1 | ///////////////////////////////////////////////////////////////////////////// |
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2 | version="version freegb.lib 4.1.1.0 Dec_2017 "; // $Id$ |
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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 approach |
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6 | AUTHORS: Viktor Levandovskyy, viktor.levandovskyy@math.rwth-aachen.de |
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7 | Grischa Studzinski, grischa.studzinski@math.rwth-aachen.de |
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8 | |
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9 | OVERVIEW: For the theory, see chapter 'Letterplace' in the @sc{Singular} Manual |
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10 | |
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11 | Support: Joint projects LE 2697/2-1 and KR 1907/3-1 of the Priority Programme SPP 1489: |
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12 | 'Algorithmische und Experimentelle Methoden in Algebra, Geometrie und Zahlentheorie' |
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13 | of the German DFG |
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14 | and Project II.6 of the transregional collaborative research centre |
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15 | SFB-TRR 195 'Symbolic Tools in Mathematics and their Application' of the German DFG |
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16 | |
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17 | PROCEDURES: |
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18 | makeLetterplaceRing(d); creates a ring with d blocks of shifted original variables |
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19 | letplaceGBasis(I); computes two-sided Groebner basis of a letterplace ideal I up to a degree bound |
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20 | lpNF(f,I); two-sided normal form of f with respect to ideal I |
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21 | setLetterplaceAttributes(R,d,b); supplies ring R with the letterplace structure |
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22 | freeGBasis(L, n); computes two-sided Groebner basis of an ideal, encoded via list L, up to degree n |
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23 | |
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24 | lpMult(f,g); letterplace multiplication of letterplace polynomials |
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25 | shiftPoly(p,i); compute the i-th shift of letterplace polynomial p |
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26 | lpPower(f,n); natural power of a letterplace polynomial |
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27 | lieBracket(a,b[, N]); compute Lie bracket ab-ba of two letterplace polynomials |
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28 | |
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29 | lp2lstr(K, s); convert a letterplace ideal into a list of modules |
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30 | lst2str(L[, n]); convert a list (of modules) into polynomials in free algebra via strings |
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31 | mod2str(M[, n]); convert a module into a polynomial in free algebra via strings |
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32 | vct2str(M[, n]); convert a vector into a word in free algebra |
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33 | |
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34 | serreRelations(A,z); compute the homogeneous part of Serre's relations associated to a generalized Cartan matrix A |
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35 | 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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36 | isVar(p); check whether p is a power of a single variable |
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37 | ademRelations(i,j); compute the ideal of Adem relations for i<2j in char 0 |
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38 | |
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39 | lpPrint(ideal I, def @r); print a letterplace ideal as an easily readable string |
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40 | |
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41 | SEE ALSO: fpadim_lib, LETTERPLACE |
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42 | "; |
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43 | |
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44 | // this library computes two-sided GB of an ideal |
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45 | // in a free associative algebra |
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46 | |
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47 | // a monomial is encoded via a vector V |
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48 | // where V[1] = coefficient |
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49 | // V[1+i] = the corresponding symbol |
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50 | |
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51 | LIB "qhmoduli.lib"; // for Max |
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52 | LIB "bfun.lib"; // for inForm |
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53 | |
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54 | proc tstfreegb() |
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55 | { |
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56 | /* tests all procs for consistency */ |
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57 | /* adding the new proc, add it here */ |
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58 | example makeLetterplaceRing; |
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59 | example letplaceGBasis; |
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60 | example lpNF; |
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61 | example freeGBasis; |
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62 | example setLetterplaceAttributes; |
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63 | /* secondary */ |
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64 | example lpMult; |
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65 | example shiftPoly; |
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66 | example lpPower; |
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67 | example lieBracket; |
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68 | example lp2lstr; |
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69 | example lst2str; |
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70 | example mod2str; |
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71 | example vct2str; |
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72 | example serreRelations; |
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73 | example fullSerreRelations; |
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74 | example isVar; |
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75 | example ademRelations; |
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76 | } |
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77 | |
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78 | proc setLetterplaceAttributes(def R, int uptodeg, int lV) |
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79 | "USAGE: setLetterplaceAttributes(R, d, b); R a ring, b,d integers |
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80 | RETURN: ring with special attributes set |
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81 | PURPOSE: sets attributes for a letterplace ring: |
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82 | @* 'isLetterplaceRing' = true, 'uptodeg' = d, 'lV' = b, where |
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83 | @* 'uptodeg' stands for the degree bound, |
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84 | @* 'lV' for the number of variables in the block 0. |
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85 | NOTE: Activate the resulting ring by using @code{setring} |
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86 | " |
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87 | { |
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88 | if (uptodeg*lV != nvars(R)) |
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89 | { |
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90 | ERROR("uptodeg and lV do not agree on the basering!"); |
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91 | } |
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92 | |
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93 | // Set letterplace-specific attributes for the output ring! |
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94 | attrib(R, "uptodeg", uptodeg); |
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95 | attrib(R, "lV", lV); |
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96 | attrib(R, "isLetterplaceRing", 1); |
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97 | return (R); |
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98 | } |
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99 | example |
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100 | { |
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101 | "EXAMPLE:"; echo = 2; |
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102 | ring r = 0,(x(1),y(1),x(2),y(2),x(3),y(3),x(4),y(4)),dp; |
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103 | def R = setLetterplaceAttributes(r, 4, 2); setring R; |
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104 | attrib(R,"isLetterplaceRing"); |
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105 | lieBracket(x(1),y(1),2); |
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106 | } |
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107 | |
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108 | |
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109 | // obsolete? |
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110 | |
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111 | static proc lshift(module M, int s, string varing, def lpring) |
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112 | { |
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113 | // FINALLY IMPLEMENTED AS A PART OT THE CODE |
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114 | // shifts a polynomial from the ring R to s positions |
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115 | // M lives in varing, the result in lpring |
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116 | // to be run from varing |
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117 | int i, j, k, sm, sv; |
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118 | vector v; |
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119 | // execute("setring "+lpring); |
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120 | setring lpring; |
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121 | poly @@p; |
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122 | ideal I; |
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123 | execute("setring "+varing); |
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124 | sm = ncols(M); |
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125 | for (i=1; i<=s; i++) |
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126 | { |
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127 | // modules, e.g. free polynomials |
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128 | for (j=1; j<=sm; j++) |
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129 | { |
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130 | //vectors, e.g. free monomials |
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131 | v = M[j]; |
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132 | sv = size(v); |
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133 | sp = "@@p = @@p + "; |
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134 | for (k=2; k<=sv; k++) |
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135 | { |
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136 | sp = sp + string(v[k])+"("+string(k-1+s)+")*"; |
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137 | } |
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138 | sp = sp + string(v[1])+";"; // coef; |
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139 | setring lpring; |
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140 | // execute("setring "+lpring); |
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141 | execute(sp); |
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142 | execute("setring "+varing); |
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143 | } |
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144 | setring lpring; |
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145 | // execute("setring "+lpring); |
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146 | I = I,@@p; |
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147 | @@p = 0; |
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148 | } |
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149 | setring lpring; |
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150 | //execute("setring "+lpring); |
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151 | export(I); |
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152 | // setring varing; |
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153 | execute("setring "+varing); |
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154 | } |
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155 | |
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156 | static proc skip0(vector v) |
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157 | { |
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158 | // skips zeros in a vector, producing another vector |
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159 | if ( (v[1]==0) || (v==0) ) { return(vector(0)); } |
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160 | int sv = nrows(v); |
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161 | int sw = size(v); |
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162 | if (sv == sw) |
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163 | { |
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164 | return(v); |
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165 | } |
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166 | int i; |
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167 | int j=1; |
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168 | vector w; |
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169 | for (i=1; i<=sv; i++) |
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170 | { |
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171 | if (v[i] != 0) |
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172 | { |
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173 | w = w + v[i]*gen(j); |
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174 | j++; |
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175 | } |
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176 | } |
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177 | return(w); |
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178 | } |
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179 | |
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180 | proc lst2str(list L, list #) |
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181 | "USAGE: lst2str(L[,n]); L a list of modules, n an optional integer |
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182 | RETURN: list (of strings) |
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183 | PURPOSE: convert a list (of modules) into polynomials in free algebra |
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184 | EXAMPLE: example lst2str; shows examples |
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185 | NOTE: if an optional integer is not 0, stars signs are used in multiplication |
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186 | " |
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187 | { |
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188 | // returns a list of strings |
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189 | // being sentences in words built from L |
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190 | // if #[1] = 1, use * between generators |
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191 | int useStar = 0; |
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192 | if ( size(#)>0 ) |
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193 | { |
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194 | if ( typeof(#[1]) != "int") |
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195 | { |
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196 | ERROR("Second argument of type int expected"); |
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197 | } |
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198 | if (#[1]) |
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199 | { |
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200 | useStar = 1; |
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201 | } |
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202 | } |
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203 | int i; |
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204 | int s = size(L); |
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205 | if (s<1) { return(list(""));} |
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206 | list N; |
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207 | for(i=1; i<=s; i++) |
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208 | { |
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209 | if ((typeof(L[i]) == "module") || (typeof(L[i]) == "matrix") ) |
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210 | { |
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211 | N[i] = mod2str(L[i],useStar); |
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212 | } |
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213 | else |
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214 | { |
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215 | "module or matrix expected in the list"; |
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216 | return(N); |
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217 | } |
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218 | } |
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219 | return(N); |
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220 | } |
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221 | example |
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222 | { |
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223 | "EXAMPLE:"; echo = 2; |
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224 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
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225 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
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226 | module N = [1,x,y,x,y],[-2,y,x,y,x],[6,x,y,y,x,y]; |
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227 | list L; L[1] = M; L[2] = N; |
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228 | lst2str(L); |
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229 | lst2str(L[1],1); |
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230 | } |
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231 | |
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232 | |
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233 | proc mod2str(module M, list #) |
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234 | "USAGE: mod2str(M[,n]); M a module, n an optional integer |
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235 | RETURN: string |
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236 | PURPOSE: convert a module into a polynomial in free algebra |
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237 | EXAMPLE: example mod2str; shows examples |
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238 | NOTE: if an optional integer is not 0, stars signs are used in multiplication |
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239 | " |
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240 | { |
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241 | if (size(M)==0) { return(""); } |
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242 | // returns a string |
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243 | // a sentence in words built from M |
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244 | // if #[1] = 1, use * between generators |
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245 | int useStar = 0; |
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246 | if ( size(#)>0 ) |
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247 | { |
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248 | if ( typeof(#[1]) != "int") |
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249 | { |
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250 | ERROR("Second argument of type int expected"); |
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251 | } |
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252 | if (#[1]) |
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253 | { |
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254 | useStar = 1; |
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255 | } |
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256 | } |
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257 | int i; |
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258 | int s = ncols(M); |
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259 | string t; |
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260 | string mp; |
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261 | for(i=1; i<=s; i++) |
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262 | { |
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263 | mp = vct2str(M[i],useStar); |
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264 | if (mp[1] == "-") |
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265 | { |
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266 | t = t + mp; |
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267 | } |
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268 | else |
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269 | { |
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270 | if (mp != "") |
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271 | { |
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272 | t = t + "+" + mp; |
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273 | } |
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274 | } |
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275 | } |
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276 | if (t[1]=="+") |
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277 | { |
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278 | t = t[2..size(t)]; // remove first "+" |
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279 | } |
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280 | return(t); |
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281 | } |
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282 | example |
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283 | { |
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284 | "EXAMPLE:"; echo = 2; |
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285 | ring r = 0,(x,y,z),(dp); |
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286 | module M = [1,x,y,x,y],[-2,y,x,y,x],[6,x,y,y,x,y]; |
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287 | mod2str(M); |
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288 | mod2str(M,1); |
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289 | } |
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290 | |
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291 | proc vct2str(vector v, list #) |
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292 | "USAGE: vct2str(v[,n]); v a vector, n an optional integer |
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293 | RETURN: string |
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294 | PURPOSE: convert a vector into a word in free algebra |
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295 | EXAMPLE: example vct2str; shows examples |
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296 | NOTE: if an optional integer is not 0, stars signs are used in multiplication |
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297 | " |
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298 | { |
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299 | if (v==0) { return(""); } |
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300 | // if #[1] = 1, use * between generators |
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301 | int useStar = 0; |
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302 | if ( size(#)>0 ) |
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303 | { |
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304 | if (#[1]) |
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305 | { |
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306 | useStar = 1; |
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307 | } |
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308 | } |
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309 | int ppl = printlevel-voice+2; |
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310 | // for a word, encoded by v |
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311 | // produces a string for it |
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312 | v = skip0(v); |
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313 | if (v==0) { return(string(""));} |
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314 | number cf = leadcoef(v[1]); |
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315 | int s = size(v); |
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316 | string vs,vv,vp,err; |
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317 | int i,j,p,q; |
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318 | for (i=1; i<=s-1; i++) |
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319 | { |
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320 | p = isVar(v[i+1]); |
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321 | if (p==0) |
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322 | { |
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323 | err = "Error: monomial expected at nonzero position " + string(i+1); |
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324 | ERROR(err+" in vct2str"); |
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325 | // dbprint(ppl,err); |
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326 | // return("_"); |
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327 | } |
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328 | if (p==1) |
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329 | { |
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330 | if (useStar && (size(vs) >0)) { vs = vs + "*"; } |
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331 | vs = vs + string(v[i+1]); |
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332 | } |
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333 | else //power |
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334 | { |
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335 | vv = string(v[i+1]); |
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336 | q = find(vv,"^"); |
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337 | if (q==0) |
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338 | { |
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339 | q = find(vv,string(p)); |
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340 | if (q==0) |
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341 | { |
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342 | err = "error in find for string "+vv; |
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343 | dbprint(ppl,err); |
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344 | return("_"); |
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345 | } |
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346 | } |
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347 | // q>0 |
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348 | vp = vv[1..q-1]; |
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349 | for(j=1;j<=p;j++) |
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350 | { |
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351 | if (useStar && (size(vs) >0)) { vs = vs + "*"; } |
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352 | vs = vs + vp; |
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353 | } |
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354 | } |
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355 | } |
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356 | string scf; |
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357 | if (cf == -1) |
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358 | { |
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359 | scf = "-"; |
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360 | } |
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361 | else |
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362 | { |
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363 | scf = string(cf); |
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364 | if ( (cf == 1) && (size(vs)>0) ) |
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365 | { |
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366 | scf = ""; |
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367 | } |
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368 | } |
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369 | if (useStar && (size(scf) >0) && (scf!="-") ) { scf = scf + "*"; } |
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370 | vs = scf + vs; |
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371 | return(vs); |
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372 | } |
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373 | example |
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374 | { |
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375 | "EXAMPLE:"; echo = 2; |
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376 | ring r = (0,a),(x,y3,z(1)),dp; |
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377 | vector v = [-7,x,y3^4,x2,z(1)^3]; |
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378 | vct2str(v); |
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379 | vct2str(v,1); |
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380 | vector w = [-7a^5+6a,x,y3,y3,x,z(1),z(1)]; |
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381 | vct2str(w); |
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382 | vct2str(w,1); |
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383 | } |
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384 | |
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385 | proc isVar(poly p) |
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386 | "USAGE: isVar(p); poly p |
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387 | RETURN: int |
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388 | PURPOSE: check, whether leading monomial of p is a power of a single variable |
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389 | @* from the basering. Returns the exponent or 0 if p is multivariate. |
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390 | EXAMPLE: example isVar; shows examples |
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391 | " |
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392 | { |
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393 | // checks whether p is a variable indeed |
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394 | // if it's a power of a variable, returns the power |
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395 | if (p==0) { return(0); } //"p=0"; |
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396 | poly q = leadmonom(p); |
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397 | if ( (p-lead(p)) !=0 ) { return(0); } // "p-lm(p)>0"; |
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398 | intvec v = leadexp(p); |
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399 | int s = size(v); |
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400 | int i=1; |
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401 | int cnt = 0; |
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402 | int pwr = 0; |
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403 | for (i=1; i<=s; i++) |
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404 | { |
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405 | if (v[i] != 0) |
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406 | { |
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407 | cnt++; |
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408 | pwr = v[i]; |
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409 | } |
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410 | } |
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411 | // "cnt:"; cnt; |
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412 | if (cnt==1) { return(pwr); } |
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413 | else { return(0); } |
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414 | } |
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415 | example |
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416 | { |
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417 | "EXAMPLE:"; echo = 2; |
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418 | ring r = 0,(x,y),dp; |
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419 | poly f = xy+1; |
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420 | isVar(f); |
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421 | poly g = y^3; |
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422 | isVar(g); |
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423 | poly h = 7*x^3; |
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424 | isVar(h); |
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425 | poly i = 1; |
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426 | isVar(i); |
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427 | } |
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428 | |
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429 | // new conversion routines |
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430 | |
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431 | static proc id2words(ideal I, int d) |
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432 | { |
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433 | // NOT FINISHED |
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434 | // input: ideal I of polys in letter-place notation |
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435 | // in the ring with d real vars |
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436 | // output: the list of strings: associative words |
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437 | // extract names of vars |
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438 | int i,m,n; string s; string place = "(1)"; |
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439 | list lv; |
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440 | for(i=1; i<=d; i++) |
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441 | { |
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442 | s = string(var(i)); |
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443 | // get rid of place |
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444 | n = find(s, place); |
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445 | if (n>0) |
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446 | { |
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447 | s = s[1..n-1]; |
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448 | } |
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449 | lv[i] = s; |
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450 | } |
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451 | poly p,q; |
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452 | for (i=1; i<=ncols(I); i++) |
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453 | { |
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454 | if (I[i] != 0) |
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455 | { |
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456 | p = I[i]; |
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457 | while (p!=0) |
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458 | { |
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459 | q = leadmonom(p); |
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460 | } |
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461 | } |
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462 | } |
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463 | |
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464 | return(lv); |
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465 | } |
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466 | example |
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467 | { |
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468 | "EXAMPLE:"; echo = 2; |
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469 | ring r = 0,(x(1),y(1),z(1),x(2),y(2),z(2)),dp; |
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470 | ideal I = x(1)*y(2) -z(1)*x(2); |
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471 | id2words(I,3); |
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472 | } |
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473 | |
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474 | static proc mono2word(poly p, int d) |
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475 | { |
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476 | } |
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477 | |
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478 | proc letplaceGBasis(ideal I) |
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479 | "USAGE: letplaceGBasis(I); I an ideal |
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480 | RETURN: ideal |
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481 | ASSUME: basering is a Letterplace ring, an ideal consists of Letterplace |
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482 | @* polynomials |
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483 | PURPOSE: compute the two-sided Groebner basis of an ideal I via Letterplace |
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484 | @* algorithm |
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485 | NOTE: the degree bound for this computation is read off the letterplace |
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486 | @* structure of basering |
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487 | EXAMPLE: example letplaceGBasis; shows examples |
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488 | " |
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489 | { |
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490 | if (lpAssumeViolation()) |
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491 | { |
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492 | ERROR("Incomplete Letterplace structure on the basering!"); |
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493 | } |
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494 | int ppl = printlevel-voice+2; |
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495 | def save = basering; |
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496 | // assumes of the ring have been checked |
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497 | // run the computation - it will test assumes on the ideal |
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498 | int uptodeg = attrib(save,"uptodeg"); |
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499 | int lV = attrib(save,"lV"); |
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500 | dbprint(ppl,"start computing GB"); |
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501 | ideal J = system("freegb",I,uptodeg,lV); |
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502 | dbprint(ppl,"finished computing GB"); |
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503 | dbprint(ppl-1,"the result is:"); |
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504 | dbprint(ppl-1,J); |
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505 | return(J); |
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506 | } |
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507 | example |
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508 | { |
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509 | "EXAMPLE:"; echo = 2; |
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510 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
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511 | int degree_bound = 5; |
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512 | def R = makeLetterplaceRing(5); |
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513 | setring R; |
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514 | ideal I = -x(1)*y(2)-7*y(1)*y(2)+3*x(1)*x(2), x(1)*y(2)*x(3)-y(1)*x(2)*y(3); |
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515 | ideal J = letplaceGBasis(I); |
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516 | J; |
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517 | // now transfom letterplace polynomials into strings of words |
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518 | lp2lstr(J,r); // export an object called @code{@LN} to the ring r |
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519 | setring r; // change to the ring r |
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520 | lst2str(@LN,1); |
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521 | } |
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522 | |
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523 | // given the element -7xy^2x, it is represented as [-7,x,y^2,x] or as [-7,x,y,y,x] |
---|
524 | // use the orig ord on (x,y,z) and expand it blockwise to (x(i),y(i),z(i)) |
---|
525 | |
---|
526 | // the correspondences: |
---|
527 | // monomial in K<x,y,z> <<--->> vector in R |
---|
528 | // polynomial in K<x,y,z> <<--->> list of vectors (matrix/module) in R |
---|
529 | // ideal in K<x,y,z> <<--->> list of matrices/modules in R |
---|
530 | |
---|
531 | |
---|
532 | // 1. form a new ring |
---|
533 | // 2. NOP |
---|
534 | // 3. compute GB -> with the kernel stuff |
---|
535 | // 4. skip shifted elts (check that no such exist?) |
---|
536 | // 5. go back to orig vars, produce strings/modules |
---|
537 | // 6. return the result |
---|
538 | |
---|
539 | proc freeGBasis(list LM, int d) |
---|
540 | "USAGE: freeGBasis(L, d); L a list of modules, d an integer |
---|
541 | RETURN: ring |
---|
542 | ASSUME: L has a special form. Namely, it is a list of modules, where |
---|
543 | |
---|
544 | - each generator of every module stands for a monomial times coefficient in |
---|
545 | @* free algebra, |
---|
546 | |
---|
547 | - in such a vector generator, the 1st entry is a nonzero coefficient from the |
---|
548 | @* ground field |
---|
549 | |
---|
550 | - and each next entry hosts a variable from the basering. |
---|
551 | PURPOSE: compute the two-sided Groebner basis of an ideal, encoded by L |
---|
552 | @* in the free associative algebra, up to degree d |
---|
553 | NOTE: Apply @code{lst2str} to the output in order to obtain a better readable |
---|
554 | @* presentation |
---|
555 | EXAMPLE: example freeGBasis; shows examples |
---|
556 | " |
---|
557 | { |
---|
558 | // d = up to degree, will be shifted to d+1 |
---|
559 | if (d<1) {"bad d"; return(0);} |
---|
560 | |
---|
561 | int ppl = printlevel-voice+2; |
---|
562 | string err = ""; |
---|
563 | |
---|
564 | int i,j,s; |
---|
565 | def save = basering; |
---|
566 | // determine max no of places in the input |
---|
567 | int slm = size(LM); // numbers of polys in the ideal |
---|
568 | int sm; |
---|
569 | intvec iv; |
---|
570 | module M; |
---|
571 | for (i=1; i<=slm; i++) |
---|
572 | { |
---|
573 | // modules, e.g. free polynomials |
---|
574 | M = LM[i]; |
---|
575 | sm = ncols(M); |
---|
576 | for (j=1; j<=sm; j++) |
---|
577 | { |
---|
578 | //vectors, e.g. free monomials |
---|
579 | iv = iv, size(M[j])-1; // 1 place is reserved by the coeff |
---|
580 | } |
---|
581 | } |
---|
582 | int D = Max(iv); // max size of input words |
---|
583 | if (d<D) {"bad d"; return(LM);} |
---|
584 | D = D + d-1; |
---|
585 | // D = d; |
---|
586 | list LR = ringlist(save); |
---|
587 | list L, tmp; |
---|
588 | L[1] = LR[1]; // ground field |
---|
589 | L[4] = LR[4]; // quotient ideal |
---|
590 | tmp = LR[2]; // varnames |
---|
591 | s = size(LR[2]); |
---|
592 | for (i=1; i<=D; i++) |
---|
593 | { |
---|
594 | for (j=1; j<=s; j++) |
---|
595 | { |
---|
596 | tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")"; |
---|
597 | } |
---|
598 | } |
---|
599 | for (i=1; i<=s; i++) |
---|
600 | { |
---|
601 | tmp[i] = string(tmp[i])+"("+string(1)+")"; |
---|
602 | } |
---|
603 | L[2] = tmp; |
---|
604 | list OrigNames = LR[2]; |
---|
605 | // ordering: d blocks of the ord on r |
---|
606 | // try to get whether the ord on r is blockord itself |
---|
607 | s = size(LR[3]); |
---|
608 | if (s==2) |
---|
609 | { |
---|
610 | // not a blockord, 1 block + module ord |
---|
611 | tmp = LR[3][s]; // module ord |
---|
612 | for (i=1; i<=D; i++) |
---|
613 | { |
---|
614 | LR[3][s-1+i] = LR[3][1]; |
---|
615 | } |
---|
616 | LR[3][s+D] = tmp; |
---|
617 | } |
---|
618 | if (s>2) |
---|
619 | { |
---|
620 | // there are s-1 blocks |
---|
621 | int nb = s-1; |
---|
622 | tmp = LR[3][s]; // module ord |
---|
623 | for (i=1; i<=D; i++) |
---|
624 | { |
---|
625 | for (j=1; j<=nb; j++) |
---|
626 | { |
---|
627 | LR[3][i*nb+j] = LR[3][j]; |
---|
628 | } |
---|
629 | } |
---|
630 | // size(LR[3]); |
---|
631 | LR[3][nb*(D+1)+1] = tmp; |
---|
632 | } |
---|
633 | L[3] = LR[3]; |
---|
634 | def @R = ring(L); |
---|
635 | setring @R; |
---|
636 | ideal I; |
---|
637 | poly @p; |
---|
638 | s = size(OrigNames); |
---|
639 | // "s:";s; |
---|
640 | // convert LM to canonical vectors (no powers) |
---|
641 | setring save; |
---|
642 | kill M; // M was defined earlier |
---|
643 | module M; |
---|
644 | slm = size(LM); // numbers of polys in the ideal |
---|
645 | int sv,k,l; |
---|
646 | vector v; |
---|
647 | // poly p; |
---|
648 | string sp; |
---|
649 | setring @R; |
---|
650 | poly @@p=0; |
---|
651 | setring save; |
---|
652 | for (l=1; l<=slm; l++) |
---|
653 | { |
---|
654 | // modules, e.g. free polynomials |
---|
655 | M = LM[l]; |
---|
656 | sm = ncols(M); // in intvec iv the sizes are stored |
---|
657 | // modules, e.g. free polynomials |
---|
658 | for (j=1; j<=sm; j++) |
---|
659 | { |
---|
660 | //vectors, e.g. free monomials |
---|
661 | v = M[j]; |
---|
662 | sv = size(v); |
---|
663 | // "sv:";sv; |
---|
664 | sp = "@@p = @@p + "; |
---|
665 | for (k=2; k<=sv; k++) |
---|
666 | { |
---|
667 | sp = sp + string(v[k])+"("+string(k-1)+")*"; |
---|
668 | } |
---|
669 | sp = sp + string(v[1])+";"; // coef; |
---|
670 | setring @R; |
---|
671 | execute(sp); |
---|
672 | setring save; |
---|
673 | } |
---|
674 | setring @R; |
---|
675 | // "@@p:"; @@p; |
---|
676 | I = I,@@p; |
---|
677 | @@p = 0; |
---|
678 | setring save; |
---|
679 | } |
---|
680 | kill sp; |
---|
681 | // 3. compute GB |
---|
682 | setring @R; |
---|
683 | dbprint(ppl,"computing GB"); |
---|
684 | ideal J = system("freegb",I,d,nvars(save)); |
---|
685 | // ideal J = slimgb(I); |
---|
686 | dbprint(ppl,J); |
---|
687 | // 4. skip shifted elts |
---|
688 | ideal K = select1(J,1..s); // s = size(OrigNames) |
---|
689 | dbprint(ppl,K); |
---|
690 | dbprint(ppl, "done with GB"); |
---|
691 | // K contains vars x(1),...z(1) = images of originals |
---|
692 | // 5. go back to orig vars, produce strings/modules |
---|
693 | if (K[1] == 0) |
---|
694 | { |
---|
695 | "no reasonable output, GB gives 0"; |
---|
696 | return(0); |
---|
697 | } |
---|
698 | int sk = size(K); |
---|
699 | int sp, sx, a, b; |
---|
700 | intvec x; |
---|
701 | poly p,q; |
---|
702 | poly pn; |
---|
703 | // vars in 'save' |
---|
704 | setring save; |
---|
705 | module N; |
---|
706 | list LN; |
---|
707 | vector V; |
---|
708 | poly pn; |
---|
709 | // test and skip exponents >=2 |
---|
710 | setring @R; |
---|
711 | for(i=1; i<=sk; i++) |
---|
712 | { |
---|
713 | p = K[i]; |
---|
714 | while (p!=0) |
---|
715 | { |
---|
716 | q = lead(p); |
---|
717 | // "processing q:";q; |
---|
718 | x = leadexp(q); |
---|
719 | sx = size(x); |
---|
720 | for(k=1; k<=sx; k++) |
---|
721 | { |
---|
722 | if ( x[k] >= 2 ) |
---|
723 | { |
---|
724 | err = "skip: the value x[k] is " + string(x[k]); |
---|
725 | dbprint(ppl,err); |
---|
726 | // return(0); |
---|
727 | K[i] = 0; |
---|
728 | p = 0; |
---|
729 | q = 0; |
---|
730 | break; |
---|
731 | } |
---|
732 | } |
---|
733 | p = p - q; |
---|
734 | } |
---|
735 | } |
---|
736 | K = simplify(K,2); |
---|
737 | sk = size(K); |
---|
738 | for(i=1; i<=sk; i++) |
---|
739 | { |
---|
740 | // setring save; |
---|
741 | // V = 0; |
---|
742 | setring @R; |
---|
743 | p = K[i]; |
---|
744 | while (p!=0) |
---|
745 | { |
---|
746 | q = lead(p); |
---|
747 | err = "processing q:" + string(q); |
---|
748 | dbprint(ppl,err); |
---|
749 | x = leadexp(q); |
---|
750 | sx = size(x); |
---|
751 | pn = leadcoef(q); |
---|
752 | setring save; |
---|
753 | pn = imap(@R,pn); |
---|
754 | V = V + leadcoef(pn)*gen(1); |
---|
755 | for(k=1; k<=sx; k++) |
---|
756 | { |
---|
757 | if (x[k] ==1) |
---|
758 | { |
---|
759 | a = k div s; // block number=a+1, a!=0 |
---|
760 | b = k % s; // remainder |
---|
761 | // printf("a: %s, b: %s",a,b); |
---|
762 | if (b == 0) |
---|
763 | { |
---|
764 | // that is it's the last var in the block |
---|
765 | b = s; |
---|
766 | a = a-1; |
---|
767 | } |
---|
768 | V = V + var(b)*gen(a+2); |
---|
769 | } |
---|
770 | // else |
---|
771 | // { |
---|
772 | // printf("error: the value x[k] is %s", x[k]); |
---|
773 | // return(0); |
---|
774 | // } |
---|
775 | } |
---|
776 | err = "V: " + string(V); |
---|
777 | dbprint(ppl,err); |
---|
778 | // printf("V: %s", string(V)); |
---|
779 | N = N,V; |
---|
780 | V = 0; |
---|
781 | setring @R; |
---|
782 | p = p - q; |
---|
783 | pn = 0; |
---|
784 | } |
---|
785 | setring save; |
---|
786 | LN[i] = simplify(N,2); |
---|
787 | N = 0; |
---|
788 | } |
---|
789 | setring save; |
---|
790 | return(LN); |
---|
791 | } |
---|
792 | example |
---|
793 | { |
---|
794 | "EXAMPLE:"; echo = 2; |
---|
795 | ring r = 0,(x,y,z),(dp(1),dp(2)); // ring r = 0,(x,y,z),(a(3,0,2), dp(2)); |
---|
796 | module M = [-1,x,y],[-7,y,y],[3,x,x]; // stands for free poly -xy - 7yy - 3xx |
---|
797 | module N = [1,x,y,x],[-1,y,x,y]; // stands for free poly xyx - yxy |
---|
798 | list L; L[1] = M; L[2] = N; // list of modules stands for an ideal in free algebra |
---|
799 | lst2str(L); // list to string conversion of input polynomials |
---|
800 | def U = freeGBasis(L,5); // 5 is the degree bound |
---|
801 | lst2str(U); |
---|
802 | } |
---|
803 | |
---|
804 | static proc crs(list LM, int d) |
---|
805 | "USAGE: crs(L, d); L a list of modules, d an integer |
---|
806 | RETURN: ring |
---|
807 | PURPOSE: create a ring and shift the ideal |
---|
808 | EXAMPLE: example crs; shows examples |
---|
809 | " |
---|
810 | { |
---|
811 | // d = up to degree, will be shifted to d+1 |
---|
812 | if (d<1) {"bad d"; return(0);} |
---|
813 | |
---|
814 | int ppl = printlevel-voice+2; |
---|
815 | string err = ""; |
---|
816 | |
---|
817 | int i,j,s; |
---|
818 | def save = basering; |
---|
819 | // determine max no of places in the input |
---|
820 | int slm = size(LM); // numbers of polys in the ideal |
---|
821 | int sm; |
---|
822 | intvec iv; |
---|
823 | module M; |
---|
824 | for (i=1; i<=slm; i++) |
---|
825 | { |
---|
826 | // modules, e.g. free polynomials |
---|
827 | M = LM[i]; |
---|
828 | sm = ncols(M); |
---|
829 | for (j=1; j<=sm; j++) |
---|
830 | { |
---|
831 | //vectors, e.g. free monomials |
---|
832 | iv = iv, size(M[j])-1; // 1 place is reserved by the coeff |
---|
833 | } |
---|
834 | } |
---|
835 | int D = Max(iv); // max size of input words |
---|
836 | if (d<D) {"bad d"; return(LM);} |
---|
837 | D = D + d-1; |
---|
838 | // D = d; |
---|
839 | list LR = ringlist(save); |
---|
840 | list L, tmp; |
---|
841 | L[1] = LR[1]; // ground field |
---|
842 | L[4] = LR[4]; // quotient ideal |
---|
843 | tmp = LR[2]; // varnames |
---|
844 | s = size(LR[2]); |
---|
845 | for (i=1; i<=D; i++) |
---|
846 | { |
---|
847 | for (j=1; j<=s; j++) |
---|
848 | { |
---|
849 | tmp[i*s+j] = string(tmp[j])+"("+string(i)+")"; |
---|
850 | } |
---|
851 | } |
---|
852 | for (i=1; i<=s; i++) |
---|
853 | { |
---|
854 | tmp[i] = string(tmp[i])+"("+string(0)+")"; |
---|
855 | } |
---|
856 | L[2] = tmp; |
---|
857 | list OrigNames = LR[2]; |
---|
858 | // ordering: d blocks of the ord on r |
---|
859 | // try to get whether the ord on r is blockord itself |
---|
860 | s = size(LR[3]); |
---|
861 | if (s==2) |
---|
862 | { |
---|
863 | // not a blockord, 1 block + module ord |
---|
864 | tmp = LR[3][s]; // module ord |
---|
865 | for (i=1; i<=D; i++) |
---|
866 | { |
---|
867 | LR[3][s-1+i] = LR[3][1]; |
---|
868 | } |
---|
869 | LR[3][s+D] = tmp; |
---|
870 | } |
---|
871 | if (s>2) |
---|
872 | { |
---|
873 | // there are s-1 blocks |
---|
874 | int nb = s-1; |
---|
875 | tmp = LR[3][s]; // module ord |
---|
876 | for (i=1; i<=D; i++) |
---|
877 | { |
---|
878 | for (j=1; j<=nb; j++) |
---|
879 | { |
---|
880 | LR[3][i*nb+j] = LR[3][j]; |
---|
881 | } |
---|
882 | } |
---|
883 | // size(LR[3]); |
---|
884 | LR[3][nb*(D+1)+1] = tmp; |
---|
885 | } |
---|
886 | L[3] = LR[3]; |
---|
887 | def @R = ring(L); |
---|
888 | setring @R; |
---|
889 | ideal I; |
---|
890 | poly @p; |
---|
891 | s = size(OrigNames); |
---|
892 | // "s:";s; |
---|
893 | // convert LM to canonical vectors (no powers) |
---|
894 | setring save; |
---|
895 | kill M; // M was defined earlier |
---|
896 | module M; |
---|
897 | slm = size(LM); // numbers of polys in the ideal |
---|
898 | int sv,k,l; |
---|
899 | vector v; |
---|
900 | // poly p; |
---|
901 | string sp; |
---|
902 | setring @R; |
---|
903 | poly @@p=0; |
---|
904 | setring save; |
---|
905 | for (l=1; l<=slm; l++) |
---|
906 | { |
---|
907 | // modules, e.g. free polynomials |
---|
908 | M = LM[l]; |
---|
909 | sm = ncols(M); // in intvec iv the sizes are stored |
---|
910 | for (i=0; i<=d-iv[l]; i++) |
---|
911 | { |
---|
912 | // modules, e.g. free polynomials |
---|
913 | for (j=1; j<=sm; j++) |
---|
914 | { |
---|
915 | //vectors, e.g. free monomials |
---|
916 | v = M[j]; |
---|
917 | sv = size(v); |
---|
918 | // "sv:";sv; |
---|
919 | sp = "@@p = @@p + "; |
---|
920 | for (k=2; k<=sv; k++) |
---|
921 | { |
---|
922 | sp = sp + string(v[k])+"("+string(k-2+i)+")*"; |
---|
923 | } |
---|
924 | sp = sp + string(v[1])+";"; // coef; |
---|
925 | setring @R; |
---|
926 | execute(sp); |
---|
927 | setring save; |
---|
928 | } |
---|
929 | setring @R; |
---|
930 | // "@@p:"; @@p; |
---|
931 | I = I,@@p; |
---|
932 | @@p = 0; |
---|
933 | setring save; |
---|
934 | } |
---|
935 | } |
---|
936 | setring @R; |
---|
937 | export I; |
---|
938 | return(@R); |
---|
939 | } |
---|
940 | example |
---|
941 | { |
---|
942 | "EXAMPLE:"; echo = 2; |
---|
943 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
944 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
---|
945 | module N = [1,x,y,x],[-1,y,x,y]; |
---|
946 | list L; L[1] = M; L[2] = N; |
---|
947 | lst2str(L); |
---|
948 | def U = crs(L,5); |
---|
949 | setring U; U; |
---|
950 | I; |
---|
951 | } |
---|
952 | |
---|
953 | static proc polylen(ideal I) |
---|
954 | { |
---|
955 | // returns the ideal of length of polys |
---|
956 | int i; |
---|
957 | intvec J; |
---|
958 | number s = 0; |
---|
959 | for(i=1;i<=ncols(I);i++) |
---|
960 | { |
---|
961 | J[i] = size(I[i]); |
---|
962 | s = s + J[i]; |
---|
963 | } |
---|
964 | printf("the sum of length %s",s); |
---|
965 | // print(s); |
---|
966 | return(J); |
---|
967 | } |
---|
968 | |
---|
969 | // new: uniting both mLR1 (homog) and mLR2 (nonhomog) |
---|
970 | proc makeLetterplaceRing(int d, list #) |
---|
971 | "USAGE: makeLetterplaceRing(d [,h]); d an integer, h an optional integer |
---|
972 | RETURN: ring |
---|
973 | PURPOSE: creates a ring with the ordering, used in letterplace computations |
---|
974 | NOTE: h = 0 (default) : Dp ordering will be used |
---|
975 | h = 2 : weights 1 used for all the variables, a tie breaker is a list of block of original ring |
---|
976 | h = 1 : the pure homogeneous letterplace block ordering (applicable in the situation of homogeneous input ideals) will be used. |
---|
977 | EXAMPLE: example makeLetterplaceRing; shows examples |
---|
978 | " |
---|
979 | { |
---|
980 | int alternativeVersion = 0; |
---|
981 | if ( size(#)>0 ) |
---|
982 | { |
---|
983 | if (typeof(#[1]) == "int") |
---|
984 | { |
---|
985 | alternativeVersion = #[1]; |
---|
986 | } |
---|
987 | } |
---|
988 | if (alternativeVersion == 1) |
---|
989 | { |
---|
990 | def @A = makeLetterplaceRing1(d); |
---|
991 | } |
---|
992 | else { |
---|
993 | if (alternativeVersion == 2) |
---|
994 | { |
---|
995 | def @A = makeLetterplaceRing2(d); |
---|
996 | } |
---|
997 | else { |
---|
998 | def @A = makeLetterplaceRing4(d); |
---|
999 | } |
---|
1000 | } |
---|
1001 | return(@A); |
---|
1002 | } |
---|
1003 | example |
---|
1004 | { |
---|
1005 | "EXAMPLE:"; echo = 2; |
---|
1006 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
1007 | def A = makeLetterplaceRing(2); |
---|
1008 | setring A; A; |
---|
1009 | attrib(A,"isLetterplaceRing"); |
---|
1010 | attrib(A,"uptodeg"); // degree bound |
---|
1011 | attrib(A,"lV"); // number of variables in the main block |
---|
1012 | setring r; def B = makeLetterplaceRing(2,1); // to compare: |
---|
1013 | setring B; B; |
---|
1014 | } |
---|
1015 | |
---|
1016 | //proc freegbRing(int d) |
---|
1017 | static proc makeLetterplaceRing1(int d) |
---|
1018 | "USAGE: makeLetterplaceRing1(d); d an integer |
---|
1019 | RETURN: ring |
---|
1020 | PURPOSE: creates a ring with a special ordering, suitable for |
---|
1021 | @* the use of homogeneous letterplace (d blocks of shifted original variables) |
---|
1022 | EXAMPLE: example makeLetterplaceRing1; shows examples |
---|
1023 | " |
---|
1024 | { |
---|
1025 | // d = up to degree, will be shifted to d+1 |
---|
1026 | if (d<1) {"bad d"; return(0);} |
---|
1027 | |
---|
1028 | int uptodeg = d; int lV = nvars(basering); |
---|
1029 | |
---|
1030 | int ppl = printlevel-voice+2; |
---|
1031 | string err = ""; |
---|
1032 | |
---|
1033 | int i,j,s; |
---|
1034 | def save = basering; |
---|
1035 | int D = d-1; |
---|
1036 | list LR = ringlist(save); |
---|
1037 | list L, tmp; |
---|
1038 | L[1] = LR[1]; // ground field |
---|
1039 | L[4] = LR[4]; // quotient ideal |
---|
1040 | tmp = LR[2]; // varnames |
---|
1041 | s = size(LR[2]); |
---|
1042 | for (i=1; i<=D; i++) |
---|
1043 | { |
---|
1044 | for (j=1; j<=s; j++) |
---|
1045 | { |
---|
1046 | tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")"; |
---|
1047 | } |
---|
1048 | } |
---|
1049 | for (i=1; i<=s; i++) |
---|
1050 | { |
---|
1051 | tmp[i] = string(tmp[i])+"("+string(1)+")"; |
---|
1052 | } |
---|
1053 | L[2] = tmp; |
---|
1054 | list OrigNames = LR[2]; |
---|
1055 | // ordering: d blocks of the ord on r |
---|
1056 | // try to get whether the ord on r is blockord itself |
---|
1057 | // TODO: make L(2) ordering! exponent is maximally 2 |
---|
1058 | s = size(LR[3]); |
---|
1059 | if (s==2) |
---|
1060 | { |
---|
1061 | // not a blockord, 1 block + module ord |
---|
1062 | tmp = LR[3][s]; // module ord |
---|
1063 | for (i=1; i<=D; i++) |
---|
1064 | { |
---|
1065 | LR[3][s-1+i] = LR[3][1]; |
---|
1066 | } |
---|
1067 | LR[3][s+D] = tmp; |
---|
1068 | } |
---|
1069 | if (s>2) |
---|
1070 | { |
---|
1071 | // there are s-1 blocks |
---|
1072 | int nb = s-1; |
---|
1073 | tmp = LR[3][s]; // module ord |
---|
1074 | for (i=1; i<=D; i++) |
---|
1075 | { |
---|
1076 | for (j=1; j<=nb; j++) |
---|
1077 | { |
---|
1078 | LR[3][i*nb+j] = LR[3][j]; |
---|
1079 | } |
---|
1080 | } |
---|
1081 | // size(LR[3]); |
---|
1082 | LR[3][nb*(D+1)+1] = tmp; |
---|
1083 | } |
---|
1084 | L[3] = LR[3]; |
---|
1085 | attrib(L,"maxExp",1); |
---|
1086 | def @R = ring(L); |
---|
1087 | // setring @R; |
---|
1088 | // int uptodeg = d; int lV = nvars(basering); // were defined before |
---|
1089 | def @@R = setLetterplaceAttributes(@R,uptodeg,lV); |
---|
1090 | return (@@R); |
---|
1091 | } |
---|
1092 | example |
---|
1093 | { |
---|
1094 | "EXAMPLE:"; echo = 2; |
---|
1095 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
1096 | def A = makeLetterplaceRing1(2); |
---|
1097 | setring A; |
---|
1098 | A; |
---|
1099 | attrib(A,"isLetterplaceRing"); |
---|
1100 | attrib(A,"uptodeg"); // degree bound |
---|
1101 | attrib(A,"lV"); // number of variables in the main block |
---|
1102 | } |
---|
1103 | |
---|
1104 | static proc makeLetterplaceRing2(int d) |
---|
1105 | "USAGE: makeLetterplaceRing2(d); d an integer |
---|
1106 | RETURN: ring |
---|
1107 | PURPOSE: creates a ring with a special ordering, suitable for |
---|
1108 | @* the use of non-homogeneous letterplace |
---|
1109 | NOTE: the matrix for the ordering looks as follows: first row is 1,1,...,1 |
---|
1110 | @* then there come 'd' blocks of shifted original variables |
---|
1111 | EXAMPLE: example makeLetterplaceRing2; shows examples |
---|
1112 | " |
---|
1113 | { |
---|
1114 | |
---|
1115 | // ToDo future: inherit positive weights in the orig ring |
---|
1116 | // complain on nonpositive ones |
---|
1117 | |
---|
1118 | // d = up to degree, will be shifted to d+1 |
---|
1119 | if (d<1) {"bad d"; return(0);} |
---|
1120 | |
---|
1121 | int uptodeg = d; int lV = nvars(basering); |
---|
1122 | |
---|
1123 | int ppl = printlevel-voice+2; |
---|
1124 | string err = ""; |
---|
1125 | |
---|
1126 | int i,j,s; |
---|
1127 | def save = basering; |
---|
1128 | int D = d-1; |
---|
1129 | list LR = ringlist(save); |
---|
1130 | list L, tmp, tmp2, tmp3; |
---|
1131 | L[1] = LR[1]; // ground field |
---|
1132 | L[4] = LR[4]; // quotient ideal |
---|
1133 | tmp = LR[2]; // varnames |
---|
1134 | s = size(LR[2]); |
---|
1135 | for (i=1; i<=D; i++) |
---|
1136 | { |
---|
1137 | for (j=1; j<=s; j++) |
---|
1138 | { |
---|
1139 | tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")"; |
---|
1140 | } |
---|
1141 | } |
---|
1142 | for (i=1; i<=s; i++) |
---|
1143 | { |
---|
1144 | tmp[i] = string(tmp[i])+"("+string(1)+")"; |
---|
1145 | } |
---|
1146 | L[2] = tmp; |
---|
1147 | list OrigNames = LR[2]; |
---|
1148 | // ordering: one 1..1 a above |
---|
1149 | // ordering: d blocks of the ord on r |
---|
1150 | // try to get whether the ord on r is blockord itself |
---|
1151 | // TODO: make L(2) ordering! exponent is maximally 2 |
---|
1152 | s = size(LR[3]); |
---|
1153 | if (s==2) |
---|
1154 | { |
---|
1155 | // not a blockord, 1 block + module ord |
---|
1156 | tmp = LR[3][s]; // module ord |
---|
1157 | for (i=1; i<=d; i++) |
---|
1158 | { |
---|
1159 | LR[3][s-1+i] = LR[3][1]; |
---|
1160 | } |
---|
1161 | // LR[3][s+D] = tmp; |
---|
1162 | LR[3][s+1+D] = tmp; |
---|
1163 | LR[3][1] = list("a",intvec(1: int(d*lV))); // deg-ord |
---|
1164 | } |
---|
1165 | if (s>2) |
---|
1166 | { |
---|
1167 | // there are s-1 blocks |
---|
1168 | int nb = s-1; |
---|
1169 | tmp = LR[3][s]; // module ord to place at the very end |
---|
1170 | tmp2 = LR[3]; tmp2 = tmp2[1..nb]; |
---|
1171 | tmp3[1] = list("a",intvec(1: int(d*lV))); // deg-ord, insert as the 1st |
---|
1172 | for (i=1; i<=d; i++) |
---|
1173 | { |
---|
1174 | tmp3 = tmp3 + tmp2; |
---|
1175 | } |
---|
1176 | tmp3 = tmp3 + list(tmp); |
---|
1177 | LR[3] = tmp3; |
---|
1178 | // for (i=1; i<=d; i++) |
---|
1179 | // { |
---|
1180 | // for (j=1; j<=nb; j++) |
---|
1181 | // { |
---|
1182 | // // LR[3][i*nb+j+1]= LR[3][j]; |
---|
1183 | // LR[3][i*nb+j+1]= tmp2[j]; |
---|
1184 | // } |
---|
1185 | // } |
---|
1186 | // // size(LR[3]); |
---|
1187 | // LR[3][(s-1)*d+2] = tmp; |
---|
1188 | // LR[3] = list("a",intvec(1: int(d*lV))) + LR[3]; // deg-ord, insert as the 1st |
---|
1189 | // remove everything behind nb*(D+1)+1 ? |
---|
1190 | // tmp = LR[3]; |
---|
1191 | // LR[3] = tmp[1..size(tmp)-1]; |
---|
1192 | } |
---|
1193 | L[3] = LR[3]; |
---|
1194 | attrib(L,"maxExp",1); |
---|
1195 | def @R = ring(L); |
---|
1196 | // setring @R; |
---|
1197 | // int uptodeg = d; int lV = nvars(basering); // were defined before |
---|
1198 | def @@R = setLetterplaceAttributes(@R,uptodeg,lV); |
---|
1199 | return (@@R); |
---|
1200 | } |
---|
1201 | example |
---|
1202 | { |
---|
1203 | "EXAMPLE:"; echo = 2; |
---|
1204 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
1205 | def A = makeLetterplaceRing2(2); |
---|
1206 | setring A; |
---|
1207 | A; |
---|
1208 | attrib(A,"isLetterplaceRing"); |
---|
1209 | attrib(A,"uptodeg"); // degree bound |
---|
1210 | attrib(A,"lV"); // number of variables in the main block |
---|
1211 | } |
---|
1212 | |
---|
1213 | static proc makeLetterplaceRing4(int d) |
---|
1214 | "USAGE: makeLetterplaceRing2(d); d an integer |
---|
1215 | RETURN: ring |
---|
1216 | PURPOSE: creates a Letterplace ring with a Dp ordering, suitable for |
---|
1217 | @* the use of non-homogeneous letterplace |
---|
1218 | NOTE: the matrix for the ordering looks as follows: first row is 1,1,...,1 |
---|
1219 | EXAMPLE: example makeLetterplaceRing2; shows examples |
---|
1220 | " |
---|
1221 | { |
---|
1222 | |
---|
1223 | // ToDo future: inherit positive weights in the orig ring |
---|
1224 | // complain on nonpositive ones |
---|
1225 | |
---|
1226 | // d = up to degree, will be shifted to d+1 |
---|
1227 | if (d<1) {"bad d"; return(0);} |
---|
1228 | |
---|
1229 | int uptodeg = d; int lV = nvars(basering); |
---|
1230 | |
---|
1231 | int ppl = printlevel-voice+2; |
---|
1232 | string err = ""; |
---|
1233 | |
---|
1234 | int i,j,s; |
---|
1235 | def save = basering; |
---|
1236 | int D = d-1; |
---|
1237 | list LR = ringlist(save); |
---|
1238 | list L, tmp, tmp2, tmp3; |
---|
1239 | L[1] = LR[1]; // ground field |
---|
1240 | L[4] = LR[4]; // quotient ideal |
---|
1241 | tmp = LR[2]; // varnames |
---|
1242 | s = size(LR[2]); |
---|
1243 | for (i=1; i<=D; i++) |
---|
1244 | { |
---|
1245 | for (j=1; j<=s; j++) |
---|
1246 | { |
---|
1247 | tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")"; |
---|
1248 | } |
---|
1249 | } |
---|
1250 | for (i=1; i<=s; i++) |
---|
1251 | { |
---|
1252 | tmp[i] = string(tmp[i])+"("+string(1)+")"; |
---|
1253 | } |
---|
1254 | L[2] = tmp; |
---|
1255 | list OrigNames = LR[2]; |
---|
1256 | |
---|
1257 | s = size(LR[3]); |
---|
1258 | list ordering; |
---|
1259 | ordering[1] = list("Dp",intvec(1: int(d*lV))); |
---|
1260 | ordering[2] = LR[3][s]; // module ord to place at the very end |
---|
1261 | LR[3] = ordering; |
---|
1262 | |
---|
1263 | L[3] = LR[3]; |
---|
1264 | attrib(L,"maxExp",1); |
---|
1265 | def @R = ring(L); |
---|
1266 | def @@R = setLetterplaceAttributes(@R,uptodeg,lV); |
---|
1267 | return (@@R); |
---|
1268 | } |
---|
1269 | example |
---|
1270 | { |
---|
1271 | "EXAMPLE:"; echo = 2; |
---|
1272 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
1273 | def A = makeLetterplaceRing2(2); |
---|
1274 | setring A; |
---|
1275 | A; |
---|
1276 | attrib(A,"isLetterplaceRing"); |
---|
1277 | attrib(A,"uptodeg"); // degree bound |
---|
1278 | attrib(A,"lV"); // number of variables in the main block |
---|
1279 | } |
---|
1280 | |
---|
1281 | // P[s;sigma] approach |
---|
1282 | static proc makeLetterplaceRing3(int d) |
---|
1283 | "USAGE: makeLetterplaceRing1(d); d an integer |
---|
1284 | RETURN: ring |
---|
1285 | PURPOSE: creates a ring with a special ordering, representing |
---|
1286 | @* the original P[s,sigma] (adds d blocks of shifted s) |
---|
1287 | ASSUME: basering is a letterplace ring |
---|
1288 | NOTE: experimental status |
---|
1289 | EXAMPLE: example makeLetterplaceRing1; shows examples |
---|
1290 | " |
---|
1291 | { |
---|
1292 | // d = up to degree, will be shifted to d+1 |
---|
1293 | if (d<1) {"bad d"; return(0);} |
---|
1294 | |
---|
1295 | int uptodeg = d; int lV = nvars(basering); |
---|
1296 | |
---|
1297 | int ppl = printlevel-voice+2; |
---|
1298 | string err = ""; |
---|
1299 | |
---|
1300 | int i,j,s; |
---|
1301 | def save = basering; |
---|
1302 | int D = d-1; |
---|
1303 | list LR = ringlist(save); |
---|
1304 | list L, tmp; |
---|
1305 | L[1] = LR[1]; // ground field |
---|
1306 | L[4] = LR[4]; // quotient ideal |
---|
1307 | tmp = LR[2]; // varnames |
---|
1308 | tmp[size(tmp)+1] = "s"; |
---|
1309 | // add s's |
---|
1310 | // string newSname = "@s"; |
---|
1311 | s = size(LR[2]); |
---|
1312 | for (i=1; i<=D; i++) |
---|
1313 | { |
---|
1314 | for (j=1; j<=s; j++) |
---|
1315 | { |
---|
1316 | tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")"; |
---|
1317 | } |
---|
1318 | } |
---|
1319 | // the final index is D*s+s = (D+1)*s = degBound*s |
---|
1320 | for (i=1; i<=d; i++) |
---|
1321 | { |
---|
1322 | tmp[FIndex + i] = string(newSname)+"("+string(i)+")"; |
---|
1323 | } |
---|
1324 | L[2] = tmp; |
---|
1325 | list OrigNames = LR[2]; |
---|
1326 | // ordering: d blocks of the MODIFIED ord on r |
---|
1327 | // try to get whether the ord on r is blockord itself |
---|
1328 | // TODO: make L(2) ordering! exponent is maximally 2 |
---|
1329 | s = size(LR[3]); |
---|
1330 | |
---|
1331 | // assume: basering was a letterplace, so get its block |
---|
1332 | tmp = LR[3][1]; // ASSUME: it's a nice block |
---|
1333 | // modify it |
---|
1334 | // add (0,..,0,1) ... as antiblock part |
---|
1335 | intvec iv; list ttmp, tmp1; |
---|
1336 | for (i=1; i<=d; i++) |
---|
1337 | { |
---|
1338 | // the position to hold 1: |
---|
1339 | iv = intvec( gen( i*(lV+1)-1 ) ); |
---|
1340 | ttmp[1] = "a"; |
---|
1341 | ttmp[2] = iv; |
---|
1342 | tmp1[i] = ttmp; |
---|
1343 | } |
---|
1344 | // finished: antiblock part //TOCONTINUE |
---|
1345 | |
---|
1346 | if (s==2) |
---|
1347 | { |
---|
1348 | // not a blockord, 1 block + module ord |
---|
1349 | tmp = LR[3][s]; // module ord |
---|
1350 | for (i=1; i<=D; i++) |
---|
1351 | { |
---|
1352 | LR[3][s-1+i] = LR[3][1]; |
---|
1353 | } |
---|
1354 | LR[3][s+D] = tmp; |
---|
1355 | } |
---|
1356 | if (s>2) |
---|
1357 | { |
---|
1358 | // there are s-1 blocks |
---|
1359 | int nb = s-1; |
---|
1360 | tmp = LR[3][s]; // module ord |
---|
1361 | for (i=1; i<=D; i++) |
---|
1362 | { |
---|
1363 | for (j=1; j<=nb; j++) |
---|
1364 | { |
---|
1365 | LR[3][i*nb+j] = LR[3][j]; |
---|
1366 | } |
---|
1367 | } |
---|
1368 | // size(LR[3]); |
---|
1369 | LR[3][nb*(D+1)+1] = tmp; |
---|
1370 | } |
---|
1371 | L[3] = LR[3]; |
---|
1372 | attrib(L,"maxExp",1); |
---|
1373 | def @R = ring(L); |
---|
1374 | // setring @R; |
---|
1375 | // int uptodeg = d; int lV = nvars(basering); // were defined before |
---|
1376 | def @@R = setLetterplaceAttributes(@R,uptodeg,lV); |
---|
1377 | return (@@R); |
---|
1378 | } |
---|
1379 | example |
---|
1380 | { |
---|
1381 | "EXAMPLE:"; echo = 2; |
---|
1382 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
1383 | def A = makeLetterplaceRing3(2); |
---|
1384 | setring A; |
---|
1385 | A; |
---|
1386 | attrib(A,"isLetterplaceRing"); |
---|
1387 | attrib(A,"uptodeg"); // degree bound |
---|
1388 | attrib(A,"lV"); // number of variables in the main block |
---|
1389 | } |
---|
1390 | |
---|
1391 | /* EXAMPLES: |
---|
1392 | |
---|
1393 | //static proc ex_shift() |
---|
1394 | { |
---|
1395 | LIB "freegb.lib"; |
---|
1396 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
1397 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
---|
1398 | module N = [1,x,y,x],[-1,y,x,y]; |
---|
1399 | list L; L[1] = M; L[2] = N; |
---|
1400 | lst2str(L); |
---|
1401 | def U = crs(L,5); |
---|
1402 | setring U; U; |
---|
1403 | I; |
---|
1404 | poly p = I[2]; // I[8]; |
---|
1405 | p; |
---|
1406 | system("stest",p,7,7,3); // error -> the world is ok |
---|
1407 | poly q1 = system("stest",p,1,7,3); //ok |
---|
1408 | poly q6 = system("stest",p,6,7,3); //ok |
---|
1409 | system("btest",p,3); //ok |
---|
1410 | system("btest",q1,3); //ok |
---|
1411 | system("btest",q6,3); //ok |
---|
1412 | } |
---|
1413 | |
---|
1414 | //static proc test_shrink() |
---|
1415 | { |
---|
1416 | LIB "freegb.lib"; |
---|
1417 | ring r =0,(x,y,z),dp; |
---|
1418 | int d = 5; |
---|
1419 | def R = makeLetterplaceRing(d); |
---|
1420 | setring R; |
---|
1421 | poly p1 = x(1)*y(2)*z(3); |
---|
1422 | poly p2 = x(1)*y(4)*z(5); |
---|
1423 | poly p3 = x(1)*y(1)*z(3); |
---|
1424 | poly p4 = x(1)*y(2)*z(2); |
---|
1425 | poly p5 = x(3)*z(5); |
---|
1426 | poly p6 = x(1)*y(1)*x(3)*z(5); |
---|
1427 | poly p7 = x(1)*y(2)*x(3)*y(4)*z(5); |
---|
1428 | poly p8 = p1+p2+p3+p4+p5 + p6 + p7; |
---|
1429 | p1; system("shrinktest",p1,3); |
---|
1430 | p2; system("shrinktest",p2,3); |
---|
1431 | p3; system("shrinktest",p3,3); |
---|
1432 | p4; system("shrinktest",p4,3); |
---|
1433 | p5; system("shrinktest",p5,3); |
---|
1434 | p6; system("shrinktest",p6,3); |
---|
1435 | p7; system("shrinktest",p7,3); |
---|
1436 | p8; system("shrinktest",p8,3); |
---|
1437 | poly p9 = p1 + 2*p2 + 5*p5 + 7*p7; |
---|
1438 | p9; system("shrinktest",p9,3); |
---|
1439 | } |
---|
1440 | |
---|
1441 | //static proc ex2() |
---|
1442 | { |
---|
1443 | option(prot); |
---|
1444 | LIB "freegb.lib"; |
---|
1445 | ring r = 0,(x,y),dp; |
---|
1446 | module M = [-1,x,y],[3,x,x]; // 3x^2 - xy |
---|
1447 | def U = freegb(M,7); |
---|
1448 | lst2str(U); |
---|
1449 | } |
---|
1450 | |
---|
1451 | //static proc ex_nonhomog() |
---|
1452 | { |
---|
1453 | option(prot); |
---|
1454 | LIB "freegb.lib"; |
---|
1455 | ring r = 0,(x,y,h),dp; |
---|
1456 | list L; |
---|
1457 | module M; |
---|
1458 | M = [-1,y,y],[1,x,x,x]; // x3-y2 |
---|
1459 | L[1] = M; |
---|
1460 | M = [1,x,h],[-1,h,x]; // xh-hx |
---|
1461 | L[2] = M; |
---|
1462 | M = [1,y,h],[-1,h,y]; // yh-hy |
---|
1463 | L[3] = M; |
---|
1464 | def U = freegb(L,4); |
---|
1465 | lst2str(U); |
---|
1466 | // strange elements in the basis |
---|
1467 | } |
---|
1468 | |
---|
1469 | //static proc ex_nonhomog_comm() |
---|
1470 | { |
---|
1471 | option(prot); |
---|
1472 | LIB "freegb.lib"; |
---|
1473 | ring r = 0,(x,y),dp; |
---|
1474 | module M = [-1,y,y],[1,x,x,x]; |
---|
1475 | def U = freegb(M,5); |
---|
1476 | lst2str(U); |
---|
1477 | } |
---|
1478 | |
---|
1479 | //static proc ex_nonhomog_h() |
---|
1480 | { |
---|
1481 | option(prot); |
---|
1482 | LIB "freegb.lib"; |
---|
1483 | ring r = 0,(x,y,h),(a(1,1),dp); |
---|
1484 | module M = [-1,y,y,h],[1,x,x,x]; // x3 - y2h |
---|
1485 | def U = freegb(M,6); |
---|
1486 | lst2str(U); |
---|
1487 | } |
---|
1488 | |
---|
1489 | //static proc ex_nonhomog_h2() |
---|
1490 | { |
---|
1491 | option(prot); |
---|
1492 | LIB "freegb.lib"; |
---|
1493 | ring r = 0,(x,y,h),(dp); |
---|
1494 | list L; |
---|
1495 | module M; |
---|
1496 | M = [-1,y,y,h],[1,x,x,x]; // x3 - y2h |
---|
1497 | L[1] = M; |
---|
1498 | M = [1,x,h],[-1,h,x]; // xh - hx |
---|
1499 | L[2] = M; |
---|
1500 | M = [1,y,h],[-1,h,y]; // yh - hy |
---|
1501 | L[3] = M; |
---|
1502 | def U = freeGBasis(L,3); |
---|
1503 | lst2str(U); |
---|
1504 | // strange answer CHECK |
---|
1505 | } |
---|
1506 | |
---|
1507 | |
---|
1508 | //static proc ex_nonhomog_3() |
---|
1509 | { |
---|
1510 | option(prot); |
---|
1511 | LIB "./freegb.lib"; |
---|
1512 | ring r = 0,(x,y,z),(dp); |
---|
1513 | list L; |
---|
1514 | module M; |
---|
1515 | M = [1,z,y],[-1,x]; // zy - x |
---|
1516 | L[1] = M; |
---|
1517 | M = [1,z,x],[-1,y]; // zx - y |
---|
1518 | L[2] = M; |
---|
1519 | M = [1,y,x],[-1,z]; // yx - z |
---|
1520 | L[3] = M; |
---|
1521 | lst2str(L); |
---|
1522 | list U = freegb(L,4); |
---|
1523 | lst2str(U); |
---|
1524 | // strange answer CHECK |
---|
1525 | } |
---|
1526 | |
---|
1527 | //static proc ex_densep_2() |
---|
1528 | { |
---|
1529 | option(prot); |
---|
1530 | LIB "freegb.lib"; |
---|
1531 | ring r = (0,a,b,c),(x,y),(Dp); // deglex |
---|
1532 | module M = [1,x,x], [a,x,y], [b,y,x], [c,y,y]; |
---|
1533 | lst2str(M); |
---|
1534 | list U = freegb(M,5); |
---|
1535 | lst2str(U); |
---|
1536 | // a=b is important -> finite basis!!! |
---|
1537 | module M = [1,x,x], [a,x,y], [a,y,x], [c,y,y]; |
---|
1538 | lst2str(M); |
---|
1539 | list U = freegb(M,5); |
---|
1540 | lst2str(U); |
---|
1541 | } |
---|
1542 | |
---|
1543 | // END COMMENTED EXAMPLES |
---|
1544 | |
---|
1545 | */ |
---|
1546 | |
---|
1547 | // 1. form a new ring |
---|
1548 | // 2. produce shifted generators |
---|
1549 | // 3. compute GB |
---|
1550 | // 4. skip shifted elts |
---|
1551 | // 5. go back to orig vars, produce strings/modules |
---|
1552 | // 6. return the result |
---|
1553 | |
---|
1554 | static proc freegbold(list LM, int d) |
---|
1555 | "USAGE: freegbold(L, d); L a list of modules, d an integer |
---|
1556 | RETURN: ring |
---|
1557 | PURPOSE: compute the two-sided Groebner basis of an ideal, encoded by L in |
---|
1558 | the free associative algebra, up to degree d |
---|
1559 | EXAMPLE: example freegbold; shows examples |
---|
1560 | " |
---|
1561 | { |
---|
1562 | // d = up to degree, will be shifted to d+1 |
---|
1563 | if (d<1) {"bad d"; return(0);} |
---|
1564 | |
---|
1565 | int ppl = printlevel-voice+2; |
---|
1566 | string err = ""; |
---|
1567 | |
---|
1568 | int i,j,s; |
---|
1569 | def save = basering; |
---|
1570 | // determine max no of places in the input |
---|
1571 | int slm = size(LM); // numbers of polys in the ideal |
---|
1572 | int sm; |
---|
1573 | intvec iv; |
---|
1574 | module M; |
---|
1575 | for (i=1; i<=slm; i++) |
---|
1576 | { |
---|
1577 | // modules, e.g. free polynomials |
---|
1578 | M = LM[i]; |
---|
1579 | sm = ncols(M); |
---|
1580 | for (j=1; j<=sm; j++) |
---|
1581 | { |
---|
1582 | //vectors, e.g. free monomials |
---|
1583 | iv = iv, size(M[j])-1; // 1 place is reserved by the coeff |
---|
1584 | } |
---|
1585 | } |
---|
1586 | int D = Max(iv); // max size of input words |
---|
1587 | if (d<D) {"bad d"; return(LM);} |
---|
1588 | D = D + d-1; |
---|
1589 | // D = d; |
---|
1590 | list LR = ringlist(save); |
---|
1591 | list L, tmp; |
---|
1592 | L[1] = LR[1]; // ground field |
---|
1593 | L[4] = LR[4]; // quotient ideal |
---|
1594 | tmp = LR[2]; // varnames |
---|
1595 | s = size(LR[2]); |
---|
1596 | for (i=1; i<=D; i++) |
---|
1597 | { |
---|
1598 | for (j=1; j<=s; j++) |
---|
1599 | { |
---|
1600 | tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")"; |
---|
1601 | } |
---|
1602 | } |
---|
1603 | for (i=1; i<=s; i++) |
---|
1604 | { |
---|
1605 | tmp[i] = string(tmp[i])+"("+string(1)+")"; |
---|
1606 | } |
---|
1607 | L[2] = tmp; |
---|
1608 | list OrigNames = LR[2]; |
---|
1609 | // ordering: d blocks of the ord on r |
---|
1610 | // try to get whether the ord on r is blockord itself |
---|
1611 | // TODO: make L(2) ordering! exponent is maximally 2 |
---|
1612 | s = size(LR[3]); |
---|
1613 | if (s==2) |
---|
1614 | { |
---|
1615 | // not a blockord, 1 block + module ord |
---|
1616 | tmp = LR[3][s]; // module ord |
---|
1617 | for (i=1; i<=D; i++) |
---|
1618 | { |
---|
1619 | LR[3][s-1+i] = LR[3][1]; |
---|
1620 | } |
---|
1621 | LR[3][s+D] = tmp; |
---|
1622 | } |
---|
1623 | if (s>2) |
---|
1624 | { |
---|
1625 | // there are s-1 blocks |
---|
1626 | int nb = s-1; |
---|
1627 | tmp = LR[3][s]; // module ord |
---|
1628 | for (i=1; i<=D; i++) |
---|
1629 | { |
---|
1630 | for (j=1; j<=nb; j++) |
---|
1631 | { |
---|
1632 | LR[3][i*nb+j] = LR[3][j]; |
---|
1633 | } |
---|
1634 | } |
---|
1635 | // size(LR[3]); |
---|
1636 | LR[3][nb*(D+1)+1] = tmp; |
---|
1637 | } |
---|
1638 | L[3] = LR[3]; |
---|
1639 | def @R = ring(L); |
---|
1640 | setring @R; |
---|
1641 | ideal I; |
---|
1642 | poly @p; |
---|
1643 | s = size(OrigNames); |
---|
1644 | // "s:";s; |
---|
1645 | // convert LM to canonical vectors (no powers) |
---|
1646 | setring save; |
---|
1647 | kill M; // M was defined earlier |
---|
1648 | module M; |
---|
1649 | slm = size(LM); // numbers of polys in the ideal |
---|
1650 | int sv,k,l; |
---|
1651 | vector v; |
---|
1652 | // poly p; |
---|
1653 | string sp; |
---|
1654 | setring @R; |
---|
1655 | poly @@p=0; |
---|
1656 | setring save; |
---|
1657 | for (l=1; l<=slm; l++) |
---|
1658 | { |
---|
1659 | // modules, e.g. free polynomials |
---|
1660 | M = LM[l]; |
---|
1661 | sm = ncols(M); // in intvec iv the sizes are stored |
---|
1662 | for (i=0; i<=d-iv[l]; i++) |
---|
1663 | { |
---|
1664 | // modules, e.g. free polynomials |
---|
1665 | for (j=1; j<=sm; j++) |
---|
1666 | { |
---|
1667 | //vectors, e.g. free monomials |
---|
1668 | v = M[j]; |
---|
1669 | sv = size(v); |
---|
1670 | // "sv:";sv; |
---|
1671 | sp = "@@p = @@p + "; |
---|
1672 | for (k=2; k<=sv; k++) |
---|
1673 | { |
---|
1674 | sp = sp + string(v[k])+"("+string(k-1+i)+")*"; |
---|
1675 | } |
---|
1676 | sp = sp + string(v[1])+";"; // coef; |
---|
1677 | setring @R; |
---|
1678 | execute(sp); |
---|
1679 | setring save; |
---|
1680 | } |
---|
1681 | setring @R; |
---|
1682 | // "@@p:"; @@p; |
---|
1683 | I = I,@@p; |
---|
1684 | @@p = 0; |
---|
1685 | setring save; |
---|
1686 | } |
---|
1687 | } |
---|
1688 | kill sp; |
---|
1689 | // 3. compute GB |
---|
1690 | setring @R; |
---|
1691 | dbprint(ppl,"computing GB"); |
---|
1692 | // ideal J = groebner(I); |
---|
1693 | ideal J = slimgb(I); |
---|
1694 | dbprint(ppl,J); |
---|
1695 | // 4. skip shifted elts |
---|
1696 | ideal K = select1(J,1..s); // s = size(OrigNames) |
---|
1697 | dbprint(ppl,K); |
---|
1698 | dbprint(ppl, "done with GB"); |
---|
1699 | // K contains vars x(1),...z(1) = images of originals |
---|
1700 | // 5. go back to orig vars, produce strings/modules |
---|
1701 | if (K[1] == 0) |
---|
1702 | { |
---|
1703 | "no reasonable output, GB gives 0"; |
---|
1704 | return(0); |
---|
1705 | } |
---|
1706 | int sk = size(K); |
---|
1707 | int sp, sx, a, b; |
---|
1708 | intvec x; |
---|
1709 | poly p,q; |
---|
1710 | poly pn; |
---|
1711 | // vars in 'save' |
---|
1712 | setring save; |
---|
1713 | module N; |
---|
1714 | list LN; |
---|
1715 | vector V; |
---|
1716 | poly pn; |
---|
1717 | // test and skip exponents >=2 |
---|
1718 | setring @R; |
---|
1719 | for(i=1; i<=sk; i++) |
---|
1720 | { |
---|
1721 | p = K[i]; |
---|
1722 | while (p!=0) |
---|
1723 | { |
---|
1724 | q = lead(p); |
---|
1725 | // "processing q:";q; |
---|
1726 | x = leadexp(q); |
---|
1727 | sx = size(x); |
---|
1728 | for(k=1; k<=sx; k++) |
---|
1729 | { |
---|
1730 | if ( x[k] >= 2 ) |
---|
1731 | { |
---|
1732 | err = "skip: the value x[k] is " + string(x[k]); |
---|
1733 | dbprint(ppl,err); |
---|
1734 | // return(0); |
---|
1735 | K[i] = 0; |
---|
1736 | p = 0; |
---|
1737 | q = 0; |
---|
1738 | break; |
---|
1739 | } |
---|
1740 | } |
---|
1741 | p = p - q; |
---|
1742 | } |
---|
1743 | } |
---|
1744 | K = simplify(K,2); |
---|
1745 | sk = size(K); |
---|
1746 | for(i=1; i<=sk; i++) |
---|
1747 | { |
---|
1748 | // setring save; |
---|
1749 | // V = 0; |
---|
1750 | setring @R; |
---|
1751 | p = K[i]; |
---|
1752 | while (p!=0) |
---|
1753 | { |
---|
1754 | q = lead(p); |
---|
1755 | err = "processing q:" + string(q); |
---|
1756 | dbprint(ppl,err); |
---|
1757 | x = leadexp(q); |
---|
1758 | sx = size(x); |
---|
1759 | pn = leadcoef(q); |
---|
1760 | setring save; |
---|
1761 | pn = imap(@R,pn); |
---|
1762 | V = V + leadcoef(pn)*gen(1); |
---|
1763 | for(k=1; k<=sx; k++) |
---|
1764 | { |
---|
1765 | if (x[k] ==1) |
---|
1766 | { |
---|
1767 | a = k div s; // block number=a+1, a!=0 |
---|
1768 | b = k % s; // remainder |
---|
1769 | // printf("a: %s, b: %s",a,b); |
---|
1770 | if (b == 0) |
---|
1771 | { |
---|
1772 | // that is it's the last var in the block |
---|
1773 | b = s; |
---|
1774 | a = a-1; |
---|
1775 | } |
---|
1776 | V = V + var(b)*gen(a+2); |
---|
1777 | } |
---|
1778 | // else |
---|
1779 | // { |
---|
1780 | // printf("error: the value x[k] is %s", x[k]); |
---|
1781 | // return(0); |
---|
1782 | // } |
---|
1783 | } |
---|
1784 | err = "V: " + string(V); |
---|
1785 | dbprint(ppl,err); |
---|
1786 | // printf("V: %s", string(V)); |
---|
1787 | N = N,V; |
---|
1788 | V = 0; |
---|
1789 | setring @R; |
---|
1790 | p = p - q; |
---|
1791 | pn = 0; |
---|
1792 | } |
---|
1793 | setring save; |
---|
1794 | LN[i] = simplify(N,2); |
---|
1795 | N = 0; |
---|
1796 | } |
---|
1797 | setring save; |
---|
1798 | return(LN); |
---|
1799 | } |
---|
1800 | example |
---|
1801 | { |
---|
1802 | "EXAMPLE:"; echo = 2; |
---|
1803 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
1804 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
---|
1805 | module N = [1,x,y,x],[-1,y,x,y]; |
---|
1806 | list L; L[1] = M; L[2] = N; |
---|
1807 | lst2str(L); |
---|
1808 | def U = freegbold(L,5); |
---|
1809 | lst2str(U); |
---|
1810 | } |
---|
1811 | |
---|
1812 | /* begin older procs and tests |
---|
1813 | |
---|
1814 | static proc sgb(ideal I, int d) |
---|
1815 | { |
---|
1816 | // new code |
---|
1817 | // map x_i to x_i(1) via map() |
---|
1818 | //LIB "freegb.lib"; |
---|
1819 | def save = basering; |
---|
1820 | //int d =7;// degree |
---|
1821 | int nv = nvars(save); |
---|
1822 | def R = makeLetterplaceRing(d); |
---|
1823 | setring R; |
---|
1824 | int i; |
---|
1825 | ideal Imap; |
---|
1826 | for (i=1; i<=nv; i++) |
---|
1827 | { |
---|
1828 | Imap[i] = var(i); |
---|
1829 | } |
---|
1830 | //ideal I = x(1)*y(2), y(1)*x(2)+z(1)*z(2); |
---|
1831 | ideal I = x(1)*x(2),x(1)*y(2) + z(1)*x(2); |
---|
1832 | option(prot); |
---|
1833 | //option(teach); |
---|
1834 | ideal J = system("freegb",I,d,nv); |
---|
1835 | } |
---|
1836 | |
---|
1837 | static proc checkCeq() |
---|
1838 | { |
---|
1839 | ring r = 0,(x,y),Dp; |
---|
1840 | def A = makeLetterplaceRing(4); |
---|
1841 | setring A; |
---|
1842 | A; |
---|
1843 | // I = x2-xy |
---|
1844 | 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); |
---|
1845 | 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); |
---|
1846 | ideal K = I,C; |
---|
1847 | groebner(K); |
---|
1848 | } |
---|
1849 | |
---|
1850 | static proc exHom1() |
---|
1851 | { |
---|
1852 | // we start with |
---|
1853 | // z*y - x, z*x - y, y*x - z |
---|
1854 | LIB "freegb.lib"; |
---|
1855 | LIB "elim.lib"; |
---|
1856 | ring r = 0,(x,y,z,h),dp; |
---|
1857 | list L; |
---|
1858 | module M; |
---|
1859 | M = [1,z,y],[-1,x,h]; // zy - xh |
---|
1860 | L[1] = M; |
---|
1861 | M = [1,z,x],[-1,y,h]; // zx - yh |
---|
1862 | L[2] = M; |
---|
1863 | M = [1,y,x],[-1,z,h]; // yx - zh |
---|
1864 | L[3] = M; |
---|
1865 | lst2str(L); |
---|
1866 | def U = crs(L,4); |
---|
1867 | setring U; |
---|
1868 | I = I, |
---|
1869 | y(2)*h(3)+z(2)*x(3), y(3)*h(4)+z(3)*x(4), |
---|
1870 | y(2)*x(3)-z(2)*h(3), y(3)*x(4)-z(3)*h(4); |
---|
1871 | I = simplify(I,2); |
---|
1872 | ring r2 = 0,(x(0..4),y(0..4),z(0..4),h(0..4)),dp; |
---|
1873 | ideal J = imap(U,I); |
---|
1874 | // ideal K = homog(J,h); |
---|
1875 | option(redSB); |
---|
1876 | option(redTail); |
---|
1877 | ideal L = groebner(J); //(K); |
---|
1878 | ideal LL = sat(L,ideal(h))[1]; |
---|
1879 | ideal M = subst(LL,h,1); |
---|
1880 | M = simplify(M,2); |
---|
1881 | setring U; |
---|
1882 | ideal M = imap(r2,M); |
---|
1883 | lst2str(U); |
---|
1884 | } |
---|
1885 | |
---|
1886 | static proc test1() |
---|
1887 | { |
---|
1888 | LIB "freegb.lib"; |
---|
1889 | ring r = 0,(x,y),Dp; |
---|
1890 | int d = 10; // degree |
---|
1891 | def R = makeLetterplaceRing(d); |
---|
1892 | setring R; |
---|
1893 | ideal I = x(1)*x(2) - y(1)*y(2); |
---|
1894 | option(prot); |
---|
1895 | option(teach); |
---|
1896 | ideal J = system("freegb",I,d,2); |
---|
1897 | J; |
---|
1898 | } |
---|
1899 | |
---|
1900 | static proc test2() |
---|
1901 | { |
---|
1902 | LIB "freegb.lib"; |
---|
1903 | ring r = 0,(x,y),Dp; |
---|
1904 | int d = 10; // degree |
---|
1905 | def R = makeLetterplaceRing(d); |
---|
1906 | setring R; |
---|
1907 | ideal I = x(1)*x(2) - x(1)*y(2); |
---|
1908 | option(prot); |
---|
1909 | option(teach); |
---|
1910 | ideal J = system("freegb",I,d,2); |
---|
1911 | J; |
---|
1912 | } |
---|
1913 | |
---|
1914 | static proc test3() |
---|
1915 | { |
---|
1916 | LIB "freegb.lib"; |
---|
1917 | ring r = 0,(x,y,z),dp; |
---|
1918 | int d =5; // degree |
---|
1919 | def R = makeLetterplaceRing(d); |
---|
1920 | setring R; |
---|
1921 | ideal I = x(1)*y(2), y(1)*x(2)+z(1)*z(2); |
---|
1922 | option(prot); |
---|
1923 | option(teach); |
---|
1924 | ideal J = system("freegb",I,d,3); |
---|
1925 | } |
---|
1926 | |
---|
1927 | static proc schur2-3() |
---|
1928 | { |
---|
1929 | // nonhomog: |
---|
1930 | // h^4-10*h^2+9,f*e-e*f+h, h*2-e*h-2*e,h*f-f*h+2*f |
---|
1931 | // homogenized with t |
---|
1932 | // 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, |
---|
1933 | // t*h - h*t, t*f - f*t, t*e - e*t |
---|
1934 | } |
---|
1935 | |
---|
1936 | end older procs and tests */ |
---|
1937 | |
---|
1938 | proc ademRelations(int i, int j) |
---|
1939 | "USAGE: ademRelations(i,j); i,j int |
---|
1940 | RETURN: ring (and exports ideal) |
---|
1941 | ASSUME: there are at least i+j variables in the basering |
---|
1942 | PURPOSE: compute the ideal of Adem relations for i<2j in characteristic 0 |
---|
1943 | @* the ideal is exported under the name AdemRel in the output ring |
---|
1944 | EXAMPLE: example ademRelations; shows examples |
---|
1945 | " |
---|
1946 | { |
---|
1947 | // produces Adem relations for i<2j in char 0 |
---|
1948 | // assume: 0<i<2j |
---|
1949 | // requires presence of vars up to i+j |
---|
1950 | if ( (i<0) || (i >= 2*j) ) |
---|
1951 | { |
---|
1952 | ERROR("arguments out of range"); return(0); |
---|
1953 | } |
---|
1954 | ring @r = 0,(s(i+j..0)),lp; |
---|
1955 | poly p,q; |
---|
1956 | number n; |
---|
1957 | int ii = i div 2; int k; |
---|
1958 | // k=0 => s(0)=1 |
---|
1959 | n = binomial(j-1,i); |
---|
1960 | q = n*s(i+j)*s(0); |
---|
1961 | // printf("k=0, term=%s",q); |
---|
1962 | p = p + q; |
---|
1963 | for (k=1; k<= ii; k++) |
---|
1964 | { |
---|
1965 | n = binomial(j-k-1,i-2*k); |
---|
1966 | q = n*s(i+j-k)*s(k);; |
---|
1967 | // printf("k=%s, term=%s",k,q); |
---|
1968 | p = p + q; |
---|
1969 | } |
---|
1970 | poly AdemRel = p; |
---|
1971 | export AdemRel; |
---|
1972 | return(@r); |
---|
1973 | } |
---|
1974 | example |
---|
1975 | { |
---|
1976 | "EXAMPLE:"; echo = 2; |
---|
1977 | def A = ademRelations(2,5); |
---|
1978 | setring A; |
---|
1979 | AdemRel; |
---|
1980 | } |
---|
1981 | |
---|
1982 | /* |
---|
1983 | 1,1: 0 |
---|
1984 | 1,2: s(3)*s(0) == s(3) -> def for s(3):=s(1)s(2) |
---|
1985 | 2,1: adm |
---|
1986 | 2,2: s(3)*s(1) == s(1)s(2)s(1) |
---|
1987 | 1,3: 0 ( since 2*s(4)*s(0) = 0 mod 2) |
---|
1988 | 3,1: adm |
---|
1989 | 2,3: s(5)*s(0)+s(4)*s(1) == s(5)+s(4)*s(1) |
---|
1990 | 3,2: 0 |
---|
1991 | 3,3: s(5)*s(1) |
---|
1992 | 1,4: 3*s(5)*s(0) == s(5) -> def for s(5):=s(1)*s(4) |
---|
1993 | 4,1: adm |
---|
1994 | 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) |
---|
1995 | 4,2: adm |
---|
1996 | 4,3: s(5)*s(2) |
---|
1997 | 3,4: s(7)*s(0)+2*s(6)*s(1) == s(7) -> def for s(7):=s(3)*s(4) |
---|
1998 | 4,4: s(7)*s(1)+s(6)*s(2) |
---|
1999 | */ |
---|
2000 | |
---|
2001 | /* s1,s2: |
---|
2002 | s1*s1 =0, s2*s2 = s1*s2*s1 |
---|
2003 | */ |
---|
2004 | |
---|
2005 | /* |
---|
2006 | try char 0: |
---|
2007 | s1,s2: |
---|
2008 | 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) |
---|
2009 | hence 2==0! only in char 2 |
---|
2010 | */ |
---|
2011 | |
---|
2012 | // Adem rels modulo 2 are interesting |
---|
2013 | |
---|
2014 | static proc stringpoly2lplace(string s) |
---|
2015 | { |
---|
2016 | // decomposes sentence into terms |
---|
2017 | s = replace(s,newline,""); // get rid of newlines |
---|
2018 | s = replace(s," ",""); // get rid of empties |
---|
2019 | //arith symbols: +,- |
---|
2020 | // decompose into words with coeffs |
---|
2021 | list LS; |
---|
2022 | int i,j,ie,je,k,cnt; |
---|
2023 | // s[1]="-" situation |
---|
2024 | if (s[1]=="-") |
---|
2025 | { |
---|
2026 | LS = stringpoly2lplace(string(s[2..size(s)])); |
---|
2027 | LS[1] = string("-"+string(LS[1])); |
---|
2028 | return(LS); |
---|
2029 | } |
---|
2030 | i = find(s,"-",2); |
---|
2031 | // i==1 might happen if the 1st symbol coeff is negative |
---|
2032 | j = find(s,"+"); |
---|
2033 | list LL; |
---|
2034 | if (i==j) |
---|
2035 | { |
---|
2036 | "return a monomial"; |
---|
2037 | // that is both are 0 -> s is a monomial |
---|
2038 | LS[1] = s; |
---|
2039 | return(LS); |
---|
2040 | } |
---|
2041 | if (i==0) |
---|
2042 | { |
---|
2043 | "i==0 situation"; |
---|
2044 | // no minuses at all => pluses only |
---|
2045 | cnt++; |
---|
2046 | LS[cnt] = string(s[1..j-1]); |
---|
2047 | s = s[j+1..size(s)]; |
---|
2048 | while (s!= "") |
---|
2049 | { |
---|
2050 | j = find(s,"+"); |
---|
2051 | cnt++; |
---|
2052 | if (j==0) |
---|
2053 | { |
---|
2054 | LS[cnt] = string(s); |
---|
2055 | s = ""; |
---|
2056 | } |
---|
2057 | else |
---|
2058 | { |
---|
2059 | LS[cnt] = string(s[1..j-1]); |
---|
2060 | s = s[j+1..size(s)]; |
---|
2061 | } |
---|
2062 | } |
---|
2063 | return(LS); |
---|
2064 | } |
---|
2065 | if (j==0) |
---|
2066 | { |
---|
2067 | "j==0 situation"; |
---|
2068 | // no pluses at all except the lead coef => the rest are minuses only |
---|
2069 | cnt++; |
---|
2070 | LS[cnt] = string(s[1..i-1]); |
---|
2071 | s = s[i..size(s)]; |
---|
2072 | while (s!= "") |
---|
2073 | { |
---|
2074 | i = find(s,"-",2); |
---|
2075 | cnt++; |
---|
2076 | if (i==0) |
---|
2077 | { |
---|
2078 | LS[cnt] = string(s); |
---|
2079 | s = ""; |
---|
2080 | } |
---|
2081 | else |
---|
2082 | { |
---|
2083 | LS[cnt] = string(s[1..i-1]); |
---|
2084 | s = s[i..size(s)]; |
---|
2085 | } |
---|
2086 | } |
---|
2087 | return(LS); |
---|
2088 | } |
---|
2089 | // now i, j are nonzero |
---|
2090 | if (i>j) |
---|
2091 | { |
---|
2092 | "i>j situation"; |
---|
2093 | // + comes first, at place j |
---|
2094 | cnt++; |
---|
2095 | // "cnt:"; cnt; "j:"; j; |
---|
2096 | LS[cnt] = string(s[1..j-1]); |
---|
2097 | s = s[j+1..size(s)]; |
---|
2098 | LL = stringpoly2lplace(s); |
---|
2099 | LS = LS + LL; |
---|
2100 | kill LL; |
---|
2101 | return(LS); |
---|
2102 | } |
---|
2103 | else |
---|
2104 | { |
---|
2105 | "j>i situation"; |
---|
2106 | // - might come first, at place i |
---|
2107 | if (i>1) |
---|
2108 | { |
---|
2109 | cnt++; |
---|
2110 | LS[cnt] = string(s[1..i-1]); |
---|
2111 | s = s[i..size(s)]; |
---|
2112 | } |
---|
2113 | else |
---|
2114 | { |
---|
2115 | // i==1-> minus at leadcoef |
---|
2116 | ie = find(s,"-",i+1); |
---|
2117 | je = find(s,"+",i+1); |
---|
2118 | if (je == ie) |
---|
2119 | { |
---|
2120 | "ie=je situation"; |
---|
2121 | //monomial |
---|
2122 | cnt++; |
---|
2123 | LS[cnt] = s; |
---|
2124 | return(LS); |
---|
2125 | } |
---|
2126 | if (je < ie) |
---|
2127 | { |
---|
2128 | "je<ie situation"; |
---|
2129 | // + comes first |
---|
2130 | cnt++; |
---|
2131 | LS[cnt] = s[1..je-1]; |
---|
2132 | s = s[je+1..size(s)]; |
---|
2133 | } |
---|
2134 | else |
---|
2135 | { |
---|
2136 | // ie < je |
---|
2137 | "ie<je situation"; |
---|
2138 | cnt++; |
---|
2139 | LS[cnt] = s[1..ie-1]; |
---|
2140 | s = s[ie..size(s)]; |
---|
2141 | } |
---|
2142 | } |
---|
2143 | "going into recursion with "+s; |
---|
2144 | LL = stringpoly2lplace(s); |
---|
2145 | LS = LS + LL; |
---|
2146 | return(LS); |
---|
2147 | } |
---|
2148 | } |
---|
2149 | example |
---|
2150 | { |
---|
2151 | "EXAMPLE:"; echo = 2; |
---|
2152 | string s = "x*y+y*z+z*t"; // + only |
---|
2153 | stringpoly2lplace(s); |
---|
2154 | string s2 = "x*y - y*z-z*t*w*w"; // +1, - only |
---|
2155 | stringpoly2lplace(s2); |
---|
2156 | string s3 = "-x*y + y - 2*x +7*w*w*w"; |
---|
2157 | stringpoly2lplace(s3); |
---|
2158 | } |
---|
2159 | |
---|
2160 | static proc addplaces(list L) |
---|
2161 | { |
---|
2162 | // adds places to the list of strings |
---|
2163 | // according to their order in the list |
---|
2164 | int s = size(L); |
---|
2165 | int i; |
---|
2166 | for (i=1; i<=s; i++) |
---|
2167 | { |
---|
2168 | if (typeof(L[i]) == "string") |
---|
2169 | { |
---|
2170 | L[i] = L[i] + "(" + string(i) + ")"; |
---|
2171 | } |
---|
2172 | else |
---|
2173 | { |
---|
2174 | ERROR("entry of type string expected"); |
---|
2175 | return(0); |
---|
2176 | } |
---|
2177 | } |
---|
2178 | return(L); |
---|
2179 | } |
---|
2180 | example |
---|
2181 | { |
---|
2182 | "EXAMPLE:"; echo = 2; |
---|
2183 | string a = "f1"; string b = "f2"; |
---|
2184 | list L = a,b,a; |
---|
2185 | addplaces(L); |
---|
2186 | } |
---|
2187 | |
---|
2188 | static proc sent2lplace(string s) |
---|
2189 | { |
---|
2190 | // SENTence of words TO LetterPLACE |
---|
2191 | list L = stringpoly2lplace(s); |
---|
2192 | int i; int ss = size(L); |
---|
2193 | for(i=1; i<=ss; i++) |
---|
2194 | { |
---|
2195 | L[i] = str2lplace(L[i]); |
---|
2196 | } |
---|
2197 | return(L); |
---|
2198 | } |
---|
2199 | example |
---|
2200 | { |
---|
2201 | "EXAMPLE:"; echo = 2; |
---|
2202 | ring r = 0,(f2,f1),dp; |
---|
2203 | string s = "f2*f1*f1 - 2*f1*f2*f1+ f1*f1*f2"; |
---|
2204 | sent2lplace(s); |
---|
2205 | } |
---|
2206 | |
---|
2207 | static proc testnumber(string s) |
---|
2208 | { |
---|
2209 | string t; |
---|
2210 | if (s[1]=="-") |
---|
2211 | { |
---|
2212 | // two situations: either there's a negative number |
---|
2213 | t = s[2..size(s)]; |
---|
2214 | if (testnumber(t)) |
---|
2215 | { |
---|
2216 | //a negative number |
---|
2217 | } |
---|
2218 | else |
---|
2219 | { |
---|
2220 | // a variable times -1 |
---|
2221 | } |
---|
2222 | // or just a "-" for -1 |
---|
2223 | } |
---|
2224 | t = "ring @r=("; |
---|
2225 | t = t + charstr(basering)+"),"; |
---|
2226 | t = t + string(var(1))+",dp;"; |
---|
2227 | // write(":w tstnum.tst",t); |
---|
2228 | t = t+ "number @@Nn = " + s + ";"+"$"; |
---|
2229 | write(":w tstnum.tst",t); |
---|
2230 | string runsing = system("Singular"); |
---|
2231 | int k; |
---|
2232 | t = runsing+ " -teq <tstnum.tst >tstnum.out"; |
---|
2233 | k = system("sh",t); |
---|
2234 | if (k!=0) |
---|
2235 | { |
---|
2236 | ERROR("Problems running Singular"); |
---|
2237 | } |
---|
2238 | int i = system("sh", "grep error tstnum.out > /dev/NULL"); |
---|
2239 | if (i!=0) |
---|
2240 | { |
---|
2241 | // no error: s is a number |
---|
2242 | i = 1; |
---|
2243 | } |
---|
2244 | k = system("sh","rm tstnum.tst tstnum.out > /dev/NULL"); |
---|
2245 | return(i); |
---|
2246 | } |
---|
2247 | example |
---|
2248 | { |
---|
2249 | "EXAMPLE:"; echo = 2; |
---|
2250 | ring r = (0,a),x,dp; |
---|
2251 | string s = "a^2+7*a-2"; |
---|
2252 | testnumber(s); |
---|
2253 | s = "b+a"; |
---|
2254 | testnumber(s); |
---|
2255 | } |
---|
2256 | |
---|
2257 | static proc str2lplace(string s) |
---|
2258 | { |
---|
2259 | // converts a word (monomial) with coeff into letter-place |
---|
2260 | // string: coef*var1^exp1*var2^exp2*...varN^expN |
---|
2261 | s = strpower2rep(s); // expand powers |
---|
2262 | if (size(s)==0) { return(0); } |
---|
2263 | int i,j,k,insC; |
---|
2264 | string a,b,c,d,t; |
---|
2265 | // 1. get coeff |
---|
2266 | i = find(s,"*"); |
---|
2267 | if (i==0) { return(s); } |
---|
2268 | list VN; |
---|
2269 | c = s[1..i-1]; // incl. the case like (-a^2+1) |
---|
2270 | int tn = testnumber(c); |
---|
2271 | if (tn == 0) |
---|
2272 | { |
---|
2273 | // failed test |
---|
2274 | if (c[1]=="-") |
---|
2275 | { |
---|
2276 | // two situations: either there's a negative number |
---|
2277 | t = c[2..size(c)]; |
---|
2278 | if (testnumber(t)) |
---|
2279 | { |
---|
2280 | //a negative number |
---|
2281 | // nop here |
---|
2282 | } |
---|
2283 | else |
---|
2284 | { |
---|
2285 | // a variable times -1 |
---|
2286 | c = "-1"; |
---|
2287 | j++; VN[j] = t; //string(c[2..size(c)]); |
---|
2288 | insC = 1; |
---|
2289 | } |
---|
2290 | } |
---|
2291 | else |
---|
2292 | { |
---|
2293 | // just a variable with coeff 1 |
---|
2294 | j++; VN[j] = string(c); |
---|
2295 | c = "1"; |
---|
2296 | insC = 1; |
---|
2297 | } |
---|
2298 | } |
---|
2299 | // get vars |
---|
2300 | t = s; |
---|
2301 | // t = s[i+1..size(s)]; |
---|
2302 | k = size(t); //j = 0; |
---|
2303 | while (k>0) |
---|
2304 | { |
---|
2305 | t = t[i+1..size(t)]; //next part |
---|
2306 | i = find(t,"*"); // next * |
---|
2307 | if (i==0) |
---|
2308 | { |
---|
2309 | // last monomial |
---|
2310 | j++; |
---|
2311 | VN[j] = t; |
---|
2312 | k = size(t); |
---|
2313 | break; |
---|
2314 | } |
---|
2315 | b = t[1..i-1]; |
---|
2316 | // print(b); |
---|
2317 | j++; |
---|
2318 | VN[j] = b; |
---|
2319 | k = size(t); |
---|
2320 | } |
---|
2321 | VN = addplaces(VN); |
---|
2322 | VN[size(VN)+1] = string(c); |
---|
2323 | return(VN); |
---|
2324 | } |
---|
2325 | example |
---|
2326 | { |
---|
2327 | "EXAMPLE:"; echo = 2; |
---|
2328 | ring r = (0,a),(f2,f1),dp; |
---|
2329 | str2lplace("-2*f2^2*f1^2*f2"); |
---|
2330 | str2lplace("-f1*f2"); |
---|
2331 | str2lplace("(-a^2+7a)*f1*f2"); |
---|
2332 | } |
---|
2333 | |
---|
2334 | static proc strpower2rep(string s) |
---|
2335 | { |
---|
2336 | // makes x*x*x*x out of x^4 ., rep statys for repetitions |
---|
2337 | // looks for "-" problem |
---|
2338 | // exception: "-" as coeff |
---|
2339 | string ex,t; |
---|
2340 | int i,j,k; |
---|
2341 | |
---|
2342 | i = find(s,"^"); // first ^ |
---|
2343 | if (i==0) { return(s); } // no ^ signs |
---|
2344 | |
---|
2345 | if (s[1] == "-") |
---|
2346 | { |
---|
2347 | // either -coef or -1 |
---|
2348 | // got the coeff: |
---|
2349 | i = find(s,"*"); |
---|
2350 | if (i==0) |
---|
2351 | { |
---|
2352 | // no *'s => coef == -1 or s == -23 |
---|
2353 | i = size(s)+1; |
---|
2354 | } |
---|
2355 | t = string(s[2..i-1]); // without "-" |
---|
2356 | if ( testnumber(t) ) |
---|
2357 | { |
---|
2358 | // a good number |
---|
2359 | t = strpower2rep(string(s[2..size(s)])); |
---|
2360 | t = "-"+t; |
---|
2361 | return(t); |
---|
2362 | } |
---|
2363 | else |
---|
2364 | { |
---|
2365 | // a variable |
---|
2366 | t = strpower2rep(string(s[2..size(s)])); |
---|
2367 | t = "-1*"+ t; |
---|
2368 | return(t); |
---|
2369 | } |
---|
2370 | } |
---|
2371 | // the case when leadcoef is a number in () |
---|
2372 | if (s[1] == "(") |
---|
2373 | { |
---|
2374 | i = find(s,")",2); // must be nonzero |
---|
2375 | t = s[2..i-1]; |
---|
2376 | if ( testnumber(t) ) |
---|
2377 | { |
---|
2378 | // a good number |
---|
2379 | } |
---|
2380 | else {"strpower2rep: bad number as coef";} |
---|
2381 | ex = string(s[i+2..size(s)]); // 2 because of * |
---|
2382 | ex = strpower2rep(ex); |
---|
2383 | t = "("+t+")*"+ex; |
---|
2384 | return(t); |
---|
2385 | } |
---|
2386 | |
---|
2387 | i = find(s,"^"); // first ^ |
---|
2388 | j = find(s,"*",i+1); // next * == end of ^ |
---|
2389 | if (j==0) |
---|
2390 | { |
---|
2391 | ex = s[i+1..size(s)]; |
---|
2392 | } |
---|
2393 | else |
---|
2394 | { |
---|
2395 | ex = s[i+1..j-1]; |
---|
2396 | } |
---|
2397 | execute("int @exp = " + ex + ";"); //@exp = exponent |
---|
2398 | // got varname |
---|
2399 | for (k=i-1; k>0; k--) |
---|
2400 | { |
---|
2401 | if (s[k] == "*") break; |
---|
2402 | } |
---|
2403 | string varn = s[k+1..i-1]; |
---|
2404 | // "varn:"; varn; |
---|
2405 | string pref; |
---|
2406 | if (k>0) |
---|
2407 | { |
---|
2408 | pref = s[1..k]; // with * on the k-th place |
---|
2409 | } |
---|
2410 | // "pref:"; pref; |
---|
2411 | string suf; |
---|
2412 | if ( (j>0) && (j+1 <= size(s)) ) |
---|
2413 | { |
---|
2414 | suf = s[j+1..size(s)]; // without * on the 1st place |
---|
2415 | } |
---|
2416 | // "suf:"; suf; |
---|
2417 | string toins; |
---|
2418 | for (k=1; k<=@exp; k++) |
---|
2419 | { |
---|
2420 | toins = toins + varn+"*"; |
---|
2421 | } |
---|
2422 | // "toins: "; toins; |
---|
2423 | if (size(suf) == 0) |
---|
2424 | { |
---|
2425 | toins = toins[1..size(toins)-1]; // get rid of trailing * |
---|
2426 | } |
---|
2427 | else |
---|
2428 | { |
---|
2429 | suf = strpower2rep(suf); |
---|
2430 | } |
---|
2431 | ex = pref + toins + suf; |
---|
2432 | return(ex); |
---|
2433 | // return(strpower2rep(ex)); |
---|
2434 | } |
---|
2435 | example |
---|
2436 | { |
---|
2437 | "EXAMPLE:"; echo = 2; |
---|
2438 | ring r = (0,a),(x,y,z,t),dp; |
---|
2439 | strpower2rep("-x^4"); |
---|
2440 | strpower2rep("-2*x^4*y^3*z*t^2"); |
---|
2441 | strpower2rep("-a^2*x^4"); |
---|
2442 | } |
---|
2443 | |
---|
2444 | proc lieBracket(poly a, poly b, list #) |
---|
2445 | "USAGE: lieBracket(a,b[,N]); a,b letterplace polynomials, N an optional integer |
---|
2446 | RETURN: poly |
---|
2447 | ASSUME: basering has a letterplace ring structure |
---|
2448 | PURPOSE:compute the Lie bracket [a,b] = ab - ba between letterplace polynomials |
---|
2449 | NOTE: if N>1 is specified, then the left normed bracket [a,[...[a,b]]]] is |
---|
2450 | @* computed. |
---|
2451 | EXAMPLE: example lieBracket; shows examples |
---|
2452 | " |
---|
2453 | { |
---|
2454 | if (lpAssumeViolation()) |
---|
2455 | { |
---|
2456 | // ERROR("Either 'uptodeg' or 'lV' global variables are not set!"); |
---|
2457 | ERROR("Incomplete Letterplace structure on the basering!"); |
---|
2458 | } |
---|
2459 | // alias ppLiebr; |
---|
2460 | //if int N is given compute [a,[...[a,b]]]] left normed bracket |
---|
2461 | poly q; |
---|
2462 | int N=1; |
---|
2463 | if (size(#)>0) |
---|
2464 | { |
---|
2465 | if (typeof(#[1])=="int") |
---|
2466 | { |
---|
2467 | N = int(#[1]); |
---|
2468 | } |
---|
2469 | } |
---|
2470 | if (N<=0) { return(q); } |
---|
2471 | while (b!=0) |
---|
2472 | { |
---|
2473 | q = q + pmLiebr(a,lead(b)); |
---|
2474 | b = b - lead(b); |
---|
2475 | } |
---|
2476 | int i; |
---|
2477 | if (N >1) |
---|
2478 | { |
---|
2479 | for(i=1; i<=N; i++) |
---|
2480 | { |
---|
2481 | q = lieBracket(a,q); |
---|
2482 | } |
---|
2483 | } |
---|
2484 | return(q); |
---|
2485 | } |
---|
2486 | example |
---|
2487 | { |
---|
2488 | "EXAMPLE:"; echo = 2; |
---|
2489 | ring r = 0,(x(1),y(1),x(2),y(2),x(3),y(3),x(4),y(4)),dp; |
---|
2490 | def R = setLetterplaceAttributes(r,4,2); // supply R with letterplace structure |
---|
2491 | setring R; |
---|
2492 | poly a = x(1)*y(2); poly b = y(1); |
---|
2493 | lieBracket(a,b); |
---|
2494 | lieBracket(x(1),y(1),2); |
---|
2495 | } |
---|
2496 | |
---|
2497 | static proc pmLiebr(poly a, poly b) |
---|
2498 | { |
---|
2499 | // a poly, b mono |
---|
2500 | poly s; |
---|
2501 | while (a!=0) |
---|
2502 | { |
---|
2503 | s = s + mmLiebr(lead(a),lead(b)); |
---|
2504 | a = a - lead(a); |
---|
2505 | } |
---|
2506 | return(s); |
---|
2507 | } |
---|
2508 | |
---|
2509 | proc shiftPoly(poly a, int i) |
---|
2510 | "USAGE: shiftPoly(p,i); p letterplace poly, i int |
---|
2511 | RETURN: poly |
---|
2512 | ASSUME: basering has letterplace ring structure |
---|
2513 | PURPOSE: compute the i-th shift of letterplace polynomial p |
---|
2514 | EXAMPLE: example shiftPoly; shows examples |
---|
2515 | " |
---|
2516 | { |
---|
2517 | // shifts a monomial a by i |
---|
2518 | // calls pLPshift(p,sh,uptodeg,lVblock); |
---|
2519 | if (lpAssumeViolation()) |
---|
2520 | { |
---|
2521 | ERROR("Incomplete Letterplace structure on the basering!"); |
---|
2522 | } |
---|
2523 | int uptodeg = attrib(basering,"uptodeg"); |
---|
2524 | int lV = attrib(basering,"lV"); |
---|
2525 | if (deg(a) + i > uptodeg) |
---|
2526 | { |
---|
2527 | ERROR("degree bound violated by the shift!"); |
---|
2528 | } |
---|
2529 | return(system("stest",a,i,uptodeg,lV)); |
---|
2530 | } |
---|
2531 | example |
---|
2532 | { |
---|
2533 | "EXAMPLE:"; echo = 2; |
---|
2534 | ring r = 0,(x,y,z),dp; |
---|
2535 | int uptodeg = 5; int lV = 3; |
---|
2536 | def R = makeLetterplaceRing(uptodeg); |
---|
2537 | setring R; |
---|
2538 | poly f = x(1)*z(2)*y(3) - 2*z(1)*y(2) + 3*x(1); |
---|
2539 | shiftPoly(f,1); |
---|
2540 | shiftPoly(f,2); |
---|
2541 | } |
---|
2542 | |
---|
2543 | |
---|
2544 | static proc mmLiebr(poly a, poly b) |
---|
2545 | { |
---|
2546 | // a,b, monomials |
---|
2547 | a = lead(a); |
---|
2548 | b = lead(b); |
---|
2549 | int sa = deg(a); |
---|
2550 | int sb = deg(b); |
---|
2551 | poly v = a*shiftPoly(b,sa) - b*shiftPoly(a,sb); |
---|
2552 | return(v); |
---|
2553 | } |
---|
2554 | |
---|
2555 | static proc test_shift() |
---|
2556 | { |
---|
2557 | LIB "freegb.lib"; |
---|
2558 | ring r = 0,(a,b),dp; |
---|
2559 | int d =5; |
---|
2560 | def R = makeLetterplaceRing(d); |
---|
2561 | setring R; |
---|
2562 | int uptodeg = d; |
---|
2563 | int lV = 2; |
---|
2564 | def R = setLetterplaceAttributes(r,uptodeg,2); // supply R with letterplace structure |
---|
2565 | setring R; |
---|
2566 | poly p = mmLiebr(a(1),b(1)); |
---|
2567 | poly p = lieBracket(a(1),b(1)); |
---|
2568 | } |
---|
2569 | |
---|
2570 | proc serreRelations(intmat A, int zu) |
---|
2571 | "USAGE: serreRelations(A,z); A an intmat, z an int |
---|
2572 | RETURN: ideal |
---|
2573 | ASSUME: basering has a letterplace ring structure and |
---|
2574 | @* A is a generalized Cartan matrix with integer entries |
---|
2575 | PURPOSE: compute the ideal of Serre's relations associated to A |
---|
2576 | EXAMPLE: example serreRelations; shows examples |
---|
2577 | " |
---|
2578 | { |
---|
2579 | // zu = 1 -> with commutators [f_i,f_j]; zu == 0 without them |
---|
2580 | // suppose that A is cartan matrix |
---|
2581 | // then Serre's relations are |
---|
2582 | // (ad f_j)^{1-A_{ij}} ( f_i) |
---|
2583 | int ppl = printlevel-voice+2; |
---|
2584 | int n = ncols(A); // hence n variables |
---|
2585 | int i,j,k,el; |
---|
2586 | poly p,q; |
---|
2587 | ideal I; |
---|
2588 | for (i=1; i<=n; i++) |
---|
2589 | { |
---|
2590 | for (j=1; j<=n; j++) |
---|
2591 | { |
---|
2592 | el = 1 - A[i,j]; |
---|
2593 | // printf("i:%s, j: %s, l: %s",i,j,l); |
---|
2594 | dbprint(ppl,"i, j, l: ",i,j,el); |
---|
2595 | // if ((i!=j) && (l >0)) |
---|
2596 | // if ( (i!=j) && ( ((zu ==0) && (l >=2)) || ((zu ==1) && (l >=1)) ) ) |
---|
2597 | if ((i!=j) && (el >0)) |
---|
2598 | { |
---|
2599 | q = lieBracket(var(j),var(i)); |
---|
2600 | dbprint(ppl,"first bracket: ",q); |
---|
2601 | // if (l >=2) |
---|
2602 | // { |
---|
2603 | for (k=1; k<=el-1; k++) |
---|
2604 | { |
---|
2605 | q = lieBracket(var(j),q); |
---|
2606 | dbprint(ppl,"further bracket:",q); |
---|
2607 | } |
---|
2608 | // } |
---|
2609 | } |
---|
2610 | if (q!=0) { I = I,q; q=0;} |
---|
2611 | } |
---|
2612 | } |
---|
2613 | I = simplify(I,2); |
---|
2614 | return(I); |
---|
2615 | } |
---|
2616 | example |
---|
2617 | { |
---|
2618 | "EXAMPLE:"; echo = 2; |
---|
2619 | intmat A[3][3] = |
---|
2620 | 2, -1, 0, |
---|
2621 | -1, 2, -3, |
---|
2622 | 0, -1, 2; // G^1_2 Cartan matrix |
---|
2623 | ring r = 0,(f1,f2,f3),dp; |
---|
2624 | int uptodeg = 5; |
---|
2625 | def R = makeLetterplaceRing(uptodeg); |
---|
2626 | setring R; |
---|
2627 | ideal I = serreRelations(A,1); I = simplify(I,1+2+8); |
---|
2628 | I; |
---|
2629 | } |
---|
2630 | |
---|
2631 | /* setup for older example: |
---|
2632 | intmat A[2][2] = 2, -1, -1, 2; // sl_3 == A_2 |
---|
2633 | ring r = 0,(f1,f2),dp; |
---|
2634 | int uptodeg = 5; int lV = 2; |
---|
2635 | */ |
---|
2636 | |
---|
2637 | proc fullSerreRelations(intmat A, ideal rNegative, ideal rCartan, ideal rPositive, int uptodeg) |
---|
2638 | "USAGE: fullSerreRelations(A,N,C,P,d); A an intmat, N,C,P ideals, d an int |
---|
2639 | RETURN: ring (and ideal) |
---|
2640 | PURPOSE: compute the inhomogeneous Serre's relations associated to A in given |
---|
2641 | @* variable names |
---|
2642 | ASSUME: three ideals in the input are of the same sizes and contain merely |
---|
2643 | @* variables which are interpreted as follows: N resp. P stand for negative |
---|
2644 | @* resp. positive roots, C stand for Cartan elements. d is the degree bound for |
---|
2645 | @* letterplace ring, which will be returned. |
---|
2646 | @* The matrix A is a generalized Cartan matrix with integer entries |
---|
2647 | @* The result is the ideal called 'fsRel' in the returned ring. |
---|
2648 | EXAMPLE: example fullSerreRelations; shows examples |
---|
2649 | " |
---|
2650 | { |
---|
2651 | /* SerreRels on rNeg and rPos plus Cartans etc. */ |
---|
2652 | int ppl = printlevel -voice+2; |
---|
2653 | /* ideals must be written in variables: assume each term is of degree 1 */ |
---|
2654 | int i,j,k; |
---|
2655 | int N = nvars(basering); |
---|
2656 | def save = basering; |
---|
2657 | int comFlag = 0; |
---|
2658 | /* assume: (size(rNegative) == size(rPositive)) */ |
---|
2659 | /* assume: (size(rNegative) == size(rCartan)) i.e. nonsimple Cartans */ |
---|
2660 | if ( (size(rNegative) != size(rPositive)) || (size(rNegative) != size(rCartan)) ) |
---|
2661 | { |
---|
2662 | ERROR("All input ideals must be of the same size"); |
---|
2663 | } |
---|
2664 | |
---|
2665 | // if (size(rNegative) != size(rPositive)) |
---|
2666 | // { |
---|
2667 | // ERROR("The 1st and the 3rd input ideals must be of the same size"); |
---|
2668 | // } |
---|
2669 | |
---|
2670 | /* assume: 2*size(rNegative) + size(rCartan) >= nvars(basering) */ |
---|
2671 | i = 2*size(rNegative) + size(rCartan); |
---|
2672 | if (i>N) |
---|
2673 | { |
---|
2674 | string s1="The total number of elements in input ideals"; |
---|
2675 | string s2="must not exceed the dimension of the ground ring"; |
---|
2676 | ERROR(s1+s2); |
---|
2677 | } |
---|
2678 | if (i < N) |
---|
2679 | { |
---|
2680 | comFlag = N-i; // so many elements will commute |
---|
2681 | "Warning: some elements will be treated as mutually commuting"; |
---|
2682 | } |
---|
2683 | /* extract varnames from input ideals */ |
---|
2684 | intvec iNeg = varIdeal2intvec(rNegative); |
---|
2685 | intvec iCartan = varIdeal2intvec(rCartan); |
---|
2686 | intvec iPos = varIdeal2intvec(rPositive); |
---|
2687 | /* for each vector in rNeg and rPositive, go into the corr. ring and create SerreRels */ |
---|
2688 | /* rNegative: */ |
---|
2689 | list L = ringlist(save); |
---|
2690 | def LPsave = makeLetterplaceRing2(uptodeg); setring save; |
---|
2691 | list LNEG = L; list tmp; |
---|
2692 | /* L[1] field as is; L[2] vars: a subset; L[3] ordering: dp, L[4] as is */ |
---|
2693 | for (i=1; i<=size(iNeg); i++) |
---|
2694 | { |
---|
2695 | tmp[i] = string(var(iNeg[i])); |
---|
2696 | } |
---|
2697 | LNEG[2] = tmp; LNEG[3] = list(list("dp",intvec(1:size(iNeg))), list("C",0)); |
---|
2698 | def RNEG = ring(LNEG); setring RNEG; |
---|
2699 | def RRNEG = makeLetterplaceRing2(uptodeg); |
---|
2700 | setring RRNEG; |
---|
2701 | ideal I = serreRelations(A,1); I = simplify(I,1+2+8); |
---|
2702 | setring LPsave; |
---|
2703 | ideal srNeg = imap(RRNEG,I); |
---|
2704 | dbprint(ppl,"0-1 ideal of negative relations is ready"); |
---|
2705 | dbprint(ppl-1,srNeg); |
---|
2706 | setring save; kill L,tmp,RRNEG,RNEG, LNEG; |
---|
2707 | /* rPositive: */ |
---|
2708 | list L = ringlist(save); |
---|
2709 | list LPOS = L; list tmp; |
---|
2710 | /* L[1] field as is; L[2] vars: a subset; L[3] ordering: dp, L[4] as is */ |
---|
2711 | for (i=1; i<=size(iPos); i++) |
---|
2712 | { |
---|
2713 | tmp[i] = string(var(iPos[i])); |
---|
2714 | } |
---|
2715 | LPOS[2] = tmp; LPOS[3] = list(list("dp",intvec(1:size(iPos))), list("C",0)); |
---|
2716 | def RPOS = ring(LPOS); setring RPOS; |
---|
2717 | def RRPOS = makeLetterplaceRing2(uptodeg); |
---|
2718 | setring RRPOS; |
---|
2719 | ideal I = serreRelations(A,1); I = simplify(I,1+2+8); |
---|
2720 | setring LPsave; |
---|
2721 | ideal srPos = imap(RRPOS,I); |
---|
2722 | dbprint(ppl,"0-2 ideal of positive relations is ready"); |
---|
2723 | dbprint(ppl-1,srPos); |
---|
2724 | setring save; kill L,tmp,RRPOS,RPOS, LPOS; |
---|
2725 | string sMap = "ideal Mmap ="; |
---|
2726 | for (i=1; i<=nvars(save); i++) |
---|
2727 | { |
---|
2728 | sMap = sMap + string(var(i)) +"(1),"; |
---|
2729 | } |
---|
2730 | sMap[size(sMap)] = ";"; |
---|
2731 | /* cartans: h_j h_i = h_i h_j */ |
---|
2732 | setring LPsave; |
---|
2733 | ideal ComCartan; |
---|
2734 | for (i=1; i<size(iCartan); i++) |
---|
2735 | { |
---|
2736 | for (j=i+1; j<=size(iCartan); j++) |
---|
2737 | { |
---|
2738 | ComCartan = ComCartan + lieBracket(var(iCartan[j]),var(iCartan[i])); |
---|
2739 | } |
---|
2740 | } |
---|
2741 | ComCartan = simplify(ComCartan,1+2+8); |
---|
2742 | execute(sMap); // defines an ideal Mmap |
---|
2743 | map F = save, Mmap; |
---|
2744 | dbprint(ppl,"1. commuting Cartans: "); |
---|
2745 | dbprint(ppl-1,ComCartan); |
---|
2746 | /* [e_i, f_j] =0 if i<>j */ |
---|
2747 | ideal ComPosNeg; // assume: #Neg=#Pos |
---|
2748 | for (i=1; i<size(iPos); i++) |
---|
2749 | { |
---|
2750 | for (j=1; j<=size(iPos); j++) |
---|
2751 | { |
---|
2752 | if (j !=i) |
---|
2753 | { |
---|
2754 | ComPosNeg = ComPosNeg + lieBracket(var(iPos[i]),var(iNeg[j])); |
---|
2755 | ComPosNeg = ComPosNeg + lieBracket(var(iPos[j]),var(iNeg[i])); |
---|
2756 | } |
---|
2757 | } |
---|
2758 | } |
---|
2759 | ComPosNeg = simplify(ComPosNeg,1+2+8); |
---|
2760 | dbprint(ppl,"2. commuting Positive and Negative:"); |
---|
2761 | dbprint(ppl-1,ComPosNeg); |
---|
2762 | /* [e_i, f_i] = h_i */ |
---|
2763 | poly tempo; |
---|
2764 | for (i=1; i<=size(iCartan); i++) |
---|
2765 | { |
---|
2766 | tempo = lieBracket(var(iPos[i]),var(iNeg[i])) - var(iCartan[i]); |
---|
2767 | ComPosNeg = ComPosNeg + tempo; |
---|
2768 | } |
---|
2769 | // ComPosNeg = simplify(ComPosNeg,1+2+8); |
---|
2770 | dbprint(ppl,"3. added sl2 triples [e_i,f_i]=h_i"); |
---|
2771 | dbprint(ppl-1,ComPosNeg); |
---|
2772 | |
---|
2773 | /* [h_i, e_j] = A_ij e_j */ |
---|
2774 | /* [h_i, f_j] = -A_ij f_j */ |
---|
2775 | ideal ActCartan; // assume: #Neg=#Pos |
---|
2776 | for (i=1; i<=size(iCartan); i++) |
---|
2777 | { |
---|
2778 | for (j=1; j<=size(iCartan); j++) |
---|
2779 | { |
---|
2780 | tempo = lieBracket(var(iCartan[i]),var(iPos[j])) - A[i,j]*var(iPos[j]); |
---|
2781 | ActCartan = ActCartan + tempo; |
---|
2782 | tempo = lieBracket(var(iCartan[i]),var(iNeg[j])) + A[i,j]*var(iNeg[j]); |
---|
2783 | ActCartan = ActCartan + tempo; |
---|
2784 | } |
---|
2785 | } |
---|
2786 | ActCartan = simplify(ActCartan,1+2+8); |
---|
2787 | dbprint(ppl,"4. actions of Cartan:"); |
---|
2788 | dbprint(ppl-1, ActCartan); |
---|
2789 | |
---|
2790 | /* final part: prepare the output */ |
---|
2791 | setring LPsave; |
---|
2792 | ideal fsRel = srNeg, srPos, ComPosNeg, ComCartan, ActCartan; |
---|
2793 | export fsRel; |
---|
2794 | setring save; |
---|
2795 | return(LPsave); |
---|
2796 | } |
---|
2797 | example |
---|
2798 | { |
---|
2799 | "EXAMPLE:"; echo = 2; |
---|
2800 | intmat A[2][2] = |
---|
2801 | 2, -1, |
---|
2802 | -1, 2; // A_2 = sl_3 Cartan matrix |
---|
2803 | ring r = 0,(f1,f2,h1,h2,e1,e2),dp; |
---|
2804 | ideal negroots = f1,f2; ideal cartans = h1,h2; ideal posroots = e1,e2; |
---|
2805 | int uptodeg = 5; |
---|
2806 | def RS = fullSerreRelations(A,negroots,cartans,posroots,uptodeg); |
---|
2807 | setring RS; fsRel; |
---|
2808 | } |
---|
2809 | |
---|
2810 | static proc varIdeal2intvec(ideal I) |
---|
2811 | { |
---|
2812 | // used in SerreRelations |
---|
2813 | /* assume1: input ideal is a list of variables of the ground ring */ |
---|
2814 | int i,j; intvec V; |
---|
2815 | for (i=1; i<= size(I); i++) |
---|
2816 | { |
---|
2817 | j = univariate(I[i]); |
---|
2818 | if (j<=0) |
---|
2819 | { |
---|
2820 | ERROR("input ideal must contain only variables"); |
---|
2821 | } |
---|
2822 | V[i] = j; |
---|
2823 | } |
---|
2824 | dbprint(printlevel-voice+2,V); |
---|
2825 | /* now we make a smaller list of non-repeating entries */ |
---|
2826 | ideal iW = simplify(ideal(V),2+4); // no zeros, no repetitions |
---|
2827 | if (size(iW) < size(V)) |
---|
2828 | { |
---|
2829 | /* extract intvec from iW */ |
---|
2830 | intvec inW; |
---|
2831 | for(j=1; j<=size(iW); j++) |
---|
2832 | { |
---|
2833 | inW[j] = int(leadcoef(iW[j])); |
---|
2834 | } |
---|
2835 | return(inW); |
---|
2836 | } |
---|
2837 | return(V); |
---|
2838 | } |
---|
2839 | example |
---|
2840 | { |
---|
2841 | "EXAMPLE:"; echo = 2; |
---|
2842 | ring r = 0,(x,y,z),dp; |
---|
2843 | ideal I = x,z; |
---|
2844 | varIdeal2intvec(I); |
---|
2845 | varIdeal2intvec(ideal(x2,y^3,x+1)); |
---|
2846 | varIdeal2intvec(ideal(x*y,y,x+1)); |
---|
2847 | } |
---|
2848 | |
---|
2849 | proc lp2lstr(ideal K, def save) |
---|
2850 | "USAGE: lp2lstr(K,s); K an ideal, s a ring name |
---|
2851 | RETURN: nothing (exports object @LN into the ring named s) |
---|
2852 | ASSUME: basering has a letterplace ring structure |
---|
2853 | PURPOSE: converts letterplace ideal to list of modules |
---|
2854 | NOTE: useful as preprocessing to 'lst2str' |
---|
2855 | EXAMPLE: example lp2lstr; shows examples |
---|
2856 | " |
---|
2857 | { |
---|
2858 | def @R = basering; |
---|
2859 | string err; |
---|
2860 | int s = nvars(save); |
---|
2861 | int i,j,k; |
---|
2862 | // K contains vars x(1),...z(1) = images of originals |
---|
2863 | // 5. go back to orig vars, produce strings/modules |
---|
2864 | int sk = size(K); |
---|
2865 | int sp, sx, a, b; |
---|
2866 | intvec x; |
---|
2867 | poly p,q; |
---|
2868 | poly pn; |
---|
2869 | // vars in 'save' |
---|
2870 | setring save; |
---|
2871 | module N; |
---|
2872 | list LN; |
---|
2873 | vector V; |
---|
2874 | poly pn; |
---|
2875 | // test and skip exponents >=2 |
---|
2876 | setring @R; |
---|
2877 | for(i=1; i<=sk; i++) |
---|
2878 | { |
---|
2879 | p = K[i]; |
---|
2880 | while (p!=0) |
---|
2881 | { |
---|
2882 | q = lead(p); |
---|
2883 | // "processing q:";q; |
---|
2884 | x = leadexp(q); |
---|
2885 | sx = size(x); |
---|
2886 | for(k=1; k<=sx; k++) |
---|
2887 | { |
---|
2888 | if ( x[k] >= 2 ) |
---|
2889 | { |
---|
2890 | err = "skip: the value x[k] is " + string(x[k]); |
---|
2891 | dbprint(ppl,err); |
---|
2892 | // return(0); |
---|
2893 | K[i] = 0; |
---|
2894 | p = 0; |
---|
2895 | q = 0; |
---|
2896 | break; |
---|
2897 | } |
---|
2898 | } |
---|
2899 | p = p - q; |
---|
2900 | } |
---|
2901 | } |
---|
2902 | K = simplify(K,2); |
---|
2903 | sk = size(K); |
---|
2904 | for(i=1; i<=sk; i++) |
---|
2905 | { |
---|
2906 | // setring save; |
---|
2907 | // V = 0; |
---|
2908 | setring @R; |
---|
2909 | p = K[i]; |
---|
2910 | while (p!=0) |
---|
2911 | { |
---|
2912 | q = lead(p); |
---|
2913 | err = "processing q:" + string(q); |
---|
2914 | dbprint(ppl,err); |
---|
2915 | x = leadexp(q); |
---|
2916 | sx = size(x); |
---|
2917 | pn = leadcoef(q); |
---|
2918 | setring save; |
---|
2919 | pn = imap(@R,pn); |
---|
2920 | V = V + leadcoef(pn)*gen(1); |
---|
2921 | for(k=1; k<=sx; k++) |
---|
2922 | { |
---|
2923 | if (x[k] ==1) |
---|
2924 | { |
---|
2925 | a = k div s; // block number=a+1, a!=0 |
---|
2926 | b = k % s; // remainder |
---|
2927 | // printf("a: %s, b: %s",a,b); |
---|
2928 | if (b == 0) |
---|
2929 | { |
---|
2930 | // that is it's the last var in the block |
---|
2931 | b = s; |
---|
2932 | a = a-1; |
---|
2933 | } |
---|
2934 | V = V + var(b)*gen(a+2); |
---|
2935 | } |
---|
2936 | } |
---|
2937 | err = "V: " + string(V); |
---|
2938 | dbprint(ppl,err); |
---|
2939 | // printf("V: %s", string(V)); |
---|
2940 | N = N,V; |
---|
2941 | V = 0; |
---|
2942 | setring @R; |
---|
2943 | p = p - q; |
---|
2944 | pn = 0; |
---|
2945 | } |
---|
2946 | setring save; |
---|
2947 | LN[i] = simplify(N,2); |
---|
2948 | N = 0; |
---|
2949 | } |
---|
2950 | setring save; |
---|
2951 | list @LN = LN; |
---|
2952 | export @LN; |
---|
2953 | // return(LN); |
---|
2954 | } |
---|
2955 | example |
---|
2956 | { |
---|
2957 | "EXAMPLE:"; echo = 2; |
---|
2958 | intmat A[2][2] = 2, -1, -1, 2; // sl_3 == A_2 |
---|
2959 | ring r = 0,(f1,f2),dp; |
---|
2960 | def R = makeLetterplaceRing(3); |
---|
2961 | setring R; |
---|
2962 | ideal I = serreRelations(A,1); |
---|
2963 | lp2lstr(I,r); |
---|
2964 | setring r; |
---|
2965 | lst2str(@LN,1); |
---|
2966 | } |
---|
2967 | |
---|
2968 | static proc strList2poly(list L) |
---|
2969 | { |
---|
2970 | // list L comes from sent2lplace (which takes a polynomial as input) |
---|
2971 | // each entry of L is a sublist with the coef on the last place |
---|
2972 | int s = size(L); int t; |
---|
2973 | int i,j; |
---|
2974 | list M; |
---|
2975 | poly p,q; |
---|
2976 | string Q; |
---|
2977 | for(i=1; i<=s; i++) |
---|
2978 | { |
---|
2979 | M = L[i]; |
---|
2980 | t = size(M); |
---|
2981 | // q = M[t]; // a constant |
---|
2982 | Q = string(M[t]); |
---|
2983 | for(j=1; j<t; j++) |
---|
2984 | { |
---|
2985 | // q = q*M[j]; |
---|
2986 | Q = Q+"*"+string(M[j]); |
---|
2987 | } |
---|
2988 | execute("q="+Q+";"); |
---|
2989 | // q; |
---|
2990 | p = p + q; |
---|
2991 | } |
---|
2992 | kill Q; |
---|
2993 | return(p); |
---|
2994 | } |
---|
2995 | example |
---|
2996 | { |
---|
2997 | "EXAMPLE:"; echo = 2; |
---|
2998 | ring r =0,(x,y,z,t),Dp; |
---|
2999 | def A = makeLetterplaceRing(4); |
---|
3000 | setring A; |
---|
3001 | string t = "-2*y*z*y*z + y*t*z*z - z*x*x*y + 2*z*y*z*y"; |
---|
3002 | list L = sent2lplace(t); |
---|
3003 | L; |
---|
3004 | poly p = strList2poly(L); |
---|
3005 | p; |
---|
3006 | } |
---|
3007 | |
---|
3008 | static proc file2lplace(string fname) |
---|
3009 | "USAGE: file2lplace(fnm); fnm a string |
---|
3010 | RETURN: ideal |
---|
3011 | PURPOSE: convert the contents of the file fnm into ideal of polynomials in free algebra |
---|
3012 | EXAMPLE: example file2lplace; shows examples |
---|
3013 | " |
---|
3014 | { |
---|
3015 | // format: from the usual string to letterplace |
---|
3016 | string s = read(fname); |
---|
3017 | // assume: file is a comma-sep list of polys |
---|
3018 | // the vars are declared before |
---|
3019 | // the file ends with ";" |
---|
3020 | string t; int i; |
---|
3021 | ideal I; |
---|
3022 | list tst; |
---|
3023 | while (s!="") |
---|
3024 | { |
---|
3025 | i = find(s,","); |
---|
3026 | "i"; i; |
---|
3027 | if (i==0) |
---|
3028 | { |
---|
3029 | i = find(s,";"); |
---|
3030 | if (i==0) |
---|
3031 | { |
---|
3032 | // no ; ?? |
---|
3033 | "no colon or semicolon found anymore"; |
---|
3034 | return(I); |
---|
3035 | } |
---|
3036 | // no "," but ";" on the i-th place |
---|
3037 | t = s[1..i-1]; |
---|
3038 | s = ""; |
---|
3039 | "processing: "; t; |
---|
3040 | tst = sent2lplace(t); |
---|
3041 | tst; |
---|
3042 | I = I, strList2poly(tst); |
---|
3043 | return(I); |
---|
3044 | } |
---|
3045 | // here i !=0 |
---|
3046 | t = s[1..i-1]; |
---|
3047 | s = s[i+1..size(s)]; |
---|
3048 | "processing: "; t; |
---|
3049 | tst = sent2lplace(t); |
---|
3050 | tst; |
---|
3051 | I = I, strList2poly(tst); |
---|
3052 | } |
---|
3053 | return(I); |
---|
3054 | } |
---|
3055 | example |
---|
3056 | { |
---|
3057 | "EXAMPLE:"; echo = 2; |
---|
3058 | ring r =0,(x,y,z,t),dp; |
---|
3059 | def A = makeLetterplaceRing(4); |
---|
3060 | setring A; |
---|
3061 | string fn = "myfile"; |
---|
3062 | string s1 = "z*y*y*y - 3*y*z*x*y + 3*y*y*z*y - y*x*y*z,"; |
---|
3063 | string s2 = "-2*y*x*y*z + y*y*z*z - z*z*y*y + 2*z*y*z*y,"; |
---|
3064 | 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;"; |
---|
3065 | write(":w "+fn,s1); write(":a "+fn,s2); write(":a "+fn,s3); |
---|
3066 | read(fn); |
---|
3067 | ideal I = file2lplace(fn); |
---|
3068 | I; |
---|
3069 | } |
---|
3070 | |
---|
3071 | /* EXAMPLES AGAIN: |
---|
3072 | //static proc get_ls3nilp() |
---|
3073 | { |
---|
3074 | //first app of file2lplace |
---|
3075 | ring r =0,(x,y,z,t),dp; |
---|
3076 | int d = 10; |
---|
3077 | def A = makeLetterplaceRing(d); |
---|
3078 | setring A; |
---|
3079 | ideal I = file2lplace("./ls3nilp.bg"); |
---|
3080 | // and now test the correctness: go back from lplace to strings |
---|
3081 | lp2lstr(I,r); |
---|
3082 | setring r; |
---|
3083 | lst2str(@LN,1); // agree! |
---|
3084 | } |
---|
3085 | |
---|
3086 | //static proc doc_example() |
---|
3087 | { |
---|
3088 | LIB "freegb.lib"; |
---|
3089 | ring r = 0,(x,y,z),dp; |
---|
3090 | int d =4; // degree bound |
---|
3091 | def R = makeLetterplaceRing(d); |
---|
3092 | setring R; |
---|
3093 | 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); |
---|
3094 | option(redSB);option(redTail); |
---|
3095 | ideal J = system("freegb",I,d,nvars(r)); |
---|
3096 | J; |
---|
3097 | // visualization: |
---|
3098 | lp2lstr(J,r); // export an object called @LN to the ring r |
---|
3099 | setring r; // change to the ring r |
---|
3100 | lst2str(@LN,1); // output the strings |
---|
3101 | } |
---|
3102 | |
---|
3103 | */ |
---|
3104 | |
---|
3105 | static proc lpMultX(poly f, poly g) |
---|
3106 | { |
---|
3107 | /* multiplies two polys in a very general setting correctly */ |
---|
3108 | /* alternative to lpMult, possibly better at non-positive orderings */ |
---|
3109 | |
---|
3110 | if (lpAssumeViolation()) |
---|
3111 | { |
---|
3112 | ERROR("Incomplete Letterplace structure on the basering!"); |
---|
3113 | } |
---|
3114 | // decompose f,g into graded pieces with inForm: need dmodapp.lib |
---|
3115 | int b = attrib(basering,"lV"); // the length of the block |
---|
3116 | intvec w; // inherit the graded on the oridinal ring |
---|
3117 | int i; |
---|
3118 | for(i=1; i<=b; i++) |
---|
3119 | { |
---|
3120 | w[i] = deg(var(i)); |
---|
3121 | } |
---|
3122 | intvec v = w; |
---|
3123 | for(i=1; i< attrib(basering,"uptodeg"); i++) |
---|
3124 | { |
---|
3125 | v = v,w; |
---|
3126 | } |
---|
3127 | w = v; |
---|
3128 | poly p,q,s, result; |
---|
3129 | s = g; |
---|
3130 | while (f!=0) |
---|
3131 | { |
---|
3132 | p = inForm(f,w)[1]; |
---|
3133 | f = f - p; |
---|
3134 | s = g; |
---|
3135 | while (s!=0) |
---|
3136 | { |
---|
3137 | q = inForm(s,w)[1]; |
---|
3138 | s = s - q; |
---|
3139 | result = result + lpMult(p,q); |
---|
3140 | } |
---|
3141 | } |
---|
3142 | // shrinking |
---|
3143 | // result; |
---|
3144 | return( system("shrinktest",result,attrib(basering, "lV")) ); |
---|
3145 | } |
---|
3146 | example |
---|
3147 | { |
---|
3148 | "EXAMPLE:"; echo = 2; |
---|
3149 | // define a ring in letterplace form as follows: |
---|
3150 | ring r = 0,(x(1),y(1),x(2),y(2),x(3),y(3),x(4),y(4)),dp; |
---|
3151 | def R = setLetterplaceAttributes(r,4,2); // supply R with letterplace structure |
---|
3152 | setring R; |
---|
3153 | poly a = x(1)*y(2)+x(1)+y(1); poly b = y(1)+3; |
---|
3154 | lpMultX(b,a); |
---|
3155 | lpMultX(a,b); |
---|
3156 | } |
---|
3157 | |
---|
3158 | // multiply two letterplace polynomials, lpMult: done |
---|
3159 | // reduction/ Normalform? needs kernel stuff |
---|
3160 | |
---|
3161 | |
---|
3162 | proc lpMult(poly f, poly g) |
---|
3163 | "USAGE: lpMult(f,g); f,g letterplace polynomials |
---|
3164 | RETURN: poly |
---|
3165 | ASSUME: basering has a letterplace ring structure |
---|
3166 | PURPOSE: compute the letterplace form of f*g |
---|
3167 | EXAMPLE: example lpMult; shows examples |
---|
3168 | " |
---|
3169 | { |
---|
3170 | |
---|
3171 | // changelog: |
---|
3172 | // VL oct 2010: deg -> deg(_,w) for the length |
---|
3173 | // shrink the result => don't need to decompose polys |
---|
3174 | // since the shift is big enough |
---|
3175 | |
---|
3176 | // indeed it's better to have that |
---|
3177 | // ASSUME: both f and g are quasi-homogeneous |
---|
3178 | |
---|
3179 | if (lpAssumeViolation()) |
---|
3180 | { |
---|
3181 | ERROR("Incomplete Letterplace structure on the basering!"); |
---|
3182 | } |
---|
3183 | intvec w = 1:nvars(basering); |
---|
3184 | int sf = deg(f,w); // VL Oct 2010: we need rather length than degree |
---|
3185 | int sg = deg(g,w); // esp. in the case of weighted ordering |
---|
3186 | int uptodeg = attrib(basering, "uptodeg"); |
---|
3187 | if (sf+sg > uptodeg) |
---|
3188 | { |
---|
3189 | ERROR("degree bound violated by the product!"); |
---|
3190 | } |
---|
3191 | // if (sf>1) { sf = sf -1; } |
---|
3192 | poly v = f*shiftPoly(g,sf); |
---|
3193 | // bug, reported by Simon King: in nonhomog case [solved] |
---|
3194 | // we need to shrink |
---|
3195 | return( system("shrinktest",v,attrib(basering, "lV")) ); |
---|
3196 | } |
---|
3197 | example |
---|
3198 | { |
---|
3199 | "EXAMPLE:"; echo = 2; |
---|
3200 | // define a ring in letterplace form as follows: |
---|
3201 | ring r = 0,(x(1),y(1),x(2),y(2),x(3),y(3),x(4),y(4)),dp; |
---|
3202 | def R = setLetterplaceAttributes(r,4,2); // supply R with letterplace structure |
---|
3203 | setring R; |
---|
3204 | poly a = x(1)*y(2)+x(1)+y(1); poly b = y(1)+3; |
---|
3205 | lpMult(b,a); |
---|
3206 | lpMult(a,b); |
---|
3207 | } |
---|
3208 | |
---|
3209 | proc lpPower(poly f, int n) |
---|
3210 | "USAGE: lpPower(f,n); f letterplace polynomial, int n |
---|
3211 | RETURN: poly |
---|
3212 | ASSUME: basering has a letterplace ring structure |
---|
3213 | PURPOSE: compute the letterplace form of f^n |
---|
3214 | EXAMPLE: example lpPower; shows examples |
---|
3215 | " |
---|
3216 | { |
---|
3217 | if (n<0) { ERROR("the power must be a natural number!"); } |
---|
3218 | if (n==0) { return(poly(1)); } |
---|
3219 | if (n==1) { return(f); } |
---|
3220 | int i; |
---|
3221 | poly p = 1; |
---|
3222 | for(i=1; i<= n; i++) |
---|
3223 | { |
---|
3224 | p = lpMult(p,f); |
---|
3225 | } |
---|
3226 | return(p); |
---|
3227 | } |
---|
3228 | example |
---|
3229 | { |
---|
3230 | "EXAMPLE:"; echo = 2; |
---|
3231 | // define a ring in letterplace form as follows: |
---|
3232 | ring r = 0,(x(1),y(1),x(2),y(2),x(3),y(3),x(4),y(4)),dp; |
---|
3233 | def R = setLetterplaceAttributes(r,4,2); // supply R with letterplace structure |
---|
3234 | setring R; |
---|
3235 | poly a = x(1)*y(2) + y(1); poly b = y(1) - 1; |
---|
3236 | lpPower(a,2); |
---|
3237 | lpPower(b,4); |
---|
3238 | } |
---|
3239 | |
---|
3240 | // new: lp normal from by using shift-invariant data by Grischa Studzinski |
---|
3241 | |
---|
3242 | ///////////////////////////////////////////////////////// |
---|
3243 | // ASSUMPTIONS: every polynomial is an element of V', |
---|
3244 | //@* else there wouldn't be an dvec representation |
---|
3245 | |
---|
3246 | //Main procedure for the user |
---|
3247 | |
---|
3248 | proc lpNF(poly p, ideal G) |
---|
3249 | "USAGE: lpNF(p,G); f letterplace polynomial, ideal I |
---|
3250 | RETURN: poly |
---|
3251 | PURPOSE: computation of the normal form of p with respect to G |
---|
3252 | ASSUME: p is a Letterplace polynomial, G is a set Letterplace polynomials, |
---|
3253 | being a Letterplace Groebner basis (no check for this will be done) |
---|
3254 | NOTE: Strategy: take the smallest monomial wrt ordering for reduction |
---|
3255 | - For homogenous ideals the shift does not matter |
---|
3256 | - For non-homogenous ideals the first shift will be the smallest monomial |
---|
3257 | EXAMPLE: example lpNF; shows examples |
---|
3258 | " |
---|
3259 | {if ((p==0) || (size(G) == 0)){return(p);} |
---|
3260 | checkAssumptions(p,G); |
---|
3261 | G = sort(G)[1]; |
---|
3262 | list L = makeDVecI(G); |
---|
3263 | return(lpNormalForm2(p,G,L)); |
---|
3264 | } |
---|
3265 | example |
---|
3266 | { |
---|
3267 | "EXAMPLE:"; echo = 2; |
---|
3268 | ring r = 0,(x,y,z),dp; |
---|
3269 | int d =5; // degree |
---|
3270 | def R = makeLetterplaceRing(d); |
---|
3271 | setring R; |
---|
3272 | ideal I = y(1)*x(2)*y(3) - z(1)*y(2)*z(3), x(1)*y(2)*x(3) - z(1)*x(2)*y(3), z(1)*x(2)*z(3) - y(1)*z(2)*x(3), x(1)*x(2)*x(3) + y(1)*y(2)*y(3) + z(1)*z(2)*z(3) + x(1)*y(2)*z(3); |
---|
3273 | ideal J = letplaceGBasis(I); // compute a Letterplace Groebner basis |
---|
3274 | poly p = y(1)*x(2)*y(3)*z(4)*y(5) - y(1)*z(2)*z(3)*y(4) + z(1)*y(2)*z(3); |
---|
3275 | poly q = z(1)*x(2)*z(3)*y(4)*z(5) - y(1)*z(2)*x(3)*y(4)*z(5); |
---|
3276 | lpNF(p,J); |
---|
3277 | lpNF(q,J); |
---|
3278 | } |
---|
3279 | |
---|
3280 | //procedures to convert monomials into the DVec representation, all static |
---|
3281 | //////////////////////////////////////////////////////// |
---|
3282 | |
---|
3283 | |
---|
3284 | static proc getExpVecs(ideal G) |
---|
3285 | "USUAGE: getExpVecs(G); |
---|
3286 | RETURN: list of intvecs |
---|
3287 | PURPOSE: convert G into a list of intvecs, corresponding to the exponent vector |
---|
3288 | of the leading monomials of G |
---|
3289 | " |
---|
3290 | {int i; list L; |
---|
3291 | for (i = 1; i <= size(G); i++) {L[i] = leadexp(G[i]); } |
---|
3292 | return(L); |
---|
3293 | } |
---|
3294 | |
---|
3295 | static proc delSupZero(intvec I) |
---|
3296 | "USUAGE:delSupZero(I); |
---|
3297 | RETURN: intvec |
---|
3298 | PURPOSE: Deletes superfluous zero blocks of an exponent vector |
---|
3299 | ASSUME: Intvec is an exponent vector of a letterplace monomial contained in V' |
---|
3300 | " |
---|
3301 | {if (I==intvec(0)) {return(intvec(0));} |
---|
3302 | int j,k,l; |
---|
3303 | int n = attrib(basering,"lV"); int d = attrib(basering,"uptodeg"); |
---|
3304 | intvec w; j = 1; |
---|
3305 | while (j <= d) |
---|
3306 | {w = I[1..n]; |
---|
3307 | if (w<>intvec(0)){break;} |
---|
3308 | else {I = I[(n+1)..(n*d)]; d = d-1; j++;} |
---|
3309 | } |
---|
3310 | for (j = 1; j <= d; j++) |
---|
3311 | {l=(j-1)*n+1; k= j*n; |
---|
3312 | w = I[l..k]; |
---|
3313 | if (w==intvec(0)){w = I[1..(l-1)]; return(w);}//if a zero block is found there are only zero blocks left, |
---|
3314 | //otherwise there would be a hole in the monomial |
---|
3315 | // shrink should take care that this will not happen |
---|
3316 | } |
---|
3317 | return(I); |
---|
3318 | } |
---|
3319 | |
---|
3320 | static proc delSupZeroList(list L) |
---|
3321 | "USUAGE:delSupZeroList(L); L a list, containing intvecs |
---|
3322 | RETURN: list, containing intvecs |
---|
3323 | PURPOSE: Deletes all superfluous zero blocks for a list of exponent vectors |
---|
3324 | ASSUME: All intvecs are exponent vectors of letterplace monomials contained in V' |
---|
3325 | " |
---|
3326 | {int i; |
---|
3327 | for (i = size(L); 0 < i; i--){L[i] = delSupZero(L[i]);} |
---|
3328 | return(L); |
---|
3329 | } |
---|
3330 | |
---|
3331 | |
---|
3332 | static proc makeDVec(intvec V) |
---|
3333 | "USUAGE:makeDVec(V); |
---|
3334 | RETURN: intvec |
---|
3335 | PURPOSE: Converts an modified exponent vector into an Dvec |
---|
3336 | NOTE: Superfluos zero blocks must have been deleted befor using this procedure |
---|
3337 | " |
---|
3338 | {int i,j,k,r1,r2; intvec D; |
---|
3339 | int n = attrib(basering,"lV"); |
---|
3340 | k = size(V) div n; r1 = 0; r2 = 0; |
---|
3341 | for (i=1; i<= k; i++) |
---|
3342 | {for (j=(1+((i-1)*n)); j <= (i*n); j++) |
---|
3343 | {if (V[j]>0){r2 = j - ((i-1)*n); j = (j mod n); break;} |
---|
3344 | } |
---|
3345 | D[size(D)+1] = r1+r2; |
---|
3346 | if (j == 0) {r1 = 0;} else{r1= n-j;} |
---|
3347 | } |
---|
3348 | D = D[2..size(D)]; |
---|
3349 | return(D); |
---|
3350 | } |
---|
3351 | |
---|
3352 | static proc makeDVecL(list L) |
---|
3353 | "USUAGE:makeDVecL(L); L, a list containing intvecs |
---|
3354 | RETURN: list, containing intvecs |
---|
3355 | ASSUME: |
---|
3356 | " |
---|
3357 | {int i; list R; |
---|
3358 | for (i=1; i <= size(L); i++) {R[i] = makeDVec(L[i]);} |
---|
3359 | return(R); |
---|
3360 | } |
---|
3361 | |
---|
3362 | static proc makeDVecI(ideal G) |
---|
3363 | "USUAGE:makeDVecI(G); |
---|
3364 | RETURN:list, containing intvecs |
---|
3365 | PURPOSE:computing the DVec representation for lead(G) |
---|
3366 | ASSUME: |
---|
3367 | " |
---|
3368 | {list L = delSupZeroList(getExpVecs(G)); |
---|
3369 | return(makeDVecL(L)); |
---|
3370 | } |
---|
3371 | |
---|
3372 | |
---|
3373 | //procedures, which are dealing with the DVec representation, all static |
---|
3374 | |
---|
3375 | static proc dShiftDiv(intvec V, intvec W) |
---|
3376 | "USUAGE: dShiftDiv(V,W); |
---|
3377 | RETURN: a list,containing integers, or -1, if no shift of W divides V |
---|
3378 | PURPOSE: find all possible shifts s, such that s.W|V |
---|
3379 | ASSUME: V,W are DVecs of monomials contained in V' |
---|
3380 | " |
---|
3381 | {if(size(V)<size(W)){return(list(-1));} |
---|
3382 | |
---|
3383 | int i,j,r; intvec T; list R; |
---|
3384 | int n = attrib(basering,"lV"); |
---|
3385 | int k = size(V) - size(W) + 1; |
---|
3386 | if (intvec(V[1..size(W)])-W == 0){R[1]=0;} |
---|
3387 | for (i =2; i <=k; i++) |
---|
3388 | {r = 0; kill T; intvec T; |
---|
3389 | for (j =1; j <= i; j++) {r = r + V[j];} |
---|
3390 | //if (i==1) {T[1] = r-(i-1)*n;} else |
---|
3391 | T[1] = r-(i-1)*n; if (size(W)>1) {T[2..size(W)] = V[(i+1)..(size(W)+i-1)];} |
---|
3392 | if (T-W == 0) {R[size(R)+1] = i-1;} |
---|
3393 | } |
---|
3394 | if (size(R)>0) {return(R);} |
---|
3395 | else {return(list(-1));} |
---|
3396 | } |
---|
3397 | |
---|
3398 | //the first normal form procedure, if a user want not to presort the ideal, just make it not static |
---|
3399 | |
---|
3400 | static proc lpNormalForm1(poly p, ideal G, list L) |
---|
3401 | "USUAGE:lpNormalForm1(p,G); |
---|
3402 | RETURN:poly |
---|
3403 | PURPOSE:computation of the normalform of p w.r.t. G |
---|
3404 | ASSUME: p is a Letterplace polynomial, G is a set of Letterplace polynomials |
---|
3405 | NOTE: Taking the first possible reduction |
---|
3406 | " |
---|
3407 | { |
---|
3408 | if (deg(p) <1) {return(p);} |
---|
3409 | else |
---|
3410 | { |
---|
3411 | int i; int s; |
---|
3412 | intvec V = makeDVec(delSupZero(leadexp(p))); |
---|
3413 | for (i = 1; i <= size(L); i++) |
---|
3414 | {s = dShiftDiv(V, L[i])[1]; |
---|
3415 | if (s <> -1) |
---|
3416 | {p = lpReduce(p,G[i],s); |
---|
3417 | p = lpNormalForm1(p,G,L); |
---|
3418 | break; |
---|
3419 | } |
---|
3420 | } |
---|
3421 | p = p[1] + lpNormalForm1(p-p[1],G,L); |
---|
3422 | return(p); |
---|
3423 | } |
---|
3424 | } |
---|
3425 | |
---|
3426 | |
---|
3427 | // new VL; called from lpNF |
---|
3428 | static proc lpNormalForm2(poly pp, ideal G, list L) |
---|
3429 | "USUAGE:lpNormalForm2(p,G); |
---|
3430 | RETURN:poly |
---|
3431 | PURPOSE:computation of the normal form of p w.r.t. G |
---|
3432 | ASSUME: p is a Letterplace polynomial, G is a set of Letterplace polynomials |
---|
3433 | NOTE: Taking the first possible reduction |
---|
3434 | " |
---|
3435 | { |
---|
3436 | poly one = 1; |
---|
3437 | if ( (pp == 0) || (leadmonom(pp) == one) ) { return(pp); } |
---|
3438 | poly p = pp; poly q; |
---|
3439 | int i; int s; intvec V; |
---|
3440 | while ( (p != 0) && (leadmonom(p) != one) ) |
---|
3441 | { |
---|
3442 | //"entered while with p="; p; |
---|
3443 | V = makeDVec(delSupZero(leadexp(p))); |
---|
3444 | i = 0; |
---|
3445 | s = -1; |
---|
3446 | //"look for divisor"; |
---|
3447 | while ( (s == -1) && (i<size(L)) ) |
---|
3448 | { |
---|
3449 | i = i+1; |
---|
3450 | s = dShiftDiv(V, L[i])[1]; |
---|
3451 | } |
---|
3452 | // now, out of here: either i=size(L) and s==-1 => no reduction |
---|
3453 | // otherwise: i<=size(L) and s!= -1 => reduction |
---|
3454 | //"out of divisor search: s="; s; "i="; i; |
---|
3455 | if (s != -1) |
---|
3456 | { |
---|
3457 | //"start reducing with G[i]:"; |
---|
3458 | p = lpReduce(p,G[i],s); // lm-reduction |
---|
3459 | //"reduced to p="; p; |
---|
3460 | } |
---|
3461 | else |
---|
3462 | { |
---|
3463 | // ie no lm-reduction possible; proceed with the tail reduction |
---|
3464 | q = p-lead(p); |
---|
3465 | p = lead(p); |
---|
3466 | if (q!=0) |
---|
3467 | { |
---|
3468 | p = p + lpNormalForm2(q,G,L); |
---|
3469 | } |
---|
3470 | return(p); |
---|
3471 | } |
---|
3472 | } |
---|
3473 | // out of while when p==0 or p == const |
---|
3474 | return(p); |
---|
3475 | } |
---|
3476 | |
---|
3477 | |
---|
3478 | |
---|
3479 | |
---|
3480 | //secundary procedures, all static |
---|
3481 | |
---|
3482 | static proc getlpCoeffs(poly q, poly p) |
---|
3483 | {list R; poly m; intvec cq,t,lv,rv,bla; |
---|
3484 | int n = attrib(basering,"lV"); int d = attrib(basering,"uptodeg"); |
---|
3485 | int i; |
---|
3486 | m = p/q; |
---|
3487 | cq = leadexp(m); |
---|
3488 | for (i = 1; i<= d; i++) |
---|
3489 | {bla = cq[((i-1)*n+1)..(i*n)]; |
---|
3490 | if (bla == 0) {lv = cq[1..i*n]; cq = cq[(i*n+1)..(d*n)]; break;} |
---|
3491 | } |
---|
3492 | |
---|
3493 | d = size(cq) div n; |
---|
3494 | for (i = 1; i<= d; i++) |
---|
3495 | {bla = cq[((i-1)*n+1)..(i*n)]; |
---|
3496 | if (bla <> 0){rv = cq[((i-1)*n+1)..(d*n)]; break;} |
---|
3497 | } |
---|
3498 | return(list(monomial(lv),monomial(rv))); |
---|
3499 | } |
---|
3500 | |
---|
3501 | static proc lpReduce(poly p, poly g, int s) |
---|
3502 | "NOTE: shift can not exceed the degree bound, because s*g | p |
---|
3503 | " |
---|
3504 | {poly l,r,qt; int i; |
---|
3505 | list K = getlpCoeffs(lead(shiftPoly(g,s)), lead(p)); |
---|
3506 | l = K[1]; r = K[2]; |
---|
3507 | kill K; |
---|
3508 | for (i = 1; i <= size(g); i++) |
---|
3509 | { |
---|
3510 | qt = qt + lpMult(lpMult(l,g[i]),r); |
---|
3511 | } |
---|
3512 | return(p - leadcoef(p)*normalize(qt)); |
---|
3513 | } |
---|
3514 | |
---|
3515 | |
---|
3516 | static proc lpShrink(poly p) |
---|
3517 | " |
---|
3518 | " |
---|
3519 | {int n; |
---|
3520 | if (typeof(attrib(basering,"isLetterplaceRing"))=="int") |
---|
3521 | {n = attrib(basering,"lV"); |
---|
3522 | return(system("shrinktest",p,n)); |
---|
3523 | } |
---|
3524 | else {ERROR("Basering is not a Letterplace ring!");} |
---|
3525 | } |
---|
3526 | |
---|
3527 | static proc checkAssumptions(poly p, ideal G) |
---|
3528 | " |
---|
3529 | " |
---|
3530 | {checkLPRing(); |
---|
3531 | checkAssumptionPoly(p); |
---|
3532 | checkAssumptionIdeal(G); |
---|
3533 | return(); |
---|
3534 | } |
---|
3535 | |
---|
3536 | static proc checkLPRing(); |
---|
3537 | " |
---|
3538 | " |
---|
3539 | {if (typeof(attrib(basering,"isLetterplaceRing"))=="string") {ERROR("Basering is not a Letterplace ring!");} |
---|
3540 | return(); |
---|
3541 | } |
---|
3542 | |
---|
3543 | static proc checkAssumptionIdeal(ideal G) |
---|
3544 | "PURPOSE:Check if all elements of ideal are elements of V' |
---|
3545 | " |
---|
3546 | {ideal L = lead(normalize(G)); |
---|
3547 | int i; |
---|
3548 | for (i = 1; i <= ncols(G); i++) {if (!isContainedInVp(G[i])) {ERROR("Ideal containes elements not contained in V'");}} |
---|
3549 | return(); |
---|
3550 | } |
---|
3551 | |
---|
3552 | static proc checkAssumptionPoly(poly p) |
---|
3553 | "PURPOSE:Check if p is an element of V' |
---|
3554 | " |
---|
3555 | {poly l = lead(normalize(p)); |
---|
3556 | if (!isContainedInVp(l)) {ERROR("Polynomial is not contained in V'");} |
---|
3557 | return(); |
---|
3558 | } |
---|
3559 | |
---|
3560 | static proc isContainedInVp(poly p) |
---|
3561 | "PURPOSE: Check monomial for holes in the places |
---|
3562 | " |
---|
3563 | {int r = 0; intvec w; |
---|
3564 | intvec l = leadexp(p); |
---|
3565 | int n = attrib(basering,"lV"); int d = attrib(basering,"uptodeg"); |
---|
3566 | int i,j,c,c1; |
---|
3567 | while (1 <= d) |
---|
3568 | {w = l[1..n]; |
---|
3569 | if (w<>intvec(0)){break;} |
---|
3570 | else {l = l[(n+1)..(n*d)]; d = d-1;} |
---|
3571 | } |
---|
3572 | |
---|
3573 | while (1 <= d) |
---|
3574 | {for (j = 1; j <= n; j++) |
---|
3575 | {if (l[j]<>0) |
---|
3576 | {if (c1<>0){return(0);} |
---|
3577 | if (c<>0){return(0);} |
---|
3578 | if (l[j]<>1){return(0);} |
---|
3579 | c=1; |
---|
3580 | } |
---|
3581 | } |
---|
3582 | if (c == 0){c1=1;if (1 < d){l = l[(n+1)..(n*d)]; d = d-1;} else {d = d -1;}} |
---|
3583 | else {c = 0; if (1 < d){l = l[(n+1)..(n*d)]; d = d-1;} else {d = d -1;}} |
---|
3584 | } |
---|
3585 | return(1); |
---|
3586 | } |
---|
3587 | |
---|
3588 | // under development for Roberto |
---|
3589 | static proc extractLinearPart(module M) |
---|
3590 | { |
---|
3591 | /* returns vectors from a module whose max leadexp is 1 */ |
---|
3592 | /* does not take place nonlinearity into account yet */ |
---|
3593 | /* use rather kernel function isinV to get really nonlinear things */ |
---|
3594 | int i; int s = ncols(M); |
---|
3595 | int answer = 1; |
---|
3596 | vector v; module Ret; |
---|
3597 | for(i=1; i<=s; i++) |
---|
3598 | { |
---|
3599 | if ( isLinearVector(M[i]) ) |
---|
3600 | { |
---|
3601 | Ret = Ret, M[i]; |
---|
3602 | } |
---|
3603 | } |
---|
3604 | Ret = simplify(Ret,2); |
---|
3605 | return(Ret); |
---|
3606 | } |
---|
3607 | |
---|
3608 | // under development for Roberto |
---|
3609 | static proc isLinearVector(vector v) |
---|
3610 | { |
---|
3611 | /* returns true iff max leadexp is 1 */ |
---|
3612 | int i,j,k; |
---|
3613 | intvec w; |
---|
3614 | int s = size(v); |
---|
3615 | poly p; |
---|
3616 | int answer = 1; |
---|
3617 | for(i=1; i<=s; i++) |
---|
3618 | { |
---|
3619 | p = v[i]; |
---|
3620 | while (p != 0) |
---|
3621 | { |
---|
3622 | w = leadexp(p); |
---|
3623 | j = Max(w); |
---|
3624 | if (j >=2) |
---|
3625 | { |
---|
3626 | answer = 0; |
---|
3627 | return(answer); |
---|
3628 | } |
---|
3629 | p = p-lead(p); |
---|
3630 | } |
---|
3631 | } |
---|
3632 | return(answer); |
---|
3633 | } |
---|
3634 | |
---|
3635 | |
---|
3636 | // // the following is to determine a shift of a mono/poly from the |
---|
3637 | // // interface |
---|
3638 | |
---|
3639 | // static proc whichshift(poly p, int numvars) |
---|
3640 | // { |
---|
3641 | // // numvars = number of vars of the orig free algebra |
---|
3642 | // // assume: we are in the letterplace ring |
---|
3643 | // // takes monomial on the input |
---|
3644 | // poly q = lead(p); |
---|
3645 | // intvec v = leadexp(v); |
---|
3646 | // if (v==0) { return(int(0)); } |
---|
3647 | // int sv = size(v); |
---|
3648 | // int i=1; |
---|
3649 | // while ( (v[i]==0) && (i<sv) ) { i++; } |
---|
3650 | // i = sv div i; |
---|
3651 | // return(i); |
---|
3652 | // } |
---|
3653 | |
---|
3654 | |
---|
3655 | // LIB "qhmoduli.lib"; |
---|
3656 | // static proc polyshift(poly p, int numvars) |
---|
3657 | // { |
---|
3658 | // poly q = p; int i = 0; |
---|
3659 | // while (q!=0) |
---|
3660 | // { |
---|
3661 | // i = Max(i, whichshift(q,numvars)); |
---|
3662 | // q = q - lead(q); |
---|
3663 | // } |
---|
3664 | // return(q); |
---|
3665 | // } |
---|
3666 | |
---|
3667 | static proc lpAssumeViolation() |
---|
3668 | { |
---|
3669 | // checks whether the global vars |
---|
3670 | // uptodeg and lV are defined |
---|
3671 | // returns Boolean : yes/no [for assume violation] |
---|
3672 | def lpring = attrib(basering,"isLetterplaceRing"); |
---|
3673 | if ( typeof(lpring)!="int" ) |
---|
3674 | { |
---|
3675 | // if ( typeof(lpring)=="string" ) ?? |
---|
3676 | // basering is NOT lp Ring |
---|
3677 | return(1); |
---|
3678 | } |
---|
3679 | def uptodeg = attrib(basering,"uptodeg"); |
---|
3680 | if ( typeof(uptodeg)!="int" ) |
---|
3681 | { |
---|
3682 | return(1); |
---|
3683 | } |
---|
3684 | def lV = attrib(basering,"lV"); |
---|
3685 | if ( typeof(lV)!="int" ) |
---|
3686 | { |
---|
3687 | return(1); |
---|
3688 | } |
---|
3689 | // int i = ( defined(uptodeg) && (defined(lV)) ); |
---|
3690 | // return ( !i ); |
---|
3691 | return(0); |
---|
3692 | } |
---|
3693 | |
---|
3694 | static proc bugSKing() |
---|
3695 | { |
---|
3696 | LIB "freegb.lib"; |
---|
3697 | ring r=0,(a,b),dp; |
---|
3698 | def R = makeLetterplaceRing(5); |
---|
3699 | setring R; |
---|
3700 | poly p = a(1); |
---|
3701 | poly q = b(1); |
---|
3702 | poly p2 = lpPower(p,2); |
---|
3703 | lpMult(p2+q,q)-lpMult(p2,q)-lpMult(q,q); // now its 0 |
---|
3704 | } |
---|
3705 | |
---|
3706 | static proc bugRucker() |
---|
3707 | { |
---|
3708 | // needs unstatic lpMultX |
---|
3709 | LIB "freegb.lib"; |
---|
3710 | ring r=0,(a,b,c,d,p,q,r,s,t,u,v,w),(a(7,1,1,7),dp); |
---|
3711 | def R=makeLetterplaceRing(20,1); |
---|
3712 | setring R; |
---|
3713 | option(redSB); option(redTail); |
---|
3714 | ideal I=a(1)*b(2)*c(3)-p(1)*q(2)*r(3)*s(4)*t(5)*u(6),b(1)*c(2)*d(3)-v(1)*w(2); |
---|
3715 | poly ttt = a(1)*v(2)*w(3)-p(1)*q(2)*r(3)*s(4)*t(5)*u(6)*d(7); |
---|
3716 | // with lpMult |
---|
3717 | lpMult(I[1],d(1)) - lpMult(a(1),I[2]); // spoly; has been incorrect before |
---|
3718 | _ - ttt; |
---|
3719 | // with lpMultX |
---|
3720 | lpMultX(I[1],d(1)) - lpMultX(a(1),I[2]); // spoly; has been incorrect before |
---|
3721 | _ - ttt; |
---|
3722 | } |
---|
3723 | |
---|
3724 | static proc checkWeightedExampleLP() |
---|
3725 | { |
---|
3726 | ring r = 0,(x(1),y(1),x(2),y(2),x(3),y(3),x(4),y(4)),wp(2,1,2,1,2,1,2,1); |
---|
3727 | def R = setLetterplaceAttributes(r,4,2); // supply R with letterplace structure |
---|
3728 | setring R; |
---|
3729 | poly a = x(1)*y(2)+x(1)+y(1); poly b = y(1)+3; |
---|
3730 | lpMultX(b,a); |
---|
3731 | lpMultX(a,b); // seems to work properly |
---|
3732 | } |
---|
3733 | |
---|
3734 | proc lpPrint(ideal I, def @r) |
---|
3735 | "USAGE: lpPrint(I, r); I an ideal, r a ring |
---|
3736 | RETURN: list of strings |
---|
3737 | PURPOSE: represent Letterplace ideal in the form of words |
---|
3738 | ASSUME: - basering is a Letterplace ring, r is the commutative ring |
---|
3739 | from which basering has been built |
---|
3740 | EXAMPLE: example lpPrint; shows example |
---|
3741 | " |
---|
3742 | { |
---|
3743 | def save = basering; |
---|
3744 | lp2lstr(I,@r); // export an object called @code{@LN} to the ring r |
---|
3745 | setring @r; // change to the ring r |
---|
3746 | list @L = lst2str(@LN,1); |
---|
3747 | export @L; |
---|
3748 | setring save; |
---|
3749 | list @@L = @L; |
---|
3750 | setring @r; |
---|
3751 | kill @L; |
---|
3752 | kill @LN; |
---|
3753 | setring save; |
---|
3754 | return(@@L); |
---|
3755 | } |
---|
3756 | example |
---|
3757 | { |
---|
3758 | "EXAMPLE:"; echo = 2; |
---|
3759 | ring r = (0,a,b,g),(x,y),Dp; |
---|
3760 | def R = makeLetterplaceRing(4); // constructs a Letterplace ring |
---|
3761 | setring R; // downup algebra A |
---|
3762 | ideal J = x(1)*x(2)*y(3)-a*x(1)*y(2)*x(3) - b*y(1)*x(2)*x(3) - g*x(1), |
---|
3763 | x(1)*y(2)*y(3)-a*y(1)*x(2)*y(3) - b*y(1)*y(2)*x(3) - g*y(1); |
---|
3764 | list L = lpPrint(J,r); |
---|
3765 | L; |
---|
3766 | } |
---|
3767 | |
---|
3768 | /* THE FOLLOWING ARE UNDER DEVELOPMENT |
---|
3769 | // copied following from freegb_wrkcp.lib by Karim Abou Zeid on 07.04.2017: |
---|
3770 | // makeLetterplaceRingElim(int d) |
---|
3771 | // makeLetterplaceRingNDO(int d) |
---|
3772 | // setLetterplaceAttributesElim(def R, int uptodeg, int lV) |
---|
3773 | // lpElimIdeal(ideal I) |
---|
3774 | // makeLetterplaceRingWt(int d, intvec W) |
---|
3775 | |
---|
3776 | static proc makeLetterplaceRingElim(int d) |
---|
3777 | "USAGE: makeLetterplaceRingElim(d); d integers |
---|
3778 | RETURN: ring |
---|
3779 | PURPOSE: creates a ring with an elimination ordering |
---|
3780 | NOTE: the matrix for the ordering looks as follows: first row is 1,..,0,1,0,.. |
---|
3781 | @* then 0,1,0,...,0,0,1,0... and so on, lastly its lp |
---|
3782 | @* this ordering is only correct if only polys with same shift are compared |
---|
3783 | EXAMPLE: example makeLetterplaceRingElim; shows examples |
---|
3784 | " |
---|
3785 | { |
---|
3786 | |
---|
3787 | // ToDo future: inherit positive weights in the orig ring |
---|
3788 | // complain on nonpositive ones |
---|
3789 | |
---|
3790 | // d = up to degree, will be shifted to d+1 |
---|
3791 | if (d<1) {"bad d"; return(0);} |
---|
3792 | |
---|
3793 | int uptodeg = d; int lV = nvars(basering); |
---|
3794 | |
---|
3795 | int ppl = printlevel-voice+2; |
---|
3796 | string err = ""; |
---|
3797 | |
---|
3798 | int i,j,s; intvec iV,iVl; |
---|
3799 | def save = basering; |
---|
3800 | int D = d-1; |
---|
3801 | list LR = ringlist(save); |
---|
3802 | list L, tmp, tmp2, tmp3; |
---|
3803 | L[1] = LR[1]; // ground field |
---|
3804 | L[4] = LR[4]; // quotient ideal |
---|
3805 | tmp = LR[2]; // varnames |
---|
3806 | s = size(LR[2]); |
---|
3807 | for (i=1; i<=D; i++) |
---|
3808 | { |
---|
3809 | for (j=1; j<=s; j++) |
---|
3810 | { |
---|
3811 | tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")"; |
---|
3812 | } |
---|
3813 | } |
---|
3814 | for (i=1; i<=s; i++) |
---|
3815 | { |
---|
3816 | tmp[i] = string(tmp[i])+"("+string(1)+")"; |
---|
3817 | } |
---|
3818 | L[2] = tmp; |
---|
3819 | L[3] = list(); |
---|
3820 | list OrigNames = LR[2]; |
---|
3821 | s = size(LR[3]); |
---|
3822 | //creation of first block |
---|
3823 | |
---|
3824 | if (s==2) |
---|
3825 | { |
---|
3826 | // not a blockord, 1 block + module ord |
---|
3827 | tmp = LR[3][s]; // module ord |
---|
3828 | for (i = 1; i <= lV; i++) |
---|
3829 | { |
---|
3830 | iV = (0: lV); |
---|
3831 | iV[i] = 1; |
---|
3832 | iVl = iV; |
---|
3833 | for (j = 1; j <= D; j++) |
---|
3834 | { iVl = iVl,iV; } |
---|
3835 | L[3][i] = list("a",iVl); |
---|
3836 | } |
---|
3837 | // for (i=1; i<=d; i++) |
---|
3838 | // { |
---|
3839 | // LR[3][s-1+i] = LR[3][1]; |
---|
3840 | // } |
---|
3841 | // LR[3][s+D] = tmp; |
---|
3842 | //iV = (1:(d*lV)); |
---|
3843 | L[3][lV+1] = list("lp",(1:(d*lV))); |
---|
3844 | L[3][lV+2] = tmp; |
---|
3845 | } |
---|
3846 | else {ERROR("Please set the ordering of basering to dp");} |
---|
3847 | // if (s>2) |
---|
3848 | // { |
---|
3849 | // // there are s-1 blocks |
---|
3850 | // int nb = s-1; |
---|
3851 | // tmp = LR[3][s]; // module ord to place at the very end |
---|
3852 | // tmp2 = LR[3]; tmp2 = tmp2[1..nb]; |
---|
3853 | // LR[3][1] = list("a",LTO); |
---|
3854 | // //tmp3[1] = list("a",intvec(1: int(d*lV))); // deg-ord, insert as the 1st |
---|
3855 | // for (i=1; i<=d; i++) |
---|
3856 | // { |
---|
3857 | // tmp3 = tmp3 + tmp2; |
---|
3858 | // } |
---|
3859 | // tmp3 = tmp3 + list(tmp); |
---|
3860 | // LR[3] = tmp3; |
---|
3861 | // for (i=1; i<=d; i++) |
---|
3862 | // { |
---|
3863 | // for (j=1; j<=nb; j++) |
---|
3864 | // { |
---|
3865 | // // LR[3][i*nb+j+1]= LR[3][j]; |
---|
3866 | // LR[3][i*nb+j+1]= tmp2[j]; |
---|
3867 | // } |
---|
3868 | // } |
---|
3869 | // // size(LR[3]); |
---|
3870 | // LR[3][(s-1)*d+2] = tmp; |
---|
3871 | // LR[3] = list("a",intvec(1: int(d*lV))) + LR[3]; // deg-ord, insert as the 1st |
---|
3872 | // remove everything behind nb*(D+1)+1 ? |
---|
3873 | // tmp = LR[3]; |
---|
3874 | // LR[3] = tmp[1..size(tmp)-1]; |
---|
3875 | // } |
---|
3876 | // L[3] = LR[3]; |
---|
3877 | def @R = ring(L); |
---|
3878 | // setring @R; |
---|
3879 | // int uptodeg = d; int lV = nvars(basering); // were defined before |
---|
3880 | def @@R = setLetterplaceAttributesElim(@R,uptodeg,lV); |
---|
3881 | return (@@R); |
---|
3882 | } |
---|
3883 | example |
---|
3884 | { |
---|
3885 | "EXAMPLE:"; echo = 2; |
---|
3886 | ring r = 0,(x,y,z),lp; |
---|
3887 | def A = makeLetterplaceRingElim(2); |
---|
3888 | setring A; |
---|
3889 | A; |
---|
3890 | attrib(A,"isLetterplaceRing"); |
---|
3891 | attrib(A,"uptodeg"); // degree bound |
---|
3892 | attrib(A,"lV"); // number of variables in the main block |
---|
3893 | } |
---|
3894 | |
---|
3895 | |
---|
3896 | |
---|
3897 | static proc makeLetterplaceRingNDO(int d) |
---|
3898 | "USAGE: makeLetterplaceRingNDO(d); d an integer |
---|
3899 | RETURN: ring |
---|
3900 | PURPOSE: creates a ring with a non-degree first ordering, suitable for |
---|
3901 | @* the use of non-homogeneous letterplace |
---|
3902 | NOTE: the matrix for the ordering looks as follows: |
---|
3903 | @* 'd' blocks of shifted original variables |
---|
3904 | EXAMPLE: example makeLetterplaceRingNDO; shows examples |
---|
3905 | " |
---|
3906 | { |
---|
3907 | |
---|
3908 | // ToDo future: inherit positive weights in the orig ring |
---|
3909 | // complain on nonpositive ones |
---|
3910 | |
---|
3911 | // d = up to degree, will be shifted to d+1 |
---|
3912 | if (d<1) {"bad d"; return(0);} |
---|
3913 | |
---|
3914 | int uptodeg = d; int lV = nvars(basering); |
---|
3915 | |
---|
3916 | int ppl = printlevel-voice+2; |
---|
3917 | string err = ""; |
---|
3918 | |
---|
3919 | int i,j,s; |
---|
3920 | def save = basering; |
---|
3921 | int D = d-1; |
---|
3922 | list LR = ringlist(save); |
---|
3923 | list L, tmp, tmp2, tmp3; |
---|
3924 | L[1] = LR[1]; // ground field |
---|
3925 | L[4] = LR[4]; // quotient ideal |
---|
3926 | tmp = LR[2]; // varnames |
---|
3927 | s = size(LR[2]); |
---|
3928 | for (i=1; i<=D; i++) |
---|
3929 | { |
---|
3930 | for (j=1; j<=s; j++) |
---|
3931 | { |
---|
3932 | tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")"; |
---|
3933 | } |
---|
3934 | } |
---|
3935 | for (i=1; i<=s; i++) |
---|
3936 | { |
---|
3937 | tmp[i] = string(tmp[i])+"("+string(1)+")"; |
---|
3938 | } |
---|
3939 | L[2] = tmp; |
---|
3940 | list OrigNames = LR[2]; |
---|
3941 | // ordering: one 1..1 a above |
---|
3942 | // ordering: d blocks of the ord on r |
---|
3943 | // try to get whether the ord on r is blockord itself |
---|
3944 | // TODO: make L(2) ordering! exponent is maximally 2 |
---|
3945 | s = size(LR[3]); |
---|
3946 | if (s==2) |
---|
3947 | { |
---|
3948 | // not a blockord, 1 block + module ord |
---|
3949 | tmp = LR[3][s]; // module ord |
---|
3950 | for (i=1; i<=d; i++) |
---|
3951 | { |
---|
3952 | LR[3][i] = LR[3][1]; |
---|
3953 | } |
---|
3954 | // LR[3][s+D] = tmp; |
---|
3955 | LR[3][d+1] = tmp; |
---|
3956 | //LR[3][1] = list("a",intvec(1: int(d*lV))); // deg-ord not wanted here |
---|
3957 | } |
---|
3958 | if (s>2) |
---|
3959 | { |
---|
3960 | // there are s-1 blocks |
---|
3961 | int nb = s-1; |
---|
3962 | tmp = LR[3][s]; // module ord to place at the very end |
---|
3963 | tmp2 = LR[3]; tmp2 = tmp2[1..nb]; |
---|
3964 | //tmp3[1] = list("a",intvec(1: int(d*lV))); // deg-ord not wanted here |
---|
3965 | for (i=1; i<=d; i++) |
---|
3966 | { |
---|
3967 | tmp3 = tmp3 + tmp2; |
---|
3968 | } |
---|
3969 | tmp3 = tmp3 + list(tmp); |
---|
3970 | LR[3] = tmp3; |
---|
3971 | // for (i=1; i<=d; i++) |
---|
3972 | // { |
---|
3973 | // for (j=1; j<=nb; j++) |
---|
3974 | // { |
---|
3975 | // // LR[3][i*nb+j+1]= LR[3][j]; |
---|
3976 | // LR[3][i*nb+j+1]= tmp2[j]; |
---|
3977 | // } |
---|
3978 | // } |
---|
3979 | // // size(LR[3]); |
---|
3980 | // LR[3][(s-1)*d+2] = tmp; |
---|
3981 | // LR[3] = list("a",intvec(1: int(d*lV))) + LR[3]; // deg-ord, insert as the 1st |
---|
3982 | // remove everything behind nb*(D+1)+1 ? |
---|
3983 | // tmp = LR[3]; |
---|
3984 | // LR[3] = tmp[1..size(tmp)-1]; |
---|
3985 | } |
---|
3986 | L[3] = LR[3]; |
---|
3987 | def @R = ring(L); |
---|
3988 | // setring @R; |
---|
3989 | // int uptodeg = d; int lV = nvars(basering); // were defined before |
---|
3990 | def @@R = setLetterplaceAttributes(@R,uptodeg,lV); |
---|
3991 | return (@@R); |
---|
3992 | } |
---|
3993 | example |
---|
3994 | { |
---|
3995 | "EXAMPLE:"; echo = 2; |
---|
3996 | ring r = 0,(x,y,z),lp; |
---|
3997 | def A = makeLetterplaceRingNDO(2); |
---|
3998 | setring A; |
---|
3999 | A; |
---|
4000 | attrib(A,"isLetterplaceRing"); |
---|
4001 | attrib(A,"uptodeg"); // degree bound |
---|
4002 | attrib(A,"lV"); // number of variables in the main block |
---|
4003 | } |
---|
4004 | |
---|
4005 | static proc setLetterplaceAttributesElim(def R, int uptodeg, int lV) |
---|
4006 | "USAGE: setLetterplaceAttributesElim(R, d, b, eV); R a ring, b,d, eV integers |
---|
4007 | RETURN: ring with special attributes set |
---|
4008 | PURPOSE: sets attributes for a letterplace ring: |
---|
4009 | @* 'isLetterplaceRing' = true, 'uptodeg' = d, 'lV' = b, 'eV' = eV, where |
---|
4010 | @* 'uptodeg' stands for the degree bound, |
---|
4011 | @* 'lV' for the number of variables in the block 0 |
---|
4012 | @* 'eV' for the number of elimination variables |
---|
4013 | NOTE: Activate the resulting ring by using @code{setring} |
---|
4014 | " |
---|
4015 | { |
---|
4016 | if (uptodeg*lV != nvars(R)) |
---|
4017 | { |
---|
4018 | ERROR("uptodeg and lV do not agree on the basering!"); |
---|
4019 | } |
---|
4020 | |
---|
4021 | |
---|
4022 | // Set letterplace-specific attributes for the output ring! |
---|
4023 | attrib(R, "uptodeg", uptodeg); |
---|
4024 | attrib(R, "lV", lV); |
---|
4025 | attrib(R, "isLetterplaceRing", 1); |
---|
4026 | attrib(R, "HasElimOrd", 1); |
---|
4027 | return (R); |
---|
4028 | } |
---|
4029 | example |
---|
4030 | { |
---|
4031 | "EXAMPLE:"; echo = 2; |
---|
4032 | ring r = 0,(x(1),y(1),x(2),y(2),x(3),y(3),x(4),y(4)),dp; |
---|
4033 | def R = setLetterplaceAttributesElim(r, 4, 2, 1); setring R; |
---|
4034 | attrib(R,"isLetterplaceRing"); |
---|
4035 | lieBracket(x(1),y(1),2); |
---|
4036 | } |
---|
4037 | |
---|
4038 | |
---|
4039 | static proc lpElimIdeal(ideal I) |
---|
4040 | " |
---|
4041 | does not work for degree reasons (deg function does not work for lp rings -> newone!) |
---|
4042 | " |
---|
4043 | { |
---|
4044 | def lpring = attrib(basering,"isLetterplaceRing"); |
---|
4045 | def lpEO = attrib(basering,"HasElimOrd"); |
---|
4046 | if ( typeof(lpring)!="int" && typeof(lpEO)!="int") |
---|
4047 | { |
---|
4048 | ERROR("Ring is not a lp-ring with an elimination ordering"); |
---|
4049 | } |
---|
4050 | |
---|
4051 | //int nE = attrib(basering, "eV"); |
---|
4052 | |
---|
4053 | return(letplaceGBasis(I)); |
---|
4054 | } |
---|
4055 | |
---|
4056 | |
---|
4057 | static proc makeLetterplaceRingWt(int d, intvec W) |
---|
4058 | "USAGE: makeLetterplaceRingWt(d,W); d an integer, W a vector of positive integers |
---|
4059 | RETURN: ring |
---|
4060 | PURPOSE: creates a ring with a special ordering, suitable for |
---|
4061 | @* the use of non-homogeneous letterplace |
---|
4062 | NOTE: the matrix for the ordering looks as follows: first row is W,W,W,... |
---|
4063 | @* then there come 'd' blocks of shifted original variables |
---|
4064 | EXAMPLE: example makeLetterplaceRing2; shows examples |
---|
4065 | " |
---|
4066 | { |
---|
4067 | |
---|
4068 | // ToDo future: inherit positive weights in the orig ring |
---|
4069 | // complain on nonpositive ones |
---|
4070 | |
---|
4071 | // d = up to degree, will be shifted to d+1 |
---|
4072 | if (d<1) {"bad d"; return(0);} |
---|
4073 | |
---|
4074 | int uptodeg = d; int lV = nvars(basering); |
---|
4075 | |
---|
4076 | //check weightvector |
---|
4077 | if (size(W) <> lV) {"bad weights"; return(0);} |
---|
4078 | |
---|
4079 | int i; |
---|
4080 | for (i = 1; i <= size(W); i++) {if (W[i] < 0) {"bad weights"; return(0);}} |
---|
4081 | intvec Wt = W; |
---|
4082 | for (i = 2; i <= d; i++) {Wt = Wt, W;} |
---|
4083 | kill i; |
---|
4084 | |
---|
4085 | int ppl = printlevel-voice+2; |
---|
4086 | string err = ""; |
---|
4087 | |
---|
4088 | int i,j,s; |
---|
4089 | def save = basering; |
---|
4090 | int D = d-1; |
---|
4091 | list LR = ringlist(save); |
---|
4092 | list L, tmp, tmp2, tmp3; |
---|
4093 | L[1] = LR[1]; // ground field |
---|
4094 | L[4] = LR[4]; // quotient ideal |
---|
4095 | tmp = LR[2]; // varnames |
---|
4096 | s = size(LR[2]); |
---|
4097 | for (i=1; i<=D; i++) |
---|
4098 | { |
---|
4099 | for (j=1; j<=s; j++) |
---|
4100 | { |
---|
4101 | tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")"; |
---|
4102 | } |
---|
4103 | } |
---|
4104 | for (i=1; i<=s; i++) |
---|
4105 | { |
---|
4106 | tmp[i] = string(tmp[i])+"("+string(1)+")"; |
---|
4107 | } |
---|
4108 | L[2] = tmp; |
---|
4109 | list OrigNames = LR[2]; |
---|
4110 | // ordering: one 1..1 a above |
---|
4111 | // ordering: d blocks of the ord on r |
---|
4112 | // try to get whether the ord on r is blockord itself |
---|
4113 | // TODO: make L(2) ordering! exponent is maximally 2 |
---|
4114 | s = size(LR[3]); |
---|
4115 | if (s==2) |
---|
4116 | { |
---|
4117 | // not a blockord, 1 block + module ord |
---|
4118 | tmp = LR[3][s]; // module ord |
---|
4119 | for (i=1; i<=d; i++) |
---|
4120 | { |
---|
4121 | LR[3][s-1+i] = LR[3][1]; |
---|
4122 | } |
---|
4123 | // LR[3][s+D] = tmp; |
---|
4124 | LR[3][s+1+D] = tmp; |
---|
4125 | LR[3][1] = list("a",Wt); // deg-ord |
---|
4126 | } |
---|
4127 | if (s>2) |
---|
4128 | { |
---|
4129 | // there are s-1 blocks |
---|
4130 | int nb = s-1; |
---|
4131 | tmp = LR[3][s]; // module ord to place at the very end |
---|
4132 | tmp2 = LR[3]; tmp2 = tmp2[1..nb]; |
---|
4133 | tmp3[1] = list("a",Wt); // deg-ord, insert as the 1st |
---|
4134 | for (i=1; i<=d; i++) |
---|
4135 | { |
---|
4136 | tmp3 = tmp3 + tmp2; |
---|
4137 | } |
---|
4138 | tmp3 = tmp3 + list(tmp); |
---|
4139 | LR[3] = tmp3; |
---|
4140 | |
---|
4141 | } |
---|
4142 | L[3] = LR[3]; |
---|
4143 | def @R = ring(L); |
---|
4144 | // setring @R; |
---|
4145 | // int uptodeg = d; int lV = nvars(basering); // were defined before |
---|
4146 | def @@R = setLetterplaceAttributes(@R,uptodeg,lV); |
---|
4147 | return (@@R); |
---|
4148 | } |
---|
4149 | example |
---|
4150 | { |
---|
4151 | "EXAMPLE:"; echo = 2; |
---|
4152 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
4153 | def A = makeLetterplaceRingWt(2,intvec(1,2,3)); |
---|
4154 | setring A; |
---|
4155 | A; |
---|
4156 | attrib(A,"isLetterplaceRing"); |
---|
4157 | attrib(A,"uptodeg"); // degree bound |
---|
4158 | attrib(A,"lV"); // number of variables in the main block |
---|
4159 | } |
---|
4160 | */ |
---|