | |
---|
1275 | //static proc ex_densep_2() |
---|
1276 | { |
---|
1277 | option(prot); |
---|
1278 | LIB "freegb.lib"; |
---|
1279 | ring r = (0,a,b,c),(x,y),(Dp); // deglex |
---|
1280 | module M = [1,x,x], [a,x,y], [b,y,x], [c,y,y]; |
---|
1281 | lst2str(M); |
---|
1282 | list U = freegb(M,5); |
---|
1283 | lst2str(U); |
---|
1284 | // a=b is important -> finite basis!!! |
---|
1285 | module M = [1,x,x], [a,x,y], [a,y,x], [c,y,y]; |
---|
1286 | lst2str(M); |
---|
1287 | list U = freegb(M,5); |
---|
1288 | lst2str(U); |
---|
1289 | } |
---|
1290 | |
---|
1291 | // END COMMENTED EXAMPLES |
---|
1292 | |
---|
1293 | */ |
---|
1294 | |
---|
1295 | // 1. form a new ring |
---|
1296 | // 2. produce shifted generators |
---|
1297 | // 3. compute GB |
---|
1298 | // 4. skip shifted elts |
---|
1299 | // 5. go back to orig vars, produce strings/modules |
---|
1300 | // 6. return the result |
---|
1301 | |
---|
1302 | static proc freegbold(list LM, int d) |
---|
1303 | "USAGE: freegbold(L, d); L a list of modules, d an integer |
---|
1304 | RETURN: ring |
---|
1305 | PURPOSE: compute the two-sided Groebner basis of an ideal, encoded by L in |
---|
1306 | the free associative algebra, up to degree d |
---|
1307 | EXAMPLE: example freegbold; shows examples |
---|
1308 | " |
---|
1309 | { |
---|
1310 | // d = up to degree, will be shifted to d+1 |
---|
1311 | if (d<1) {"bad d"; return(0);} |
---|
1312 | |
---|
1313 | int ppl = printlevel-voice+2; |
---|
1314 | string err = ""; |
---|
1315 | |
---|
1316 | int i,j,s; |
---|
1317 | def save = basering; |
---|
1318 | // determine max no of places in the input |
---|
1319 | int slm = size(LM); // numbers of polys in the ideal |
---|
1320 | int sm; |
---|
1321 | intvec iv; |
---|
1322 | module M; |
---|
1323 | for (i=1; i<=slm; i++) |
---|
1324 | { |
---|
1325 | // modules, e.g. free polynomials |
---|
1326 | M = LM[i]; |
---|
1327 | sm = ncols(M); |
---|
1328 | for (j=1; j<=sm; j++) |
---|
1329 | { |
---|
1330 | //vectors, e.g. free monomials |
---|
1331 | iv = iv, size(M[j])-1; // 1 place is reserved by the coeff |
---|
1332 | } |
---|
1333 | } |
---|
1334 | int D = Max(iv); // max size of input words |
---|
1335 | if (d<D) {"bad d"; return(LM);} |
---|
1336 | D = D + d-1; |
---|
1337 | // D = d; |
---|
1338 | list LR = ringlist(save); |
---|
1339 | list L, tmp; |
---|
1340 | L[1] = LR[1]; // ground field |
---|
1341 | L[4] = LR[4]; // quotient ideal |
---|
1342 | tmp = LR[2]; // varnames |
---|
1343 | s = size(LR[2]); |
---|
1344 | for (i=1; i<=D; i++) |
---|
1345 | { |
---|
1346 | for (j=1; j<=s; j++) |
---|
1347 | { |
---|
1348 | tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")"; |
---|
1349 | } |
---|
1350 | } |
---|
1351 | for (i=1; i<=s; i++) |
---|
1352 | { |
---|
1353 | tmp[i] = string(tmp[i])+"("+string(1)+")"; |
---|
1354 | } |
---|
1355 | L[2] = tmp; |
---|
1356 | list OrigNames = LR[2]; |
---|
1357 | // ordering: d blocks of the ord on r |
---|
1358 | // try to get whether the ord on r is blockord itself |
---|
1359 | // TODO: make L(2) ordering! exponent is maximally 2 |
---|
1360 | s = size(LR[3]); |
---|
1361 | if (s==2) |
---|
1362 | { |
---|
1363 | // not a blockord, 1 block + module ord |
---|
1364 | tmp = LR[3][s]; // module ord |
---|
1365 | for (i=1; i<=D; i++) |
---|
1366 | { |
---|
1367 | LR[3][s-1+i] = LR[3][1]; |
---|
1368 | } |
---|
1369 | LR[3][s+D] = tmp; |
---|
1370 | } |
---|
1371 | if (s>2) |
---|
1372 | { |
---|
1373 | // there are s-1 blocks |
---|
1374 | int nb = s-1; |
---|
1375 | tmp = LR[3][s]; // module ord |
---|
1376 | for (i=1; i<=D; i++) |
---|
1377 | { |
---|
1378 | for (j=1; j<=nb; j++) |
---|
1379 | { |
---|
1380 | LR[3][i*nb+j] = LR[3][j]; |
---|
1381 | } |
---|
1382 | } |
---|
1383 | // size(LR[3]); |
---|
1384 | LR[3][nb*(D+1)+1] = tmp; |
---|
1385 | } |
---|
1386 | L[3] = LR[3]; |
---|
1387 | def @R = ring(L); |
---|
1388 | setring @R; |
---|
1389 | ideal I; |
---|
1390 | poly @p; |
---|
1391 | s = size(OrigNames); |
---|
1392 | // "s:";s; |
---|
1393 | // convert LM to canonical vectors (no powers) |
---|
1394 | setring save; |
---|
1395 | kill M; // M was defined earlier |
---|
1396 | module M; |
---|
1397 | slm = size(LM); // numbers of polys in the ideal |
---|
1398 | int sv,k,l; |
---|
1399 | vector v; |
---|
1400 | // poly p; |
---|
1401 | string sp; |
---|
1402 | setring @R; |
---|
1403 | poly @@p=0; |
---|
1404 | setring save; |
---|
1405 | for (l=1; l<=slm; l++) |
---|
1406 | { |
---|
1407 | // modules, e.g. free polynomials |
---|
1408 | M = LM[l]; |
---|
1409 | sm = ncols(M); // in intvec iv the sizes are stored |
---|
1410 | for (i=0; i<=d-iv[l]; i++) |
---|
1411 | { |
---|
1412 | // modules, e.g. free polynomials |
---|
1413 | for (j=1; j<=sm; j++) |
---|
1414 | { |
---|
1415 | //vectors, e.g. free monomials |
---|
1416 | v = M[j]; |
---|
1417 | sv = size(v); |
---|
1418 | // "sv:";sv; |
---|
1419 | sp = "@@p = @@p + "; |
---|
1420 | for (k=2; k<=sv; k++) |
---|
1421 | { |
---|
1422 | sp = sp + string(v[k])+"("+string(k-1+i)+")*"; |
---|
1423 | } |
---|
1424 | sp = sp + string(v[1])+";"; // coef; |
---|
1425 | setring @R; |
---|
1426 | execute(sp); |
---|
1427 | setring save; |
---|
1428 | } |
---|
1429 | setring @R; |
---|
1430 | // "@@p:"; @@p; |
---|
1431 | I = I,@@p; |
---|
1432 | @@p = 0; |
---|
1433 | setring save; |
---|
1434 | } |
---|
1435 | } |
---|
1436 | kill sp; |
---|
1437 | // 3. compute GB |
---|
1438 | setring @R; |
---|
1439 | dbprint(ppl,"computing GB"); |
---|
1440 | // ideal J = groebner(I); |
---|
1441 | ideal J = slimgb(I); |
---|
1442 | dbprint(ppl,J); |
---|
1443 | // 4. skip shifted elts |
---|
1444 | ideal K = select1(J,1..s); // s = size(OrigNames) |
---|
1445 | dbprint(ppl,K); |
---|
1446 | dbprint(ppl, "done with GB"); |
---|
1447 | // K contains vars x(1),...z(1) = images of originals |
---|
1448 | // 5. go back to orig vars, produce strings/modules |
---|
1449 | if (K[1] == 0) |
---|
1450 | { |
---|
1451 | "no reasonable output, GB gives 0"; |
---|
1452 | return(0); |
---|
1453 | } |
---|
1454 | int sk = size(K); |
---|
1455 | int sp, sx, a, b; |
---|
1456 | intvec x; |
---|
1457 | poly p,q; |
---|
1458 | poly pn; |
---|
1459 | // vars in 'save' |
---|
1460 | setring save; |
---|
1461 | module N; |
---|
1462 | list LN; |
---|
1463 | vector V; |
---|
1464 | poly pn; |
---|
1465 | // test and skip exponents >=2 |
---|
1466 | setring @R; |
---|
1467 | for(i=1; i<=sk; i++) |
---|
1468 | { |
---|
1469 | p = K[i]; |
---|
1470 | while (p!=0) |
---|
1471 | { |
---|
1472 | q = lead(p); |
---|
1473 | // "processing q:";q; |
---|
1474 | x = leadexp(q); |
---|
1475 | sx = size(x); |
---|
1476 | for(k=1; k<=sx; k++) |
---|
1477 | { |
---|
1478 | if ( x[k] >= 2 ) |
---|
1479 | { |
---|
1480 | err = "skip: the value x[k] is " + string(x[k]); |
---|
1481 | dbprint(ppl,err); |
---|
1482 | // return(0); |
---|
1483 | K[i] = 0; |
---|
1484 | p = 0; |
---|
1485 | q = 0; |
---|
1486 | break; |
---|
1487 | } |
---|
1488 | } |
---|
1489 | p = p - q; |
---|
1490 | } |
---|
1491 | } |
---|
1492 | K = simplify(K,2); |
---|
1493 | sk = size(K); |
---|
1494 | for(i=1; i<=sk; i++) |
---|
1495 | { |
---|
1496 | // setring save; |
---|
1497 | // V = 0; |
---|
1498 | setring @R; |
---|
1499 | p = K[i]; |
---|
1500 | while (p!=0) |
---|
1501 | { |
---|
1502 | q = lead(p); |
---|
1503 | err = "processing q:" + string(q); |
---|
1504 | dbprint(ppl,err); |
---|
1505 | x = leadexp(q); |
---|
1506 | sx = size(x); |
---|
1507 | pn = leadcoef(q); |
---|
1508 | setring save; |
---|
1509 | pn = imap(@R,pn); |
---|
1510 | V = V + leadcoef(pn)*gen(1); |
---|
1511 | for(k=1; k<=sx; k++) |
---|
1512 | { |
---|
1513 | if (x[k] ==1) |
---|
1514 | { |
---|
1515 | a = k / s; // block number=a+1, a!=0 |
---|
1516 | b = k % s; // remainder |
---|
1517 | // printf("a: %s, b: %s",a,b); |
---|
1518 | if (b == 0) |
---|
1519 | { |
---|
1520 | // that is it's the last var in the block |
---|
1521 | b = s; |
---|
1522 | a = a-1; |
---|
1523 | } |
---|
1524 | V = V + var(b)*gen(a+2); |
---|
1525 | } |
---|
1526 | // else |
---|
1527 | // { |
---|
1528 | // printf("error: the value x[k] is %s", x[k]); |
---|
1529 | // return(0); |
---|
1530 | // } |
---|
1531 | } |
---|
1532 | err = "V: " + string(V); |
---|
1533 | dbprint(ppl,err); |
---|
1534 | // printf("V: %s", string(V)); |
---|
1535 | N = N,V; |
---|
1536 | V = 0; |
---|
1537 | setring @R; |
---|
1538 | p = p - q; |
---|
1539 | pn = 0; |
---|
1540 | } |
---|
1541 | setring save; |
---|
1542 | LN[i] = simplify(N,2); |
---|
1543 | N = 0; |
---|
1544 | } |
---|
1545 | setring save; |
---|
1546 | return(LN); |
---|
1547 | } |
---|
1548 | example |
---|
1549 | { |
---|
1550 | "EXAMPLE:"; echo = 2; |
---|
1551 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
1552 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
---|
1553 | module N = [1,x,y,x],[-1,y,x,y]; |
---|
1554 | list L; L[1] = M; L[2] = N; |
---|
1555 | lst2str(L); |
---|
1556 | def U = freegbold(L,5); |
---|
1557 | lst2str(U); |
---|
1558 | } |
---|
1559 | |
---|
1560 | /* begin older procs and tests |
---|
1561 | |
---|
1562 | static proc sgb(ideal I, int d) |
---|
1563 | { |
---|
1564 | // new code |
---|
1565 | // map x_i to x_i(1) via map() |
---|
1566 | //LIB "freegb.lib"; |
---|
1567 | def save = basering; |
---|
1568 | //int d =7;// degree |
---|
1569 | int nv = nvars(save); |
---|
1570 | def R = makeLetterplaceRing(d); |
---|
1571 | setring R; |
---|
1572 | int i; |
---|
1573 | ideal Imap; |
---|
1574 | for (i=1; i<=nv; i++) |
---|
1575 | { |
---|
1576 | Imap[i] = var(i); |
---|
1577 | } |
---|
1578 | //ideal I = x(1)*y(2), y(1)*x(2)+z(1)*z(2); |
---|
1579 | ideal I = x(1)*x(2),x(1)*y(2) + z(1)*x(2); |
---|
1580 | option(prot); |
---|
1581 | //option(teach); |
---|
1582 | ideal J = system("freegb",I,d,nv); |
---|
1583 | } |
---|
1584 | |
---|
1585 | static proc checkCeq() |
---|
1586 | { |
---|
1587 | ring r = 0,(x,y),Dp; |
---|
1588 | def A = makeLetterplaceRing(4); |
---|
1589 | setring A; |
---|
1590 | A; |
---|
1591 | // I = x2-xy |
---|
1592 | 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); |
---|
1593 | 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); |
---|
1594 | ideal K = I,C; |
---|
1595 | groebner(K); |
---|
1596 | } |
---|
1597 | |
---|
1598 | static proc exHom1() |
---|
1599 | { |
---|
1600 | // we start with |
---|
1601 | // z*y - x, z*x - y, y*x - z |
---|
1602 | LIB "freegb.lib"; |
---|
1603 | LIB "elim.lib"; |
---|
1604 | ring r = 0,(x,y,z,h),dp; |
---|
1605 | list L; |
---|
1606 | module M; |
---|
1607 | M = [1,z,y],[-1,x,h]; // zy - xh |
---|
1608 | L[1] = M; |
---|
1609 | M = [1,z,x],[-1,y,h]; // zx - yh |
---|
1610 | L[2] = M; |
---|
1611 | M = [1,y,x],[-1,z,h]; // yx - zh |
---|
1612 | L[3] = M; |
---|
1613 | lst2str(L); |
---|
1614 | def U = crs(L,4); |
---|
1615 | setring U; |
---|
1616 | I = I, |
---|
1617 | y(2)*h(3)+z(2)*x(3), y(3)*h(4)+z(3)*x(4), |
---|
1618 | y(2)*x(3)-z(2)*h(3), y(3)*x(4)-z(3)*h(4); |
---|
1619 | I = simplify(I,2); |
---|
1620 | ring r2 = 0,(x(0..4),y(0..4),z(0..4),h(0..4)),dp; |
---|
1621 | ideal J = imap(U,I); |
---|
1622 | // ideal K = homog(J,h); |
---|
1623 | option(redSB); |
---|
1624 | option(redTail); |
---|
1625 | ideal L = groebner(J); //(K); |
---|
1626 | ideal LL = sat(L,ideal(h))[1]; |
---|
1627 | ideal M = subst(LL,h,1); |
---|
1628 | M = simplify(M,2); |
---|
1629 | setring U; |
---|
1630 | ideal M = imap(r2,M); |
---|
1631 | lst2str(U); |
---|
1632 | } |
---|
1633 | |
---|
1634 | static proc test1() |
---|
1635 | { |
---|
1636 | LIB "freegb.lib"; |
---|
1637 | ring r = 0,(x,y),Dp; |
---|
1638 | int d = 10; // degree |
---|
1639 | def R = makeLetterplaceRing(d); |
---|
1640 | setring R; |
---|
1641 | ideal I = x(1)*x(2) - y(1)*y(2); |
---|
1642 | option(prot); |
---|
1643 | option(teach); |
---|
1644 | ideal J = system("freegb",I,d,2); |
---|
1645 | J; |
---|
1646 | } |
---|
1647 | |
---|
1648 | static proc test2() |
---|
1649 | { |
---|
1650 | LIB "freegb.lib"; |
---|
1651 | ring r = 0,(x,y),Dp; |
---|
1652 | int d = 10; // degree |
---|
1653 | def R = makeLetterplaceRing(d); |
---|
1654 | setring R; |
---|
1655 | ideal I = x(1)*x(2) - x(1)*y(2); |
---|
1656 | option(prot); |
---|
1657 | option(teach); |
---|
1658 | ideal J = system("freegb",I,d,2); |
---|
1659 | J; |
---|
1660 | } |
---|
1661 | |
---|
1662 | static proc test3() |
---|
1663 | { |
---|
1664 | LIB "freegb.lib"; |
---|
1665 | ring r = 0,(x,y,z),dp; |
---|
1666 | int d =5; // degree |
---|
1667 | def R = makeLetterplaceRing(d); |
---|
1668 | setring R; |
---|
1669 | ideal I = x(1)*y(2), y(1)*x(2)+z(1)*z(2); |
---|
1670 | option(prot); |
---|
1671 | option(teach); |
---|
1672 | ideal J = system("freegb",I,d,3); |
---|
1673 | } |
---|
1674 | |
---|
1675 | static proc schur2-3() |
---|
1676 | { |
---|
1677 | // nonhomog: |
---|
1678 | // h^4-10*h^2+9,f*e-e*f+h, h*2-e*h-2*e,h*f-f*h+2*f |
---|
1679 | // homogenized with t |
---|
1680 | // 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, |
---|
1681 | // t*h - h*t, t*f - f*t, t*e - e*t |
---|
1682 | } |
---|
1683 | |
---|
1684 | end older procs and tests */ |
---|
1685 | |
---|
1686 | proc ademRelations(int i, int j) |
---|
1687 | "USAGE: ademRelations(i,j); i,j int |
---|
1688 | RETURN: ring (and exports ideal) |
---|
1689 | ASSUME: there are at least i+j variables in the basering |
---|
1690 | PURPOSE: compute the ideal of Adem relations for i<2j in characteristic 0 |
---|
1691 | @* the ideal is exported under the name AdemRel in the output ring |
---|
1692 | EXAMPLE: example ademRelations; shows examples |
---|
1693 | " |
---|
1694 | { |
---|
1695 | // produces Adem relations for i<2j in char 0 |
---|
1696 | // assume: 0<i<2j |
---|
1697 | // requires presence of vars up to i+j |
---|
1698 | if ( (i<0) || (i >= 2*j) ) |
---|
1699 | { |
---|
1700 | ERROR("arguments out of range"); return(0); |
---|
1701 | } |
---|
1702 | ring @r = 0,(s(i+j..0)),lp; |
---|
1703 | poly p,q; |
---|
1704 | number n; |
---|
1705 | int ii = i div 2; int k; |
---|
1706 | // k=0 => s(0)=1 |
---|
1707 | n = binomial(j-1,i); |
---|
1708 | q = n*s(i+j)*s(0); |
---|
1709 | // printf("k=0, term=%s",q); |
---|
1710 | p = p + q; |
---|
1711 | for (k=1; k<= ii; k++) |
---|
1712 | { |
---|
1713 | n = binomial(j-k-1,i-2*k); |
---|
1714 | q = n*s(i+j-k)*s(k);; |
---|
1715 | // printf("k=%s, term=%s",k,q); |
---|
1716 | p = p + q; |
---|
1717 | } |
---|
1718 | poly AdemRel = p; |
---|
1719 | export AdemRel; |
---|
1720 | return(@r); |
---|
1721 | } |
---|
1722 | example |
---|
1723 | { |
---|
1724 | "EXAMPLE:"; echo = 2; |
---|
1725 | def A = ademRelations(2,5); |
---|
1726 | setring A; |
---|
1727 | AdemRel; |
---|
1728 | } |
---|
1729 | |
---|
1730 | /* |
---|
1731 | 1,1: 0 |
---|
1732 | 1,2: s(3)*s(0) == s(3) -> def for s(3):=s(1)s(2) |
---|
1733 | 2,1: adm |
---|
1734 | 2,2: s(3)*s(1) == s(1)s(2)s(1) |
---|
1735 | 1,3: 0 ( since 2*s(4)*s(0) = 0 mod 2) |
---|
1736 | 3,1: adm |
---|
1737 | 2,3: s(5)*s(0)+s(4)*s(1) == s(5)+s(4)*s(1) |
---|
1738 | 3,2: 0 |
---|
1739 | 3,3: s(5)*s(1) |
---|
1740 | 1,4: 3*s(5)*s(0) == s(5) -> def for s(5):=s(1)*s(4) |
---|
1741 | 4,1: adm |
---|
1742 | 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) |
---|
1743 | 4,2: adm |
---|
1744 | 4,3: s(5)*s(2) |
---|
1745 | 3,4: s(7)*s(0)+2*s(6)*s(1) == s(7) -> def for s(7):=s(3)*s(4) |
---|
1746 | 4,4: s(7)*s(1)+s(6)*s(2) |
---|
1747 | */ |
---|
1748 | |
---|
1749 | /* s1,s2: |
---|
1750 | s1*s1 =0, s2*s2 = s1*s2*s1 |
---|
1751 | */ |
---|
1752 | |
---|
1753 | /* |
---|
1754 | try char 0: |
---|
1755 | s1,s2: |
---|
1756 | 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) |
---|
1757 | hence 2==0! only in char 2 |
---|
1758 | */ |
---|
1759 | |
---|
1760 | // Adem rels modulo 2 are interesting |
---|
1761 | |
---|
1762 | static proc stringpoly2lplace(string s) |
---|
1763 | { |
---|
1764 | // decomposes sentence into terms |
---|
1765 | s = replace(s,newline,""); // get rid of newlines |
---|
1766 | s = replace(s," ",""); // get rid of empties |
---|
1767 | //arith symbols: +,- |
---|
1768 | // decompose into words with coeffs |
---|
1769 | list LS; |
---|
1770 | int i,j,ie,je,k,cnt; |
---|
1771 | // s[1]="-" situation |
---|
1772 | if (s[1]=="-") |
---|
1773 | { |
---|
1774 | LS = stringpoly2lplace(string(s[2..size(s)])); |
---|
1775 | LS[1] = string("-"+string(LS[1])); |
---|
1776 | return(LS); |
---|
1777 | } |
---|
1778 | i = find(s,"-",2); |
---|
1779 | // i==1 might happen if the 1st symbol coeff is negative |
---|
1780 | j = find(s,"+"); |
---|
1781 | list LL; |
---|
1782 | if (i==j) |
---|
1783 | { |
---|
1784 | "return a monomial"; |
---|
1785 | // that is both are 0 -> s is a monomial |
---|
1786 | LS[1] = s; |
---|
1787 | return(LS); |
---|
1788 | } |
---|
1789 | if (i==0) |
---|
1790 | { |
---|
1791 | "i==0 situation"; |
---|
1792 | // no minuses at all => pluses only |
---|
1793 | cnt++; |
---|
1794 | LS[cnt] = string(s[1..j-1]); |
---|
1795 | s = s[j+1..size(s)]; |
---|
1796 | while (s!= "") |
---|
1797 | { |
---|
1798 | j = find(s,"+"); |
---|
1799 | cnt++; |
---|
1800 | if (j==0) |
---|
1801 | { |
---|
1802 | LS[cnt] = string(s); |
---|
1803 | s = ""; |
---|
1804 | } |
---|
1805 | else |
---|
1806 | { |
---|
1807 | LS[cnt] = string(s[1..j-1]); |
---|
1808 | s = s[j+1..size(s)]; |
---|
1809 | } |
---|
1810 | } |
---|
1811 | return(LS); |
---|
1812 | } |
---|
1813 | if (j==0) |
---|
1814 | { |
---|
1815 | "j==0 situation"; |
---|
1816 | // no pluses at all except the lead coef => the rest are minuses only |
---|
1817 | cnt++; |
---|
1818 | LS[cnt] = string(s[1..i-1]); |
---|
1819 | s = s[i..size(s)]; |
---|
1820 | while (s!= "") |
---|
1821 | { |
---|
1822 | i = find(s,"-",2); |
---|
1823 | cnt++; |
---|
1824 | if (i==0) |
---|
1825 | { |
---|
1826 | LS[cnt] = string(s); |
---|
1827 | s = ""; |
---|
1828 | } |
---|
1829 | else |
---|
1830 | { |
---|
1831 | LS[cnt] = string(s[1..i-1]); |
---|
1832 | s = s[i..size(s)]; |
---|
1833 | } |
---|
1834 | } |
---|
1835 | return(LS); |
---|
1836 | } |
---|
1837 | // now i, j are nonzero |
---|
1838 | if (i>j) |
---|
1839 | { |
---|
1840 | "i>j situation"; |
---|
1841 | // + comes first, at place j |
---|
1842 | cnt++; |
---|
1843 | // "cnt:"; cnt; "j:"; j; |
---|
1844 | LS[cnt] = string(s[1..j-1]); |
---|
1845 | s = s[j+1..size(s)]; |
---|
1846 | LL = stringpoly2lplace(s); |
---|
1847 | LS = LS + LL; |
---|
1848 | kill LL; |
---|
1849 | return(LS); |
---|
1850 | } |
---|
1851 | else |
---|
1852 | { |
---|
1853 | "j>i situation"; |
---|
1854 | // - might come first, at place i |
---|
1855 | if (i>1) |
---|
1856 | { |
---|
1857 | cnt++; |
---|
1858 | LS[cnt] = string(s[1..i-1]); |
---|
1859 | s = s[i..size(s)]; |
---|
1860 | } |
---|
1861 | else |
---|
1862 | { |
---|
1863 | // i==1-> minus at leadcoef |
---|
1864 | ie = find(s,"-",i+1); |
---|
1865 | je = find(s,"+",i+1); |
---|
1866 | if (je == ie) |
---|
1867 | { |
---|
1868 | "ie=je situation"; |
---|
1869 | //monomial |
---|
1870 | cnt++; |
---|
1871 | LS[cnt] = s; |
---|
1872 | return(LS); |
---|
1873 | } |
---|
1874 | if (je < ie) |
---|
1875 | { |
---|
1876 | "je<ie situation"; |
---|
1877 | // + comes first |
---|
1878 | cnt++; |
---|
1879 | LS[cnt] = s[1..je-1]; |
---|
1880 | s = s[je+1..size(s)]; |
---|
1881 | } |
---|
1882 | else |
---|
1883 | { |
---|
1884 | // ie < je |
---|
1885 | "ie<je situation"; |
---|
1886 | cnt++; |
---|
1887 | LS[cnt] = s[1..ie-1]; |
---|
1888 | s = s[ie..size(s)]; |
---|
1889 | } |
---|
1890 | } |
---|
1891 | "going into recursion with "+s; |
---|
1892 | LL = stringpoly2lplace(s); |
---|
1893 | LS = LS + LL; |
---|
1894 | return(LS); |
---|
1895 | } |
---|
1896 | } |
---|
1897 | example |
---|
1898 | { |
---|
1899 | "EXAMPLE:"; echo = 2; |
---|
1900 | string s = "x*y+y*z+z*t"; // + only |
---|
1901 | stringpoly2lplace(s); |
---|
1902 | string s2 = "x*y - y*z-z*t*w*w"; // +1, - only |
---|
1903 | stringpoly2lplace(s2); |
---|
1904 | string s3 = "-x*y + y - 2*x +7*w*w*w"; |
---|
1905 | stringpoly2lplace(s3); |
---|
1906 | } |
---|
1907 | |
---|
1908 | static proc addplaces(list L) |
---|
1909 | { |
---|
1910 | // adds places to the list of strings |
---|
1911 | // according to their order in the list |
---|
1912 | int s = size(L); |
---|
1913 | int i; |
---|
1914 | for (i=1; i<=s; i++) |
---|
1915 | { |
---|
1916 | if (typeof(L[i]) == "string") |
---|
1917 | { |
---|
1918 | L[i] = L[i] + "(" + string(i) + ")"; |
---|
1919 | } |
---|
1920 | else |
---|
1921 | { |
---|
1922 | ERROR("entry of type string expected"); |
---|
1923 | return(0); |
---|
1924 | } |
---|
1925 | } |
---|
1926 | return(L); |
---|
1927 | } |
---|
1928 | example |
---|
1929 | { |
---|
1930 | "EXAMPLE:"; echo = 2; |
---|
1931 | string a = "f1"; string b = "f2"; |
---|
1932 | list L = a,b,a; |
---|
1933 | addplaces(L); |
---|
1934 | } |
---|
1935 | |
---|
1936 | static proc sent2lplace(string s) |
---|
1937 | { |
---|
1938 | // SENTence of words TO LetterPLACE |
---|
1939 | list L = stringpoly2lplace(s); |
---|
1940 | int i; int ss = size(L); |
---|
1941 | for(i=1; i<=ss; i++) |
---|
1942 | { |
---|
1943 | L[i] = str2lplace(L[i]); |
---|
1944 | } |
---|
1945 | return(L); |
---|
1946 | } |
---|
1947 | example |
---|
1948 | { |
---|
1949 | "EXAMPLE:"; echo = 2; |
---|
1950 | ring r = 0,(f2,f1),dp; |
---|
1951 | string s = "f2*f1*f1 - 2*f1*f2*f1+ f1*f1*f2"; |
---|
1952 | sent2lplace(s); |
---|
1953 | } |
---|
1954 | |
---|
1955 | static proc testnumber(string s) |
---|
1956 | { |
---|
1957 | string t; |
---|
1958 | if (s[1]=="-") |
---|
1959 | { |
---|
1960 | // two situations: either there's a negative number |
---|
1961 | t = s[2..size(s)]; |
---|
1962 | if (testnumber(t)) |
---|
1963 | { |
---|
1964 | //a negative number |
---|
1965 | } |
---|
1966 | else |
---|
1967 | { |
---|
1968 | // a variable times -1 |
---|
1969 | } |
---|
1970 | // or just a "-" for -1 |
---|
1971 | } |
---|
1972 | t = "ring @r=("; |
---|
1973 | t = t + charstr(basering)+"),"; |
---|
1974 | t = t + string(var(1))+",dp;"; |
---|
1975 | // write(":w tstnum.tst",t); |
---|
1976 | t = t+ "number @@Nn = " + s + ";"+"$"; |
---|
1977 | write(":w tstnum.tst",t); |
---|
1978 | string runsing = system("Singular"); |
---|
1979 | int k; |
---|
1980 | t = runsing+ " -teq <tstnum.tst >tstnum.out"; |
---|
1981 | k = system("sh",t); |
---|
1982 | if (k!=0) |
---|
1983 | { |
---|
1984 | ERROR("Problems running Singular"); |
---|
1985 | } |
---|
1986 | int i = system("sh", "grep error tstnum.out > /dev/NULL"); |
---|
1987 | if (i!=0) |
---|
1988 | { |
---|
1989 | // no error: s is a number |
---|
1990 | i = 1; |
---|
1991 | } |
---|
1992 | k = system("sh","rm tstnum.tst tstnum.out > /dev/NULL"); |
---|
1993 | return(i); |
---|
1994 | } |
---|
1995 | example |
---|
1996 | { |
---|
1997 | "EXAMPLE:"; echo = 2; |
---|
1998 | ring r = (0,a),x,dp; |
---|
1999 | string s = "a^2+7*a-2"; |
---|
2000 | testnumber(s); |
---|
2001 | s = "b+a"; |
---|
2002 | testnumber(s); |
---|
2003 | } |
---|
2004 | |
---|
2005 | static proc str2lplace(string s) |
---|
2006 | { |
---|
2007 | // converts a word (monomial) with coeff into letter-place |
---|
2008 | // string: coef*var1^exp1*var2^exp2*...varN^expN |
---|
2009 | s = strpower2rep(s); // expand powers |
---|
2010 | if (size(s)==0) { return(0); } |
---|
2011 | int i,j,k,insC; |
---|
2012 | string a,b,c,d,t; |
---|
2013 | // 1. get coeff |
---|
2014 | i = find(s,"*"); |
---|
2015 | if (i==0) { return(s); } |
---|
2016 | list VN; |
---|
2017 | c = s[1..i-1]; // incl. the case like (-a^2+1) |
---|
2018 | int tn = testnumber(c); |
---|
2019 | if (tn == 0) |
---|
2020 | { |
---|
2021 | // failed test |
---|
2022 | if (c[1]=="-") |
---|
2023 | { |
---|
2024 | // two situations: either there's a negative number |
---|
2025 | t = c[2..size(c)]; |
---|
2026 | if (testnumber(t)) |
---|
2027 | { |
---|
2028 | //a negative number |
---|
2029 | // nop here |
---|
2030 | } |
---|
2031 | else |
---|
2032 | { |
---|
2033 | // a variable times -1 |
---|
2034 | c = "-1"; |
---|
2035 | j++; VN[j] = t; //string(c[2..size(c)]); |
---|
2036 | insC = 1; |
---|
2037 | } |
---|
2038 | } |
---|
2039 | else |
---|
2040 | { |
---|
2041 | // just a variable with coeff 1 |
---|
2042 | j++; VN[j] = string(c); |
---|
2043 | c = "1"; |
---|
2044 | insC = 1; |
---|
2045 | } |
---|
2046 | } |
---|
2047 | // get vars |
---|
2048 | t = s; |
---|
2049 | // t = s[i+1..size(s)]; |
---|
2050 | k = size(t); //j = 0; |
---|
2051 | while (k>0) |
---|
2052 | { |
---|
2053 | t = t[i+1..size(t)]; //next part |
---|
2054 | i = find(t,"*"); // next * |
---|
2055 | if (i==0) |
---|
2056 | { |
---|
2057 | // last monomial |
---|
2058 | j++; |
---|
2059 | VN[j] = t; |
---|
2060 | k = size(t); |
---|
2061 | break; |
---|
2062 | } |
---|
2063 | b = t[1..i-1]; |
---|
2064 | // print(b); |
---|
2065 | j++; |
---|
2066 | VN[j] = b; |
---|
2067 | k = size(t); |
---|
2068 | } |
---|
2069 | VN = addplaces(VN); |
---|
2070 | VN[size(VN)+1] = string(c); |
---|
2071 | return(VN); |
---|
2072 | } |
---|
2073 | example |
---|
2074 | { |
---|
2075 | "EXAMPLE:"; echo = 2; |
---|
2076 | ring r = (0,a),(f2,f1),dp; |
---|
2077 | str2lplace("-2*f2^2*f1^2*f2"); |
---|
2078 | str2lplace("-f1*f2"); |
---|
2079 | str2lplace("(-a^2+7a)*f1*f2"); |
---|
2080 | } |
---|
2081 | |
---|
2082 | static proc strpower2rep(string s) |
---|
2083 | { |
---|
2084 | // makes x*x*x*x out of x^4 ., rep statys for repetitions |
---|
2085 | // looks for "-" problem |
---|
2086 | // exception: "-" as coeff |
---|
2087 | string ex,t; |
---|
2088 | int i,j,k; |
---|
2089 | |
---|
2090 | i = find(s,"^"); // first ^ |
---|
2091 | if (i==0) { return(s); } // no ^ signs |
---|
2092 | |
---|
2093 | if (s[1] == "-") |
---|
2094 | { |
---|
2095 | // either -coef or -1 |
---|
2096 | // got the coeff: |
---|
2097 | i = find(s,"*"); |
---|
2098 | if (i==0) |
---|
2099 | { |
---|
2100 | // no *'s => coef == -1 or s == -23 |
---|
2101 | i = size(s)+1; |
---|
2102 | } |
---|
2103 | t = string(s[2..i-1]); // without "-" |
---|
2104 | if ( testnumber(t) ) |
---|
2105 | { |
---|
2106 | // a good number |
---|
2107 | t = strpower2rep(string(s[2..size(s)])); |
---|
2108 | t = "-"+t; |
---|
2109 | return(t); |
---|
2110 | } |
---|
2111 | else |
---|
2112 | { |
---|
2113 | // a variable |
---|
2114 | t = strpower2rep(string(s[2..size(s)])); |
---|
2115 | t = "-1*"+ t; |
---|
2116 | return(t); |
---|
2117 | } |
---|
2118 | } |
---|
2119 | // the case when leadcoef is a number in () |
---|
2120 | if (s[1] == "(") |
---|
2121 | { |
---|
2122 | i = find(s,")",2); // must be nonzero |
---|
2123 | t = s[2..i-1]; |
---|
2124 | if ( testnumber(t) ) |
---|
2125 | { |
---|
2126 | // a good number |
---|
2127 | } |
---|
2128 | else {"strpower2rep: bad number as coef";} |
---|
2129 | ex = string(s[i+2..size(s)]); // 2 because of * |
---|
2130 | ex = strpower2rep(ex); |
---|
2131 | t = "("+t+")*"+ex; |
---|
2132 | return(t); |
---|
2133 | } |
---|
2134 | |
---|
2135 | i = find(s,"^"); // first ^ |
---|
2136 | j = find(s,"*",i+1); // next * == end of ^ |
---|
2137 | if (j==0) |
---|
2138 | { |
---|
2139 | ex = s[i+1..size(s)]; |
---|
2140 | } |
---|
2141 | else |
---|
2142 | { |
---|
2143 | ex = s[i+1..j-1]; |
---|
2144 | } |
---|
2145 | execute("int @exp = " + ex + ";"); //@exp = exponent |
---|
2146 | // got varname |
---|
2147 | for (k=i-1; k>0; k--) |
---|
2148 | { |
---|
2149 | if (s[k] == "*") break; |
---|
2150 | } |
---|
2151 | string varn = s[k+1..i-1]; |
---|
2152 | // "varn:"; varn; |
---|
2153 | string pref; |
---|
2154 | if (k>0) |
---|
2155 | { |
---|
2156 | pref = s[1..k]; // with * on the k-th place |
---|
2157 | } |
---|
2158 | // "pref:"; pref; |
---|
2159 | string suf; |
---|
2160 | if ( (j>0) && (j+1 <= size(s)) ) |
---|
2161 | { |
---|
2162 | suf = s[j+1..size(s)]; // without * on the 1st place |
---|
2163 | } |
---|
2164 | // "suf:"; suf; |
---|
2165 | string toins; |
---|
2166 | for (k=1; k<=@exp; k++) |
---|
2167 | { |
---|
2168 | toins = toins + varn+"*"; |
---|
2169 | } |
---|
2170 | // "toins: "; toins; |
---|
2171 | if (size(suf) == 0) |
---|
2172 | { |
---|
2173 | toins = toins[1..size(toins)-1]; // get rid of trailing * |
---|
2174 | } |
---|
2175 | else |
---|
2176 | { |
---|
2177 | suf = strpower2rep(suf); |
---|
2178 | } |
---|
2179 | ex = pref + toins + suf; |
---|
2180 | return(ex); |
---|
2181 | // return(strpower2rep(ex)); |
---|
2182 | } |
---|
2183 | example |
---|
2184 | { |
---|
2185 | "EXAMPLE:"; echo = 2; |
---|
2186 | ring r = (0,a),(x,y,z,t),dp; |
---|
2187 | strpower2rep("-x^4"); |
---|
2188 | strpower2rep("-2*x^4*y^3*z*t^2"); |
---|
2189 | strpower2rep("-a^2*x^4"); |
---|
2190 | } |
---|
2191 | |
---|
2192 | proc lieBracket(poly a, poly b, list #) |
---|
2193 | "USAGE: lieBracket(a,b[,N]); a,b letterplace polynomials, N an optional integer |
---|
2194 | RETURN: poly |
---|
2195 | ASSUME: basering has a letterplace ring structure |
---|
2196 | PURPOSE: compute the Lie bracket [a,b] = ab - ba between letterplace polynomials |
---|
2197 | NOTE: if N>1 is specified, then the left normed bracket [a,[...[a,b]]]] is computed. |
---|
2198 | EXAMPLE: example lieBracket; shows examples |
---|
2199 | " |
---|
2200 | { |
---|
2201 | if (lpAssumeViolation()) |
---|
2202 | { |
---|
2203 | // ERROR("Either 'uptodeg' or 'lV' global variables are not set!"); |
---|
2204 | ERROR("Incomplete Letterplace structure on the basering!"); |
---|
2205 | } |
---|
2206 | // alias ppLiebr; |
---|
2207 | //if int N is given compute [a,[...[a,b]]]] left normed bracket |
---|
2208 | poly q; |
---|
2209 | int N=1; |
---|
2210 | if (size(#)>0) |
---|
2211 | { |
---|
2212 | if (typeof(#[1])=="int") |
---|
2213 | { |
---|
2214 | N = int(#[1]); |
---|
2215 | } |
---|
2216 | } |
---|
2217 | if (N<=0) { return(q); } |
---|
2218 | while (b!=0) |
---|
2219 | { |
---|
2220 | q = q + pmLiebr(a,lead(b)); |
---|
2221 | b = b - lead(b); |
---|
2222 | } |
---|
2223 | int i; |
---|
2224 | if (N >1) |
---|
2225 | { |
---|
2226 | for(i=1; i<=N; i++) |
---|
2227 | { |
---|
2228 | q = lieBracket(a,q); |
---|
2229 | } |
---|
2230 | } |
---|
2231 | return(q); |
---|
2232 | } |
---|
2233 | example |
---|
2234 | { |
---|
2235 | "EXAMPLE:"; echo = 2; |
---|
2236 | ring r = 0,(x(1),y(1),x(2),y(2),x(3),y(3),x(4),y(4)),dp; |
---|
2237 | def R = setLetterplaceAttributes(r,4,2); // supply R with letterplace structure |
---|
2238 | setring R; |
---|
2239 | poly a = x(1)*y(2); poly b = y(1); |
---|
2240 | lieBracket(a,b); |
---|
2241 | lieBracket(x(1),y(1),2); |
---|
2242 | } |
---|
2243 | |
---|
2244 | static proc pmLiebr(poly a, poly b) |
---|
2245 | { |
---|
2246 | // a poly, b mono |
---|
2247 | poly s; |
---|
2248 | while (a!=0) |
---|
2249 | { |
---|
2250 | s = s + mmLiebr(lead(a),lead(b)); |
---|
2251 | a = a - lead(a); |
---|
2252 | } |
---|
2253 | return(s); |
---|
2254 | } |
---|
2255 | |
---|
2256 | proc shiftPoly(poly a, int i) |
---|
2257 | "USAGE: shiftPoly(p,i); p letterplace poly, i int |
---|
2258 | RETURN: poly |
---|
2259 | ASSUME: basering has letterplace ring structure |
---|
2260 | PURPOSE: compute the i-th shift of letterplace polynomial p |
---|
2261 | EXAMPLE: example shiftPoly; shows examples |
---|
2262 | " |
---|
2263 | { |
---|
2264 | // shifts a monomial a by i |
---|
2265 | // calls pLPshift(p,sh,uptodeg,lVblock); |
---|
2266 | if (lpAssumeViolation()) |
---|
2267 | { |
---|
2268 | ERROR("Incomplete Letterplace structure on the basering!"); |
---|
2269 | } |
---|
2270 | int uptodeg = attrib(basering,"uptodeg"); |
---|
2271 | int lV = attrib(basering,"lV"); |
---|
2272 | if (deg(a) + i > uptodeg) |
---|
2273 | { |
---|
2274 | ERROR("degree bound violated by the shift!"); |
---|
2275 | } |
---|
2276 | return(system("stest",a,i,uptodeg,lV)); |
---|
2277 | } |
---|
2278 | example |
---|
2279 | { |
---|
2280 | "EXAMPLE:"; echo = 2; |
---|
2281 | ring r = 0,(x,y,z),dp; |
---|
2282 | int uptodeg = 5; int lV = 3; |
---|
2283 | def R = makeLetterplaceRing(uptodeg); |
---|
2284 | setring R; |
---|
2285 | poly f = x(1)*z(2)*y(3) - 2*z(1)*y(2) + 3*x(1); |
---|
2286 | shiftPoly(f,1); |
---|
2287 | shiftPoly(f,2); |
---|
2288 | } |
---|
2289 | |
---|
2290 | |
---|
2291 | static proc mmLiebr(poly a, poly b) |
---|
2292 | { |
---|
2293 | // a,b, monomials |
---|
2294 | a = lead(a); |
---|
2295 | b = lead(b); |
---|
2296 | int sa = deg(a); |
---|
2297 | int sb = deg(b); |
---|
2298 | poly v = a*shiftPoly(b,sa) - b*shiftPoly(a,sb); |
---|
2299 | return(v); |
---|
2300 | } |
---|
2301 | |
---|
2302 | static proc test_shift() |
---|
2303 | { |
---|
2304 | LIB "freegb.lib"; |
---|
2305 | ring r = 0,(a,b),dp; |
---|
2306 | int d =5; |
---|
2307 | def R = makeLetterplaceRing(d); |
---|
2308 | setring R; |
---|
2309 | int uptodeg = d; |
---|
2310 | int lV = 2; |
---|
2311 | def R = setLetterplaceAttributes(r,uptodeg,2); // supply R with letterplace structure |
---|
2312 | setring R; |
---|
2313 | poly p = mmLiebr(a(1),b(1)); |
---|
2314 | poly p = lieBracket(a(1),b(1)); |
---|
2315 | } |
---|
2316 | |
---|
2317 | proc serreRelations(intmat A, int zu) |
---|
2318 | "USAGE: serreRelations(A,z); A an intmat, z an int |
---|
2319 | RETURN: ideal |
---|
2320 | ASSUME: basering has a letterplace ring structure and |
---|
2321 | @* A is a generalized Cartan matrix with integer entries |
---|
2322 | PURPOSE: compute the ideal of Serre's relations associated to A |
---|
2323 | EXAMPLE: example serreRelations; shows examples |
---|
2324 | " |
---|
2325 | { |
---|
2326 | // zu = 1 -> with commutators [f_i,f_j]; zu == 0 without them |
---|
2327 | // suppose that A is cartan matrix |
---|
2328 | // then Serre's relations are |
---|
2329 | // (ad f_j)^{1-A_{ij}} ( f_i) |
---|
2330 | int ppl = printlevel-voice+2; |
---|
2331 | int n = ncols(A); // hence n variables |
---|
2332 | int i,j,k,el; |
---|
2333 | poly p,q; |
---|
2334 | ideal I; |
---|
2335 | for (i=1; i<=n; i++) |
---|
2336 | { |
---|
2337 | for (j=1; j<=n; j++) |
---|
2338 | { |
---|
2339 | el = 1 - A[i,j]; |
---|
2340 | // printf("i:%s, j: %s, l: %s",i,j,l); |
---|
2341 | dbprint(ppl,"i, j, l: ",i,j,el); |
---|
2342 | // if ((i!=j) && (l >0)) |
---|
2343 | // if ( (i!=j) && ( ((zu ==0) && (l >=2)) || ((zu ==1) && (l >=1)) ) ) |
---|
2344 | if ((i!=j) && (el >0)) |
---|
2345 | { |
---|
2346 | q = lieBracket(var(j),var(i)); |
---|
2347 | dbprint(ppl,"first bracket: ",q); |
---|
2348 | // if (l >=2) |
---|
2349 | // { |
---|
2350 | for (k=1; k<=el-1; k++) |
---|
2351 | { |
---|
2352 | q = lieBracket(var(j),q); |
---|
2353 | dbprint(ppl,"further bracket:",q); |
---|
2354 | } |
---|
2355 | // } |
---|
2356 | } |
---|
2357 | if (q!=0) { I = I,q; q=0;} |
---|
2358 | } |
---|
2359 | } |
---|
2360 | I = simplify(I,2); |
---|
2361 | return(I); |
---|
2362 | } |
---|
2363 | example |
---|
2364 | { |
---|
2365 | "EXAMPLE:"; echo = 2; |
---|
2366 | intmat A[3][3] = |
---|
2367 | 2, -1, 0, |
---|
2368 | -1, 2, -3, |
---|
2369 | 0, -1, 2; // G^1_2 Cartan matrix |
---|
2370 | ring r = 0,(f1,f2,f3),dp; |
---|
2371 | int uptodeg = 5; |
---|
2372 | def R = makeLetterplaceRing(uptodeg); |
---|
2373 | setring R; |
---|
2374 | ideal I = serreRelations(A,1); I = simplify(I,1+2+8); |
---|
2375 | I; |
---|
2376 | } |
---|
2377 | |
---|
2378 | /* setup for older example: |
---|
2379 | intmat A[2][2] = 2, -1, -1, 2; // sl_3 == A_2 |
---|
2380 | ring r = 0,(f1,f2),dp; |
---|
2381 | int uptodeg = 5; int lV = 2; |
---|
2382 | */ |
---|
2383 | |
---|
2384 | proc fullSerreRelations(intmat A, ideal rNegative, ideal rCartan, ideal rPositive, int uptodeg) |
---|
2385 | "USAGE: fullSerreRelations(A,N,C,P,d); A an intmat, N,C,P ideals, d an int |
---|
2386 | RETURN: ring (and ideal) |
---|
2387 | PURPOSE: compute the inhomogeneous Serre's relations associated to A in given variable names |
---|
2388 | ASSUME: three ideals in the input are of the same sizes and contain merely variables |
---|
2389 | @* which are interpreted as follows: N resp. P stand for negative resp. positive roots, |
---|
2390 | @* C stand for Cartan elements. d is the degree bound for letterplace ring, which will be returned. |
---|
2391 | @* The matrix A is a generalized Cartan matrix with integer entries |
---|
2392 | @* The result is the ideal called 'fsRel' in the returned ring. |
---|
2393 | EXAMPLE: example fullSerreRelations; shows examples |
---|
2394 | " |
---|
2395 | { |
---|
2396 | /* SerreRels on rNeg and rPos plus Cartans etc. */ |
---|
2397 | int ppl = printlevel -voice+2; |
---|
2398 | /* ideals must be written in variables: assume each term is of degree 1 */ |
---|
2399 | int i,j,k; |
---|
2400 | int N = nvars(basering); |
---|
2401 | def save = basering; |
---|
2402 | int comFlag = 0; |
---|
2403 | /* assume: (size(rNegative) == size(rPositive)) */ |
---|
2404 | /* assume: (size(rNegative) == size(rCartan)) i.e. nonsimple Cartans */ |
---|
2405 | if ( (size(rNegative) != size(rPositive)) || (size(rNegative) != size(rCartan)) ) |
---|
2406 | { |
---|
2407 | ERROR("All input ideals must be of the same size"); |
---|
2408 | } |
---|
2409 | |
---|
2410 | // if (size(rNegative) != size(rPositive)) |
---|
2411 | // { |
---|
2412 | // ERROR("The 1st and the 3rd input ideals must be of the same size"); |
---|
2413 | // } |
---|
2414 | |
---|
2415 | /* assume: 2*size(rNegative) + size(rCartan) >= nvars(basering) */ |
---|
2416 | i = 2*size(rNegative) + size(rCartan); |
---|
2417 | if (i>N) |
---|
2418 | { |
---|
2419 | ERROR("The total number of elements in input ideals must not exceed the dimension of the ground ring"); |
---|
2420 | } |
---|
2421 | if (i < N) |
---|
2422 | { |
---|
2423 | comFlag = N-i; // so many elements will commute |
---|
2424 | "Warning: some elements will be treated as mutually commuting"; |
---|
2425 | } |
---|
2426 | /* extract varnames from input ideals */ |
---|
2427 | intvec iNeg = varIdeal2intvec(rNegative); |
---|
2428 | intvec iCartan = varIdeal2intvec(rCartan); |
---|
2429 | intvec iPos = varIdeal2intvec(rPositive); |
---|
2430 | /* for each vector in rNeg and rPositive, go into the corr. ring and create SerreRels */ |
---|
2431 | /* rNegative: */ |
---|
2432 | list L = ringlist(save); |
---|
2433 | def LPsave = makeLetterplaceRing2(uptodeg); setring save; |
---|
2434 | list LNEG = L; list tmp; |
---|
2435 | /* L[1] field as is; L[2] vars: a subset; L[3] ordering: dp, L[4] as is */ |
---|
2436 | for (i=1; i<=size(iNeg); i++) |
---|
2437 | { |
---|
2438 | tmp[i] = string(var(iNeg[i])); |
---|
2439 | } |
---|
2440 | LNEG[2] = tmp; LNEG[3] = list(list("dp",intvec(1:size(iNeg))), list("C",0)); |
---|
2441 | def RNEG = ring(LNEG); setring RNEG; |
---|
2442 | def RRNEG = makeLetterplaceRing2(uptodeg); |
---|
2443 | setring RRNEG; |
---|
2444 | ideal I = serreRelations(A,1); I = simplify(I,1+2+8); |
---|
2445 | setring LPsave; |
---|
2446 | ideal srNeg = imap(RRNEG,I); |
---|
2447 | dbprint(ppl,"0-1 ideal of negative relations is ready"); |
---|
2448 | dbprint(ppl-1,srNeg); |
---|
2449 | setring save; kill L,tmp,RRNEG,RNEG, LNEG; |
---|
2450 | /* rPositive: */ |
---|
2451 | list L = ringlist(save); |
---|
2452 | list LPOS = L; list tmp; |
---|
2453 | /* L[1] field as is; L[2] vars: a subset; L[3] ordering: dp, L[4] as is */ |
---|
2454 | for (i=1; i<=size(iPos); i++) |
---|
2455 | { |
---|
2456 | tmp[i] = string(var(iPos[i])); |
---|
2457 | } |
---|
2458 | LPOS[2] = tmp; LPOS[3] = list(list("dp",intvec(1:size(iPos))), list("C",0)); |
---|
2459 | def RPOS = ring(LPOS); setring RPOS; |
---|
2460 | def RRPOS = makeLetterplaceRing2(uptodeg); |
---|
2461 | setring RRPOS; |
---|
2462 | ideal I = serreRelations(A,1); I = simplify(I,1+2+8); |
---|
2463 | setring LPsave; |
---|
2464 | ideal srPos = imap(RRPOS,I); |
---|
2465 | dbprint(ppl,"0-2 ideal of positive relations is ready"); |
---|
2466 | dbprint(ppl-1,srPos); |
---|
2467 | setring save; kill L,tmp,RRPOS,RPOS, LPOS; |
---|
2468 | string sMap = "ideal Mmap ="; |
---|
2469 | for (i=1; i<=nvars(save); i++) |
---|
2470 | { |
---|
2471 | sMap = sMap + string(var(i)) +"(1),"; |
---|
2472 | } |
---|
2473 | sMap[size(sMap)] = ";"; |
---|
2474 | /* cartans: h_j h_i = h_i h_j */ |
---|
2475 | setring LPsave; |
---|
2476 | ideal ComCartan; |
---|
2477 | for (i=1; i<size(iCartan); i++) |
---|
2478 | { |
---|
2479 | for (j=i+1; j<=size(iCartan); j++) |
---|
2480 | { |
---|
2481 | ComCartan = ComCartan + lieBracket(var(iCartan[j]),var(iCartan[i])); |
---|
2482 | } |
---|
2483 | } |
---|
2484 | ComCartan = simplify(ComCartan,1+2+8); |
---|
2485 | execute(sMap); // defines an ideal Mmap |
---|
2486 | map F = save, Mmap; |
---|
2487 | dbprint(ppl,"1. commuting Cartans: "); |
---|
2488 | dbprint(ppl-1,ComCartan); |
---|
2489 | /* [e_i, f_j] =0 if i<>j */ |
---|
2490 | ideal ComPosNeg; // assume: #Neg=#Pos |
---|
2491 | for (i=1; i<size(iPos); i++) |
---|
2492 | { |
---|
2493 | for (j=1; j<=size(iPos); j++) |
---|
2494 | { |
---|
2495 | if (j !=i) |
---|
2496 | { |
---|
2497 | ComPosNeg = ComPosNeg + lieBracket(var(iPos[i]),var(iNeg[j])); |
---|
2498 | ComPosNeg = ComPosNeg + lieBracket(var(iPos[j]),var(iNeg[i])); |
---|
2499 | } |
---|
2500 | } |
---|
2501 | } |
---|
2502 | ComPosNeg = simplify(ComPosNeg,1+2+8); |
---|
2503 | dbprint(ppl,"2. commuting Positive and Negative:"); |
---|
2504 | dbprint(ppl-1,ComPosNeg); |
---|
2505 | /* [e_i, f_i] = h_i */ |
---|
2506 | poly tempo; |
---|
2507 | for (i=1; i<=size(iCartan); i++) |
---|
2508 | { |
---|
2509 | tempo = lieBracket(var(iPos[i]),var(iNeg[i])) - var(iCartan[i]); |
---|
2510 | ComPosNeg = ComPosNeg + tempo; |
---|
2511 | } |
---|
2512 | // ComPosNeg = simplify(ComPosNeg,1+2+8); |
---|
2513 | dbprint(ppl,"3. added sl2 triples [e_i,f_i]=h_i"); |
---|
2514 | dbprint(ppl-1,ComPosNeg); |
---|
2515 | |
---|
2516 | /* [h_i, e_j] = A_ij e_j */ |
---|
2517 | /* [h_i, f_j] = -A_ij f_j */ |
---|
2518 | ideal ActCartan; // assume: #Neg=#Pos |
---|
2519 | for (i=1; i<=size(iCartan); i++) |
---|
2520 | { |
---|
2521 | for (j=1; j<=size(iCartan); j++) |
---|
2522 | { |
---|
2523 | tempo = lieBracket(var(iCartan[i]),var(iPos[j])) - A[i,j]*var(iPos[j]); |
---|
2524 | ActCartan = ActCartan + tempo; |
---|
2525 | tempo = lieBracket(var(iCartan[i]),var(iNeg[j])) + A[i,j]*var(iNeg[j]); |
---|
2526 | ActCartan = ActCartan + tempo; |
---|
2527 | } |
---|
2528 | } |
---|
2529 | ActCartan = simplify(ActCartan,1+2+8); |
---|
2530 | dbprint(ppl,"4. actions of Cartan:"); |
---|
2531 | dbprint(ppl-1, ActCartan); |
---|
2532 | |
---|
2533 | /* final part: prepare the output */ |
---|
2534 | setring LPsave; |
---|
2535 | ideal fsRel = srNeg, srPos, ComPosNeg, ComCartan, ActCartan; |
---|
2536 | export fsRel; |
---|
2537 | setring save; |
---|
2538 | return(LPsave); |
---|
2539 | } |
---|
2540 | example |
---|
2541 | { |
---|
2542 | "EXAMPLE:"; echo = 2; |
---|
2543 | intmat A[2][2] = |
---|
2544 | 2, -1, |
---|
2545 | -1, 2; // A_2 = sl_3 Cartan matrix |
---|
2546 | ring r = 0,(f1,f2,h1,h2,e1,e2),dp; |
---|
2547 | ideal negroots = f1,f2; ideal cartans = h1,h2; ideal posroots = e1,e2; |
---|
2548 | int uptodeg = 5; |
---|
2549 | def RS = fullSerreRelations(A,negroots,cartans,posroots,uptodeg); |
---|
2550 | setring RS; fsRel; |
---|
2551 | } |
---|
2552 | |
---|
2553 | proc varIdeal2intvec(ideal I) |
---|
2554 | { |
---|
2555 | /* assume1: input ideal is a list of variables of the ground ring */ |
---|
2556 | int i,j; intvec V; |
---|
2557 | for (i=1; i<= size(I); i++) |
---|
2558 | { |
---|
2559 | j = univariate(I[i]); |
---|
2560 | if (j<=0) |
---|
2561 | { |
---|
2562 | ERROR("input ideal must contain only variables"); |
---|
2563 | } |
---|
2564 | V[i] = j; |
---|
2565 | } |
---|
2566 | dbprint(printlevel-voice+2,V); |
---|
2567 | /* now we make a smaller list of non-repeating entries */ |
---|
2568 | ideal iW = simplify(ideal(V),2+4); // no zeros, no repetitions |
---|
2569 | if (size(iW) < size(V)) |
---|
2570 | { |
---|
2571 | /* extract intvec from iW */ |
---|
2572 | intvec inW; |
---|
2573 | for(j=1; j<=size(iW); j++) |
---|
2574 | { |
---|
2575 | inW[j] = int(leadcoef(iW[j])); |
---|
2576 | } |
---|
2577 | return(inW); |
---|
2578 | } |
---|
2579 | return(V); |
---|
2580 | } |
---|
2581 | example |
---|
2582 | { |
---|
2583 | "EXAMPLE:"; echo = 2; |
---|
2584 | ring r = 0,(x,y,z),dp; |
---|
2585 | ideal I = x,z; |
---|
2586 | varIdeal2intvec(I); |
---|
2587 | varIdeal2intvec(ideal(x2,y^3,x+1)); |
---|
2588 | varIdeal2intvec(ideal(x*y,y,x+1)); |
---|
2589 | } |
---|
2590 | |
---|
2591 | proc lp2lstr(ideal K, def save) |
---|
2592 | "USAGE: lp2lstr(K,s); K an ideal, s a ring name |
---|
2593 | RETURN: nothing (exports object @LN into the ring named s) |
---|
2594 | ASSUME: basering has a letterplace ring structure |
---|
2595 | PURPOSE: converts letterplace ideal to list of modules |
---|
2596 | NOTE: useful as preprocessing to 'lst2str' |
---|
2597 | EXAMPLE: example lp2lstr; shows examples |
---|
2598 | " |
---|
2599 | { |
---|
2600 | def @R = basering; |
---|
2601 | string err; |
---|
2602 | int s = nvars(save); |
---|
2603 | int i,j,k; |
---|
2604 | // K contains vars x(1),...z(1) = images of originals |
---|
2605 | // 5. go back to orig vars, produce strings/modules |
---|
2606 | int sk = size(K); |
---|
2607 | int sp, sx, a, b; |
---|
2608 | intvec x; |
---|
2609 | poly p,q; |
---|
2610 | poly pn; |
---|
2611 | // vars in 'save' |
---|
2612 | setring save; |
---|
2613 | module N; |
---|
2614 | list LN; |
---|
2615 | vector V; |
---|
2616 | poly pn; |
---|
2617 | // test and skip exponents >=2 |
---|
2618 | setring @R; |
---|
2619 | for(i=1; i<=sk; i++) |
---|
2620 | { |
---|
2621 | p = K[i]; |
---|
2622 | while (p!=0) |
---|
2623 | { |
---|
2624 | q = lead(p); |
---|
2625 | // "processing q:";q; |
---|
2626 | x = leadexp(q); |
---|
2627 | sx = size(x); |
---|
2628 | for(k=1; k<=sx; k++) |
---|
2629 | { |
---|
2630 | if ( x[k] >= 2 ) |
---|
2631 | { |
---|
2632 | err = "skip: the value x[k] is " + string(x[k]); |
---|
2633 | dbprint(ppl,err); |
---|
2634 | // return(0); |
---|
2635 | K[i] = 0; |
---|
2636 | p = 0; |
---|
2637 | q = 0; |
---|
2638 | break; |
---|
2639 | } |
---|
2640 | } |
---|
2641 | p = p - q; |
---|
2642 | } |
---|
2643 | } |
---|
2644 | K = simplify(K,2); |
---|
2645 | sk = size(K); |
---|
2646 | for(i=1; i<=sk; i++) |
---|
2647 | { |
---|
2648 | // setring save; |
---|
2649 | // V = 0; |
---|
2650 | setring @R; |
---|
2651 | p = K[i]; |
---|
2652 | while (p!=0) |
---|
2653 | { |
---|
2654 | q = lead(p); |
---|
2655 | err = "processing q:" + string(q); |
---|
2656 | dbprint(ppl,err); |
---|
2657 | x = leadexp(q); |
---|
2658 | sx = size(x); |
---|
2659 | pn = leadcoef(q); |
---|
2660 | setring save; |
---|
2661 | pn = imap(@R,pn); |
---|
2662 | V = V + leadcoef(pn)*gen(1); |
---|
2663 | for(k=1; k<=sx; k++) |
---|
2664 | { |
---|
2665 | if (x[k] ==1) |
---|
2666 | { |
---|
2667 | a = k / s; // block number=a+1, a!=0 |
---|
2668 | b = k % s; // remainder |
---|
2669 | // printf("a: %s, b: %s",a,b); |
---|
2670 | if (b == 0) |
---|
2671 | { |
---|
2672 | // that is it's the last var in the block |
---|
2673 | b = s; |
---|
2674 | a = a-1; |
---|
2675 | } |
---|
2676 | V = V + var(b)*gen(a+2); |
---|
2677 | } |
---|
2678 | } |
---|
2679 | err = "V: " + string(V); |
---|
2680 | dbprint(ppl,err); |
---|
2681 | // printf("V: %s", string(V)); |
---|
2682 | N = N,V; |
---|
2683 | V = 0; |
---|
2684 | setring @R; |
---|
2685 | p = p - q; |
---|
2686 | pn = 0; |
---|
2687 | } |
---|
2688 | setring save; |
---|
2689 | LN[i] = simplify(N,2); |
---|
2690 | N = 0; |
---|
2691 | } |
---|
2692 | setring save; |
---|
2693 | list @LN = LN; |
---|
2694 | export @LN; |
---|
2695 | // return(LN); |
---|
2696 | } |
---|
2697 | example |
---|
2698 | { |
---|
2699 | "EXAMPLE:"; echo = 2; |
---|
2700 | intmat A[2][2] = 2, -1, -1, 2; // sl_3 == A_2 |
---|
2701 | ring r = 0,(f1,f2),dp; |
---|
2702 | def R = makeLetterplaceRing(3); |
---|
2703 | setring R; |
---|
2704 | ideal I = serreRelations(A,1); |
---|
2705 | lp2lstr(I,r); |
---|
2706 | setring r; |
---|
2707 | lst2str(@LN,1); |
---|
2708 | } |
---|
2709 | |
---|
2710 | static proc strList2poly(list L) |
---|
2711 | { |
---|
2712 | // list L comes from sent2lplace (which takes a polynomial as input) |
---|
2713 | // each entry of L is a sublist with the coef on the last place |
---|
2714 | int s = size(L); int t; |
---|
2715 | int i,j; |
---|
2716 | list M; |
---|
2717 | poly p,q; |
---|
2718 | string Q; |
---|
2719 | for(i=1; i<=s; i++) |
---|
2720 | { |
---|
2721 | M = L[i]; |
---|
2722 | t = size(M); |
---|
2723 | // q = M[t]; // a constant |
---|
2724 | Q = string(M[t]); |
---|
2725 | for(j=1; j<t; j++) |
---|
2726 | { |
---|
2727 | // q = q*M[j]; |
---|
2728 | Q = Q+"*"+string(M[j]); |
---|
2729 | } |
---|
2730 | execute("q="+Q+";"); |
---|
2731 | // q; |
---|
2732 | p = p + q; |
---|
2733 | } |
---|
2734 | kill Q; |
---|
2735 | return(p); |
---|
2736 | } |
---|
2737 | example |
---|
2738 | { |
---|
2739 | "EXAMPLE:"; echo = 2; |
---|
2740 | ring r =0,(x,y,z,t),Dp; |
---|
2741 | def A = makeLetterplaceRing(4); |
---|
2742 | setring A; |
---|
2743 | string t = "-2*y*z*y*z + y*t*z*z - z*x*x*y + 2*z*y*z*y"; |
---|
2744 | list L = sent2lplace(t); |
---|
2745 | L; |
---|
2746 | poly p = strList2poly(L); |
---|
2747 | p; |
---|
2748 | } |
---|
2749 | |
---|
2750 | static proc file2lplace(string fname) |
---|
2751 | "USAGE: file2lplace(fnm); fnm a string |
---|
2752 | RETURN: ideal |
---|
2753 | PURPOSE: convert the contents of the file fnm into ideal of polynomials in free algebra |
---|
2754 | EXAMPLE: example file2lplace; shows examples |
---|
2755 | " |
---|
2756 | { |
---|
2757 | // format: from the usual string to letterplace |
---|
2758 | string s = read(fname); |
---|
2759 | // assume: file is a comma-sep list of polys |
---|
2760 | // the vars are declared before |
---|
2761 | // the file ends with ";" |
---|
2762 | string t; int i; |
---|
2763 | ideal I; |
---|
2764 | list tst; |
---|
2765 | while (s!="") |
---|
2766 | { |
---|
2767 | i = find(s,","); |
---|
2768 | "i"; i; |
---|
2769 | if (i==0) |
---|
2770 | { |
---|
2771 | i = find(s,";"); |
---|
2772 | if (i==0) |
---|
2773 | { |
---|
2774 | // no ; ?? |
---|
2775 | "no colon or semicolon found anymore"; |
---|
2776 | return(I); |
---|
2777 | } |
---|
2778 | // no "," but ";" on the i-th place |
---|
2779 | t = s[1..i-1]; |
---|
2780 | s = ""; |
---|
2781 | "processing: "; t; |
---|
2782 | tst = sent2lplace(t); |
---|
2783 | tst; |
---|
2784 | I = I, strList2poly(tst); |
---|
2785 | return(I); |
---|
2786 | } |
---|
2787 | // here i !=0 |
---|
2788 | t = s[1..i-1]; |
---|
2789 | s = s[i+1..size(s)]; |
---|
2790 | "processing: "; t; |
---|
2791 | tst = sent2lplace(t); |
---|
2792 | tst; |
---|
2793 | I = I, strList2poly(tst); |
---|
2794 | } |
---|
2795 | return(I); |
---|
2796 | } |
---|
2797 | example |
---|
2798 | { |
---|
2799 | "EXAMPLE:"; echo = 2; |
---|
2800 | ring r =0,(x,y,z,t),dp; |
---|
2801 | def A = makeLetterplaceRing(4); |
---|
2802 | setring A; |
---|
2803 | string fn = "myfile"; |
---|
2804 | string s1 = "z*y*y*y - 3*y*z*x*y + 3*y*y*z*y - y*x*y*z,"; |
---|
2805 | string s2 = "-2*y*x*y*z + y*y*z*z - z*z*y*y + 2*z*y*z*y,"; |
---|
2806 | 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;"; |
---|
2807 | write(":w "+fn,s1); write(":a "+fn,s2); write(":a "+fn,s3); |
---|
2808 | read(fn); |
---|
2809 | ideal I = file2lplace(fn); |
---|
2810 | I; |
---|
2811 | } |
---|
2812 | |
---|
2813 | /* EXAMPLES AGAIN: |
---|
2814 | //static proc get_ls3nilp() |
---|
2815 | { |
---|
2816 | //first app of file2lplace |
---|
2817 | ring r =0,(x,y,z,t),dp; |
---|
2818 | int d = 10; |
---|
2819 | def A = makeLetterplaceRing(d); |
---|
2820 | setring A; |
---|
2821 | ideal I = file2lplace("./ls3nilp.bg"); |
---|
2822 | // and now test the correctness: go back from lplace to strings |
---|
2823 | lp2lstr(I,r); |
---|
2824 | setring r; |
---|
2825 | lst2str(@LN,1); // agree! |
---|
2826 | } |
---|
2827 | |
---|
2828 | //static proc doc_example() |
---|
2829 | { |
---|
2830 | LIB "freegb.lib"; |
---|
2831 | ring r = 0,(x,y,z),dp; |
---|
2832 | int d =4; // degree bound |
---|
2833 | def R = makeLetterplaceRing(d); |
---|
2834 | setring R; |
---|
2835 | 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); |
---|
2836 | option(redSB);option(redTail); |
---|
2837 | ideal J = system("freegb",I,d,nvars(r)); |
---|
2838 | J; |
---|
2839 | // visualization: |
---|
2840 | lp2lstr(J,r); // export an object called @LN to the ring r |
---|
2841 | setring r; // change to the ring r |
---|
2842 | lst2str(@LN,1); // output the strings |
---|
2843 | } |
---|
2844 | |
---|
2845 | */ |
---|
2846 | |
---|
2847 | // TODO: |
---|
2848 | // multiply two letterplace polynomials, lpMult: done |
---|
2849 | // reduction/ Normalform? needs kernel stuff |
---|
2850 | |
---|
2851 | proc lpMult(poly f, poly g) |
---|
2852 | "USAGE: lpMult(f,g); f,g letterplace polynomials |
---|
2853 | RETURN: poly |
---|
2854 | ASSUME: basering has a letterplace ring structure |
---|
2855 | PURPOSE: compute the letterplace form of f*g |
---|
2856 | EXAMPLE: example lpMult; shows examples |
---|
2857 | " |
---|
2858 | { |
---|
2859 | if (lpAssumeViolation()) |
---|
2860 | { |
---|
2861 | ERROR("Incomplete Letterplace structure on the basering!"); |
---|
2862 | } |
---|
2863 | int sf = deg(f); |
---|
2864 | int sg = deg(g); |
---|
2865 | int uptodeg = attrib(basering, "uptodeg"); |
---|
2866 | if (sf+sg > uptodeg) |
---|
2867 | { |
---|
2868 | ERROR("degree bound violated by the product!"); |
---|
2869 | } |
---|
2870 | // if (sf>1) { sf = sf -1; } |
---|
2871 | poly v = f*shiftPoly(g,sf); |
---|
2872 | return(v); |
---|
2873 | } |
---|
2874 | example |
---|
2875 | { |
---|
2876 | "EXAMPLE:"; echo = 2; |
---|
2877 | // define a ring in letterplace form as follows: |
---|
2878 | ring r = 0,(x(1),y(1),x(2),y(2),x(3),y(3),x(4),y(4)),dp; |
---|
2879 | def R = setLetterplaceAttributes(r,4,2); // supply R with letterplace structure |
---|
2880 | setring R; |
---|
2881 | poly a = x(1)*y(2); poly b = y(1); |
---|
2882 | lpMult(b,a); |
---|
2883 | lpMult(a,b); |
---|
2884 | } |
---|
2885 | |
---|
2886 | proc lpPower(poly f, int n) |
---|
2887 | "USAGE: lpPower(f,n); f letterplace polynomial, int n |
---|
2888 | RETURN: poly |
---|
2889 | ASSUME: basering has a letterplace ring structure |
---|
2890 | PURPOSE: compute the letterplace form of f^n |
---|
2891 | EXAMPLE: example lpPower; shows examples |
---|
2892 | " |
---|
2893 | { |
---|
2894 | if (n<0) { ERROR("the power must be a natural number!"); } |
---|
2895 | if (n==0) { return(poly(1)); } |
---|
2896 | if (n==1) { return(f); } |
---|
2897 | int i; |
---|
2898 | poly p = 1; |
---|
2899 | for(i=1; i<= n; i++) |
---|
2900 | { |
---|
2901 | p = lpMult(p,f); |
---|
2902 | } |
---|
2903 | return(p); |
---|
2904 | } |
---|
2905 | example |
---|
2906 | { |
---|
2907 | "EXAMPLE:"; echo = 2; |
---|
2908 | // define a ring in letterplace form as follows: |
---|
2909 | ring r = 0,(x(1),y(1),x(2),y(2),x(3),y(3),x(4),y(4)),dp; |
---|
2910 | def R = setLetterplaceAttributes(r,4,2); // supply R with letterplace structure |
---|
2911 | setring R; |
---|
2912 | poly a = x(1)*y(2); poly b = y(1); |
---|
2913 | lpPower(a,2); |
---|
2914 | lpPower(b,4); |
---|
2915 | } |
---|
2916 | |
---|
2917 | // under development for Roberto |
---|
2918 | proc extractLinearPart(module M) |
---|
2919 | { |
---|
2920 | /* returns vectors from a module whose max leadexp is 1 */ |
---|
2921 | /* does not take place nonlinearity into account yet */ |
---|
2922 | /* use rather kernel function isinV to get really nonlinear things */ |
---|
2923 | int i; int s = ncols(M); |
---|
2924 | int answer = 1; |
---|
2925 | vector v; module Ret; |
---|
2926 | for(i=1; i<=s; i++) |
---|
2927 | { |
---|
2928 | if ( isLinearVector(M[i]) ) |
---|
2929 | { |
---|
2930 | Ret = Ret, M[i]; |
---|
2931 | } |
---|
2932 | } |
---|
2933 | Ret = simplify(Ret,2); |
---|
2934 | return(Ret); |
---|
2935 | } |
---|
2936 | |
---|
2937 | // under development for Roberto |
---|
2938 | proc isLinearVector(vector v) |
---|
2939 | { |
---|
2940 | /* returns true iff max leadexp is 1 */ |
---|
2941 | int i,j,k; |
---|
2942 | intvec w; |
---|
2943 | int s = size(v); |
---|
2944 | poly p; |
---|
2945 | int answer = 1; |
---|
2946 | for(i=1; i<=s; i++) |
---|
2947 | { |
---|
2948 | p = v[i]; |
---|
2949 | while (p != 0) |
---|
2950 | { |
---|
2951 | w = leadexp(p); |
---|
2952 | j = Max(w); |
---|
2953 | if (j >=2) |
---|
2954 | { |
---|
2955 | answer = 0; |
---|
2956 | return(answer); |
---|
2957 | } |
---|
2958 | p = p-lead(p); |
---|
2959 | } |
---|
2960 | } |
---|
2961 | return(answer); |
---|
2962 | } |
---|
2963 | |
---|
2964 | |
---|
2965 | // // the following is to determine a shift of a mono/poly from the |
---|
2966 | // // interface |
---|
2967 | |
---|
2968 | // proc whichshift(poly p, int numvars) |
---|
2969 | // { |
---|
2970 | // // numvars = number of vars of the orig free algebra |
---|
2971 | // // assume: we are in the letterplace ring |
---|
2972 | // // takes monomial on the input |
---|
2973 | // poly q = lead(p); |
---|
2974 | // intvec v = leadexp(v); |
---|
2975 | // if (v==0) { return(int(0)); } |
---|
2976 | // int sv = size(v); |
---|
2977 | // int i=1; |
---|
2978 | // while ( (v[i]==0) && (i<sv) ) { i++; } |
---|
2979 | // i = sv div i; |
---|
2980 | // return(i); |
---|
2981 | // } |
---|
2982 | |
---|
2983 | |
---|
2984 | |
---|
2985 | // LIB "qhmoduli.lib"; |
---|
2986 | // proc polyshift(poly p, int numvars) |
---|
2987 | // { |
---|
2988 | // poly q = p; int i = 0; |
---|
2989 | // while (q!=0) |
---|
2990 | // { |
---|
2991 | // i = Max(i, whichshift(q,numvars)); |
---|
2992 | // q = q - lead(q); |
---|
2993 | // } |
---|
2994 | // return(q); |
---|
2995 | // } |
---|
2996 | |
---|
2997 | static proc lpAssumeViolation() |
---|
2998 | { |
---|
2999 | // checks whether the global vars |
---|
3000 | // uptodeg and lV are defined |
---|
3001 | // returns Boolean : yes/no [for assume violation] |
---|
3002 | def lpring = attrib(basering,"isLetterplaceRing"); |
---|
3003 | if ( typeof(lpring)!="int" ) |
---|
3004 | { |
---|
3005 | // if ( typeof(lpring)=="string" ) ?? |
---|
3006 | // basering is NOT lp Ring |
---|
3007 | |
---|
3008 | return(1); |
---|
3009 | } |
---|
3010 | def uptodeg = attrib(basering,"uptodeg"); |
---|
3011 | if ( typeof(uptodeg)!="int" ) |
---|
3012 | { |
---|
3013 | return(1); |
---|
3014 | } |
---|
3015 | def lV = attrib(basering,"lV"); |
---|
3016 | if ( typeof(lV)!="int" ) |
---|
3017 | { |
---|
3018 | return(1); |
---|
3019 | } |
---|
3020 | // int i = ( defined(uptodeg) && (defined(lV)) ); |
---|
3021 | // return ( !i ); |
---|
3022 | return(0); |
---|
3023 | } |
---|