1 | ///////////////////////////////////////////////////////////////////////////// |
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2 | version="version freegb.lib 4.1.2.0 Feb_2019 "; // $Id$ |
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3 | category="Noncommutative"; |
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4 | info=" |
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5 | LIBRARY: freegb.lib Two-sided Groebner bases in free algebras and tools via Letterplace approach |
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6 | AUTHORS: Viktor Levandovskyy, viktor.levandovskyy at math.rwth-aachen.de |
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7 | @* Karim Abou Zeid, karim.abou.zeid at rwth-aachen.de |
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8 | @* Grischa Studzinski, grischa.studzinski at math.rwth-aachen.de |
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9 | |
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10 | OVERVIEW: For the theory, see chapter 'Letterplace' in the Singular Manual. |
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11 | |
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12 | This library provides access to kernel functions and also contains legacy code (partially as |
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13 | static procedures) for compatibility reasons. |
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14 | |
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15 | KEYWORDS: free associative algebra; tensor algebra; free noncommutative Groebner basis; |
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16 | Letterplace Groebner basis; finitely presented algebra |
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17 | |
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18 | Support: Joint projects LE 2697/2-1 and KR 1907/3-1 of the Priority Programme SPP 1489: |
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19 | 'Algorithmische und Experimentelle Methoden in Algebra, Geometrie und Zahlentheorie' |
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20 | of the German DFG |
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21 | and Project II.6 of the transregional collaborative research centre |
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22 | SFB-TRR 195 'Symbolic Tools in Mathematics and their Application' of the German DFG |
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23 | |
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24 | PROCEDURES: |
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25 | isFreeAlgebra(r); check whether r is a letterplace ring (free algebra) |
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26 | lpDegBound(R); returns the degree bound of a letterplace ring |
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27 | lpVarBlockSize(R); returns the size of the letterplace blocks |
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28 | lpNcgenCount(R); returns the number of ncgen variables |
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29 | lpDivision(f,I); two-sided division with remainder |
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30 | lpGBPres2Poly(L,I); reconstructs a polynomial from the output of lpDivision |
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31 | isOrderingShiftInvariant(i); tests shift-invariance of the monomial ordering |
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32 | makeLetterplaceRing(d); (deprecated, use freeAlgebra) creates a Letterplace ring out of given data |
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33 | letplaceGBasis(I); (deprecated, use twostd) two-sided Groebner basis of a letterplace ideal I |
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34 | lieBracket(a,b[, N]); (deprecated, use bracket) iterated Lie bracket of two letterplace polynomials |
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35 | setLetterplaceAttributes(R,d,b); (for testing purposes) supplies ring R with the letterplace structure |
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36 | testLift(M,T); verify the output of lift |
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37 | testSyz(M,S); verify the output of syz |
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38 | |
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39 | SEE ALSO: fpadim_lib, fpaprops_lib, fpalgebras_lib, LETTERPLACE |
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40 | "; |
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41 | |
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42 | // Remark Oct 2018: iv2lp, lp2iv etc are NOT IN HEADER because |
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43 | // they should not be used anymore |
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44 | |
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45 | /* more legacy: |
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46 | lpPrint(I, r); represents Letterplace ideal in the form of words (legacy routine) |
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47 | freeGBasis(L, n); computes two-sided Groebner basis of an ideal, encoded via list L (legacy routine) |
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48 | lp2lstr(K, s); convert a letterplace ideal into a list of modules |
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49 | lst2str(L[, n]); convert a list (of modules) into polynomials in free algebra via strings |
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50 | mod2str(M[, n]); convert a module into a polynomial in free algebra via strings |
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51 | vct2str(M[, n]); convert a vector into a word in free algebra |
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52 | //also, there were |
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53 | shiftPoly; |
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54 | lpPower; |
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55 | */ |
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56 | |
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57 | LIB "qhmoduli.lib"; // for Max |
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58 | LIB "fpalgebras.lib"; // for compatibility |
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59 | |
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60 | /* very fast and cheap test of consistency and functionality |
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61 | DO NOT make it static ! |
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62 | after adding the new proc, add it here */ |
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63 | proc tstfreegb() |
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64 | { |
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65 | example makeLetterplaceRing; |
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66 | example letplaceGBasis; |
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67 | example lpNF; |
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68 | example lpDivision; |
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69 | example lpGBPres2Poly; |
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70 | example freeGBasis; |
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71 | example setLetterplaceAttributes; |
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72 | example isOrderingShiftInvariant; |
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73 | /* secondary */ |
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74 | example lieBracket; |
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75 | example lpPrint; |
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76 | |
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77 | example ivL2lpI; |
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78 | example iv2lp; |
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79 | example iv2lpList; |
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80 | example iv2lpMat; |
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81 | example lp2iv; |
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82 | example lp2ivId; |
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83 | example lpId2ivLi; |
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84 | } |
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85 | |
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86 | proc setLetterplaceAttributes(def R, int uptodeg, int lV) |
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87 | "USAGE: setLetterplaceAttributes(R, d, b); R a ring, b,d integers |
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88 | RETURN: ring with special attributes set |
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89 | PURPOSE: sets attributes for a letterplace ring: |
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90 | 'isLetterplaceRing' = 'lV' = b, 'uptodeg' = d, where |
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91 | 'uptodeg' stands for the degree bound, |
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92 | 'lV' for the number of variables in the block 0. |
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93 | NOTE: Activate the resulting ring by using @code{setring} |
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94 | " |
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95 | { |
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96 | if (uptodeg*lV != nvars(R)) |
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97 | { |
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98 | ERROR("uptodeg and lV do not agree on the basering!"); |
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99 | } |
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100 | |
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101 | // Set letterplace-specific attributes for the output ring! |
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102 | // a kind of dirty hack, getting the ringlist again |
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103 | list RL = ringlist(R); |
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104 | attrib(RL, "isLetterplaceRing", lV); |
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105 | attrib(RL, "maxExp", 1); |
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106 | def @R = ring(RL); |
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107 | //attrib(@R, "uptodeg", uptodeg); // no longer needed |
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108 | attrib(@R, "isLetterplaceRing", lV); |
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109 | return (@R); |
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110 | } |
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111 | example |
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112 | { |
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113 | "EXAMPLE:"; echo = 2; |
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114 | ring r = 0,(x(1),y(1),x(2),y(2),x(3),y(3),x(4),y(4)),dp; |
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115 | def R = setLetterplaceAttributes(r, 4, 2); setring R; |
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116 | lpVarBlockSize(R); |
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117 | lieBracket(x(1),y(1),2); |
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118 | } |
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119 | |
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120 | |
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121 | static proc lst2str(list L, list #) |
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122 | "USAGE: lst2str(L[,n]); L a list of modules, n an optional integer |
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123 | RETURN: list (of strings) |
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124 | PURPOSE: convert a list (of modules) into polynomials in free algebra |
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125 | EXAMPLE: example lst2str; shows examples |
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126 | NOTE: if an optional integer is not 0, stars signs are used in multiplication |
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127 | " |
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128 | { |
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129 | // returns a list of strings |
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130 | // being sentences in words built from L |
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131 | // if #[1] = 1, use * between generators |
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132 | int useStar = 0; |
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133 | if ( size(#)>0 ) |
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134 | { |
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135 | if ( typeof(#[1]) != "int") |
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136 | { |
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137 | ERROR("Second argument of type int expected"); |
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138 | } |
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139 | if (#[1]) |
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140 | { |
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141 | useStar = 1; |
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142 | } |
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143 | } |
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144 | int i; |
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145 | int s = size(L); |
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146 | if (s<1) { return(list(""));} |
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147 | list N; |
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148 | for(i=1; i<=s; i++) |
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149 | { |
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150 | if ((typeof(L[i]) == "module") || (typeof(L[i]) == "matrix") ) |
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151 | { |
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152 | N[i] = mod2str(L[i],useStar); |
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153 | } |
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154 | else |
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155 | { |
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156 | "module or matrix expected in the list"; |
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157 | return(N); |
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158 | } |
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159 | } |
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160 | return(N); |
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161 | } |
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162 | example |
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163 | { |
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164 | "EXAMPLE:"; echo = 2; |
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165 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
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166 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
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167 | module N = [1,x,y,x,y],[-2,y,x,y,x],[6,x,y,y,x,y]; |
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168 | list L; L[1] = M; L[2] = N; |
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169 | lst2str(L); |
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170 | lst2str(L[1],1); |
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171 | } |
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172 | |
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173 | |
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174 | static proc mod2str(module M, list #) |
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175 | "USAGE: mod2str(M[,n]); M a module, n an optional integer |
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176 | RETURN: string |
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177 | PURPOSE: convert a module into a polynomial in free algebra |
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178 | EXAMPLE: example mod2str; shows examples |
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179 | NOTE: if an optional integer is not 0, stars signs are used in multiplication |
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180 | " |
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181 | { |
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182 | if (size(M)==0) { return(""); } |
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183 | // returns a string |
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184 | // a sentence in words built from M |
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185 | // if #[1] = 1, use * between generators |
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186 | int useStar = 0; |
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187 | if ( size(#)>0 ) |
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188 | { |
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189 | if ( typeof(#[1]) != "int") |
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190 | { |
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191 | ERROR("Second argument of type int expected"); |
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192 | } |
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193 | if (#[1]) |
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194 | { |
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195 | useStar = 1; |
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196 | } |
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197 | } |
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198 | int i; |
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199 | int s = ncols(M); |
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200 | string t; |
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201 | string mp; |
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202 | for(i=1; i<=s; i++) |
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203 | { |
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204 | mp = vct2str(M[i],useStar); |
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205 | if (mp[1] == "-") |
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206 | { |
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207 | t = t + mp; |
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208 | } |
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209 | else |
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210 | { |
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211 | if (mp != "") |
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212 | { |
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213 | t = t + "+" + mp; |
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214 | } |
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215 | } |
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216 | } |
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217 | if (t[1]=="+") |
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218 | { |
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219 | t = t[2..size(t)]; // remove first "+" |
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220 | } |
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221 | return(t); |
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222 | } |
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223 | example |
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224 | { |
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225 | "EXAMPLE:"; echo = 2; |
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226 | ring r = 0,(x,y,z),(dp); |
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227 | module M = [1,x,y,x,y],[-2,y,x,y,x],[6,x,y,y,x,y]; |
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228 | mod2str(M); |
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229 | mod2str(M,1); |
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230 | } |
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231 | |
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232 | static proc vct2str(vector v, list #) |
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233 | "USAGE: vct2str(v[,n]); v a vector, n an optional integer |
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234 | RETURN: string |
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235 | PURPOSE: convert a vector into a word in free algebra |
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236 | EXAMPLE: example vct2str; shows examples |
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237 | NOTE: if an optional integer is not 0, stars signs are used in multiplication |
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238 | " |
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239 | { |
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240 | if (v==0) { return(""); } |
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241 | // if #[1] = 1, use * between generators |
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242 | int useStar = 0; |
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243 | if ( size(#)>0 ) |
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244 | { |
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245 | if (#[1]) |
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246 | { |
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247 | useStar = 1; |
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248 | } |
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249 | } |
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250 | int ppl = printlevel-voice+2; |
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251 | // for a word, encoded by v |
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252 | // produces a string for it |
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253 | v = skip0(v); |
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254 | if (v==0) { return(string(""));} |
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255 | number cf = leadcoef(v[1]); |
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256 | int s = size(v); |
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257 | string vs,vv,vp,err; |
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258 | int i,j,p,q; |
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259 | for (i=1; i<=s-1; i++) |
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260 | { |
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261 | p = isVar(v[i+1]); |
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262 | if (p==0) |
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263 | { |
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264 | err = "Error: monomial expected at nonzero position " + string(i+1); |
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265 | ERROR(err+" in vct2str"); |
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266 | // dbprint(ppl,err); |
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267 | // return("_"); |
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268 | } |
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269 | if (p==1) |
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270 | { |
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271 | if (useStar && (size(vs) >0)) { vs = vs + "*"; } |
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272 | vs = vs + string(v[i+1]); |
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273 | } |
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274 | else //power |
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275 | { |
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276 | vv = string(v[i+1]); |
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277 | q = find(vv,"^"); |
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278 | if (q==0) |
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279 | { |
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280 | q = find(vv,string(p)); |
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281 | if (q==0) |
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282 | { |
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283 | err = "error in find for string "+vv; |
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284 | dbprint(ppl,err); |
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285 | return("_"); |
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286 | } |
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287 | } |
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288 | // q>0 |
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289 | vp = vv[1..q-1]; |
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290 | for(j=1;j<=p;j++) |
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291 | { |
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292 | if (useStar && (size(vs) >0)) { vs = vs + "*"; } |
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293 | vs = vs + vp; |
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294 | } |
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295 | } |
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296 | } |
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297 | string scf; |
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298 | if (cf == -1) |
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299 | { |
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300 | scf = "-"; |
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301 | } |
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302 | else |
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303 | { |
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304 | scf = string(cf); |
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305 | if ( (cf == 1) && (size(vs)>0) ) |
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306 | { |
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307 | scf = ""; |
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308 | } |
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309 | } |
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310 | if (useStar && (size(scf) >0) && (scf!="-") ) { scf = scf + "*"; } |
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311 | vs = scf + vs; |
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312 | return(vs); |
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313 | } |
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314 | example |
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315 | { |
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316 | "EXAMPLE:"; echo = 2; |
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317 | ring r = (0,a),(x,y3,z(1)),dp; |
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318 | vector v = [-7,x,y3^4,x2,z(1)^3]; |
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319 | vct2str(v); |
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320 | vct2str(v,1); |
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321 | vector w = [-7a^5+6a,x,y3,y3,x,z(1),z(1)]; |
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322 | vct2str(w); |
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323 | vct2str(w,1); |
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324 | } |
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325 | |
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326 | static proc isVar(poly p) |
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327 | "USAGE: isVar(p); poly p |
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328 | RETURN: int |
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329 | PURPOSE: check, whether leading monomial of p is a power of a single variable |
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330 | from the basering. Returns the exponent or 0 if p is multivariate. |
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331 | EXAMPLE: example isVar; shows examples |
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332 | " |
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333 | { |
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334 | // checks whether p is a variable indeed |
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335 | // if it's a power of a variable, returns the power |
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336 | if (p==0) { return(0); } //"p=0"; |
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337 | poly q = leadmonom(p); |
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338 | if ( (p-lead(p)) !=0 ) { return(0); } // "p-lm(p)>0"; |
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339 | intvec v = leadexp(p); |
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340 | int s = size(v); |
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341 | int i=1; |
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342 | int cnt = 0; |
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343 | int pwr = 0; |
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344 | for (i=1; i<=s; i++) |
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345 | { |
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346 | if (v[i] != 0) |
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347 | { |
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348 | cnt++; |
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349 | pwr = v[i]; |
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350 | } |
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351 | } |
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352 | // "cnt:"; cnt; |
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353 | if (cnt==1) { return(pwr); } |
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354 | else { return(0); } |
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355 | } |
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356 | example |
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357 | { |
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358 | "EXAMPLE:"; echo = 2; |
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359 | ring r = 0,(x,y),dp; |
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360 | poly f = xy+1; |
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361 | isVar(f); |
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362 | poly g = y^3; |
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363 | isVar(g); |
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364 | poly h = 7*x^3; |
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365 | isVar(h); |
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366 | poly i = 1; |
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367 | isVar(i); |
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368 | } |
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369 | |
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370 | proc letplaceGBasis(def I) |
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371 | "USAGE: letplaceGBasis(I); I an ideal/module |
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372 | RETURN: ideal/module |
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373 | ASSUME: basering is a Letterplace ring, input consists of Letterplace polynomials |
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374 | PURPOSE: compute the two-sided Groebner basis of I via Letterplace algorithm (legacy routine) |
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375 | NOTE: the degree bound for this computation is read off the letterplace |
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376 | structure of basering |
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377 | EXAMPLE: example letplaceGBasis; shows examples |
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378 | " |
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379 | { |
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380 | return(std(I)); |
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381 | } |
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382 | example |
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383 | { |
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384 | "EXAMPLE:"; echo = 2; |
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385 | ring r = 0,(x,y,z),Dp; |
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386 | int degree_bound = 5; |
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387 | def R = freeAlgebra(r, 5); |
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388 | setring R; |
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389 | ideal I = -x*y-7*y*y+3*x*x, x*y*x-y*x*y; |
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390 | ideal J = letplaceGBasis(I); |
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391 | J; |
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392 | } |
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393 | |
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394 | /* // temporary name for testing */ |
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395 | /* proc lpRightStd(ideal F, ideal Q) */ |
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396 | /* { */ |
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397 | /* return (system("rightgb", F, Q)); */ |
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398 | /* } */ |
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399 | |
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400 | /* |
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401 | //// this was the part of example in the old good Letterplace |
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402 | // now transform letterplace polynomials into strings of words |
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403 | lp2lstr(J,r); // export an object called @code{@LN} to the ring r |
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404 | setring r; // change to the ring r |
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405 | lst2str(@LN,1); |
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406 | */ |
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407 | |
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408 | |
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409 | proc lieBracket(poly a, poly b, list #) |
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410 | "USAGE: lieBracket(a,b[,N]); a,b letterplace polynomials, N an optional integer |
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411 | RETURN: poly |
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412 | ASSUME: basering has a letterplace ring structure |
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413 | PURPOSE:compute the Lie bracket [a,b] = ab - ba between letterplace polynomials |
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414 | NOTE: if N>1 is specified, then the left normed bracket [a,[...[a,b]]]] is |
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415 | computed. |
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416 | EXAMPLE: example lieBracket; shows examples |
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417 | " |
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418 | { |
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419 | int N=1; |
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420 | if (size(#)>0) |
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421 | { |
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422 | if (typeof(#[1])=="int") |
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423 | { |
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424 | N = int(#[1]); |
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425 | } |
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426 | } |
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427 | return (bracket(a,b,N)); |
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428 | |
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429 | /* if (lpAssumeViolation()) */ |
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430 | /* { */ |
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431 | /* // ERROR("Either 'uptodeg' or 'lV' global variables are not set!"); */ |
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432 | /* ERROR("Incomplete Letterplace structure on the basering!"); */ |
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433 | /* } */ |
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434 | /* // alias ppLiebr; */ |
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435 | /* //if int N is given compute [a,[...[a,b]]]] right normed bracket */ |
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436 | /* int N=1; */ |
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437 | /* if (size(#)>0) */ |
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438 | /* { */ |
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439 | /* if (typeof(#[1])=="int") */ |
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440 | /* { */ |
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441 | /* N = int(#[1]); */ |
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442 | /* } */ |
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443 | /* } */ |
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444 | /* if (N<=0) { return(q); } */ |
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445 | /* poly q = a*b - b*a; */ |
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446 | /* if (N >1) */ |
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447 | /* { */ |
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448 | /* for(int i=1; i<=N-1; i++) */ |
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449 | /* { */ |
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450 | /* q = lieBracket(a,q); */ |
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451 | /* } */ |
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452 | /* } */ |
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453 | /* return(q); */ |
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454 | } |
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455 | example |
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456 | { |
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457 | "EXAMPLE:"; echo = 2; |
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458 | ring r = 0,(x,y),dp; |
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459 | ring R = freeAlgebra(r, 4); |
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460 | poly a = x*y; poly b = y; |
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461 | lieBracket(a,b); |
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462 | lieBracket(x,y,2); |
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463 | } |
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464 | |
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465 | proc lpPrint(ideal I, def @r) |
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466 | "USAGE: lpPrint(I, r); I an ideal, r a ring |
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467 | RETURN: list of strings |
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468 | PURPOSE: represent Letterplace ideal in the form of words (legacy routine) |
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469 | ASSUME: - basering is a Letterplace ring, r is the commutative ring |
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470 | from which basering has been built |
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471 | EXAMPLE: example lpPrint; shows example |
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472 | " |
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473 | { |
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474 | def save = basering; |
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475 | lp2lstr(I,@r); // export an object called @code{@LN} to the ring r |
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476 | setring @r; // change to the ring r |
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477 | list @L = lst2str(@LN,1); |
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478 | export @L; |
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479 | setring save; |
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480 | list @@L = @L; |
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481 | setring @r; |
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482 | kill @L; |
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483 | kill @LN; |
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484 | setring save; |
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485 | return(@@L); |
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486 | } |
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487 | example |
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488 | { |
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489 | "EXAMPLE:"; echo = 2; |
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490 | ring r = (0,a,b,g),(x,y),Dp; |
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491 | ring R = freeAlgebra(r, 4); // downup algebra A |
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492 | ideal J = x*x*y-a*x*y*x - b*y*x*x - g*x, |
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493 | x*y*y-a*y*x*y - b*y*y*x - g*y; |
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494 | list L = lpPrint(J,r); |
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495 | L; |
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496 | } |
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497 | |
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498 | /* HISTORICAL STUFF from 2007 |
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499 | // given the element -7xy^2x, it is represented as [-7,x,y^2,x] or as [-7,x,y,y,x] |
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500 | // use the orig ord on (x,y,z) and expand it blockwise to (x(i),y(i),z(i)) |
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501 | |
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502 | // the correspondences: |
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503 | // monomial in K<x,y,z> <<--->> vector in R |
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504 | // polynomial in K<x,y,z> <<--->> list of vectors (matrix/module) in R |
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505 | // ideal in K<x,y,z> <<--->> list of matrices/modules in R |
---|
506 | |
---|
507 | |
---|
508 | // 1. form a new ring |
---|
509 | // 2. NOP |
---|
510 | // 3. compute GB -> with the kernel stuff |
---|
511 | // 4. skip shifted elts (check that no such exist?) |
---|
512 | // 5. go back to orig vars, produce strings/modules |
---|
513 | // 6. return the result |
---|
514 | */ |
---|
515 | |
---|
516 | proc freeGBasis(list LM, int d) |
---|
517 | "USAGE: freeGBasis(L, d); L a list of modules, d an integer |
---|
518 | RETURN: ring |
---|
519 | ASSUME: L has a special form. Namely, it is a list of modules, where |
---|
520 | - each generator of every module stands for a monomial times coefficient in |
---|
521 | free algebra, |
---|
522 | - in such a vector generator, the 1st entry is a nonzero coefficient from the |
---|
523 | ground field |
---|
524 | - and each next entry hosts a variable from the basering. |
---|
525 | PURPOSE: compute the two-sided Groebner basis of an ideal, encoded by L |
---|
526 | in the free associative algebra, up to degree d |
---|
527 | NOTE: Apply @code{lst2str} to the output in order to obtain a better readable |
---|
528 | presentation |
---|
529 | EXAMPLE: example freeGBasis; shows examples |
---|
530 | " |
---|
531 | { |
---|
532 | // d = up to degree, will be shifted to d+1 |
---|
533 | if (d<1) {"bad d"; return(0);} |
---|
534 | |
---|
535 | int ppl = printlevel-voice+2; |
---|
536 | string err = ""; |
---|
537 | |
---|
538 | int i,j,s; |
---|
539 | def save = basering; |
---|
540 | // determine max no of places in the input |
---|
541 | int slm = size(LM); // numbers of polys in the ideal |
---|
542 | int sm; |
---|
543 | intvec iv; |
---|
544 | module M; |
---|
545 | for (i=1; i<=slm; i++) |
---|
546 | { |
---|
547 | // modules, e.g. free polynomials |
---|
548 | M = LM[i]; |
---|
549 | sm = ncols(M); |
---|
550 | for (j=1; j<=sm; j++) |
---|
551 | { |
---|
552 | //vectors, e.g. free monomials |
---|
553 | iv = iv, size(M[j])-1; // 1 place is reserved by the coeff |
---|
554 | } |
---|
555 | } |
---|
556 | int D = Max(iv); // max size of input words |
---|
557 | if (d<D) {"bad d"; return(LM);} |
---|
558 | D = D + d-1; |
---|
559 | // D = d; |
---|
560 | list LR = ringlist(save); |
---|
561 | list L, tmp; |
---|
562 | L[1] = LR[1]; // ground field |
---|
563 | L[4] = LR[4]; // quotient ideal |
---|
564 | tmp = LR[2]; // varnames |
---|
565 | s = size(LR[2]); |
---|
566 | for (i=1; i<=D; i++) |
---|
567 | { |
---|
568 | for (j=1; j<=s; j++) |
---|
569 | { |
---|
570 | tmp[i*s+j] = tmp[j]; |
---|
571 | } |
---|
572 | } |
---|
573 | L[2] = tmp; |
---|
574 | list OrigNames = LR[2]; |
---|
575 | // ordering: d blocks of the ord on r |
---|
576 | // try to get whether the ord on r is blockord itself |
---|
577 | s = size(LR[3]); |
---|
578 | if (s==2) |
---|
579 | { |
---|
580 | // not a blockord, 1 block + module ord |
---|
581 | tmp = LR[3][s]; // module ord |
---|
582 | for (i=1; i<=D; i++) |
---|
583 | { |
---|
584 | LR[3][s-1+i] = LR[3][1]; |
---|
585 | } |
---|
586 | LR[3][s+D] = tmp; |
---|
587 | } |
---|
588 | if (s>2) |
---|
589 | { |
---|
590 | // there are s-1 blocks |
---|
591 | int nb = s-1; |
---|
592 | tmp = LR[3][s]; // module ord |
---|
593 | for (i=1; i<=D; i++) |
---|
594 | { |
---|
595 | for (j=1; j<=nb; j++) |
---|
596 | { |
---|
597 | LR[3][i*nb+j] = LR[3][j]; |
---|
598 | } |
---|
599 | } |
---|
600 | // size(LR[3]); |
---|
601 | LR[3][nb*(D+1)+1] = tmp; |
---|
602 | } |
---|
603 | L[3] = LR[3]; |
---|
604 | attrib(L,"isLetterplaceRing",s); |
---|
605 | attrib(L, "maxExp", 1); |
---|
606 | def @R = ring(L); |
---|
607 | @R = setLetterplaceAttributes(@R, D+1, nvars(save)); |
---|
608 | setring @R; |
---|
609 | ideal I; |
---|
610 | poly @p; |
---|
611 | s = size(OrigNames); |
---|
612 | // "s:";s; |
---|
613 | // convert LM to canonical vectors (no powers) |
---|
614 | setring save; |
---|
615 | kill M; // M was defined earlier |
---|
616 | module M; |
---|
617 | slm = size(LM); // numbers of polys in the ideal |
---|
618 | int sv,k,l; |
---|
619 | vector v; |
---|
620 | // poly p; |
---|
621 | string sp; |
---|
622 | setring @R; |
---|
623 | poly @@p=0; |
---|
624 | setring save; |
---|
625 | for (l=1; l<=slm; l++) |
---|
626 | { |
---|
627 | // modules, e.g. free polynomials |
---|
628 | M = LM[l]; |
---|
629 | sm = ncols(M); // in intvec iv the sizes are stored |
---|
630 | // modules, e.g. free polynomials |
---|
631 | for (j=1; j<=sm; j++) |
---|
632 | { |
---|
633 | //vectors, e.g. free monomials |
---|
634 | v = M[j]; |
---|
635 | sv = size(v); |
---|
636 | // "sv:";sv; |
---|
637 | sp = "@@p = @@p + "; |
---|
638 | for (k=2; k<=sv; k++) |
---|
639 | { |
---|
640 | sp = sp + string(v[k])+"*"; |
---|
641 | } |
---|
642 | sp = sp + string(v[1])+";"; // coef; |
---|
643 | setring @R; |
---|
644 | execute(sp); |
---|
645 | setring save; |
---|
646 | } |
---|
647 | setring @R; |
---|
648 | // "@@p:"; @@p; |
---|
649 | I = I,@@p; |
---|
650 | @@p = 0; |
---|
651 | setring save; |
---|
652 | } |
---|
653 | kill sp; |
---|
654 | // 3. compute GB |
---|
655 | setring @R; |
---|
656 | dbprint(ppl,"computing GB"); |
---|
657 | ideal J = std(I); |
---|
658 | // ideal J = slimgb(I); |
---|
659 | dbprint(ppl,J); |
---|
660 | // 4. skip shifted elts |
---|
661 | attrib(@R, "isLetterplaceRing", 0); // select1 doesn't want to work with letterplace enabled |
---|
662 | ideal K = select1(J,1..s); // s = size(OrigNames) |
---|
663 | dbprint(ppl,K); |
---|
664 | dbprint(ppl, "done with GB"); |
---|
665 | // K contains vars x(1),...z(1) = images of originals |
---|
666 | // 5. go back to orig vars, produce strings/modules |
---|
667 | if (K[1] == 0) |
---|
668 | { |
---|
669 | "no reasonable output, GB gives 0"; |
---|
670 | return(0); |
---|
671 | } |
---|
672 | int sk = size(K); |
---|
673 | int sp, sx, a, b; |
---|
674 | intvec x; |
---|
675 | poly p,q; |
---|
676 | poly pn; |
---|
677 | // vars in 'save' |
---|
678 | setring save; |
---|
679 | module N; |
---|
680 | list LN; |
---|
681 | vector V; |
---|
682 | poly pn; |
---|
683 | // test and skip exponents >=2 |
---|
684 | setring @R; |
---|
685 | for(i=1; i<=sk; i++) |
---|
686 | { |
---|
687 | p = K[i]; |
---|
688 | while (p!=0) |
---|
689 | { |
---|
690 | q = lead(p); |
---|
691 | // "processing q:";q; |
---|
692 | x = leadexp(q); |
---|
693 | sx = size(x); |
---|
694 | for(k=1; k<=sx; k++) |
---|
695 | { |
---|
696 | if ( x[k] >= 2 ) |
---|
697 | { |
---|
698 | err = "skip: the value x[k] is " + string(x[k]); |
---|
699 | dbprint(ppl,err); |
---|
700 | // return(0); |
---|
701 | K[i] = 0; |
---|
702 | p = 0; |
---|
703 | q = 0; |
---|
704 | break; |
---|
705 | } |
---|
706 | } |
---|
707 | p = p - q; |
---|
708 | } |
---|
709 | } |
---|
710 | K = simplify(K,2); |
---|
711 | sk = size(K); |
---|
712 | for(i=1; i<=sk; i++) |
---|
713 | { |
---|
714 | // setring save; |
---|
715 | // V = 0; |
---|
716 | setring @R; |
---|
717 | p = K[i]; |
---|
718 | while (p!=0) |
---|
719 | { |
---|
720 | q = lead(p); |
---|
721 | err = "processing q:" + string(q); |
---|
722 | dbprint(ppl,err); |
---|
723 | x = leadexp(q); |
---|
724 | sx = size(x); |
---|
725 | pn = leadcoef(q); |
---|
726 | setring save; |
---|
727 | pn = imap(@R,pn); |
---|
728 | V = V + leadcoef(pn)*gen(1); |
---|
729 | for(k=1; k<=sx; k++) |
---|
730 | { |
---|
731 | if (x[k] ==1) |
---|
732 | { |
---|
733 | a = k div s; // block number=a+1, a!=0 |
---|
734 | b = k % s; // remainder |
---|
735 | // printf("a: %s, b: %s",a,b); |
---|
736 | if (b == 0) |
---|
737 | { |
---|
738 | // that is it's the last var in the block |
---|
739 | b = s; |
---|
740 | a = a-1; |
---|
741 | } |
---|
742 | V = V + var(b)*gen(a+2); |
---|
743 | } |
---|
744 | // else |
---|
745 | // { |
---|
746 | // printf("error: the value x[k] is %s", x[k]); |
---|
747 | // return(0); |
---|
748 | // } |
---|
749 | } |
---|
750 | err = "V: " + string(V); |
---|
751 | dbprint(ppl,err); |
---|
752 | // printf("V: %s", string(V)); |
---|
753 | N = N,V; |
---|
754 | V = 0; |
---|
755 | setring @R; |
---|
756 | p = p - q; |
---|
757 | pn = 0; |
---|
758 | } |
---|
759 | setring save; |
---|
760 | LN[i] = simplify(N,2); |
---|
761 | N = 0; |
---|
762 | } |
---|
763 | setring save; |
---|
764 | return(LN); |
---|
765 | } |
---|
766 | example |
---|
767 | { |
---|
768 | "EXAMPLE:"; echo = 2; |
---|
769 | ring r = 0,(x,y,z),(dp(1),dp(2)); // ring r = 0,(x,y,z),(a(3,0,2), dp(2)); |
---|
770 | module M = [-1,x,y],[-7,y,y],[3,x,x]; // stands for free poly -xy - 7yy - 3xx |
---|
771 | module N = [1,x,y,x],[-1,y,x,y]; // stands for free poly xyx - yxy |
---|
772 | list L; L[1] = M; L[2] = N; // list of modules stands for an ideal in free algebra |
---|
773 | lst2str(L); // list to string conversion of input polynomials |
---|
774 | def U = freeGBasis(L,5); // 5 is the degree bound |
---|
775 | lst2str(U); |
---|
776 | } |
---|
777 | |
---|
778 | static proc crs(list LM, int d) |
---|
779 | "USAGE: crs(L, d); L a list of modules, d an integer |
---|
780 | RETURN: ring |
---|
781 | PURPOSE: create a ring and shift the ideal |
---|
782 | EXAMPLE: example crs; shows examples |
---|
783 | " |
---|
784 | { |
---|
785 | // d = up to degree, will be shifted to d+1 |
---|
786 | if (d<1) {"bad d"; return(0);} |
---|
787 | |
---|
788 | int ppl = printlevel-voice+2; |
---|
789 | string err = ""; |
---|
790 | |
---|
791 | int i,j,s; |
---|
792 | def save = basering; |
---|
793 | // determine max no of places in the input |
---|
794 | int slm = size(LM); // numbers of polys in the ideal |
---|
795 | int sm; |
---|
796 | intvec iv; |
---|
797 | module M; |
---|
798 | for (i=1; i<=slm; i++) |
---|
799 | { |
---|
800 | // modules, e.g. free polynomials |
---|
801 | M = LM[i]; |
---|
802 | sm = ncols(M); |
---|
803 | for (j=1; j<=sm; j++) |
---|
804 | { |
---|
805 | //vectors, e.g. free monomials |
---|
806 | iv = iv, size(M[j])-1; // 1 place is reserved by the coeff |
---|
807 | } |
---|
808 | } |
---|
809 | int D = Max(iv); // max size of input words |
---|
810 | if (d<D) {"bad d"; return(LM);} |
---|
811 | D = D + d-1; |
---|
812 | // D = d; |
---|
813 | list LR = ringlist(save); |
---|
814 | list L, tmp; |
---|
815 | L[1] = LR[1]; // ground field |
---|
816 | L[4] = LR[4]; // quotient ideal |
---|
817 | tmp = LR[2]; // varnames |
---|
818 | s = size(LR[2]); |
---|
819 | for (i=1; i<=D; i++) |
---|
820 | { |
---|
821 | for (j=1; j<=s; j++) |
---|
822 | { |
---|
823 | tmp[i*s+j] = string(tmp[j])+"("+string(i)+")"; |
---|
824 | } |
---|
825 | } |
---|
826 | for (i=1; i<=s; i++) |
---|
827 | { |
---|
828 | tmp[i] = string(tmp[i])+"("+string(0)+")"; |
---|
829 | } |
---|
830 | L[2] = tmp; |
---|
831 | list OrigNames = LR[2]; |
---|
832 | // ordering: d blocks of the ord on r |
---|
833 | // try to get whether the ord on r is blockord itself |
---|
834 | s = size(LR[3]); |
---|
835 | if (s==2) |
---|
836 | { |
---|
837 | // not a blockord, 1 block + module ord |
---|
838 | tmp = LR[3][s]; // module ord |
---|
839 | for (i=1; i<=D; i++) |
---|
840 | { |
---|
841 | LR[3][s-1+i] = LR[3][1]; |
---|
842 | } |
---|
843 | LR[3][s+D] = tmp; |
---|
844 | } |
---|
845 | if (s>2) |
---|
846 | { |
---|
847 | // there are s-1 blocks |
---|
848 | int nb = s-1; |
---|
849 | tmp = LR[3][s]; // module ord |
---|
850 | for (i=1; i<=D; i++) |
---|
851 | { |
---|
852 | for (j=1; j<=nb; j++) |
---|
853 | { |
---|
854 | LR[3][i*nb+j] = LR[3][j]; |
---|
855 | } |
---|
856 | } |
---|
857 | // size(LR[3]); |
---|
858 | LR[3][nb*(D+1)+1] = tmp; |
---|
859 | } |
---|
860 | L[3] = LR[3]; |
---|
861 | def @R = ring(L); |
---|
862 | setring @R; |
---|
863 | ideal I; |
---|
864 | poly @p; |
---|
865 | s = size(OrigNames); |
---|
866 | // "s:";s; |
---|
867 | // convert LM to canonical vectors (no powers) |
---|
868 | setring save; |
---|
869 | kill M; // M was defined earlier |
---|
870 | module M; |
---|
871 | slm = size(LM); // numbers of polys in the ideal |
---|
872 | int sv,k,l; |
---|
873 | vector v; |
---|
874 | // poly p; |
---|
875 | string sp; |
---|
876 | setring @R; |
---|
877 | poly @@p=0; |
---|
878 | setring save; |
---|
879 | for (l=1; l<=slm; l++) |
---|
880 | { |
---|
881 | // modules, e.g. free polynomials |
---|
882 | M = LM[l]; |
---|
883 | sm = ncols(M); // in intvec iv the sizes are stored |
---|
884 | for (i=0; i<=d-iv[l]; i++) |
---|
885 | { |
---|
886 | // modules, e.g. free polynomials |
---|
887 | for (j=1; j<=sm; j++) |
---|
888 | { |
---|
889 | //vectors, e.g. free monomials |
---|
890 | v = M[j]; |
---|
891 | sv = size(v); |
---|
892 | // "sv:";sv; |
---|
893 | sp = "@@p = @@p + "; |
---|
894 | for (k=2; k<=sv; k++) |
---|
895 | { |
---|
896 | sp = sp + string(v[k])+"("+string(k-2+i)+")*"; |
---|
897 | } |
---|
898 | sp = sp + string(v[1])+";"; // coef; |
---|
899 | setring @R; |
---|
900 | execute(sp); |
---|
901 | setring save; |
---|
902 | } |
---|
903 | setring @R; |
---|
904 | // "@@p:"; @@p; |
---|
905 | I = I,@@p; |
---|
906 | @@p = 0; |
---|
907 | setring save; |
---|
908 | } |
---|
909 | } |
---|
910 | setring @R; |
---|
911 | export I; |
---|
912 | return(@R); |
---|
913 | } |
---|
914 | example |
---|
915 | { |
---|
916 | "EXAMPLE:"; echo = 2; |
---|
917 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
918 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
---|
919 | module N = [1,x,y,x],[-1,y,x,y]; |
---|
920 | list L; L[1] = M; L[2] = N; |
---|
921 | lst2str(L); |
---|
922 | def U = crs(L,5); |
---|
923 | setring U; U; |
---|
924 | I; |
---|
925 | } |
---|
926 | |
---|
927 | static proc polylen(ideal I) |
---|
928 | { |
---|
929 | // returns the ideal of length of polys |
---|
930 | int i; |
---|
931 | intvec J; |
---|
932 | number s = 0; |
---|
933 | for(i=1;i<=ncols(I);i++) |
---|
934 | { |
---|
935 | J[i] = size(I[i]); |
---|
936 | s = s + J[i]; |
---|
937 | } |
---|
938 | printf("the sum of length %s",s); |
---|
939 | // print(s); |
---|
940 | return(J); |
---|
941 | } |
---|
942 | |
---|
943 | proc lpDegBound(def R) |
---|
944 | "USAGE: lpDegBound(R); R a letterplace ring |
---|
945 | RETURN: int |
---|
946 | PURPOSE: returns the degree bound of the letterplace ring |
---|
947 | EXAMPLE: example lpDegBound; shows examples |
---|
948 | " |
---|
949 | { |
---|
950 | int lV = attrib(R, "isLetterplaceRing"); |
---|
951 | if (lV < 1) { |
---|
952 | ERROR("not a letterplace ring"); |
---|
953 | } |
---|
954 | return (nvars(R) div lV); |
---|
955 | } |
---|
956 | example |
---|
957 | { |
---|
958 | "EXAMPLE:"; echo = 2; |
---|
959 | ring r = 0,(x,y,z),dp; |
---|
960 | def R = freeAlgebra(r, 7); |
---|
961 | lpDegBound(R); |
---|
962 | } |
---|
963 | |
---|
964 | proc lpVarBlockSize(def R) |
---|
965 | "USAGE: lpVarBlockSize(R); R a letterplace ring |
---|
966 | RETURN: int |
---|
967 | PURPOSE: returns the variable block size of the letterplace ring, that is the number of variables of the original ring. |
---|
968 | EXAMPLE: example lpVarBlockSize; shows examples |
---|
969 | " |
---|
970 | { |
---|
971 | int lV = attrib(R, "isLetterplaceRing"); |
---|
972 | if (lV < 1) { |
---|
973 | ERROR("not a letterplace ring"); |
---|
974 | } |
---|
975 | return (lV); |
---|
976 | } |
---|
977 | example |
---|
978 | { |
---|
979 | "EXAMPLE:"; echo = 2; |
---|
980 | ring r = 0,(x,y,z),dp; |
---|
981 | ring R = freeAlgebra(r, 7); |
---|
982 | lpVarBlockSize(R); |
---|
983 | } |
---|
984 | |
---|
985 | proc isFreeAlgebra(def r) |
---|
986 | "USAGE: isFreeAlgebra(r); r a ring |
---|
987 | RETURN: boolean |
---|
988 | PURPOSE: check whether R is a letterplace ring (free algebra) |
---|
989 | EXAMPLE: example isFreeAlgebra; shows examples |
---|
990 | " |
---|
991 | { |
---|
992 | int lV = attrib(r, "isLetterplaceRing"); |
---|
993 | if (lV < 1) { |
---|
994 | return (0); |
---|
995 | } |
---|
996 | return (1); |
---|
997 | } |
---|
998 | example |
---|
999 | { |
---|
1000 | "EXAMPLE:"; echo = 2; |
---|
1001 | ring r = 0,(x,y,z),dp; |
---|
1002 | isFreeAlgebra(r); |
---|
1003 | ring R = freeAlgebra(r, 7); |
---|
1004 | isFreeAlgebra(R); |
---|
1005 | } |
---|
1006 | |
---|
1007 | proc lpNcgenCount(def R) |
---|
1008 | "USAGE: lpNcgenCount(R); R a letterplace ring |
---|
1009 | RETURN: int |
---|
1010 | PURPOSE: returns the number of ncgen variables in the letterplace ring. |
---|
1011 | EXAMPLE: example ncgenCount; shows examples |
---|
1012 | " |
---|
1013 | { |
---|
1014 | return(attrib(R, "ncgenCount")); |
---|
1015 | } |
---|
1016 | example |
---|
1017 | { |
---|
1018 | "EXAMPLE:"; echo = 2; |
---|
1019 | ring r = 0,(x,y,z),dp; |
---|
1020 | ring R = freeAlgebra(r, 7, 3); |
---|
1021 | lpNcgenCount(R); // should be 3 |
---|
1022 | } |
---|
1023 | |
---|
1024 | // united all previous makes, including mLR1 (homog) and mLR2 (nonhomog) |
---|
1025 | proc makeLetterplaceRing(int d, list #) |
---|
1026 | "USAGE: makeLetterplaceRing(d [,h]); d an integer, h an optional integer (deprecated, use freeAlgebra instead) |
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1027 | RETURN: ring |
---|
1028 | PURPOSE: creates a ring with the ordering, used in letterplace computations |
---|
1029 | NOTE: h = -1 (default) : the ordering of the current ring will be used |
---|
1030 | h = 0 : Dp ordering will be used |
---|
1031 | h = 2 : weights 1 used for all the variables, a tie breaker is a list of block of original ring |
---|
1032 | h = 1 : the pure homogeneous letterplace block ordering (applicable in the situation of homogeneous input ideals) will be used. |
---|
1033 | EXAMPLE: example makeLetterplaceRing; shows examples |
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1034 | " |
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1035 | { |
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1036 | int alternativeVersion = -1; |
---|
1037 | if ( size(#)>0 ) |
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1038 | { |
---|
1039 | if (typeof(#[1]) == "int") |
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1040 | { |
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1041 | alternativeVersion = #[1]; |
---|
1042 | } |
---|
1043 | } |
---|
1044 | if (alternativeVersion == 1) |
---|
1045 | { |
---|
1046 | def @A = makeLetterplaceRing1(d); |
---|
1047 | } |
---|
1048 | else { |
---|
1049 | if (alternativeVersion == 2) |
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1050 | { |
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1051 | def @A = makeLetterplaceRing2(d); |
---|
1052 | } |
---|
1053 | else { |
---|
1054 | if (alternativeVersion == 0) |
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1055 | { |
---|
1056 | def @A = makeLetterplaceRing4(d); |
---|
1057 | } |
---|
1058 | else { |
---|
1059 | def @A = freeAlgebra(basering, d); |
---|
1060 | } |
---|
1061 | } |
---|
1062 | } |
---|
1063 | return(@A); |
---|
1064 | } |
---|
1065 | example |
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1066 | { |
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1067 | "EXAMPLE:"; echo = 2; |
---|
1068 | ring r = 0,(x,y,z),Dp; |
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1069 | def A = makeLetterplaceRing(2); // same as makeLetterplaceRing(2,0) |
---|
1070 | setring A; A; |
---|
1071 | lpVarBlockSize(A); |
---|
1072 | lpDegBound(A); // degree bound |
---|
1073 | setring r; def B = makeLetterplaceRing(2,1); // to compare: |
---|
1074 | setring B; B; |
---|
1075 | lpVarBlockSize(B); |
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1076 | lpDegBound(B); // degree bound |
---|
1077 | setring r; def C = makeLetterplaceRing(2,2); // to compare: |
---|
1078 | setring C; C; |
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1079 | lpDegBound(C); |
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1080 | lpDegBound(C); // degree bound |
---|
1081 | } |
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1082 | |
---|
1083 | static proc makeLetterplaceRing1(int d) |
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1084 | "USAGE: makeLetterplaceRing1(d); d an integer |
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1085 | RETURN: ring |
---|
1086 | PURPOSE: creates a ring with a special ordering, suitable for |
---|
1087 | @* the use of homogeneous letterplace (d blocks of shifted original variables) |
---|
1088 | EXAMPLE: example makeLetterplaceRing1; shows examples |
---|
1089 | " |
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1090 | { |
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1091 | // d = up to degree, will be shifted to d+1 |
---|
1092 | if (d<1) {"bad d"; return(0);} |
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1093 | |
---|
1094 | int uptodeg = d; int lV = nvars(basering); |
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1095 | |
---|
1096 | int ppl = printlevel-voice+2; |
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1097 | string err = ""; |
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1098 | |
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1099 | int i,j,s; |
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1100 | def save = basering; |
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1101 | int D = d-1; |
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1102 | list LR = ringlist(save); |
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1103 | list L, tmp; |
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1104 | L[1] = LR[1]; // ground field |
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1105 | L[4] = LR[4]; // quotient ideal |
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1106 | tmp = LR[2]; // varnames |
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1107 | s = size(LR[2]); |
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1108 | for (i=1; i<=D; i++) |
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1109 | { |
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1110 | for (j=1; j<=s; j++) |
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1111 | { |
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1112 | tmp[i*s+j] = tmp[j]; |
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1113 | } |
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1114 | } |
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1115 | L[2] = tmp; |
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1116 | list OrigNames = LR[2]; |
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1117 | // ordering: d blocks of the ord on r |
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1118 | // try to get whether the ord on r is blockord itself |
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1119 | // TODO: make L(2) ordering! exponent is maximally 2 |
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1120 | s = size(LR[3]); |
---|
1121 | if (s==2) |
---|
1122 | { |
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1123 | // not a blockord, 1 block + module ord |
---|
1124 | tmp = LR[3][s]; // module ord |
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1125 | for (i=1; i<=D; i++) |
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1126 | { |
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1127 | LR[3][s-1+i] = LR[3][1]; |
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1128 | } |
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1129 | LR[3][s+D] = tmp; |
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1130 | } |
---|
1131 | if (s>2) |
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1132 | { |
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1133 | // there are s-1 blocks |
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1134 | int nb = s-1; |
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1135 | tmp = LR[3][s]; // module ord |
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1136 | for (i=1; i<=D; i++) |
---|
1137 | { |
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1138 | for (j=1; j<=nb; j++) |
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1139 | { |
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1140 | LR[3][i*nb+j] = LR[3][j]; |
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1141 | } |
---|
1142 | } |
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1143 | // size(LR[3]); |
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1144 | LR[3][nb*(D+1)+1] = tmp; |
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1145 | } |
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1146 | L[3] = LR[3]; |
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1147 | attrib(L,"maxExp",1); |
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1148 | attrib(L,"isLetterplaceRing",lV); |
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1149 | def @R = ring(L); |
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1150 | // setring @R; |
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1151 | // int uptodeg = d; int lV = nvars(basering); // were defined before |
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1152 | def @@R = setLetterplaceAttributes(@R,uptodeg,lV); |
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1153 | return (@@R); |
---|
1154 | } |
---|
1155 | example |
---|
1156 | { |
---|
1157 | "EXAMPLE:"; echo = 2; |
---|
1158 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
1159 | def A = makeLetterplaceRing1(2); |
---|
1160 | setring A; |
---|
1161 | A; |
---|
1162 | lpVarBlockSize(A);// number of variables in the main block |
---|
1163 | lpDegBound(A); // degree bound |
---|
1164 | } |
---|
1165 | |
---|
1166 | static proc makeLetterplaceRing2(int d) |
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1167 | "USAGE: makeLetterplaceRing2(d); d an integer |
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1168 | RETURN: ring |
---|
1169 | PURPOSE: creates a ring with a special ordering, suitable for |
---|
1170 | @* the use of non-homogeneous letterplace |
---|
1171 | NOTE: the matrix for the ordering looks as follows: first row is 1,1,...,1 |
---|
1172 | @* then there come 'd' blocks of shifted original variables |
---|
1173 | EXAMPLE: example makeLetterplaceRing2; shows examples |
---|
1174 | " |
---|
1175 | { |
---|
1176 | |
---|
1177 | // ToDo future: inherit positive weights in the orig ring |
---|
1178 | // complain on nonpositive ones |
---|
1179 | |
---|
1180 | // d = up to degree, will be shifted to d+1 |
---|
1181 | if (d<1) {"bad d"; return(0);} |
---|
1182 | |
---|
1183 | int uptodeg = d; int lV = nvars(basering); |
---|
1184 | |
---|
1185 | int ppl = printlevel-voice+2; |
---|
1186 | string err = ""; |
---|
1187 | |
---|
1188 | int i,j,s; |
---|
1189 | def save = basering; |
---|
1190 | int D = d-1; |
---|
1191 | list LR = ringlist(save); |
---|
1192 | list L, tmp, tmp2, tmp3; |
---|
1193 | L[1] = LR[1]; // ground field |
---|
1194 | L[4] = LR[4]; // quotient ideal |
---|
1195 | tmp = LR[2]; // varnames |
---|
1196 | s = size(LR[2]); |
---|
1197 | for (i=1; i<=D; i++) |
---|
1198 | { |
---|
1199 | for (j=1; j<=s; j++) |
---|
1200 | { |
---|
1201 | tmp[i*s+j] =tmp[j]; |
---|
1202 | } |
---|
1203 | } |
---|
1204 | L[2] = tmp; |
---|
1205 | list OrigNames = LR[2]; |
---|
1206 | // ordering: one 1..1 a above |
---|
1207 | // ordering: d blocks of the ord on r |
---|
1208 | // try to get whether the ord on r is blockord itself |
---|
1209 | // TODO: make L(2) ordering! exponent is maximally 2 |
---|
1210 | s = size(LR[3]); |
---|
1211 | if (s==2) |
---|
1212 | { |
---|
1213 | // not a blockord, 1 block + module ord |
---|
1214 | tmp = LR[3][s]; // module ord |
---|
1215 | for (i=1; i<=d; i++) |
---|
1216 | { |
---|
1217 | LR[3][s-1+i] = LR[3][1]; |
---|
1218 | } |
---|
1219 | // LR[3][s+D] = tmp; |
---|
1220 | LR[3][s+1+D] = tmp; |
---|
1221 | LR[3][1] = list("a",intvec(1: int(d*lV))); // deg-ord |
---|
1222 | } |
---|
1223 | if (s>2) |
---|
1224 | { |
---|
1225 | // there are s-1 blocks |
---|
1226 | int nb = s-1; |
---|
1227 | tmp = LR[3][s]; // module ord to place at the very end |
---|
1228 | tmp2 = LR[3]; tmp2 = tmp2[1..nb]; |
---|
1229 | tmp3[1] = list("a",intvec(1: int(d*lV))); // deg-ord, insert as the 1st |
---|
1230 | for (i=1; i<=d; i++) |
---|
1231 | { |
---|
1232 | tmp3 = tmp3 + tmp2; |
---|
1233 | } |
---|
1234 | tmp3 = tmp3 + list(tmp); |
---|
1235 | LR[3] = tmp3; |
---|
1236 | // for (i=1; i<=d; i++) |
---|
1237 | // { |
---|
1238 | // for (j=1; j<=nb; j++) |
---|
1239 | // { |
---|
1240 | // // LR[3][i*nb+j+1]= LR[3][j]; |
---|
1241 | // LR[3][i*nb+j+1]= tmp2[j]; |
---|
1242 | // } |
---|
1243 | // } |
---|
1244 | // // size(LR[3]); |
---|
1245 | // LR[3][(s-1)*d+2] = tmp; |
---|
1246 | // LR[3] = list("a",intvec(1: int(d*lV))) + LR[3]; // deg-ord, insert as the 1st |
---|
1247 | // remove everything behind nb*(D+1)+1 ? |
---|
1248 | // tmp = LR[3]; |
---|
1249 | // LR[3] = tmp[1..size(tmp)-1]; |
---|
1250 | } |
---|
1251 | L[3] = LR[3]; |
---|
1252 | attrib(L,"maxExp",1); |
---|
1253 | attrib(L,"isLetterplaceRing",lV); |
---|
1254 | def @R = ring(L); |
---|
1255 | // setring @R; |
---|
1256 | // int uptodeg = d; int lV = nvars(basering); // were defined before |
---|
1257 | def @@R = setLetterplaceAttributes(@R,uptodeg,lV); |
---|
1258 | return (@@R); |
---|
1259 | } |
---|
1260 | example |
---|
1261 | { |
---|
1262 | "EXAMPLE:"; echo = 2; |
---|
1263 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
1264 | def A = makeLetterplaceRing2(2); |
---|
1265 | setring A; |
---|
1266 | A; |
---|
1267 | lpVarBlockSize(A); // number of variables in the main block |
---|
1268 | lpDegBound(A); // degree bound |
---|
1269 | } |
---|
1270 | |
---|
1271 | static proc makeLetterplaceRing4(int d) |
---|
1272 | "USAGE: makeLetterplaceRing4(d); d an integer |
---|
1273 | RETURN: ring |
---|
1274 | PURPOSE: creates a Letterplace ring with a Dp ordering, suitable for |
---|
1275 | @* the use of non-homogeneous letterplace |
---|
1276 | NOTE: the matrix for the ordering looks as follows: first row is 1,1,...,1 |
---|
1277 | EXAMPLE: example makeLetterplaceRing4; shows examples |
---|
1278 | " |
---|
1279 | { |
---|
1280 | |
---|
1281 | // ToDo future: inherit positive weights in the orig ring |
---|
1282 | // complain on nonpositive ones |
---|
1283 | |
---|
1284 | // d = up to degree, will be shifted to d+1 |
---|
1285 | if (d<1) {"bad d"; return(0);} |
---|
1286 | |
---|
1287 | int uptodeg = d; int lV = nvars(basering); |
---|
1288 | |
---|
1289 | int ppl = printlevel-voice+2; |
---|
1290 | string err = ""; |
---|
1291 | |
---|
1292 | int i,j,s; |
---|
1293 | def save = basering; |
---|
1294 | int D = d-1; |
---|
1295 | list LR = ringlist(save); |
---|
1296 | list L, tmp, tmp2, tmp3; |
---|
1297 | L[1] = LR[1]; // ground field |
---|
1298 | L[4] = LR[4]; // quotient ideal |
---|
1299 | tmp = LR[2]; // varnames |
---|
1300 | s = size(LR[2]); |
---|
1301 | for (i=1; i<=D; i++) |
---|
1302 | { |
---|
1303 | for (j=1; j<=s; j++) |
---|
1304 | { |
---|
1305 | tmp[i*s+j] =tmp[j]; |
---|
1306 | } |
---|
1307 | } |
---|
1308 | L[2] = tmp; |
---|
1309 | list OrigNames = LR[2]; |
---|
1310 | |
---|
1311 | s = size(LR[3]); |
---|
1312 | list ordering; |
---|
1313 | ordering[1] = list("Dp",intvec(1: int(d*lV))); |
---|
1314 | ordering[2] = LR[3][s]; // module ord to place at the very end |
---|
1315 | LR[3] = ordering; |
---|
1316 | |
---|
1317 | L[3] = LR[3]; |
---|
1318 | attrib(L,"maxExp",1); |
---|
1319 | attrib(L,"isLetterplaceRing",lV); |
---|
1320 | def @R = ring(L); |
---|
1321 | def @@R = setLetterplaceAttributes(@R,uptodeg,lV); |
---|
1322 | return (@@R); |
---|
1323 | } |
---|
1324 | example |
---|
1325 | { |
---|
1326 | "EXAMPLE:"; echo = 2; |
---|
1327 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
1328 | def A = makeLetterplaceRing4(2); |
---|
1329 | setring A; |
---|
1330 | A; |
---|
1331 | lpVarBlockSize(A); // number of variables in the main block |
---|
1332 | lpDegBound(A); // degree bound |
---|
1333 | } |
---|
1334 | |
---|
1335 | // P[s;sigma] approach |
---|
1336 | static proc makeLetterplaceRing3(int d) |
---|
1337 | "USAGE: makeLetterplaceRing3(d); d an integer |
---|
1338 | RETURN: ring |
---|
1339 | PURPOSE: creates a ring with a special ordering, representing |
---|
1340 | @* the original P[s,sigma] (adds d blocks of shifted s) |
---|
1341 | ASSUME: basering is a letterplace ring |
---|
1342 | NOTE: experimental status |
---|
1343 | EXAMPLE: example makeLetterplaceRing3; shows examples |
---|
1344 | " |
---|
1345 | { |
---|
1346 | // d = up to degree, will be shifted to d+1 |
---|
1347 | if (d<1) {"bad d"; return(0);} |
---|
1348 | |
---|
1349 | int uptodeg = d; int lV = nvars(basering); |
---|
1350 | |
---|
1351 | int ppl = printlevel-voice+2; |
---|
1352 | string err = ""; |
---|
1353 | |
---|
1354 | int i,j,s; |
---|
1355 | def save = basering; |
---|
1356 | int D = d-1; |
---|
1357 | list LR = ringlist(save); |
---|
1358 | list L, tmp; |
---|
1359 | L[1] = LR[1]; // ground field |
---|
1360 | L[4] = LR[4]; // quotient ideal |
---|
1361 | tmp = LR[2]; // varnames |
---|
1362 | tmp[size(tmp)+1] = "s"; |
---|
1363 | // add s's |
---|
1364 | // string newSname = "@s"; |
---|
1365 | s = size(LR[2]); |
---|
1366 | for (i=1; i<=D; i++) |
---|
1367 | { |
---|
1368 | for (j=1; j<=s; j++) |
---|
1369 | { |
---|
1370 | tmp[i*s+j] = tmp[j]; |
---|
1371 | } |
---|
1372 | } |
---|
1373 | // the final index is D*s+s = (D+1)*s = degBound*s |
---|
1374 | L[2] = tmp; |
---|
1375 | list OrigNames = LR[2]; |
---|
1376 | // ordering: d blocks of the MODIFIED ord on r |
---|
1377 | // try to get whether the ord on r is blockord itself |
---|
1378 | // TODO: make L(2) ordering! exponent is maximally 2 |
---|
1379 | s = size(LR[3]); |
---|
1380 | |
---|
1381 | // assume: basering was a letterplace, so get its block |
---|
1382 | tmp = LR[3][1]; // ASSUME: it's a nice block |
---|
1383 | // modify it |
---|
1384 | // add (0,..,0,1) ... as antiblock part |
---|
1385 | intvec iv; list ttmp, tmp1; |
---|
1386 | for (i=1; i<=d; i++) |
---|
1387 | { |
---|
1388 | // the position to hold 1: |
---|
1389 | iv = intvec( gen( i*(lV+1)-1 ) ); |
---|
1390 | ttmp[1] = "a"; |
---|
1391 | ttmp[2] = iv; |
---|
1392 | tmp1[i] = ttmp; |
---|
1393 | } |
---|
1394 | // finished: antiblock part //TOCONTINUE |
---|
1395 | |
---|
1396 | if (s==2) |
---|
1397 | { |
---|
1398 | // not a blockord, 1 block + module ord |
---|
1399 | tmp = LR[3][s]; // module ord |
---|
1400 | for (i=1; i<=D; i++) |
---|
1401 | { |
---|
1402 | LR[3][s-1+i] = LR[3][1]; |
---|
1403 | } |
---|
1404 | LR[3][s+D] = tmp; |
---|
1405 | } |
---|
1406 | if (s>2) |
---|
1407 | { |
---|
1408 | // there are s-1 blocks |
---|
1409 | int nb = s-1; |
---|
1410 | tmp = LR[3][s]; // module ord |
---|
1411 | for (i=1; i<=D; i++) |
---|
1412 | { |
---|
1413 | for (j=1; j<=nb; j++) |
---|
1414 | { |
---|
1415 | LR[3][i*nb+j] = LR[3][j]; |
---|
1416 | } |
---|
1417 | } |
---|
1418 | // size(LR[3]); |
---|
1419 | LR[3][nb*(D+1)+1] = tmp; |
---|
1420 | } |
---|
1421 | L[3] = LR[3]; |
---|
1422 | attrib(L,"maxExp",1); |
---|
1423 | attrib(L,"isLetterplaceRing",lV); |
---|
1424 | def @R = ring(L); |
---|
1425 | // setring @R; |
---|
1426 | // int uptodeg = d; int lV = nvars(basering); // were defined before |
---|
1427 | def @@R = setLetterplaceAttributes(@R,uptodeg,lV); |
---|
1428 | return (@@R); |
---|
1429 | } |
---|
1430 | example |
---|
1431 | { |
---|
1432 | "EXAMPLE:"; echo = 2; |
---|
1433 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
1434 | def A = makeLetterplaceRing3(2); |
---|
1435 | setring A; |
---|
1436 | A; |
---|
1437 | lpVarBlockSize(A); // number of variables in the main block |
---|
1438 | lpDegBound(A); // degree bound |
---|
1439 | } |
---|
1440 | |
---|
1441 | /* EXAMPLES: |
---|
1442 | |
---|
1443 | //static proc ex_shift() |
---|
1444 | { |
---|
1445 | LIB "freegb.lib"; |
---|
1446 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
1447 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
---|
1448 | module N = [1,x,y,x],[-1,y,x,y]; |
---|
1449 | list L; L[1] = M; L[2] = N; |
---|
1450 | lst2str(L); |
---|
1451 | def U = crs(L,5); |
---|
1452 | setring U; U; |
---|
1453 | I; |
---|
1454 | poly p = I[2]; // I[8]; |
---|
1455 | p; |
---|
1456 | stest(p,7,7,3); // error -> the world is ok |
---|
1457 | poly q1 = stest(p,1,7,3); //ok |
---|
1458 | poly q6 = stest(p,6,7,3); //ok |
---|
1459 | btest(p,3); //ok |
---|
1460 | btest(q1,3); //ok |
---|
1461 | btest(q6,3); //ok |
---|
1462 | } |
---|
1463 | |
---|
1464 | //static proc test_shrink() |
---|
1465 | { |
---|
1466 | LIB "freegb.lib"; |
---|
1467 | ring r =0,(x,y,z),dp; |
---|
1468 | int d = 5; |
---|
1469 | def R = freeAlgebra(r, d); |
---|
1470 | setring R; |
---|
1471 | poly p1 = x(1)*y(2)*z(3); |
---|
1472 | poly p2 = x(1)*y(4)*z(5); |
---|
1473 | poly p3 = x(1)*y(1)*z(3); |
---|
1474 | poly p4 = x(1)*y(2)*z(2); |
---|
1475 | poly p5 = x(3)*z(5); |
---|
1476 | poly p6 = x(1)*y(1)*x(3)*z(5); |
---|
1477 | poly p7 = x(1)*y(2)*x(3)*y(4)*z(5); |
---|
1478 | poly p8 = p1+p2+p3+p4+p5 + p6 + p7; |
---|
1479 | p1; system("shrinktest",p1,3); |
---|
1480 | p2; system("shrinktest",p2,3); |
---|
1481 | p3; system("shrinktest",p3,3); |
---|
1482 | p4; system("shrinktest",p4,3); |
---|
1483 | p5; system("shrinktest",p5,3); |
---|
1484 | p6; system("shrinktest",p6,3); |
---|
1485 | p7; system("shrinktest",p7,3); |
---|
1486 | p8; system("shrinktest",p8,3); |
---|
1487 | poly p9 = p1 + 2*p2 + 5*p5 + 7*p7; |
---|
1488 | p9; system("shrinktest",p9,3); |
---|
1489 | } |
---|
1490 | |
---|
1491 | //static proc ex2() |
---|
1492 | { |
---|
1493 | option(prot); |
---|
1494 | LIB "freegb.lib"; |
---|
1495 | ring r = 0,(x,y),dp; |
---|
1496 | module M = [-1,x,y],[3,x,x]; // 3x^2 - xy |
---|
1497 | def U = freegb(M,7); |
---|
1498 | lst2str(U); |
---|
1499 | } |
---|
1500 | |
---|
1501 | //static proc ex_nonhomog() |
---|
1502 | { |
---|
1503 | option(prot); |
---|
1504 | LIB "freegb.lib"; |
---|
1505 | ring r = 0,(x,y,h),dp; |
---|
1506 | list L; |
---|
1507 | module M; |
---|
1508 | M = [-1,y,y],[1,x,x,x]; // x3-y2 |
---|
1509 | L[1] = M; |
---|
1510 | M = [1,x,h],[-1,h,x]; // xh-hx |
---|
1511 | L[2] = M; |
---|
1512 | M = [1,y,h],[-1,h,y]; // yh-hy |
---|
1513 | L[3] = M; |
---|
1514 | def U = freegb(L,4); |
---|
1515 | lst2str(U); |
---|
1516 | // strange elements in the basis |
---|
1517 | } |
---|
1518 | |
---|
1519 | //static proc ex_nonhomog_comm() |
---|
1520 | { |
---|
1521 | option(prot); |
---|
1522 | LIB "freegb.lib"; |
---|
1523 | ring r = 0,(x,y),dp; |
---|
1524 | module M = [-1,y,y],[1,x,x,x]; |
---|
1525 | def U = freegb(M,5); |
---|
1526 | lst2str(U); |
---|
1527 | } |
---|
1528 | |
---|
1529 | //static proc ex_nonhomog_h() |
---|
1530 | { |
---|
1531 | option(prot); |
---|
1532 | LIB "freegb.lib"; |
---|
1533 | ring r = 0,(x,y,h),(a(1,1),dp); |
---|
1534 | module M = [-1,y,y,h],[1,x,x,x]; // x3 - y2h |
---|
1535 | def U = freegb(M,6); |
---|
1536 | lst2str(U); |
---|
1537 | } |
---|
1538 | |
---|
1539 | //static proc ex_nonhomog_h2() |
---|
1540 | { |
---|
1541 | option(prot); |
---|
1542 | LIB "freegb.lib"; |
---|
1543 | ring r = 0,(x,y,h),(dp); |
---|
1544 | list L; |
---|
1545 | module M; |
---|
1546 | M = [-1,y,y,h],[1,x,x,x]; // x3 - y2h |
---|
1547 | L[1] = M; |
---|
1548 | M = [1,x,h],[-1,h,x]; // xh - hx |
---|
1549 | L[2] = M; |
---|
1550 | M = [1,y,h],[-1,h,y]; // yh - hy |
---|
1551 | L[3] = M; |
---|
1552 | def U = freeGBasis(L,3); |
---|
1553 | lst2str(U); |
---|
1554 | // strange answer CHECK |
---|
1555 | } |
---|
1556 | |
---|
1557 | //static proc ex_nonhomog_3() |
---|
1558 | { |
---|
1559 | option(prot); |
---|
1560 | LIB "./freegb.lib"; |
---|
1561 | ring r = 0,(x,y,z),(dp); |
---|
1562 | list L; |
---|
1563 | module M; |
---|
1564 | M = [1,z,y],[-1,x]; // zy - x |
---|
1565 | L[1] = M; |
---|
1566 | M = [1,z,x],[-1,y]; // zx - y |
---|
1567 | L[2] = M; |
---|
1568 | M = [1,y,x],[-1,z]; // yx - z |
---|
1569 | L[3] = M; |
---|
1570 | lst2str(L); |
---|
1571 | list U = freegb(L,4); |
---|
1572 | lst2str(U); |
---|
1573 | // strange answer CHECK |
---|
1574 | } |
---|
1575 | |
---|
1576 | //static proc ex_densep_2() |
---|
1577 | { |
---|
1578 | option(prot); |
---|
1579 | LIB "freegb.lib"; |
---|
1580 | ring r = (0,a,b,c),(x,y),(Dp); // deglex |
---|
1581 | module M = [1,x,x], [a,x,y], [b,y,x], [c,y,y]; |
---|
1582 | lst2str(M); |
---|
1583 | list U = freegb(M,5); |
---|
1584 | lst2str(U); |
---|
1585 | // a=b is important -> finite basis!!! |
---|
1586 | module M = [1,x,x], [a,x,y], [a,y,x], [c,y,y]; |
---|
1587 | lst2str(M); |
---|
1588 | list U = freegb(M,5); |
---|
1589 | lst2str(U); |
---|
1590 | } |
---|
1591 | |
---|
1592 | // END COMMENTED EXAMPLES |
---|
1593 | |
---|
1594 | */ |
---|
1595 | |
---|
1596 | // 1. form a new ring |
---|
1597 | // 2. produce shifted generators |
---|
1598 | // 3. compute GB |
---|
1599 | // 4. skip shifted elts |
---|
1600 | // 5. go back to orig vars, produce strings/modules |
---|
1601 | // 6. return the result |
---|
1602 | |
---|
1603 | static proc freegbold(list LM, int d) |
---|
1604 | "USAGE: freegbold(L, d); L a list of modules, d an integer |
---|
1605 | RETURN: ring |
---|
1606 | PURPOSE: compute the two-sided Groebner basis of an ideal, encoded by L in |
---|
1607 | the free associative algebra, up to degree d |
---|
1608 | EXAMPLE: example freegbold; shows examples |
---|
1609 | " |
---|
1610 | { |
---|
1611 | // d = up to degree, will be shifted to d+1 |
---|
1612 | if (d<1) {"bad d"; return(0);} |
---|
1613 | |
---|
1614 | int ppl = printlevel-voice+2; |
---|
1615 | string err = ""; |
---|
1616 | |
---|
1617 | int i,j,s; |
---|
1618 | def save = basering; |
---|
1619 | // determine max no of places in the input |
---|
1620 | int slm = size(LM); // numbers of polys in the ideal |
---|
1621 | int sm; |
---|
1622 | intvec iv; |
---|
1623 | module M; |
---|
1624 | for (i=1; i<=slm; i++) |
---|
1625 | { |
---|
1626 | // modules, e.g. free polynomials |
---|
1627 | M = LM[i]; |
---|
1628 | sm = ncols(M); |
---|
1629 | for (j=1; j<=sm; j++) |
---|
1630 | { |
---|
1631 | //vectors, e.g. free monomials |
---|
1632 | iv = iv, size(M[j])-1; // 1 place is reserved by the coeff |
---|
1633 | } |
---|
1634 | } |
---|
1635 | int D = Max(iv); // max size of input words |
---|
1636 | if (d<D) {"bad d"; return(LM);} |
---|
1637 | D = D + d-1; |
---|
1638 | // D = d; |
---|
1639 | list LR = ringlist(save); |
---|
1640 | list L, tmp; |
---|
1641 | L[1] = LR[1]; // ground field |
---|
1642 | L[4] = LR[4]; // quotient ideal |
---|
1643 | tmp = LR[2]; // varnames |
---|
1644 | s = size(LR[2]); |
---|
1645 | for (i=1; i<=D; i++) |
---|
1646 | { |
---|
1647 | for (j=1; j<=s; j++) |
---|
1648 | { |
---|
1649 | tmp[i*s+j] = tmp[j]; |
---|
1650 | } |
---|
1651 | } |
---|
1652 | L[2] = tmp; |
---|
1653 | list OrigNames = LR[2]; |
---|
1654 | // ordering: d blocks of the ord on r |
---|
1655 | // try to get whether the ord on r is blockord itself |
---|
1656 | // TODO: make L(2) ordering! exponent is maximally 2 |
---|
1657 | s = size(LR[3]); |
---|
1658 | if (s==2) |
---|
1659 | { |
---|
1660 | // not a blockord, 1 block + module ord |
---|
1661 | tmp = LR[3][s]; // module ord |
---|
1662 | for (i=1; i<=D; i++) |
---|
1663 | { |
---|
1664 | LR[3][s-1+i] = LR[3][1]; |
---|
1665 | } |
---|
1666 | LR[3][s+D] = tmp; |
---|
1667 | } |
---|
1668 | if (s>2) |
---|
1669 | { |
---|
1670 | // there are s-1 blocks |
---|
1671 | int nb = s-1; |
---|
1672 | tmp = LR[3][s]; // module ord |
---|
1673 | for (i=1; i<=D; i++) |
---|
1674 | { |
---|
1675 | for (j=1; j<=nb; j++) |
---|
1676 | { |
---|
1677 | LR[3][i*nb+j] = LR[3][j]; |
---|
1678 | } |
---|
1679 | } |
---|
1680 | // size(LR[3]); |
---|
1681 | LR[3][nb*(D+1)+1] = tmp; |
---|
1682 | } |
---|
1683 | L[3] = LR[3]; |
---|
1684 | attrib(L,"maxExp",1); |
---|
1685 | attrib(L,"isLetterplaceRing",s); |
---|
1686 | def @R = ring(L); |
---|
1687 | setring @R; |
---|
1688 | ideal I; |
---|
1689 | poly @p; |
---|
1690 | s = size(OrigNames); |
---|
1691 | // "s:";s; |
---|
1692 | // convert LM to canonical vectors (no powers) |
---|
1693 | setring save; |
---|
1694 | kill M; // M was defined earlier |
---|
1695 | module M; |
---|
1696 | slm = size(LM); // numbers of polys in the ideal |
---|
1697 | int sv,k,l; |
---|
1698 | vector v; |
---|
1699 | // poly p; |
---|
1700 | string sp; |
---|
1701 | setring @R; |
---|
1702 | poly @@p=0; |
---|
1703 | setring save; |
---|
1704 | for (l=1; l<=slm; l++) |
---|
1705 | { |
---|
1706 | // modules, e.g. free polynomials |
---|
1707 | M = LM[l]; |
---|
1708 | sm = ncols(M); // in intvec iv the sizes are stored |
---|
1709 | for (i=0; i<=d-iv[l]; i++) |
---|
1710 | { |
---|
1711 | // modules, e.g. free polynomials |
---|
1712 | for (j=1; j<=sm; j++) |
---|
1713 | { |
---|
1714 | //vectors, e.g. free monomials |
---|
1715 | v = M[j]; |
---|
1716 | sv = size(v); |
---|
1717 | // "sv:";sv; |
---|
1718 | sp = "@@p = @@p + "; |
---|
1719 | for (k=2; k<=sv; k++) |
---|
1720 | { |
---|
1721 | sp = sp + string(v[k])+")*"; |
---|
1722 | } |
---|
1723 | sp = sp + string(v[1])+";"; // coef; |
---|
1724 | setring @R; |
---|
1725 | execute(sp); |
---|
1726 | setring save; |
---|
1727 | } |
---|
1728 | setring @R; |
---|
1729 | // "@@p:"; @@p; |
---|
1730 | I = I,@@p; |
---|
1731 | @@p = 0; |
---|
1732 | setring save; |
---|
1733 | } |
---|
1734 | } |
---|
1735 | kill sp; |
---|
1736 | // 3. compute GB |
---|
1737 | setring @R; |
---|
1738 | dbprint(ppl,"computing GB"); |
---|
1739 | // ideal J = groebner(I); |
---|
1740 | ideal J = slimgb(I); |
---|
1741 | dbprint(ppl,J); |
---|
1742 | // 4. skip shifted elts |
---|
1743 | ideal K = select1(J,1..s); // s = size(OrigNames) |
---|
1744 | dbprint(ppl,K); |
---|
1745 | dbprint(ppl, "done with GB"); |
---|
1746 | // K contains vars x(1),...z(1) = images of originals |
---|
1747 | // 5. go back to orig vars, produce strings/modules |
---|
1748 | if (K[1] == 0) |
---|
1749 | { |
---|
1750 | "no reasonable output, GB gives 0"; |
---|
1751 | return(0); |
---|
1752 | } |
---|
1753 | int sk = size(K); |
---|
1754 | int sp, sx, a, b; |
---|
1755 | intvec x; |
---|
1756 | poly p,q; |
---|
1757 | poly pn; |
---|
1758 | // vars in 'save' |
---|
1759 | setring save; |
---|
1760 | module N; |
---|
1761 | list LN; |
---|
1762 | vector V; |
---|
1763 | poly pn; |
---|
1764 | // test and skip exponents >=2 |
---|
1765 | setring @R; |
---|
1766 | for(i=1; i<=sk; i++) |
---|
1767 | { |
---|
1768 | p = K[i]; |
---|
1769 | while (p!=0) |
---|
1770 | { |
---|
1771 | q = lead(p); |
---|
1772 | // "processing q:";q; |
---|
1773 | x = leadexp(q); |
---|
1774 | sx = size(x); |
---|
1775 | for(k=1; k<=sx; k++) |
---|
1776 | { |
---|
1777 | if ( x[k] >= 2 ) |
---|
1778 | { |
---|
1779 | err = "skip: the value x[k] is " + string(x[k]); |
---|
1780 | dbprint(ppl,err); |
---|
1781 | // return(0); |
---|
1782 | K[i] = 0; |
---|
1783 | p = 0; |
---|
1784 | q = 0; |
---|
1785 | break; |
---|
1786 | } |
---|
1787 | } |
---|
1788 | p = p - q; |
---|
1789 | } |
---|
1790 | } |
---|
1791 | K = simplify(K,2); |
---|
1792 | sk = size(K); |
---|
1793 | for(i=1; i<=sk; i++) |
---|
1794 | { |
---|
1795 | // setring save; |
---|
1796 | // V = 0; |
---|
1797 | setring @R; |
---|
1798 | p = K[i]; |
---|
1799 | while (p!=0) |
---|
1800 | { |
---|
1801 | q = lead(p); |
---|
1802 | err = "processing q:" + string(q); |
---|
1803 | dbprint(ppl,err); |
---|
1804 | x = leadexp(q); |
---|
1805 | sx = size(x); |
---|
1806 | pn = leadcoef(q); |
---|
1807 | setring save; |
---|
1808 | pn = imap(@R,pn); |
---|
1809 | V = V + leadcoef(pn)*gen(1); |
---|
1810 | for(k=1; k<=sx; k++) |
---|
1811 | { |
---|
1812 | if (x[k] ==1) |
---|
1813 | { |
---|
1814 | a = k div s; // block number=a+1, a!=0 |
---|
1815 | b = k % s; // remainder |
---|
1816 | // printf("a: %s, b: %s",a,b); |
---|
1817 | if (b == 0) |
---|
1818 | { |
---|
1819 | // that is it's the last var in the block |
---|
1820 | b = s; |
---|
1821 | a = a-1; |
---|
1822 | } |
---|
1823 | V = V + var(b)*gen(a+2); |
---|
1824 | } |
---|
1825 | // else |
---|
1826 | // { |
---|
1827 | // printf("error: the value x[k] is %s", x[k]); |
---|
1828 | // return(0); |
---|
1829 | // } |
---|
1830 | } |
---|
1831 | err = "V: " + string(V); |
---|
1832 | dbprint(ppl,err); |
---|
1833 | // printf("V: %s", string(V)); |
---|
1834 | N = N,V; |
---|
1835 | V = 0; |
---|
1836 | setring @R; |
---|
1837 | p = p - q; |
---|
1838 | pn = 0; |
---|
1839 | } |
---|
1840 | setring save; |
---|
1841 | LN[i] = simplify(N,2); |
---|
1842 | N = 0; |
---|
1843 | } |
---|
1844 | setring save; |
---|
1845 | return(LN); |
---|
1846 | } |
---|
1847 | example |
---|
1848 | { |
---|
1849 | "EXAMPLE:"; echo = 2; |
---|
1850 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
1851 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
---|
1852 | module N = [1,x,y,x],[-1,y,x,y]; |
---|
1853 | list L; L[1] = M; L[2] = N; |
---|
1854 | lst2str(L); |
---|
1855 | def U = freegbold(L,5); |
---|
1856 | lst2str(U); |
---|
1857 | } |
---|
1858 | |
---|
1859 | /* begin older procs and tests |
---|
1860 | |
---|
1861 | static proc exHom1() |
---|
1862 | { |
---|
1863 | // we start with |
---|
1864 | // z*y - x, z*x - y, y*x - z |
---|
1865 | LIB "freegb.lib"; |
---|
1866 | LIB "elim.lib"; |
---|
1867 | ring r = 0,(x,y,z,h),dp; |
---|
1868 | list L; |
---|
1869 | module M; |
---|
1870 | M = [1,z,y],[-1,x,h]; // zy - xh |
---|
1871 | L[1] = M; |
---|
1872 | M = [1,z,x],[-1,y,h]; // zx - yh |
---|
1873 | L[2] = M; |
---|
1874 | M = [1,y,x],[-1,z,h]; // yx - zh |
---|
1875 | L[3] = M; |
---|
1876 | lst2str(L); |
---|
1877 | def U = crs(L,4); |
---|
1878 | setring U; |
---|
1879 | I = I, |
---|
1880 | y(2)*h(3)+z(2)*x(3), y(3)*h(4)+z(3)*x(4), |
---|
1881 | y(2)*x(3)-z(2)*h(3), y(3)*x(4)-z(3)*h(4); |
---|
1882 | I = simplify(I,2); |
---|
1883 | ring r2 = 0,(x(0..4),y(0..4),z(0..4),h(0..4)),dp; |
---|
1884 | ideal J = imap(U,I); |
---|
1885 | // ideal K = homog(J,h); |
---|
1886 | option(redSB); |
---|
1887 | option(redTail); |
---|
1888 | ideal L = groebner(J); //(K); |
---|
1889 | ideal LL = sat(L,ideal(h)); |
---|
1890 | ideal M = subst(LL,h,1); |
---|
1891 | M = simplify(M,2); |
---|
1892 | setring U; |
---|
1893 | ideal M = imap(r2,M); |
---|
1894 | lst2str(U); |
---|
1895 | } |
---|
1896 | |
---|
1897 | static proc test1() |
---|
1898 | { |
---|
1899 | LIB "freegb.lib"; |
---|
1900 | ring r = 0,(x,y),Dp; |
---|
1901 | int d = 10; // degree |
---|
1902 | def R = freeAlgebra(r, d); |
---|
1903 | setring R; |
---|
1904 | ideal I = x(1)*x(2) - y(1)*y(2); |
---|
1905 | option(prot); |
---|
1906 | option(teach); |
---|
1907 | ideal J = system("freegb",I,d,2); |
---|
1908 | J; |
---|
1909 | } |
---|
1910 | |
---|
1911 | static proc test2() |
---|
1912 | { |
---|
1913 | LIB "freegb.lib"; |
---|
1914 | ring r = 0,(x,y),Dp; |
---|
1915 | int d = 10; // degree |
---|
1916 | def R = freeAlgebra(r, d); |
---|
1917 | setring R; |
---|
1918 | ideal I = x(1)*x(2) - x(1)*y(2); |
---|
1919 | option(prot); |
---|
1920 | option(teach); |
---|
1921 | ideal J = system("freegb",I,d,2); |
---|
1922 | J; |
---|
1923 | } |
---|
1924 | |
---|
1925 | static proc test3() |
---|
1926 | { |
---|
1927 | LIB "freegb.lib"; |
---|
1928 | ring r = 0,(x,y,z),dp; |
---|
1929 | int d =5; // degree |
---|
1930 | def R = freeAlgebra(r, d); |
---|
1931 | setring R; |
---|
1932 | ideal I = x(1)*y(2), y(1)*x(2)+z(1)*z(2); |
---|
1933 | option(prot); |
---|
1934 | option(teach); |
---|
1935 | ideal J = system("freegb",I,d,3); |
---|
1936 | } |
---|
1937 | |
---|
1938 | end older procs and tests */ |
---|
1939 | |
---|
1940 | static proc stringpoly2lplace(string s) |
---|
1941 | { |
---|
1942 | // decomposes sentence into terms |
---|
1943 | s = replace(s,newline,""); // get rid of newlines |
---|
1944 | s = replace(s," ",""); // get rid of empties |
---|
1945 | //arith symbols: +,- |
---|
1946 | // decompose into words with coeffs |
---|
1947 | list LS; |
---|
1948 | int i,j,ie,je,k,cnt; |
---|
1949 | // s[1]="-" situation |
---|
1950 | if (s[1]=="-") |
---|
1951 | { |
---|
1952 | LS = stringpoly2lplace(string(s[2..size(s)])); |
---|
1953 | LS[1] = string("-"+string(LS[1])); |
---|
1954 | return(LS); |
---|
1955 | } |
---|
1956 | i = find(s,"-",2); |
---|
1957 | // i==1 might happen if the 1st symbol coeff is negative |
---|
1958 | j = find(s,"+"); |
---|
1959 | list LL; |
---|
1960 | if (i==j) |
---|
1961 | { |
---|
1962 | "return a monomial"; |
---|
1963 | // that is both are 0 -> s is a monomial |
---|
1964 | LS[1] = s; |
---|
1965 | return(LS); |
---|
1966 | } |
---|
1967 | if (i==0) |
---|
1968 | { |
---|
1969 | "i==0 situation"; |
---|
1970 | // no minuses at all => pluses only |
---|
1971 | cnt++; |
---|
1972 | LS[cnt] = string(s[1..j-1]); |
---|
1973 | s = s[j+1..size(s)]; |
---|
1974 | while (s!= "") |
---|
1975 | { |
---|
1976 | j = find(s,"+"); |
---|
1977 | cnt++; |
---|
1978 | if (j==0) |
---|
1979 | { |
---|
1980 | LS[cnt] = string(s); |
---|
1981 | s = ""; |
---|
1982 | } |
---|
1983 | else |
---|
1984 | { |
---|
1985 | LS[cnt] = string(s[1..j-1]); |
---|
1986 | s = s[j+1..size(s)]; |
---|
1987 | } |
---|
1988 | } |
---|
1989 | return(LS); |
---|
1990 | } |
---|
1991 | if (j==0) |
---|
1992 | { |
---|
1993 | "j==0 situation"; |
---|
1994 | // no pluses at all except the lead coef => the rest are minuses only |
---|
1995 | cnt++; |
---|
1996 | LS[cnt] = string(s[1..i-1]); |
---|
1997 | s = s[i..size(s)]; |
---|
1998 | while (s!= "") |
---|
1999 | { |
---|
2000 | i = find(s,"-",2); |
---|
2001 | cnt++; |
---|
2002 | if (i==0) |
---|
2003 | { |
---|
2004 | LS[cnt] = string(s); |
---|
2005 | s = ""; |
---|
2006 | } |
---|
2007 | else |
---|
2008 | { |
---|
2009 | LS[cnt] = string(s[1..i-1]); |
---|
2010 | s = s[i..size(s)]; |
---|
2011 | } |
---|
2012 | } |
---|
2013 | return(LS); |
---|
2014 | } |
---|
2015 | // now i, j are nonzero |
---|
2016 | if (i>j) |
---|
2017 | { |
---|
2018 | "i>j situation"; |
---|
2019 | // + comes first, at place j |
---|
2020 | cnt++; |
---|
2021 | // "cnt:"; cnt; "j:"; j; |
---|
2022 | LS[cnt] = string(s[1..j-1]); |
---|
2023 | s = s[j+1..size(s)]; |
---|
2024 | LL = stringpoly2lplace(s); |
---|
2025 | LS = LS + LL; |
---|
2026 | kill LL; |
---|
2027 | return(LS); |
---|
2028 | } |
---|
2029 | else |
---|
2030 | { |
---|
2031 | "j>i situation"; |
---|
2032 | // - might come first, at place i |
---|
2033 | if (i>1) |
---|
2034 | { |
---|
2035 | cnt++; |
---|
2036 | LS[cnt] = string(s[1..i-1]); |
---|
2037 | s = s[i..size(s)]; |
---|
2038 | } |
---|
2039 | else |
---|
2040 | { |
---|
2041 | // i==1-> minus at leadcoef |
---|
2042 | ie = find(s,"-",i+1); |
---|
2043 | je = find(s,"+",i+1); |
---|
2044 | if (je == ie) |
---|
2045 | { |
---|
2046 | "ie=je situation"; |
---|
2047 | //monomial |
---|
2048 | cnt++; |
---|
2049 | LS[cnt] = s; |
---|
2050 | return(LS); |
---|
2051 | } |
---|
2052 | if (je < ie) |
---|
2053 | { |
---|
2054 | "je<ie situation"; |
---|
2055 | // + comes first |
---|
2056 | cnt++; |
---|
2057 | LS[cnt] = s[1..je-1]; |
---|
2058 | s = s[je+1..size(s)]; |
---|
2059 | } |
---|
2060 | else |
---|
2061 | { |
---|
2062 | // ie < je |
---|
2063 | "ie<je situation"; |
---|
2064 | cnt++; |
---|
2065 | LS[cnt] = s[1..ie-1]; |
---|
2066 | s = s[ie..size(s)]; |
---|
2067 | } |
---|
2068 | } |
---|
2069 | "going into recursion with "+s; |
---|
2070 | LL = stringpoly2lplace(s); |
---|
2071 | LS = LS + LL; |
---|
2072 | return(LS); |
---|
2073 | } |
---|
2074 | } |
---|
2075 | example |
---|
2076 | { |
---|
2077 | "EXAMPLE:"; echo = 2; |
---|
2078 | string s = "x*y+y*z+z*t"; // + only |
---|
2079 | stringpoly2lplace(s); |
---|
2080 | string s2 = "x*y - y*z-z*t*w*w"; // +1, - only |
---|
2081 | stringpoly2lplace(s2); |
---|
2082 | string s3 = "-x*y + y - 2*x +7*w*w*w"; |
---|
2083 | stringpoly2lplace(s3); |
---|
2084 | } |
---|
2085 | |
---|
2086 | static proc addplaces(list L) |
---|
2087 | { |
---|
2088 | // adds places to the list of strings |
---|
2089 | // according to their order in the list |
---|
2090 | int s = size(L); |
---|
2091 | int i; |
---|
2092 | for (i=1; i<=s; i++) |
---|
2093 | { |
---|
2094 | if (typeof(L[i]) == "string") |
---|
2095 | { |
---|
2096 | L[i] = L[i] + "(" + string(i) + ")"; |
---|
2097 | } |
---|
2098 | else |
---|
2099 | { |
---|
2100 | ERROR("entry of type string expected"); |
---|
2101 | return(0); |
---|
2102 | } |
---|
2103 | } |
---|
2104 | return(L); |
---|
2105 | } |
---|
2106 | example |
---|
2107 | { |
---|
2108 | "EXAMPLE:"; echo = 2; |
---|
2109 | string a = "f1"; string b = "f2"; |
---|
2110 | list L = a,b,a; |
---|
2111 | addplaces(L); |
---|
2112 | } |
---|
2113 | |
---|
2114 | static proc sent2lplace(string s) |
---|
2115 | { |
---|
2116 | // SENTence of words TO LetterPLACE |
---|
2117 | list L = stringpoly2lplace(s); |
---|
2118 | int i; int ss = size(L); |
---|
2119 | for(i=1; i<=ss; i++) |
---|
2120 | { |
---|
2121 | L[i] = str2lplace(L[i]); |
---|
2122 | } |
---|
2123 | return(L); |
---|
2124 | } |
---|
2125 | example |
---|
2126 | { |
---|
2127 | "EXAMPLE:"; echo = 2; |
---|
2128 | ring r = 0,(f2,f1),dp; |
---|
2129 | string s = "f2*f1*f1 - 2*f1*f2*f1+ f1*f1*f2"; |
---|
2130 | sent2lplace(s); |
---|
2131 | } |
---|
2132 | |
---|
2133 | static proc testnumber(string s) |
---|
2134 | { |
---|
2135 | string t; |
---|
2136 | if (s[1]=="-") |
---|
2137 | { |
---|
2138 | // two situations: either there's a negative number |
---|
2139 | t = s[2..size(s)]; |
---|
2140 | if (testnumber(t)) |
---|
2141 | { |
---|
2142 | //a negative number |
---|
2143 | } |
---|
2144 | else |
---|
2145 | { |
---|
2146 | // a variable times -1 |
---|
2147 | } |
---|
2148 | // or just a "-" for -1 |
---|
2149 | } |
---|
2150 | t = "ring @r=("; |
---|
2151 | t = t + charstr(basering)+"),"; |
---|
2152 | t = t + string(var(1))+",dp;"; |
---|
2153 | // write(":w tstnum.tst",t); |
---|
2154 | t = t+ "number @@Nn = " + s + ";"+"$"; |
---|
2155 | write(":w tstnum.tst",t); |
---|
2156 | string runsing = system("Singular"); |
---|
2157 | int k; |
---|
2158 | t = runsing+ " -teq <tstnum.tst >tstnum.out"; |
---|
2159 | k = system("sh",t); |
---|
2160 | if (k!=0) |
---|
2161 | { |
---|
2162 | ERROR("Problems running Singular"); |
---|
2163 | } |
---|
2164 | int i = system("sh", "grep error tstnum.out > /dev/NULL"); |
---|
2165 | if (i!=0) |
---|
2166 | { |
---|
2167 | // no error: s is a number |
---|
2168 | i = 1; |
---|
2169 | } |
---|
2170 | k = system("sh","rm tstnum.tst tstnum.out > /dev/NULL"); |
---|
2171 | return(i); |
---|
2172 | } |
---|
2173 | example |
---|
2174 | { |
---|
2175 | "EXAMPLE:"; echo = 2; |
---|
2176 | ring r = (0,a),x,dp; |
---|
2177 | string s = "a^2+7*a-2"; |
---|
2178 | testnumber(s); |
---|
2179 | s = "b+a"; |
---|
2180 | testnumber(s); |
---|
2181 | } |
---|
2182 | |
---|
2183 | static proc str2lplace(string s) |
---|
2184 | { |
---|
2185 | // converts a word (monomial) with coeff into letter-place |
---|
2186 | // string: coef*var1^exp1*var2^exp2*...varN^expN |
---|
2187 | s = strpower2rep(s); // expand powers |
---|
2188 | if (size(s)==0) { return(0); } |
---|
2189 | int i,j,k,insC; |
---|
2190 | string a,b,c,d,t; |
---|
2191 | // 1. get coeff |
---|
2192 | i = find(s,"*"); |
---|
2193 | if (i==0) { return(s); } |
---|
2194 | list VN; |
---|
2195 | c = s[1..i-1]; // incl. the case like (-a^2+1) |
---|
2196 | int tn = testnumber(c); |
---|
2197 | if (tn == 0) |
---|
2198 | { |
---|
2199 | // failed test |
---|
2200 | if (c[1]=="-") |
---|
2201 | { |
---|
2202 | // two situations: either there's a negative number |
---|
2203 | t = c[2..size(c)]; |
---|
2204 | if (testnumber(t)) |
---|
2205 | { |
---|
2206 | //a negative number |
---|
2207 | // nop here |
---|
2208 | } |
---|
2209 | else |
---|
2210 | { |
---|
2211 | // a variable times -1 |
---|
2212 | c = "-1"; |
---|
2213 | j++; VN[j] = t; //string(c[2..size(c)]); |
---|
2214 | insC = 1; |
---|
2215 | } |
---|
2216 | } |
---|
2217 | else |
---|
2218 | { |
---|
2219 | // just a variable with coeff 1 |
---|
2220 | j++; VN[j] = string(c); |
---|
2221 | c = "1"; |
---|
2222 | insC = 1; |
---|
2223 | } |
---|
2224 | } |
---|
2225 | // get vars |
---|
2226 | t = s; |
---|
2227 | // t = s[i+1..size(s)]; |
---|
2228 | k = size(t); //j = 0; |
---|
2229 | while (k>0) |
---|
2230 | { |
---|
2231 | t = t[i+1..size(t)]; //next part |
---|
2232 | i = find(t,"*"); // next * |
---|
2233 | if (i==0) |
---|
2234 | { |
---|
2235 | // last monomial |
---|
2236 | j++; |
---|
2237 | VN[j] = t; |
---|
2238 | k = size(t); |
---|
2239 | break; |
---|
2240 | } |
---|
2241 | b = t[1..i-1]; |
---|
2242 | // print(b); |
---|
2243 | j++; |
---|
2244 | VN[j] = b; |
---|
2245 | k = size(t); |
---|
2246 | } |
---|
2247 | VN = addplaces(VN); |
---|
2248 | VN[size(VN)+1] = string(c); |
---|
2249 | return(VN); |
---|
2250 | } |
---|
2251 | example |
---|
2252 | { |
---|
2253 | "EXAMPLE:"; echo = 2; |
---|
2254 | ring r = (0,a),(f2,f1),dp; |
---|
2255 | str2lplace("-2*f2^2*f1^2*f2"); |
---|
2256 | str2lplace("-f1*f2"); |
---|
2257 | str2lplace("(-a^2+7a)*f1*f2"); |
---|
2258 | } |
---|
2259 | |
---|
2260 | static proc strpower2rep(string s) |
---|
2261 | { |
---|
2262 | // makes x*x*x*x out of x^4 ., rep statys for repetitions |
---|
2263 | // looks for "-" problem |
---|
2264 | // exception: "-" as coeff |
---|
2265 | string ex,t; |
---|
2266 | int i,j,k; |
---|
2267 | |
---|
2268 | i = find(s,"^"); // first ^ |
---|
2269 | if (i==0) { return(s); } // no ^ signs |
---|
2270 | |
---|
2271 | if (s[1] == "-") |
---|
2272 | { |
---|
2273 | // either -coef or -1 |
---|
2274 | // got the coeff: |
---|
2275 | i = find(s,"*"); |
---|
2276 | if (i==0) |
---|
2277 | { |
---|
2278 | // no *'s => coef == -1 or s == -23 |
---|
2279 | i = size(s)+1; |
---|
2280 | } |
---|
2281 | t = string(s[2..i-1]); // without "-" |
---|
2282 | if ( testnumber(t) ) |
---|
2283 | { |
---|
2284 | // a good number |
---|
2285 | t = strpower2rep(string(s[2..size(s)])); |
---|
2286 | t = "-"+t; |
---|
2287 | return(t); |
---|
2288 | } |
---|
2289 | else |
---|
2290 | { |
---|
2291 | // a variable |
---|
2292 | t = strpower2rep(string(s[2..size(s)])); |
---|
2293 | t = "-1*"+ t; |
---|
2294 | return(t); |
---|
2295 | } |
---|
2296 | } |
---|
2297 | // the case when leadcoef is a number in () |
---|
2298 | if (s[1] == "(") |
---|
2299 | { |
---|
2300 | i = find(s,")",2); // must be nonzero |
---|
2301 | t = s[2..i-1]; |
---|
2302 | if ( testnumber(t) ) |
---|
2303 | { |
---|
2304 | // a good number |
---|
2305 | } |
---|
2306 | else {"strpower2rep: bad number as coef";} |
---|
2307 | ex = string(s[i+2..size(s)]); // 2 because of * |
---|
2308 | ex = strpower2rep(ex); |
---|
2309 | t = "("+t+")*"+ex; |
---|
2310 | return(t); |
---|
2311 | } |
---|
2312 | |
---|
2313 | i = find(s,"^"); // first ^ |
---|
2314 | j = find(s,"*",i+1); // next * == end of ^ |
---|
2315 | if (j==0) |
---|
2316 | { |
---|
2317 | ex = s[i+1..size(s)]; |
---|
2318 | } |
---|
2319 | else |
---|
2320 | { |
---|
2321 | ex = s[i+1..j-1]; |
---|
2322 | } |
---|
2323 | execute("int @exp = " + ex + ";"); //@exp = exponent |
---|
2324 | // got varname |
---|
2325 | for (k=i-1; k>0; k--) |
---|
2326 | { |
---|
2327 | if (s[k] == "*") break; |
---|
2328 | } |
---|
2329 | string varn = s[k+1..i-1]; |
---|
2330 | // "varn:"; varn; |
---|
2331 | string pref; |
---|
2332 | if (k>0) |
---|
2333 | { |
---|
2334 | pref = s[1..k]; // with * on the k-th place |
---|
2335 | } |
---|
2336 | // "pref:"; pref; |
---|
2337 | string suf; |
---|
2338 | if ( (j>0) && (j+1 <= size(s)) ) |
---|
2339 | { |
---|
2340 | suf = s[j+1..size(s)]; // without * on the 1st place |
---|
2341 | } |
---|
2342 | // "suf:"; suf; |
---|
2343 | string toins; |
---|
2344 | for (k=1; k<=@exp; k++) |
---|
2345 | { |
---|
2346 | toins = toins + varn+"*"; |
---|
2347 | } |
---|
2348 | // "toins: "; toins; |
---|
2349 | if (size(suf) == 0) |
---|
2350 | { |
---|
2351 | toins = toins[1..size(toins)-1]; // get rid of trailing * |
---|
2352 | } |
---|
2353 | else |
---|
2354 | { |
---|
2355 | suf = strpower2rep(suf); |
---|
2356 | } |
---|
2357 | ex = pref + toins + suf; |
---|
2358 | return(ex); |
---|
2359 | // return(strpower2rep(ex)); |
---|
2360 | } |
---|
2361 | example |
---|
2362 | { |
---|
2363 | "EXAMPLE:"; echo = 2; |
---|
2364 | ring r = (0,a),(x,y,z,t),dp; |
---|
2365 | strpower2rep("-x^4"); |
---|
2366 | strpower2rep("-2*x^4*y^3*z*t^2"); |
---|
2367 | strpower2rep("-a^2*x^4"); |
---|
2368 | } |
---|
2369 | |
---|
2370 | |
---|
2371 | |
---|
2372 | static proc shiftPoly(poly a, int i) |
---|
2373 | "USAGE: shiftPoly(p,i); p letterplace poly, i int |
---|
2374 | RETURN: poly |
---|
2375 | ASSUME: basering has letterplace ring structure |
---|
2376 | PURPOSE: compute the i-th shift of letterplace polynomial p |
---|
2377 | EXAMPLE: example shiftPoly; shows examples |
---|
2378 | " |
---|
2379 | { |
---|
2380 | // shifts a monomial a by i |
---|
2381 | // calls pLPshift(p,sh,uptodeg,lVblock); |
---|
2382 | if (lpAssumeViolation()) |
---|
2383 | { |
---|
2384 | ERROR("Incomplete Letterplace structure on the basering!"); |
---|
2385 | } |
---|
2386 | return(stest(a,i)); |
---|
2387 | } |
---|
2388 | example |
---|
2389 | { |
---|
2390 | "EXAMPLE:"; echo = 2; |
---|
2391 | ring r = 0,(x,y,z),dp; |
---|
2392 | int uptodeg = 5; int lV = 3; |
---|
2393 | def R = freeAlgebra(r, uptodeg); |
---|
2394 | setring R; |
---|
2395 | poly f = x*z*y - 2*z*y + 3*x; |
---|
2396 | shiftPoly(f,1); |
---|
2397 | shiftPoly(f,2); |
---|
2398 | } |
---|
2399 | |
---|
2400 | static proc lastBlock(poly p) |
---|
2401 | "USAGE: lastBlock(p); p letterplace poly |
---|
2402 | RETURN: int |
---|
2403 | ASSUME: basering has letterplace ring structure |
---|
2404 | PURPOSE: get the number of the last block occurring in the poly |
---|
2405 | EXAMPLE: example lastBlock; shows examples |
---|
2406 | " |
---|
2407 | { |
---|
2408 | if (lpAssumeViolation()) |
---|
2409 | { |
---|
2410 | ERROR("Incomplete Letterplace structure on the basering!"); |
---|
2411 | } |
---|
2412 | // calls pLastVblock(p); |
---|
2413 | return(btest(p)); |
---|
2414 | } |
---|
2415 | example |
---|
2416 | { |
---|
2417 | "EXAMPLE:"; echo = 2; |
---|
2418 | ring r = 0,(x,y,z),dp; |
---|
2419 | int uptodeg = 5; |
---|
2420 | def R = freeAlgebra(r, uptodeg); |
---|
2421 | setring R; |
---|
2422 | poly f = x*z*y - 2*z*y + 3*x; |
---|
2423 | lastBlock(f); // should be 3 |
---|
2424 | } |
---|
2425 | |
---|
2426 | static proc test_shift() |
---|
2427 | { |
---|
2428 | LIB "freegb.lib"; |
---|
2429 | ring r = 0,(a,b),dp; |
---|
2430 | int d =5; |
---|
2431 | def R = freeAlgebra(r, d); |
---|
2432 | setring R; |
---|
2433 | int uptodeg = d; |
---|
2434 | int lV = 2; |
---|
2435 | def R = setLetterplaceAttributes(r,uptodeg,2); // supply R with letterplace structure |
---|
2436 | setring R; |
---|
2437 | poly p = mmLiebr(a,b); |
---|
2438 | poly p = lieBracket(a,b); |
---|
2439 | } |
---|
2440 | |
---|
2441 | proc lp2lstr(ideal K, def save) |
---|
2442 | "USAGE: lp2lstr(K,s); K an ideal, s a ring name |
---|
2443 | RETURN: nothing (exports object @LN into the ring named s) |
---|
2444 | ASSUME: basering has a letterplace ring structure |
---|
2445 | PURPOSE: converts letterplace ideal to list of modules |
---|
2446 | NOTE: useful as preprocessing to 'lst2str' |
---|
2447 | EXAMPLE: example lp2lstr; shows examples |
---|
2448 | " |
---|
2449 | { |
---|
2450 | def @R = basering; |
---|
2451 | string err; |
---|
2452 | int s = nvars(save); |
---|
2453 | int i,j,k; |
---|
2454 | // K contains vars x(1),...z(1) = images of originals |
---|
2455 | // 5. go back to orig vars, produce strings/modules |
---|
2456 | int sk = size(K); |
---|
2457 | int sp, sx, a, b; |
---|
2458 | intvec x; |
---|
2459 | poly p,q; |
---|
2460 | poly pn; |
---|
2461 | // vars in 'save' |
---|
2462 | setring save; |
---|
2463 | module N; |
---|
2464 | list LN; |
---|
2465 | vector V; |
---|
2466 | poly pn; |
---|
2467 | // test and skip exponents >=2 |
---|
2468 | setring @R; |
---|
2469 | for(i=1; i<=sk; i++) |
---|
2470 | { |
---|
2471 | p = K[i]; |
---|
2472 | while (p!=0) |
---|
2473 | { |
---|
2474 | q = lead(p); |
---|
2475 | // "processing q:";q; |
---|
2476 | x = leadexp(q); |
---|
2477 | sx = size(x); |
---|
2478 | for(k=1; k<=sx; k++) |
---|
2479 | { |
---|
2480 | if ( x[k] >= 2 ) |
---|
2481 | { |
---|
2482 | err = "skip: the value x[k] is " + string(x[k]); |
---|
2483 | dbprint(ppl,err); |
---|
2484 | // return(0); |
---|
2485 | K[i] = 0; |
---|
2486 | p = 0; |
---|
2487 | q = 0; |
---|
2488 | break; |
---|
2489 | } |
---|
2490 | } |
---|
2491 | p = p - q; |
---|
2492 | } |
---|
2493 | } |
---|
2494 | K = simplify(K,2); |
---|
2495 | sk = size(K); |
---|
2496 | for(i=1; i<=sk; i++) |
---|
2497 | { |
---|
2498 | // setring save; |
---|
2499 | // V = 0; |
---|
2500 | setring @R; |
---|
2501 | p = K[i]; |
---|
2502 | while (p!=0) |
---|
2503 | { |
---|
2504 | q = lead(p); |
---|
2505 | err = "processing q:" + string(q); |
---|
2506 | dbprint(ppl,err); |
---|
2507 | x = leadexp(q); |
---|
2508 | sx = size(x); |
---|
2509 | pn = leadcoef(q); |
---|
2510 | setring save; |
---|
2511 | pn = imap(@R,pn); |
---|
2512 | V = V + leadcoef(pn)*gen(1); |
---|
2513 | for(k=1; k<=sx; k++) |
---|
2514 | { |
---|
2515 | if (x[k] ==1) |
---|
2516 | { |
---|
2517 | a = k div s; // block number=a+1, a!=0 |
---|
2518 | b = k % s; // remainder |
---|
2519 | // printf("a: %s, b: %s",a,b); |
---|
2520 | if (b == 0) |
---|
2521 | { |
---|
2522 | // that is it's the last var in the block |
---|
2523 | b = s; |
---|
2524 | a = a-1; |
---|
2525 | } |
---|
2526 | V = V + var(b)*gen(a+2); |
---|
2527 | } |
---|
2528 | } |
---|
2529 | err = "V: " + string(V); |
---|
2530 | dbprint(ppl,err); |
---|
2531 | // printf("V: %s", string(V)); |
---|
2532 | N = N,V; |
---|
2533 | V = 0; |
---|
2534 | setring @R; |
---|
2535 | p = p - q; |
---|
2536 | pn = 0; |
---|
2537 | } |
---|
2538 | setring save; |
---|
2539 | LN[i] = simplify(N,2); |
---|
2540 | N = 0; |
---|
2541 | } |
---|
2542 | setring save; |
---|
2543 | list @LN = LN; |
---|
2544 | export @LN; |
---|
2545 | // return(LN); |
---|
2546 | } |
---|
2547 | example |
---|
2548 | { |
---|
2549 | "EXAMPLE:"; echo = 2; |
---|
2550 | intmat A[2][2] = 2, -1, -1, 2; // sl_3 == A_2 |
---|
2551 | ring r = 0,(f1,f2),dp; |
---|
2552 | def R = freeAlgebra(r, 3); |
---|
2553 | setring R; |
---|
2554 | ideal I = serreRelations(A,1); |
---|
2555 | lp2lstr(I,r); |
---|
2556 | setring r; |
---|
2557 | lst2str(@LN,1); |
---|
2558 | } |
---|
2559 | |
---|
2560 | static proc strList2poly(list L) |
---|
2561 | { |
---|
2562 | // list L comes from sent2lplace (which takes a polynomial as input) |
---|
2563 | // each entry of L is a sublist with the coef on the last place |
---|
2564 | int s = size(L); int t; |
---|
2565 | int i,j; |
---|
2566 | list M; |
---|
2567 | poly p,q; |
---|
2568 | string Q; |
---|
2569 | for(i=1; i<=s; i++) |
---|
2570 | { |
---|
2571 | M = L[i]; |
---|
2572 | t = size(M); |
---|
2573 | // q = M[t]; // a constant |
---|
2574 | Q = string(M[t]); |
---|
2575 | for(j=1; j<t; j++) |
---|
2576 | { |
---|
2577 | // q = q*M[j]; |
---|
2578 | Q = Q+"*"+string(M[j]); |
---|
2579 | } |
---|
2580 | execute("q="+Q+";"); |
---|
2581 | // q; |
---|
2582 | p = p + q; |
---|
2583 | } |
---|
2584 | kill Q; |
---|
2585 | return(p); |
---|
2586 | } |
---|
2587 | example |
---|
2588 | { |
---|
2589 | "EXAMPLE:"; echo = 2; |
---|
2590 | ring r =0,(x,y,z,t),Dp; |
---|
2591 | def A = freeAlgebra(r, 4); |
---|
2592 | setring A; |
---|
2593 | string t = "-2*y*z*y*z + y*t*z*z - z*x*x*y + 2*z*y*z*y"; |
---|
2594 | list L = sent2lplace(t); |
---|
2595 | L; |
---|
2596 | poly p = strList2poly(L); |
---|
2597 | p; |
---|
2598 | } |
---|
2599 | |
---|
2600 | static proc file2lplace(string fname) |
---|
2601 | "USAGE: file2lplace(fnm); fnm a string |
---|
2602 | RETURN: ideal |
---|
2603 | PURPOSE: convert the contents of the file fnm into ideal of polynomials in free algebra |
---|
2604 | EXAMPLE: example file2lplace; shows examples |
---|
2605 | " |
---|
2606 | { |
---|
2607 | // format: from the usual string to letterplace |
---|
2608 | string s = read(fname); |
---|
2609 | // assume: file is a comma-sep list of polys |
---|
2610 | // the vars are declared before |
---|
2611 | // the file ends with ";" |
---|
2612 | string t; int i; |
---|
2613 | ideal I; |
---|
2614 | list tst; |
---|
2615 | while (s!="") |
---|
2616 | { |
---|
2617 | i = find(s,","); |
---|
2618 | "i"; i; |
---|
2619 | if (i==0) |
---|
2620 | { |
---|
2621 | i = find(s,";"); |
---|
2622 | if (i==0) |
---|
2623 | { |
---|
2624 | // no ; ?? |
---|
2625 | "no colon or semicolon found anymore"; |
---|
2626 | return(I); |
---|
2627 | } |
---|
2628 | // no "," but ";" on the i-th place |
---|
2629 | t = s[1..i-1]; |
---|
2630 | s = ""; |
---|
2631 | "processing: "; t; |
---|
2632 | tst = sent2lplace(t); |
---|
2633 | tst; |
---|
2634 | I = I, strList2poly(tst); |
---|
2635 | return(I); |
---|
2636 | } |
---|
2637 | // here i !=0 |
---|
2638 | t = s[1..i-1]; |
---|
2639 | s = s[i+1..size(s)]; |
---|
2640 | "processing: "; t; |
---|
2641 | tst = sent2lplace(t); |
---|
2642 | tst; |
---|
2643 | I = I, strList2poly(tst); |
---|
2644 | } |
---|
2645 | return(I); |
---|
2646 | } |
---|
2647 | example |
---|
2648 | { |
---|
2649 | "EXAMPLE:"; echo = 2; |
---|
2650 | ring r =0,(x,y,z,t),dp; |
---|
2651 | def A = freeAlgebra(r, 4); |
---|
2652 | setring A; |
---|
2653 | string fn = "myfile"; |
---|
2654 | string s1 = "z*y*y*y - 3*y*z*x*y + 3*y*y*z*y - y*x*y*z,"; |
---|
2655 | string s2 = "-2*y*x*y*z + y*y*z*z - z*z*y*y + 2*z*y*z*y,"; |
---|
2656 | 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;"; |
---|
2657 | write(":w "+fn,s1); write(":a "+fn,s2); write(":a "+fn,s3); |
---|
2658 | read(fn); |
---|
2659 | ideal I = file2lplace(fn); |
---|
2660 | I; |
---|
2661 | } |
---|
2662 | |
---|
2663 | /* EXAMPLES AGAIN: |
---|
2664 | //static proc get_ls3nilp() |
---|
2665 | { |
---|
2666 | //first app of file2lplace |
---|
2667 | ring r =0,(x,y,z,t),dp; |
---|
2668 | int d = 10; |
---|
2669 | def A = freeAlgebra(r, d); |
---|
2670 | setring A; |
---|
2671 | ideal I = file2lplace("./ls3nilp.bg"); |
---|
2672 | // and now test the correctness: go back from lplace to strings |
---|
2673 | lp2lstr(I,r); |
---|
2674 | setring r; |
---|
2675 | lst2str(@LN,1); // agree! |
---|
2676 | } |
---|
2677 | |
---|
2678 | */ |
---|
2679 | |
---|
2680 | // static proc lpMultX(poly f, poly g) |
---|
2681 | // { |
---|
2682 | // /* multiplies two polys in a very general setting correctly */ |
---|
2683 | // /* alternative to lpMult, possibly better at non-positive orderings */ |
---|
2684 | // |
---|
2685 | // if (lpAssumeViolation()) |
---|
2686 | // { |
---|
2687 | // ERROR("Incomplete Letterplace structure on the basering!"); |
---|
2688 | // } |
---|
2689 | // // decompose f,g into graded pieces with inForm: need dmodapp.lib |
---|
2690 | // int b = attrib(basering,"isLetterplaceRing"); // the length of the block |
---|
2691 | // intvec w; // inherit the graded on the ordinal ring |
---|
2692 | // int i; |
---|
2693 | // for(i=1; i<=b; i++) |
---|
2694 | // { |
---|
2695 | // w[i] = deg(var(i)); |
---|
2696 | // } |
---|
2697 | // intvec v = w; |
---|
2698 | // for(i=1; i< lpDegBound(basering); i++) |
---|
2699 | // { |
---|
2700 | // v = v,w; |
---|
2701 | // } |
---|
2702 | // w = v; |
---|
2703 | // poly p,q,s, result; |
---|
2704 | // s = g; |
---|
2705 | // while (f!=0) |
---|
2706 | // { |
---|
2707 | // p = inForm(f,w)[1]; |
---|
2708 | // f = f - p; |
---|
2709 | // s = g; |
---|
2710 | // while (s!=0) |
---|
2711 | // { |
---|
2712 | // q = inForm(s,w)[1]; |
---|
2713 | // s = s - q; |
---|
2714 | // result = result + lpMult(p,q); |
---|
2715 | // } |
---|
2716 | // } |
---|
2717 | // // shrinking |
---|
2718 | // // result; |
---|
2719 | // return( system("shrinktest",result,attrib(basering, "isLetterplaceRing")) ); |
---|
2720 | // } |
---|
2721 | // example |
---|
2722 | // { |
---|
2723 | // "EXAMPLE:"; echo = 2; |
---|
2724 | // // define a ring in letterplace form as follows: |
---|
2725 | // ring r = 0,(x(1),y(1),x(2),y(2),x(3),y(3),x(4),y(4)),dp; |
---|
2726 | // def R = setLetterplaceAttributes(r,4,2); // supply R with letterplace structure |
---|
2727 | // setring R; |
---|
2728 | // poly a = x(1)*y(2)+x(1)+y(1); poly b = y(1)+3; |
---|
2729 | // lpMultX(b,a); |
---|
2730 | // lpMultX(a,b); |
---|
2731 | // } |
---|
2732 | // |
---|
2733 | // // multiply two letterplace polynomials, lpMult: done |
---|
2734 | // // reduction/ Normalform? needs kernel stuff |
---|
2735 | // |
---|
2736 | // |
---|
2737 | // proc lpMult(poly f, poly g) |
---|
2738 | // "USAGE: lpMult(f,g); f,g letterplace polynomials |
---|
2739 | // RETURN: poly |
---|
2740 | // ASSUME: basering has a letterplace ring structure |
---|
2741 | // PURPOSE: compute the letterplace form of f*g |
---|
2742 | // EXAMPLE: example lpMult; shows examples |
---|
2743 | // " |
---|
2744 | // { |
---|
2745 | // |
---|
2746 | // // changelog: |
---|
2747 | // // VL oct 2010: deg -> deg(_,w) for the length |
---|
2748 | // // shrink the result => don't need to decompose polys |
---|
2749 | // // since the shift is big enough |
---|
2750 | // |
---|
2751 | // // indeed it's better to have that |
---|
2752 | // // ASSUME: both f and g are quasi-homogeneous |
---|
2753 | // |
---|
2754 | // if (lpAssumeViolation()) |
---|
2755 | // { |
---|
2756 | // ERROR("Incomplete Letterplace structure on the basering!"); |
---|
2757 | // } |
---|
2758 | // intvec w = 1:nvars(basering); |
---|
2759 | // int sf = deg(f,w); // VL Oct 2010: we need rather length than degree |
---|
2760 | // int sg = deg(g,w); // esp. in the case of weighted ordering |
---|
2761 | // int uptodeg = attrib(basering, "uptodeg"); |
---|
2762 | // if (sf+sg > uptodeg) |
---|
2763 | // { |
---|
2764 | // ERROR("degree bound violated by the product!"); |
---|
2765 | // } |
---|
2766 | // // if (sf>1) { sf = sf -1; } |
---|
2767 | // poly v = f*shiftPoly(g,sf); |
---|
2768 | // // bug, reported by Simon King: in nonhomog case [solved] |
---|
2769 | // // we need to shrink |
---|
2770 | // return( system("shrinktest",v,attrib(basering, "isLetterplaceRing")) ); |
---|
2771 | // } |
---|
2772 | // example |
---|
2773 | // { |
---|
2774 | // "EXAMPLE:"; echo = 2; |
---|
2775 | // // define a ring in letterplace form as follows: |
---|
2776 | // ring r = 0,(x(1),y(1),x(2),y(2),x(3),y(3),x(4),y(4)),dp; |
---|
2777 | // def R = setLetterplaceAttributes(r,4,2); // supply R with letterplace structure |
---|
2778 | // setring R; |
---|
2779 | // poly a = x(1)*y(2)+x(1)+y(1); poly b = y(1)+3; |
---|
2780 | // lpMult(b,a); |
---|
2781 | // lpMult(a,b); |
---|
2782 | // } |
---|
2783 | |
---|
2784 | static proc lpPower(poly f, int n) |
---|
2785 | "USAGE: lpPower(f,n); f letterplace polynomial, int n |
---|
2786 | RETURN: poly |
---|
2787 | ASSUME: basering has a letterplace ring structure |
---|
2788 | PURPOSE: compute the letterplace form of f^n |
---|
2789 | EXAMPLE: example lpPower; shows examples |
---|
2790 | " |
---|
2791 | { |
---|
2792 | if (n<0) { ERROR("the power must be a natural number!"); } |
---|
2793 | if (n==0) { return(poly(1)); } |
---|
2794 | if (n==1) { return(f); } |
---|
2795 | poly p = 1; |
---|
2796 | for(int i = 1; i <= n; i++) |
---|
2797 | { |
---|
2798 | p = p*f; |
---|
2799 | } |
---|
2800 | return(p); |
---|
2801 | } |
---|
2802 | example |
---|
2803 | { |
---|
2804 | "EXAMPLE:"; echo = 2; |
---|
2805 | // define a ring in letterplace form as follows: |
---|
2806 | ring r = 0,(x,y),dp; |
---|
2807 | def R = freeAlgebra(r, 4,2); // supply R with letterplace structure |
---|
2808 | setring R; |
---|
2809 | poly a = x*y + y; poly b = y - 1; |
---|
2810 | lpPower(a,2); |
---|
2811 | lpPower(b,4); |
---|
2812 | } |
---|
2813 | |
---|
2814 | //Main normal form procedure for the user |
---|
2815 | // TODO Oct 18: replace by legacy call to the kernel function |
---|
2816 | proc lpNF(poly p, ideal G) |
---|
2817 | "USAGE: lpNF(p,G); poly p, ideal G (deprecated in favor of reduce(). will be removed soon) |
---|
2818 | RETURN: poly |
---|
2819 | PURPOSE: computation of the normal form of p with respect to G |
---|
2820 | ASSUME: p is a Letterplace polynomial, G is a set Letterplace polynomials, |
---|
2821 | being a Letterplace Groebner basis (no check for this will be done) |
---|
2822 | NOTE: Strategy: take the smallest monomial wrt ordering for reduction |
---|
2823 | - For homogeneous ideals the shift does not matter |
---|
2824 | - For non-homogenous ideals the first shift will be the smallest monomial |
---|
2825 | EXAMPLE: example lpNF; shows examples |
---|
2826 | " |
---|
2827 | {if ((p==0) || (size(G) == 0)){return(p);} |
---|
2828 | checkAssumptions(p,G); |
---|
2829 | G = sort(G)[1]; |
---|
2830 | list L = makeDVecI(G); |
---|
2831 | return(lpNormalForm2(p,G,L)); |
---|
2832 | } |
---|
2833 | example |
---|
2834 | { |
---|
2835 | "EXAMPLE:"; echo = 2; |
---|
2836 | ring r = 0,(x,y),dp; |
---|
2837 | def R = freeAlgebra(r, 4); setring R; |
---|
2838 | ideal I = x*x + y*y - 1; // 2D sphere |
---|
2839 | ideal J = letplaceGBasis(I); // compute a Letterplace Groebner basis |
---|
2840 | J; // it is finite and nice |
---|
2841 | poly f = lieBracket(x,y); f; |
---|
2842 | lpNF(f,J); |
---|
2843 | poly g = lieBracket(x,y*y); g; |
---|
2844 | lpNF(g,J); |
---|
2845 | } |
---|
2846 | /* old and more complicated example |
---|
2847 | { |
---|
2848 | "EXAMPLE:"; echo = 2; |
---|
2849 | ring r = 0,(x,y,z),dp; |
---|
2850 | int d = 5; // degree |
---|
2851 | ring R = freeAlgebra(r, d); |
---|
2852 | ideal I = y*x*y - z*y*z, x*y*x - z*x*y, z*x*z - y*z*x, x*x*x + y*y*y + z*z*z + x*y*z; |
---|
2853 | ideal J = letplaceGBasis(I); // compute a Letterplace Groebner basis |
---|
2854 | poly p = y*x*y*z*y - y*z*z*y + z*y*z; |
---|
2855 | poly q = z*x*z*y*z - y*z*x*y*z; |
---|
2856 | lpNF(p,J); |
---|
2857 | lpNF(q,J); |
---|
2858 | } |
---|
2859 | */ |
---|
2860 | // analog of division(); but the output HAS different structure |
---|
2861 | proc lpDivision(poly p, ideal I) |
---|
2862 | "USAGE: lpDivision(p,G); poly p, ideal G |
---|
2863 | PURPOSE: compute a two-sided division with remainder of p wrt G; two-sided noncommutative analogue of the procedure division |
---|
2864 | ASSUME: G = {g1,...,gN} is a Groebner basis, the original ring of the Letterplace ring has the name 'r' and no variable is called 'tag_i' for i in 1...N |
---|
2865 | RETURN: list L |
---|
2866 | NOTE: - L[1] is NF(p,I) |
---|
2867 | - L[2] is the list of expressions [i,l_(ij),r_(ij)] with \sum_(i,j) l_(ij) g_i r_(ij) = p - NF(p,I) |
---|
2868 | - procedure lpGBPres2Poly, applied to L, reconstructs p |
---|
2869 | EXAMPLE: example lpDivision; shows examples |
---|
2870 | " |
---|
2871 | { |
---|
2872 | if (p == 0 || size(I) == 0) { |
---|
2873 | list L = 0; |
---|
2874 | list empty; |
---|
2875 | L[2] = empty; |
---|
2876 | return (L); |
---|
2877 | } |
---|
2878 | //poly pNF = lpNF(p,I); |
---|
2879 | poly pNF = reduce(p,I); |
---|
2880 | p = p - pNF; |
---|
2881 | |
---|
2882 | // make new ring |
---|
2883 | def save = basering; |
---|
2884 | int norigvars = lpVarBlockSize(save); |
---|
2885 | def Rtagged; def temp = save; |
---|
2886 | for (int i = 1; i <= size(I); i++) { |
---|
2887 | Rtagged = temp + ("tag_" + string(i)); |
---|
2888 | temp = Rtagged; |
---|
2889 | } kill i; |
---|
2890 | // currently R + "var" doesn't preserve uptodeg |
---|
2891 | Rtagged = setLetterplaceAttributes(Rtagged, lpVarBlockSize(Rtagged), lpDegBound(save)); |
---|
2892 | setring Rtagged; |
---|
2893 | |
---|
2894 | // restore vars |
---|
2895 | poly p = imap(save, p); |
---|
2896 | poly pNF = imap(save, pNF); |
---|
2897 | ideal I = imap(save, I); |
---|
2898 | for (int i = 1; i <= size(I); i++) { |
---|
2899 | I[i] = I[i] - var(norigvars + i); |
---|
2900 | } kill i; |
---|
2901 | |
---|
2902 | list summands; |
---|
2903 | list L = pNF; |
---|
2904 | poly pTaggedNF = lpNF(p,I); |
---|
2905 | for (int i = 1; i <= size(pTaggedNF); i++) { |
---|
2906 | intvec iv = lp2iv(pTaggedNF[i]); |
---|
2907 | for (int j = 1; j <= size(iv); j++) { |
---|
2908 | if (iv[j] > norigvars) { |
---|
2909 | intvec left; |
---|
2910 | intvec right; |
---|
2911 | if (j > 1) { |
---|
2912 | left = iv[1..(j-1)]; |
---|
2913 | } |
---|
2914 | if (j < size(iv)) { |
---|
2915 | right = iv[(j+1)..size(iv)]; |
---|
2916 | } |
---|
2917 | list summand = (iv[j] - norigvars), leadcoef(pTaggedNF[i])*iv2lp(left), iv2lp(right); |
---|
2918 | summands = insert(summands, summand, size(summands)); |
---|
2919 | |
---|
2920 | kill left; |
---|
2921 | kill right; |
---|
2922 | kill summand; |
---|
2923 | break; |
---|
2924 | } |
---|
2925 | } kill j; |
---|
2926 | kill iv; |
---|
2927 | } kill i; |
---|
2928 | |
---|
2929 | L[2] = summands; |
---|
2930 | |
---|
2931 | setring save; |
---|
2932 | list L = imap(Rtagged,L); |
---|
2933 | return (L); |
---|
2934 | } |
---|
2935 | example |
---|
2936 | { |
---|
2937 | "EXAMPLE:"; echo = 2; |
---|
2938 | ring r = 0,(x,y),dp; |
---|
2939 | ring R = freeAlgebra(r, 4); |
---|
2940 | ideal I = x*x + y*y - 1; // 2D sphere |
---|
2941 | ideal J = twostd(I); // compute a two-sided Groebner basis |
---|
2942 | J; // it is finite and nice |
---|
2943 | poly h = x*x*y-y*x*x+x*y; |
---|
2944 | list L = lpDivision(h,J); L; // what means that the NF of h wrt J is x*y |
---|
2945 | h - lpNF(h,J); // and this poly has the following two-sided Groebner presentation: |
---|
2946 | -y*J[1] + J[1]*y; |
---|
2947 | lpGBPres2Poly(L,J); // reconstructs the above automatically |
---|
2948 | } |
---|
2949 | |
---|
2950 | proc lpGBPres2Poly(list L, ideal I) |
---|
2951 | "USAGE: lpGBPres2Poly(p,G); poly p, ideal G |
---|
2952 | ASSUME: L is a valid Groebner presentation like the result of lpDivision |
---|
2953 | RETURN: poly |
---|
2954 | NOTE: assembles p = \sum_(i,j) l_(ij) g_i r_(ij) + NF(p,I) = \sum_(i) L[2][i][2] I[L[2][i][1]] L[2][i][3] + L[1] |
---|
2955 | EXAMPLE: example lpGBPres2Poly; shows examples |
---|
2956 | " |
---|
2957 | { |
---|
2958 | poly p; |
---|
2959 | for (int i = 1; i <= size(L[2]); i++) { |
---|
2960 | p = p + L[2][i][2] * I[L[2][i][1]] * L[2][i][3]; |
---|
2961 | } |
---|
2962 | p = p + L[1]; |
---|
2963 | return (p); |
---|
2964 | } |
---|
2965 | example |
---|
2966 | { |
---|
2967 | "EXAMPLE:"; echo = 2; |
---|
2968 | ring r = 0,(x,y),dp; |
---|
2969 | ring R = freeAlgebra(r, 4); |
---|
2970 | ideal I = x*x + y*y - 1; // 2D sphere |
---|
2971 | ideal J = twostd(I); // compute a two-sided Groebner basis |
---|
2972 | J; // it is finite and nice |
---|
2973 | poly h = x*x*y-y*x*x+x*y; |
---|
2974 | list L = lpDivision(h,J); |
---|
2975 | L[1]; // what means that the normal form (or the remainder) of h wrt J is x*y |
---|
2976 | lpGBPres2Poly(L,J); // we see, that it is equal to h from above |
---|
2977 | } |
---|
2978 | |
---|
2979 | |
---|
2980 | //procedures to convert monomials into the DVec representation, all static |
---|
2981 | //////////////////////////////////////////////////////// |
---|
2982 | |
---|
2983 | |
---|
2984 | static proc getExpVecs(ideal G) |
---|
2985 | "USAGE: getExpVecs(G); |
---|
2986 | RETURN: list of intvecs |
---|
2987 | PURPOSE: convert G into a list of intvecs, corresponding to the exponent vector |
---|
2988 | of the leading monomials of G |
---|
2989 | " |
---|
2990 | {int i; list L; |
---|
2991 | for (i = 1; i <= size(G); i++) {L[i] = leadexp(G[i]); } |
---|
2992 | return(L); |
---|
2993 | } |
---|
2994 | |
---|
2995 | static proc delSupZero(intvec I) |
---|
2996 | "USAGE:delSupZero(I); |
---|
2997 | RETURN: intvec |
---|
2998 | PURPOSE: Deletes superfluous zero blocks of an exponent vector |
---|
2999 | ASSUME: Intvec is an exponent vector of a letterplace monomial contained in V' |
---|
3000 | " |
---|
3001 | {if (I==intvec(0)) {return(intvec(0));} |
---|
3002 | int j,k,l; |
---|
3003 | int n = lpVarBlockSize(basering); int d = lpDegBound(basering); |
---|
3004 | intvec w; j = 1; |
---|
3005 | while (j <= d) |
---|
3006 | {w = I[1..n]; |
---|
3007 | if (w<>intvec(0)){break;} |
---|
3008 | else {I = I[(n+1)..(n*d)]; d = d-1; j++;} |
---|
3009 | } |
---|
3010 | for (j = 1; j <= d; j++) |
---|
3011 | {l=(j-1)*n+1; k= j*n; |
---|
3012 | w = I[l..k]; |
---|
3013 | if (w==intvec(0)){w = I[1..(l-1)]; return(w);}//if a zero block is found there are only zero blocks left, |
---|
3014 | //otherwise there would be a hole in the monomial |
---|
3015 | // shrink should take care that this will not happen |
---|
3016 | } |
---|
3017 | return(I); |
---|
3018 | } |
---|
3019 | |
---|
3020 | static proc delSupZeroList(list L) |
---|
3021 | "USAGE:delSupZeroList(L); L a list, containing intvecs |
---|
3022 | RETURN: list, containing intvecs |
---|
3023 | PURPOSE: Deletes all superfluous zero blocks for a list of exponent vectors |
---|
3024 | ASSUME: All intvecs are exponent vectors of letterplace monomials contained in V' |
---|
3025 | " |
---|
3026 | {int i; |
---|
3027 | for (i = size(L); 0 < i; i--){L[i] = delSupZero(L[i]);} |
---|
3028 | return(L); |
---|
3029 | } |
---|
3030 | |
---|
3031 | |
---|
3032 | static proc makeDVec(intvec V) |
---|
3033 | "USAGE:makeDVec(V); |
---|
3034 | RETURN: intvec |
---|
3035 | PURPOSE: Converts an modified exponent vector into an Dvec |
---|
3036 | NOTE: Superfluous zero blocks must have been deleted before using this procedure |
---|
3037 | " |
---|
3038 | {int i,j,k,r1,r2; intvec D; |
---|
3039 | int n = lpVarBlockSize(basering); |
---|
3040 | k = size(V) div n; r1 = 0; r2 = 0; |
---|
3041 | for (i=1; i<= k; i++) |
---|
3042 | {for (j=(1+((i-1)*n)); j <= (i*n); j++) |
---|
3043 | {if (V[j]>0){r2 = j - ((i-1)*n); j = (j mod n); break;} |
---|
3044 | } |
---|
3045 | D[size(D)+1] = r1+r2; |
---|
3046 | if (j == 0) {r1 = 0;} else{r1= n-j;} |
---|
3047 | } |
---|
3048 | D = D[2..size(D)]; |
---|
3049 | return(D); |
---|
3050 | } |
---|
3051 | |
---|
3052 | static proc makeDVecL(list L) |
---|
3053 | "USAGE:makeDVecL(L); L, a list containing intvecs |
---|
3054 | RETURN: list, containing intvecs |
---|
3055 | ASSUME: |
---|
3056 | " |
---|
3057 | {int i; list R; |
---|
3058 | for (i=1; i <= size(L); i++) {R[i] = makeDVec(L[i]);} |
---|
3059 | return(R); |
---|
3060 | } |
---|
3061 | |
---|
3062 | static proc makeDVecI(ideal G) |
---|
3063 | "USAGE:makeDVecI(G); |
---|
3064 | RETURN:list, containing intvecs |
---|
3065 | PURPOSE:computing the DVec representation for lead(G) |
---|
3066 | ASSUME: |
---|
3067 | " |
---|
3068 | {list L = delSupZeroList(getExpVecs(G)); |
---|
3069 | return(makeDVecL(L)); |
---|
3070 | } |
---|
3071 | |
---|
3072 | |
---|
3073 | //procedures, which are dealing with the DVec representation, all static |
---|
3074 | |
---|
3075 | static proc dShiftDiv(intvec V, intvec W) |
---|
3076 | "USAGE: dShiftDiv(V,W); |
---|
3077 | RETURN: a list,containing integers, or -1, if no shift of W divides V |
---|
3078 | PURPOSE: find all possible shifts s, such that s.W|V |
---|
3079 | ASSUME: V,W are DVecs of monomials contained in V' |
---|
3080 | " |
---|
3081 | {if(size(V)<size(W)){return(list(-1));} |
---|
3082 | |
---|
3083 | int i,j,r; intvec T; list R; |
---|
3084 | int n = lpVarBlockSize(basering); |
---|
3085 | int k = size(V) - size(W) + 1; |
---|
3086 | if (intvec(V[1..size(W)])-W == 0){R[1]=0;} |
---|
3087 | for (i =2; i <=k; i++) |
---|
3088 | {r = 0; kill T; intvec T; |
---|
3089 | for (j =1; j <= i; j++) {r = r + V[j];} |
---|
3090 | //if (i==1) {T[1] = r-(i-1)*n;} else |
---|
3091 | T[1] = r-(i-1)*n; if (size(W)>1) {T[2..size(W)] = V[(i+1)..(size(W)+i-1)];} |
---|
3092 | if (T-W == 0) {R[size(R)+1] = i-1;} |
---|
3093 | } |
---|
3094 | if (size(R)>0) {return(R);} |
---|
3095 | else {return(list(-1));} |
---|
3096 | } |
---|
3097 | |
---|
3098 | //the first normal form procedure, if a user want not to presort the ideal, just make it not static |
---|
3099 | static proc lpNormalForm1(poly p, ideal G, list L) |
---|
3100 | "USAGE:lpNormalForm1(p,G); |
---|
3101 | RETURN:poly |
---|
3102 | PURPOSE:computation of the normalform of p w.r.t. G |
---|
3103 | ASSUME: p is a Letterplace polynomial, G is a set of Letterplace polynomials |
---|
3104 | NOTE: Taking the first possible reduction |
---|
3105 | " |
---|
3106 | { |
---|
3107 | if (deg(p) <1) {return(p);} |
---|
3108 | else |
---|
3109 | { |
---|
3110 | int i; int s; |
---|
3111 | intvec V = makeDVec(delSupZero(leadexp(p))); |
---|
3112 | for (i = 1; i <= size(L); i++) |
---|
3113 | {s = dShiftDiv(V, L[i])[1]; |
---|
3114 | if (s <> -1) |
---|
3115 | {p = lpReduce(p,G[i],s); |
---|
3116 | p = lpNormalForm1(p,G,L); |
---|
3117 | break; |
---|
3118 | } |
---|
3119 | } |
---|
3120 | p = p[1] + lpNormalForm1(p-p[1],G,L); |
---|
3121 | return(p); |
---|
3122 | } |
---|
3123 | } |
---|
3124 | |
---|
3125 | // VL; called from lpNF |
---|
3126 | static proc lpNormalForm2(poly pp, ideal G, list L) |
---|
3127 | "USAGE:lpNormalForm2(p,G); |
---|
3128 | RETURN:poly |
---|
3129 | PURPOSE:computation of the normal form of p w.r.t. G |
---|
3130 | ASSUME: p is a Letterplace polynomial, G is a set of Letterplace polynomials |
---|
3131 | NOTE: Taking the first possible reduction |
---|
3132 | " |
---|
3133 | { |
---|
3134 | poly one = 1; |
---|
3135 | if ( (pp == 0) || (leadmonom(pp) == one) ) { return(pp); } |
---|
3136 | poly p = pp; poly q; |
---|
3137 | int i; int s; intvec V; |
---|
3138 | while ( (p != 0) && (leadmonom(p) != one) ) |
---|
3139 | { |
---|
3140 | //"entered while with p="; p; |
---|
3141 | V = makeDVec(delSupZero(leadexp(p))); |
---|
3142 | i = 0; |
---|
3143 | s = -1; |
---|
3144 | //"look for divisor"; |
---|
3145 | while ( (s == -1) && (i<size(L)) ) |
---|
3146 | { |
---|
3147 | i = i+1; |
---|
3148 | s = dShiftDiv(V, L[i])[1]; |
---|
3149 | } |
---|
3150 | // now, out of here: either i=size(L) and s==-1 => no reduction |
---|
3151 | // otherwise: i<=size(L) and s!= -1 => reduction |
---|
3152 | //"out of divisor search: s="; s; "i="; i; |
---|
3153 | if (s != -1) |
---|
3154 | { |
---|
3155 | //"start reducing with G[i]:"; |
---|
3156 | p = lpReduce(p,G[i],s); // lm-reduction |
---|
3157 | //"reduced to p="; p; |
---|
3158 | } |
---|
3159 | else |
---|
3160 | { |
---|
3161 | // ie no lm-reduction possible; proceed with the tail reduction |
---|
3162 | q = p-lead(p); |
---|
3163 | p = lead(p); |
---|
3164 | if (q!=0) |
---|
3165 | { |
---|
3166 | p = p + lpNormalForm2(q,G,L); |
---|
3167 | } |
---|
3168 | return(p); |
---|
3169 | } |
---|
3170 | } |
---|
3171 | // out of while when p==0 or p == const |
---|
3172 | return(p); |
---|
3173 | } |
---|
3174 | |
---|
3175 | proc isOrderingShiftInvariant(int withHoles) |
---|
3176 | "USAGE: isOrderingShiftInvariant(b); b an integer interpreted as a boolean |
---|
3177 | RETURN: int |
---|
3178 | NOTE: Tests whether the ordering of the current ring is shift invariant, which is the case, when LM(p) > LM(p') for all p and p' where p' is p shifted by any number of places. |
---|
3179 | |
---|
3180 | If withHoles != 0 even Letterplace polynomials with holes (eg. x(1)*y(4)) are considered. |
---|
3181 | ASSUME: - basering is a Letterplace ring. |
---|
3182 | " |
---|
3183 | { |
---|
3184 | int shiftInvariant = 1; |
---|
3185 | |
---|
3186 | int d = lpDegBound(basering); |
---|
3187 | |
---|
3188 | ideal monomials; |
---|
3189 | if (withHoles) { |
---|
3190 | monomials = delete(lpMonomialsWithHoles(d-1), 1); // ignore the first element (1) |
---|
3191 | } else { |
---|
3192 | monomials = maxideal(1); |
---|
3193 | for (int i = 2; i <= d-1; i++) { |
---|
3194 | monomials = monomials, maxideal(i); |
---|
3195 | } kill i; |
---|
3196 | } |
---|
3197 | |
---|
3198 | for (int i = 1; i <= size(monomials); i++) { |
---|
3199 | poly monom = monomials[i]; |
---|
3200 | int lastblock = lastBlock(monom); |
---|
3201 | for (int s = 1; s <= d - lastblock; s++) { |
---|
3202 | for (int s2 = 0; s2 < s; s2++) { // paranoid, check every pair |
---|
3203 | poly first = shiftPoly(monom,s2); |
---|
3204 | poly second = shiftPoly(monom,s); |
---|
3205 | if (!(first > second)) { |
---|
3206 | if (printlevel >= voice) { // otherwise string() is always evaluated |
---|
3207 | dbprint(string(first) + " <= " + string(second)); |
---|
3208 | } |
---|
3209 | shiftInvariant = 0; |
---|
3210 | } |
---|
3211 | kill first; kill second; |
---|
3212 | } kill s2; |
---|
3213 | } kill s; |
---|
3214 | kill monom; kill lastblock; |
---|
3215 | } kill i; |
---|
3216 | |
---|
3217 | return(shiftInvariant); |
---|
3218 | } |
---|
3219 | example |
---|
3220 | { |
---|
3221 | "EXAMPLE:"; echo = 2; |
---|
3222 | ring r = 0,(x,y,z),dp; |
---|
3223 | def R = freeAlgebra(r, 5); |
---|
3224 | setring R; |
---|
3225 | isOrderingShiftInvariant(0);// should be 1 |
---|
3226 | |
---|
3227 | ring r2 = 0,(x,y,z),dp; |
---|
3228 | def R2 = freeAlgebra(r2, 5); |
---|
3229 | list RL = ringlist(R2); |
---|
3230 | RL[3][1][1] = "wp"; |
---|
3231 | intvec weights = 1,1,1,1,1,1,1,2,3,1,1,1,1,1,1; |
---|
3232 | RL[3][1][2] = weights; |
---|
3233 | attrib(RL,"isLetterplaceRing",3); |
---|
3234 | attrib(RL,"maxExp",1); |
---|
3235 | def Rw = setLetterplaceAttributes(ring(RL),5,3); |
---|
3236 | setring Rw; |
---|
3237 | /* printlevel = voice + 1; */ |
---|
3238 | isOrderingShiftInvariant(0); |
---|
3239 | isOrderingShiftInvariant(1); |
---|
3240 | } |
---|
3241 | |
---|
3242 | static proc lpMonomialsWithHoles(int d) |
---|
3243 | { |
---|
3244 | if (d < 0) { |
---|
3245 | ERROR("d must not be negative") |
---|
3246 | } |
---|
3247 | |
---|
3248 | ideal monomials = 1; |
---|
3249 | if (d == 0) { |
---|
3250 | return (monomials); |
---|
3251 | } |
---|
3252 | |
---|
3253 | int lV = lpVarBlockSize(basering); // variable count |
---|
3254 | ideal prevMonomials = lpMonomialsWithHoles(d - 1); |
---|
3255 | |
---|
3256 | for (int i = 1; i <= size(prevMonomials); i++) { |
---|
3257 | /* if (deg(prevMonomials[i]) >= d - 1) { */ |
---|
3258 | for (int j = 1; j <= lV; j++) { |
---|
3259 | poly m = prevMonomials[i]; |
---|
3260 | m = m * var(j + (d-1)*lV); |
---|
3261 | monomials = monomials, m; |
---|
3262 | kill m; |
---|
3263 | } kill j; |
---|
3264 | /* } */ |
---|
3265 | } kill i; |
---|
3266 | |
---|
3267 | if (d > 1) { |
---|
3268 | // removes the 1 |
---|
3269 | monomials[1] = 0; |
---|
3270 | monomials = simplify(monomials,2); |
---|
3271 | |
---|
3272 | monomials = prevMonomials, monomials; |
---|
3273 | } |
---|
3274 | return (monomials); |
---|
3275 | } |
---|
3276 | |
---|
3277 | static proc getlpCoeffs(poly q, poly p) |
---|
3278 | {list R; intvec cq,t,lv,rv,bla; |
---|
3279 | int n = lpVarBlockSize(basering); int d = lpDegBound(basering); |
---|
3280 | int i; |
---|
3281 | cq = leadexp(p)-leadexp(q); /* p/q */ |
---|
3282 | for (i = 1; i<= d; i++) |
---|
3283 | {bla = cq[((i-1)*n+1)..(i*n)]; |
---|
3284 | if (bla == 0) {lv = cq[1..i*n]; cq = cq[(i*n+1)..(d*n)]; break;} |
---|
3285 | } |
---|
3286 | |
---|
3287 | d = size(cq) div n; |
---|
3288 | for (i = 1; i<= d; i++) |
---|
3289 | {bla = cq[((i-1)*n+1)..(i*n)]; |
---|
3290 | if (bla <> 0){rv = cq[((i-1)*n+1)..(d*n)]; break;} |
---|
3291 | } |
---|
3292 | return(list(monomial(lv),monomial(rv))); |
---|
3293 | } |
---|
3294 | |
---|
3295 | static proc lpReduce(poly p, poly g, int s) |
---|
3296 | "NOTE: shift can not exceed the degree bound, because s*g | p |
---|
3297 | " |
---|
3298 | {poly l,r,qt; int i; |
---|
3299 | list K = getlpCoeffs(lead(shiftPoly(g,s)), lead(p)); |
---|
3300 | l = K[1]; r = K[2]; |
---|
3301 | kill K; |
---|
3302 | for (i = 1; i <= size(g); i++) |
---|
3303 | { |
---|
3304 | qt = qt + l*g[i]*r; |
---|
3305 | } |
---|
3306 | return(p - leadcoef(p)*normalize(qt)); |
---|
3307 | } |
---|
3308 | |
---|
3309 | static proc entryViolation(intmat M, int n) |
---|
3310 | "PURPOSE:checks, if all entries in M are variable-related |
---|
3311 | " |
---|
3312 | {int i,j; |
---|
3313 | for (i = 1; i <= nrows(M); i++) |
---|
3314 | {for (j = 1; j <= ncols(M); j++) |
---|
3315 | {if(!((1<=M[i,j])&&(M[i,j]<=n))) {return(1);}} |
---|
3316 | } |
---|
3317 | return(0); |
---|
3318 | } |
---|
3319 | |
---|
3320 | static proc checkAssumptionsLPIV(int d, list L) |
---|
3321 | "PURPOSE: Checks, if all the Assumptions are holding |
---|
3322 | " |
---|
3323 | {if (!isFreeAlgebra(basering)) {ERROR("Basering is not a Letterplace ring!");} |
---|
3324 | if (d > lpDegBound(basering)) {ERROR("Specified degree bound exceeds ring parameter!");} |
---|
3325 | int i; |
---|
3326 | for (i = 1; i <= size(L); i++) |
---|
3327 | {if (entryViolation(L[i], lpVarBlockSize(basering))) |
---|
3328 | {ERROR("Not allowed monomial/intvec found!");} |
---|
3329 | } |
---|
3330 | return(); |
---|
3331 | } |
---|
3332 | |
---|
3333 | static proc checkAssumptions(poly p, ideal G) |
---|
3334 | " |
---|
3335 | " |
---|
3336 | {checkLPRing(); |
---|
3337 | checkAssumptionPoly(p); |
---|
3338 | checkAssumptionIdeal(G); |
---|
3339 | return(); |
---|
3340 | } |
---|
3341 | |
---|
3342 | static proc checkLPRing(); |
---|
3343 | " |
---|
3344 | " |
---|
3345 | {if (!isFreeAlgebra(basering)) {ERROR("Basering is not a Letterplace ring!");} |
---|
3346 | return(); |
---|
3347 | } |
---|
3348 | |
---|
3349 | static proc checkAssumptionIdeal(ideal G) |
---|
3350 | "PURPOSE:Check if all elements of ideal are elements of V' |
---|
3351 | " |
---|
3352 | {ideal L = lead(normalize(G)); |
---|
3353 | int i; |
---|
3354 | for (i = 1; i <= ncols(G); i++) {if (!isContainedInVp(G[i])) {ERROR("Ideal contains elements not contained in V'");}} |
---|
3355 | return(); |
---|
3356 | } |
---|
3357 | |
---|
3358 | static proc checkAssumptionPoly(poly p) |
---|
3359 | "PURPOSE:Check if p is an element of V' |
---|
3360 | " |
---|
3361 | {poly l = lead(normalize(p)); |
---|
3362 | if (!isContainedInVp(l)) {ERROR("Polynomial is not contained in V'");} |
---|
3363 | return(); |
---|
3364 | } |
---|
3365 | |
---|
3366 | static proc isContainedInVp(poly p) |
---|
3367 | "PURPOSE: Check monomial for holes in the places |
---|
3368 | " |
---|
3369 | {int r = 0; intvec w; |
---|
3370 | intvec l = leadexp(p); |
---|
3371 | int n = lpVarBlockSize(basering); int d = lpDegBound(basering); |
---|
3372 | int i,j,c,c1; |
---|
3373 | while (1 <= d) |
---|
3374 | { |
---|
3375 | w = l[1..n]; |
---|
3376 | if (w<>(0:n)) {break;} |
---|
3377 | else |
---|
3378 | { |
---|
3379 | if (size(w)==size(l)) break; |
---|
3380 | l = l[(n+1)..(n*d)]; |
---|
3381 | d = d-1; |
---|
3382 | } |
---|
3383 | } |
---|
3384 | |
---|
3385 | while (1 <= d) |
---|
3386 | {for (j = 1; j <= n; j++) |
---|
3387 | {if (l[j]<>0) |
---|
3388 | {if (c1<>0){return(0);} |
---|
3389 | if (c<>0){return(0);} |
---|
3390 | if (l[j]<>1){return(0);} |
---|
3391 | c=1; |
---|
3392 | } |
---|
3393 | } |
---|
3394 | if (c == 0){c1=1;if (1 < d){l = l[(n+1)..(n*d)]; d = d-1;} else {d = d -1;}} |
---|
3395 | else {c = 0; if (1 < d){l = l[(n+1)..(n*d)]; d = d-1;} else {d = d -1;}} |
---|
3396 | } |
---|
3397 | return(1); |
---|
3398 | } |
---|
3399 | |
---|
3400 | static proc extractLinearPart(module M) |
---|
3401 | { |
---|
3402 | /* returns vectors from a module whose max leadexp is 1 */ |
---|
3403 | /* does not take nonlinearity into account yet */ |
---|
3404 | /* use rather kernel function isinV to get really nonlinear things */ |
---|
3405 | int i; int s = ncols(M); |
---|
3406 | int answer = 1; |
---|
3407 | vector v; module Ret; |
---|
3408 | for(i=1; i<=s; i++) |
---|
3409 | { |
---|
3410 | if ( isLinearVector(M[i]) ) |
---|
3411 | { |
---|
3412 | Ret = Ret, M[i]; |
---|
3413 | } |
---|
3414 | } |
---|
3415 | Ret = simplify(Ret,2); |
---|
3416 | return(Ret); |
---|
3417 | } |
---|
3418 | |
---|
3419 | static proc isLinearVector(vector v) |
---|
3420 | { |
---|
3421 | /* vector v consists of polynomials */ |
---|
3422 | /* returns true iff max leadexp is 1 */ |
---|
3423 | int i,j,k; |
---|
3424 | intvec w; |
---|
3425 | int s = size(v); |
---|
3426 | poly p; |
---|
3427 | int answer = 1; |
---|
3428 | for(i=1; i<=s; i++) |
---|
3429 | { |
---|
3430 | p = v[i]; |
---|
3431 | while (p != 0) |
---|
3432 | { |
---|
3433 | w = leadexp(p); |
---|
3434 | j = Max(w); |
---|
3435 | if (j >=2) |
---|
3436 | { |
---|
3437 | answer = 0; |
---|
3438 | return(answer); |
---|
3439 | } |
---|
3440 | p = p-lead(p); |
---|
3441 | } |
---|
3442 | } |
---|
3443 | return(answer); |
---|
3444 | } |
---|
3445 | |
---|
3446 | |
---|
3447 | // // the following is to determine a shift of a mono/poly from the |
---|
3448 | // // interface |
---|
3449 | |
---|
3450 | // static proc whichshift(poly p, int numvars) |
---|
3451 | // { |
---|
3452 | // // numvars = number of vars of the orig free algebra |
---|
3453 | // // assume: we are in the letterplace ring |
---|
3454 | // // takes monomial on the input |
---|
3455 | // poly q = lead(p); |
---|
3456 | // intvec v = leadexp(v); |
---|
3457 | // if (v==0) { return(int(0)); } |
---|
3458 | // int sv = size(v); |
---|
3459 | // int i=1; |
---|
3460 | // while ( (v[i]==0) && (i<sv) ) { i++; } |
---|
3461 | // i = sv div i; |
---|
3462 | // return(i); |
---|
3463 | // } |
---|
3464 | |
---|
3465 | |
---|
3466 | // LIB "qhmoduli.lib"; |
---|
3467 | // static proc polyshift(poly p, int numvars) |
---|
3468 | // { |
---|
3469 | // poly q = p; int i = 0; |
---|
3470 | // while (q!=0) |
---|
3471 | // { |
---|
3472 | // i = Max(i, whichshift(q,numvars)); |
---|
3473 | // q = q - lead(q); |
---|
3474 | // } |
---|
3475 | // return(q); |
---|
3476 | // } |
---|
3477 | |
---|
3478 | static proc lpAssumeViolation() |
---|
3479 | { |
---|
3480 | // checks whether the global vars |
---|
3481 | // uptodeg and lV are defined |
---|
3482 | // returns Boolean : yes/no [for assume violation] |
---|
3483 | def uptodeg = lpDegBound(basering); |
---|
3484 | if ( typeof(uptodeg)!="int" ) |
---|
3485 | { |
---|
3486 | return(1); |
---|
3487 | } |
---|
3488 | return (!isFreeAlgebra(basering)) |
---|
3489 | } |
---|
3490 | |
---|
3491 | static proc skip0(vector v) |
---|
3492 | { |
---|
3493 | // skips zeros in a vector, producing another vector |
---|
3494 | if ( (v[1]==0) || (v==0) ) { return(vector(0)); } |
---|
3495 | int sv = nrows(v); |
---|
3496 | int sw = size(v); |
---|
3497 | if (sv == sw) |
---|
3498 | { |
---|
3499 | return(v); |
---|
3500 | } |
---|
3501 | int i; |
---|
3502 | int j=1; |
---|
3503 | vector w; |
---|
3504 | for (i=1; i<=sv; i++) |
---|
3505 | { |
---|
3506 | if (v[i] != 0) |
---|
3507 | { |
---|
3508 | w = w + v[i]*gen(j); |
---|
3509 | j++; |
---|
3510 | } |
---|
3511 | } |
---|
3512 | return(w); |
---|
3513 | } |
---|
3514 | |
---|
3515 | // static proc bugSKing() |
---|
3516 | // { |
---|
3517 | // LIB "freegb.lib"; |
---|
3518 | // ring r=0,(a,b),dp; |
---|
3519 | // def R = freeAlgebra(r, 5); |
---|
3520 | // setring R; |
---|
3521 | // poly p = a(1); |
---|
3522 | // poly q = b(1); |
---|
3523 | // poly p2 = lpPower(p,2); |
---|
3524 | // lpMult(p2+q,q)-lpMult(p2,q)-lpMult(q,q); // now its 0 |
---|
3525 | // } |
---|
3526 | // |
---|
3527 | // static proc bugRucker() |
---|
3528 | // { |
---|
3529 | // // needs unstatic lpMultX |
---|
3530 | // LIB "freegb.lib"; |
---|
3531 | // ring r=0,(a,b,c,d,p,q,r,s,t,u,v,w),(a(7,1,1,7),dp); |
---|
3532 | // def R=freeAlgebra(r, 20,1); |
---|
3533 | // setring R; |
---|
3534 | // option(redSB); option(redTail); |
---|
3535 | // 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); |
---|
3536 | // poly ttt = a(1)*v(2)*w(3)-p(1)*q(2)*r(3)*s(4)*t(5)*u(6)*d(7); |
---|
3537 | // // with lpMult |
---|
3538 | // lpMult(I[1],d(1)) - lpMult(a(1),I[2]); // spoly; has been incorrect before |
---|
3539 | // _ - ttt; |
---|
3540 | // // with lpMultX |
---|
3541 | // lpMultX(I[1],d(1)) - lpMultX(a(1),I[2]); // spoly; has been incorrect before |
---|
3542 | // _ - ttt; |
---|
3543 | // } |
---|
3544 | // |
---|
3545 | // static proc checkWeightedExampleLP() |
---|
3546 | // { |
---|
3547 | // 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); |
---|
3548 | // def R = setLetterplaceAttributes(r,4,2); // supply R with letterplace structure |
---|
3549 | // setring R; |
---|
3550 | // poly a = x(1)*y(2)+x(1)+y(1); poly b = y(1)+3; |
---|
3551 | // lpMultX(b,a); |
---|
3552 | // lpMultX(a,b); // seems to work properly |
---|
3553 | // } |
---|
3554 | |
---|
3555 | // ----------------- iv2lp and lp2iv ---------------------- |
---|
3556 | proc ivL2lpI(list L) |
---|
3557 | "USAGE: ivL2lpI(L); L a list of intvecs (deprecated, will be removed soon) |
---|
3558 | RETURN: ideal |
---|
3559 | PURPOSE:Transforming a list of intvecs into an ideal of Letterplace monomials |
---|
3560 | ASSUME: - Intvec corresponds to a Letterplace monomial |
---|
3561 | @* - basering has to be a Letterplace ring |
---|
3562 | NOTE: - Assumptions will not be checked! |
---|
3563 | EXAMPLE: example ivL2lpI; shows examples |
---|
3564 | " |
---|
3565 | { |
---|
3566 | int i; ideal G; |
---|
3567 | poly p; |
---|
3568 | for (i = 1; i <= size(L); i++) |
---|
3569 | { |
---|
3570 | p = iv2lp(L[i]); |
---|
3571 | G[(size(G) + 1)] = p; |
---|
3572 | } |
---|
3573 | return(G); |
---|
3574 | } |
---|
3575 | example |
---|
3576 | { |
---|
3577 | "EXAMPLE:"; echo = 2; |
---|
3578 | ring r = 0,(x,y,z),dp; |
---|
3579 | def R = freeAlgebra(r, 5);// constructs a Letterplace ring |
---|
3580 | setring R; //sets basering to Letterplace ring |
---|
3581 | intvec u = 1,1,2; intvec v = 2,1,3; intvec w = 3,1,1; |
---|
3582 | // u = x^2y, v = yxz, w = zx^2 in intvec representation |
---|
3583 | list L = u,v,w; |
---|
3584 | ivL2lpI(L);// invokes the procedure, returns the ideal containing u,v,w |
---|
3585 | } |
---|
3586 | |
---|
3587 | proc iv2lp(intvec I) |
---|
3588 | "USAGE: iv2lp(I); I an intvec (deprecated, will be removed soon) |
---|
3589 | RETURN: poly |
---|
3590 | PURPOSE:Transforming an intvec into the corresponding Letterplace polynomial |
---|
3591 | ASSUME: - Intvec corresponds to a Letterplace monomial |
---|
3592 | @* - basering has to be a Letterplace ring |
---|
3593 | NOTE: - Assumptions will not be checked! |
---|
3594 | EXAMPLE: example iv2lp; shows examples |
---|
3595 | " |
---|
3596 | {if (I[1] == 0) {return(1);} |
---|
3597 | int i = size(I); |
---|
3598 | if (i > lpDegBound(basering)) {ERROR("polynomial exceeds degreebound");} |
---|
3599 | int j; poly p = 1; |
---|
3600 | for (j = 1; j <= i; j++) {if (I[j] > 0) { p = p*var(I[j]);}} //ignore zeroes, because they correspond to 1 |
---|
3601 | return(p); |
---|
3602 | } |
---|
3603 | example |
---|
3604 | { |
---|
3605 | "EXAMPLE:"; echo = 2; |
---|
3606 | ring r = 0,(x,y,z),dp; |
---|
3607 | def R = freeAlgebra(r, 5); // constructs a Letterplace ring |
---|
3608 | setring R; //sets basering to Letterplace ring |
---|
3609 | // u = x^2y, v = yxz, w = zx^2 in intvec representation |
---|
3610 | intvec u = 1,1,2; |
---|
3611 | iv2lp(u); // invokes the procedure and returns the corresponding poly |
---|
3612 | intvec v = 2,1,3; |
---|
3613 | iv2lp(v); |
---|
3614 | intvec w = 3,1,1; |
---|
3615 | iv2lp(w); |
---|
3616 | } |
---|
3617 | |
---|
3618 | proc iv2lpList(list L) |
---|
3619 | "USAGE: iv2lpList(L); L a list of intmats (deprecated, will be removed soon) |
---|
3620 | RETURN: ideal |
---|
3621 | PURPOSE:Converting a list of intmats into an ideal of corresponding monomials |
---|
3622 | ASSUME: - The rows of each intmat in L must correspond to a Letterplace monomial |
---|
3623 | @* - basering has to be a Letterplace ring |
---|
3624 | EXAMPLE: example iv2lpList; shows examples |
---|
3625 | " |
---|
3626 | {checkAssumptionsLPIV(0,L); |
---|
3627 | ideal G; |
---|
3628 | int i; |
---|
3629 | for (i = 1; i <= size(L); i++){G = G + iv2lpMat(L[i]);} |
---|
3630 | return(G); |
---|
3631 | } |
---|
3632 | example |
---|
3633 | { |
---|
3634 | "EXAMPLE:"; echo = 2; |
---|
3635 | ring r = 0,(x,y,z),dp; |
---|
3636 | def R = freeAlgebra(r, 5); // constructs a Letterplace ring |
---|
3637 | setring R; // sets basering to Letterplace ring |
---|
3638 | intmat u[3][1] = 1,1,2; intmat v[1][3] = 2,1,3; intmat w[2][3] = 3,1,1,2,3,1; |
---|
3639 | // defines intmats of different size containing intvec representations of |
---|
3640 | // monomials as rows |
---|
3641 | list L = u,v,w; |
---|
3642 | print(u); print(v); print(w); // shows the intmats contained in L |
---|
3643 | iv2lpList(L); // returns the corresponding monomials as an ideal |
---|
3644 | } |
---|
3645 | |
---|
3646 | |
---|
3647 | proc iv2lpMat(intmat M) |
---|
3648 | "USAGE: iv2lpMat(M); M an intmat (deprecated, will be removed soon) |
---|
3649 | RETURN: ideal |
---|
3650 | PURPOSE:Converting an intmat into an ideal of the corresponding monomials |
---|
3651 | ASSUME: - The rows of M must correspond to Letterplace monomials |
---|
3652 | @* - basering has to be a Letterplace ring |
---|
3653 | EXAMPLE: example iv2lpMat; shows examples |
---|
3654 | " |
---|
3655 | {list L = M; |
---|
3656 | checkAssumptionsLPIV(0,L); |
---|
3657 | kill L; |
---|
3658 | ideal G; poly p; |
---|
3659 | int i; intvec I; |
---|
3660 | for (i = 1; i <= nrows(M); i++) |
---|
3661 | { I = M[i,1..ncols(M)]; |
---|
3662 | p = iv2lp(I); |
---|
3663 | G[size(G)+1] = p; |
---|
3664 | } |
---|
3665 | return(G); |
---|
3666 | } |
---|
3667 | example |
---|
3668 | { |
---|
3669 | "EXAMPLE:"; echo = 2; |
---|
3670 | ring r = 0,(x,y,z),dp; |
---|
3671 | def R = freeAlgebra(r, 5); // constructs a Letterplace ring |
---|
3672 | setring R; // sets basering to Letterplace ring |
---|
3673 | intmat u[3][1] = 1,1,2; intmat v[1][3] = 2,1,3; intmat w[2][3] = 3,1,1,2,3,1; |
---|
3674 | // defines intmats of different size containing intvec representations of |
---|
3675 | // monomials as rows |
---|
3676 | iv2lpMat(u); // returns the monomials contained in u |
---|
3677 | iv2lpMat(v); // returns the monomials contained in v |
---|
3678 | iv2lpMat(w); // returns the monomials contained in w |
---|
3679 | } |
---|
3680 | |
---|
3681 | proc lpId2ivLi(ideal G) |
---|
3682 | "USAGE: lpId2ivLi(G); G an ideal (deprecated, will be removed soon) |
---|
3683 | RETURN: list |
---|
3684 | PURPOSE:Transforming an ideal into the corresponding list of intvecs |
---|
3685 | ASSUME: - basering has to be a Letterplace ring |
---|
3686 | EXAMPLE: example lpId2ivLi; shows examples |
---|
3687 | " |
---|
3688 | { |
---|
3689 | int i,j,k; |
---|
3690 | list M; |
---|
3691 | checkAssumptionsLPIV(0,M); |
---|
3692 | for (i = 1; i <= size(G); i++) {M[i] = lp2iv(G[i]);} |
---|
3693 | return(M); |
---|
3694 | } |
---|
3695 | example |
---|
3696 | { |
---|
3697 | "EXAMPLE:"; echo = 2; |
---|
3698 | ring r = 0,(x,y),dp; |
---|
3699 | def R = freeAlgebra(r, 5); // constructs a Letterplace ring |
---|
3700 | setring R; // sets basering to Letterplace ring |
---|
3701 | ideal L = x*x,y*y,x*y*x; |
---|
3702 | lpId2ivLi(L); // returns the corresponding intvecs as a list |
---|
3703 | } |
---|
3704 | |
---|
3705 | proc lp2iv(poly p) |
---|
3706 | "USAGE: lp2iv(p); p a poly (deprecated, will be removed soon) |
---|
3707 | RETURN: intvec |
---|
3708 | PURPOSE: Transforming a monomial into the corresponding intvec |
---|
3709 | ASSUME: - basering has to be a Letterplace ring |
---|
3710 | NOTE: - Assumptions will not be checked! |
---|
3711 | EXAMPLE: example lp2iv; shows examples |
---|
3712 | " |
---|
3713 | {p = normalize(lead(p)); |
---|
3714 | intvec I; |
---|
3715 | int i,j; |
---|
3716 | if (deg(p) > lpDegBound(basering)) {ERROR("Monomial exceeds degreebound");} |
---|
3717 | if (p == 1) {return(I);} |
---|
3718 | if (p == 0) {ERROR("Monomial is not allowed to equal zero");} |
---|
3719 | intvec lep = leadexp(p); |
---|
3720 | for ( i = 1; i <= lpVarBlockSize(basering); i++) {if (lep[i] == 1) {I = i; break;}} |
---|
3721 | for (i = (lpVarBlockSize(basering)+1); i <= size(lep); i++) |
---|
3722 | {if (lep[i] == 1) |
---|
3723 | { j = (i mod lpVarBlockSize(basering)); |
---|
3724 | if (j == 0) {I = I,lpVarBlockSize(basering);} |
---|
3725 | else {I = I,j;} |
---|
3726 | } |
---|
3727 | else { if (lep[i] > 1) {ERROR("monomial has a not allowed multidegree");}} |
---|
3728 | } |
---|
3729 | if (I[1] == 0) {ERROR("monomial has a not allowed multidegree");} |
---|
3730 | return(I); |
---|
3731 | } |
---|
3732 | example |
---|
3733 | { |
---|
3734 | "EXAMPLE:"; echo = 2; |
---|
3735 | ring r = 0,(x,y,z),dp; |
---|
3736 | def R = freeAlgebra(r, 5); // constructs a Letterplace ring |
---|
3737 | setring R; // sets basering to Letterplace ring |
---|
3738 | poly p = x*x*z; |
---|
3739 | lp2iv(p); // transforms p into the intvec representation |
---|
3740 | lp2iv(y*y*x*x); |
---|
3741 | lp2iv(z*y*x*z*z); |
---|
3742 | } |
---|
3743 | |
---|
3744 | proc lp2ivId(ideal G) |
---|
3745 | "USAGE: lp2ivId(G); G an ideal (deprecated, will be removed soon) |
---|
3746 | RETURN: list |
---|
3747 | PURPOSE:Converting an ideal into an list of intmats, |
---|
3748 | @* the corresponding intvecs forming the rows |
---|
3749 | ASSUME: - basering has to be a Letterplace ring |
---|
3750 | EXAMPLE: example lp2ivId; shows examples |
---|
3751 | " |
---|
3752 | {G = normalize(lead(G)); |
---|
3753 | intvec I; list L; |
---|
3754 | checkAssumptionsLPIV(0,L); |
---|
3755 | int i,md; |
---|
3756 | for (i = 1; i <= size(G); i++) { if (md <= deg(G[i])) {md = deg(G[i]);}} |
---|
3757 | while (size(G) > 0) |
---|
3758 | {ideal Gt; |
---|
3759 | for (i = 1; i <= ncols(G); i++) {if (md == deg(G[i])) {Gt = Gt + G[i]; G[i] = 0;}} |
---|
3760 | if (size(Gt) > 0) |
---|
3761 | {G = simplify(G,2); |
---|
3762 | intmat M [size(Gt)][md]; |
---|
3763 | for (i = 1; i <= size(Gt); i++) {M[i,1..md] = lp2iv(Gt[i]);} |
---|
3764 | L = insert(L,M); |
---|
3765 | kill M; kill Gt; |
---|
3766 | md = md - 1; |
---|
3767 | } |
---|
3768 | else {kill Gt; md = md - 1;} |
---|
3769 | } |
---|
3770 | return(L); |
---|
3771 | } |
---|
3772 | example |
---|
3773 | { |
---|
3774 | "EXAMPLE:"; echo = 2; |
---|
3775 | ring r = 0,(x,y,z),dp; |
---|
3776 | def R = freeAlgebra(r, 5); // constructs a Letterplace ring |
---|
3777 | setring R; // sets basering to Letterplace ring |
---|
3778 | poly p = x*x*z; |
---|
3779 | poly q = y*y*x*x; |
---|
3780 | poly w = z*y*x*z; |
---|
3781 | // p, q, w are some polynomials we want to transform into their |
---|
3782 | // intvec representation |
---|
3783 | ideal G = p,q,w; |
---|
3784 | lp2ivId(G); // returns the list of intmats for this ideal |
---|
3785 | } |
---|
3786 | |
---|
3787 | proc testLift(ideal M, matrix T) |
---|
3788 | "USAGE: testLift(M,T); module M, matrix T |
---|
3789 | RETURN: module |
---|
3790 | PURPOSE: assembles the result of the lift procedure |
---|
3791 | ASSUME: T is the lift matrix of a submodule of M |
---|
3792 | NOTE: the inverse of the lift procedure |
---|
3793 | EXAMPLE: example testLift; shows examples |
---|
3794 | " |
---|
3795 | { |
---|
3796 | ideal R; |
---|
3797 | if (ncols(M) != nrows(T)) { ERROR("cols(M) != rows(T)") } |
---|
3798 | for (int i=1; i <= nrows(T); i++) { |
---|
3799 | T = subst(T, ncgen(i), M[i]); |
---|
3800 | } kill i; |
---|
3801 | for (int i=1; i <= ncols(T); i++) { |
---|
3802 | R[i] = sum(T[i]); |
---|
3803 | } kill i; |
---|
3804 | return(R); |
---|
3805 | } |
---|
3806 | example |
---|
3807 | { |
---|
3808 | "EXAMPLE:"; echo = 2; |
---|
3809 | ring r = 0,(x,y),(c,Dp); |
---|
3810 | ring R = freeAlgebra(r, 7, 2); |
---|
3811 | ideal I = std(x*y*x + 1); |
---|
3812 | print(matrix(I)); // finite two-sided Groebner basis |
---|
3813 | ideal SI = x*I[1]*y + y*x*I[2], I[1]*y*x + I[2]*y; |
---|
3814 | matrix T = lift(I, SI); // T is the lifting matrix of SI wrt I |
---|
3815 | print(T); // |
---|
3816 | print(matrix(SI)); // the original generators of SI as a matrix |
---|
3817 | print(matrix(testLift(I,T))); // and the result of testLift |
---|
3818 | } |
---|
3819 | |
---|
3820 | proc testSyz(ideal M, module S) |
---|
3821 | "USAGE: testSyz(M,S); module M, S |
---|
3822 | RETURN: module |
---|
3823 | PURPOSE: tests the result of the syz procedure |
---|
3824 | ASSUME: S is the syzygy bimodule of M |
---|
3825 | EXAMPLE: example testSyz; shows examples |
---|
3826 | " |
---|
3827 | { |
---|
3828 | ideal R; |
---|
3829 | if (ncols(M) != nrows(S)) { ERROR("cols(M) != rows(T)") } |
---|
3830 | for (int i=1; i <= nrows(S); i++) { |
---|
3831 | S = subst(S, ncgen(i), M[i]); |
---|
3832 | } kill i; |
---|
3833 | for (int i=1; i <= ncols(S); i++) { |
---|
3834 | R[i] = sum(S[i]); |
---|
3835 | } kill i; |
---|
3836 | return(R); |
---|
3837 | } |
---|
3838 | example |
---|
3839 | { |
---|
3840 | "EXAMPLE:"; echo = 2; |
---|
3841 | ring r = 0,(x,y),(c,Dp); |
---|
3842 | ring R = freeAlgebra(r, 7, 2); |
---|
3843 | ideal I = twostd(x*y*x + 1); |
---|
3844 | print(matrix(I)); |
---|
3845 | module S = syz(I); |
---|
3846 | print(S); |
---|
3847 | testSyz(I,S); // returns zero |
---|
3848 | } |
---|
3849 | |
---|
3850 | static proc mod_init() |
---|
3851 | { |
---|
3852 | LIB"freealgebra.so"; |
---|
3853 | } |
---|