1 | /////////////////////////////////////////////////////////////////////////// |
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2 | version="version schreyer.lib 4.0.0.0 Jun_2013 "; // $Id$ |
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3 | category="General purpose"; |
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4 | info=" |
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5 | LIBRARY: schreyer.lib Helpers for computing a Schreyer resolution in derham.lib |
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6 | AUTHOR: Oleksandr Motsak <U@D>, where U={motsak}, D={mathematik.uni-kl.de} |
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7 | |
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8 | PROCEDURES: |
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9 | Sres(M,len) compute Schreyer resolution of module M of maximal length len |
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10 | Ssyz(M) compute Schreyer resolution of module M of length 1 |
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11 | Scontinue(len) extend currently active resolution by (at most) len syszygies |
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12 | |
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13 | KEYWORDS: Schreyer ordering; Schreyer resolution; syzygy |
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14 | NOTE: requires the dynamic or built-in module: syzextra |
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15 | "; |
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16 | |
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17 | static proc prepareSyz( module I, list # ) |
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18 | { |
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19 | int i; |
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20 | int k = 0; |
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21 | int r = nrows(I); |
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22 | int c = ncols(I); |
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23 | |
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24 | |
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25 | if( size(#) > 0 ) |
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26 | { |
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27 | if( typeof(#[1]) == "int" || typeof(#[1]) == "bigint" ) |
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28 | { |
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29 | k = #[1]; |
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30 | } |
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31 | } |
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32 | |
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33 | if( k < r ) |
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34 | { |
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35 | "// *** Wrong k: ", k, " < nrows: ", r, " => setting k = r = ", r; |
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36 | k = r; |
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37 | } |
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38 | |
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39 | // "k: ", k; "c: ", c; "I: ", I; |
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40 | |
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41 | for( i = c; i > 0; i-- ) |
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42 | { |
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43 | I[i] = I[i] + gen(k + i); |
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44 | } |
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45 | |
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46 | // DetailedPrint(I); |
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47 | |
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48 | return(I); |
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49 | } |
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50 | |
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51 | static proc separateSyzGB( module J, int c ) |
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52 | { |
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53 | module II, G; vector v; int i; |
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54 | |
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55 | J = simplify(J, 2); |
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56 | |
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57 | for( i = ncols(J); i > 0; i-- ) |
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58 | { |
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59 | v = J[i]; |
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60 | if( leadcomp(v) > c ) |
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61 | { |
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62 | II[i] = v; |
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63 | } else |
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64 | { |
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65 | G[i] = v; // leave only gen(i): i <= c |
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66 | } |
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67 | } |
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68 | |
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69 | II = simplify(II, 2); |
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70 | G = simplify(G, 2); |
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71 | |
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72 | return (list(G, II)); |
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73 | } |
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74 | |
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75 | static proc splitSyzGB( module J, int c ) |
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76 | { |
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77 | module JJ; vector v, vv; int i; |
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78 | |
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79 | for( i = ncols(J); i > 0; i-- ) |
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80 | { |
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81 | v = J[i]; |
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82 | |
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83 | vv = 0; |
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84 | |
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85 | while( leadcomp(v) <= c ) |
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86 | { |
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87 | vv = vv + lead(v); |
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88 | v = v - lead(v); |
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89 | } |
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90 | |
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91 | J[i] = vv; |
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92 | JJ[i] = v; |
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93 | } |
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94 | |
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95 | J = simplify(J, 2); |
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96 | JJ = simplify(JJ, 2); |
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97 | |
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98 | return (list(J, JJ)); |
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99 | } |
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100 | |
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101 | |
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102 | static proc Sinit(module M) |
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103 | { |
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104 | def @save = basering; |
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105 | |
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106 | int @DEBUG = !system("with", "ndebug"); |
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107 | if( @DEBUG ) |
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108 | { |
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109 | "Sinit::Input"; |
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110 | type(M); |
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111 | DetailedPrint(M); |
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112 | attrib(M); |
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113 | } |
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114 | |
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115 | int @RANK = nrows(M); int @SIZE = ncols(M); |
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116 | |
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117 | int @IS_A_SB = attrib(M, "isSB"); // ??? only if all weights were zero?! |
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118 | |
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119 | if( !@IS_A_SB ) |
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120 | { |
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121 | M = std(M); // this should be faster than computing std in S (later on) |
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122 | } |
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123 | |
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124 | def S = MakeInducedSchreyerOrdering(1); // 1 puts history terms to the back |
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125 | // TODO: NOTE: +1 causes trouble to Singular interpreter!!!??? |
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126 | setring S; // a new ring with a Schreyer ordering |
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127 | |
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128 | if( @DEBUG ) |
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129 | { |
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130 | "Sinit::StartingISRing"; |
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131 | basering; |
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132 | // DetailedPrint(basering); |
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133 | } |
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134 | |
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135 | // Setup the leading syzygy^{-1} module to zero: |
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136 | module Z = 0; Z[@RANK] = 0; attrib(Z, "isHomog", intvec(0)); |
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137 | |
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138 | module MRES = Z; |
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139 | |
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140 | list RES; RES[1] = Z; |
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141 | |
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142 | module F = freemodule(@RANK); |
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143 | intvec @V = deg(F[1..@RANK]); |
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144 | |
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145 | module M = imap(@save, M); |
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146 | |
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147 | attrib(M, "isHomog", @V); |
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148 | attrib(M, "isSB", 1); |
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149 | |
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150 | |
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151 | if( @DEBUG ) |
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152 | { |
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153 | "Sinit::SB_Input: "; |
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154 | type(M); |
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155 | attrib(M); |
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156 | attrib(M, "isHomog"); |
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157 | DetailedPrint(M); |
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158 | } |
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159 | |
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160 | if( @DEBUG ) |
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161 | { |
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162 | // 0^th syz. property |
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163 | if( size(module(transpose( transpose(M) * transpose(MRES) ))) > 0 ) |
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164 | { |
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165 | transpose( transpose(M) * transpose(MRES) ); |
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166 | "transpose( transpose(M) * transpose(MRES) ) != 0!!!"; |
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167 | $ |
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168 | } |
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169 | } |
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170 | |
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171 | RES[size(RES)+1] = M; // list of all syzygy modules |
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172 | MRES = MRES, M; |
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173 | |
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174 | attrib(MRES, "isHomog", @V); |
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175 | |
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176 | attrib(S, "InducionLeads", lead(M)); |
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177 | attrib(S, "InducionStart", @RANK); |
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178 | |
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179 | if( @DEBUG ) |
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180 | { |
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181 | "Sinit::MRES"; |
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182 | DetailedPrint(MRES); |
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183 | attrib(MRES, "isHomog"); |
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184 | attrib(S); |
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185 | } |
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186 | |
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187 | export RES; |
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188 | export MRES; |
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189 | return (S); |
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190 | } |
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191 | |
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192 | static proc Sstep() |
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193 | { |
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194 | int @DEBUG = !system("with", "ndebug"); |
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195 | |
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196 | if( @DEBUG ) |
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197 | { |
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198 | "Sstep::NextInducedRing"; |
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199 | DetailedPrint(basering); |
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200 | |
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201 | attrib(basering, "InducionLeads"); |
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202 | attrib(basering, "InducionStart"); |
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203 | |
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204 | GetInducedData(); |
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205 | } |
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206 | |
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207 | // syzygy step: |
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208 | |
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209 | /* |
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210 | // is initial weights are all zeroes! |
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211 | def L = lead(M); |
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212 | intvec @V = deg(M[1..ncols(M)]); @W; @V; @W = @V; attrib(L, "isHomog", @W); |
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213 | SetInducedReferrence(L, @RANK, 0); |
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214 | */ |
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215 | |
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216 | // def L = lead(MRES); |
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217 | // @W = @W, @V; |
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218 | // attrib(L, "isHomog", @W); |
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219 | |
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220 | |
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221 | // General setting: |
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222 | // SetInducedReferrence(MRES, 0, 0); // limit: 0! |
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223 | int @l = size(RES); |
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224 | |
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225 | module M = RES[@l]; |
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226 | |
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227 | module L = attrib(basering, "InducionLeads"); |
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228 | int limit = attrib(basering, "InducionStart"); |
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229 | |
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230 | // L; limit; |
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231 | |
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232 | int @RANK = ncols(MRES) - ncols(M); // nrows(M); // what if M is zero?! |
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233 | |
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234 | /* |
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235 | if( @RANK != nrows(M) ) |
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236 | { |
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237 | type(MRES); |
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238 | @RANK; |
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239 | type(M); |
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240 | pause(); |
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241 | } |
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242 | */ |
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243 | |
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244 | intvec @W = attrib(M, "isHomog"); |
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245 | intvec @V = deg(M[1..ncols(M)]); |
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246 | @V = @W, @V; |
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247 | |
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248 | if( @DEBUG ) |
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249 | { |
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250 | "Sstep::NextInput: "; |
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251 | M; |
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252 | deg(M[1..ncols(M)]); // no use of @W :(? |
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253 | @RANK; |
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254 | DetailedPrint(MRES); |
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255 | attrib(MRES, "isHomog"); @W; |
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256 | deg(MRES[1..ncols(MRES)]); |
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257 | } |
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258 | |
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259 | |
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260 | |
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261 | SetInducedReferrence(L, limit, 0); |
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262 | |
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263 | def K = prepareSyz(M, @RANK); |
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264 | // K; |
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265 | |
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266 | // attrib(K, "isHomog", @V); DetailedPrint(K, 1000); |
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267 | |
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268 | // pause(); |
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269 | |
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270 | K = idPrepare(K, @RANK); // std(K); // ? |
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271 | K = simplify(K, 2); |
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272 | |
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273 | // K; |
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274 | |
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275 | module N = separateSyzGB(K, @RANK)[2]; // 1^st syz. module: vectors which start in lower part (comp >= @RANK) |
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276 | |
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277 | // "N_0: "; N; DetailedPrint(N, 10); |
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278 | |
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279 | // basering; print(@V); type(N); |
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280 | // attrib(N, "isHomog", @V); // TODO: fix "wrong weights"!!!? deg is wrong :((( |
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281 | N = std(N); |
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282 | attrib(N, "isHomog", @V); |
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283 | |
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284 | // N; |
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285 | |
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286 | if( @DEBUG ) |
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287 | { |
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288 | if( size(N) > 0 ) |
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289 | { |
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290 | // next syz. property |
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291 | if( size(module(transpose( transpose(N) * transpose(MRES) ))) > 0 ) |
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292 | { |
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293 | MRES; |
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294 | |
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295 | "N: "; N; DetailedPrint(N, 10); |
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296 | |
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297 | "K:"; K; DetailedPrint(K, 10); |
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298 | |
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299 | "RANKS: ", @RANK; |
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300 | |
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301 | "transpose( transpose(N) * transpose(MRES) ) != 0!!!"; |
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302 | transpose( transpose(N) * transpose(MRES) ); |
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303 | |
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304 | "transpose(N) * transpose(MRES): "; |
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305 | transpose(N) * transpose(MRES); |
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306 | DetailedPrint(module(_), 2); |
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307 | $ |
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308 | } |
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309 | } |
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310 | } |
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311 | |
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312 | RES[@l + 1] = N; // list of all syzygy modules |
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313 | |
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314 | MRES = MRES, N; |
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315 | attrib(MRES, "isHomog", @V); |
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316 | |
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317 | |
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318 | L = L, lead(N); |
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319 | attrib(basering, "InducionLeads", L); |
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320 | |
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321 | if( @DEBUG ) |
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322 | { |
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323 | "Sstep::NextSyzOutput: "; |
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324 | DetailedPrint(N); |
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325 | attrib(N, "isHomog"); |
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326 | } |
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327 | |
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328 | } |
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329 | |
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330 | proc Scontinue(int l) |
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331 | "USAGE: Scontinue(int len) |
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332 | RETURN: nothing, instead it changes the currently active resolution |
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333 | PURPOSE: extends the currently active resolution by at most len syzygies |
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334 | NOTE: must be used within a ring returned by Sres or Ssyz |
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335 | EXAMPLE: example Scontinue; shows an example |
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336 | " |
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337 | { |
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338 | def data = GetInducedData(); |
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339 | |
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340 | if( (!defined(RES)) || (!defined(MRES)) || (typeof(data) != "list") || (size(data) != 2) ) |
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341 | { |
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342 | ERROR("Sorry, but basering does not seem to be returned by Sres or Ssyz"); |
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343 | } |
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344 | for (; (l != 0) && (size(RES[size(RES)]) > 0); l-- ) |
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345 | { |
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346 | Sstep(); |
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347 | } |
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348 | } |
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349 | example |
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350 | { "EXAMPLE:"; echo = 2; |
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351 | ring r; |
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352 | module M = maxideal(1); M; |
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353 | def S = Ssyz(M); setring S; S; |
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354 | "Only the first syzygy: "; |
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355 | RES; MRES; |
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356 | "More syzygies: "; |
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357 | Scontinue(10); |
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358 | RES; MRES; |
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359 | } |
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360 | |
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361 | proc Ssyz(module M) |
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362 | "USAGE: Ssyz(module M) |
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363 | RETURN: ring, containing a Schreyer resolution |
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364 | PURPOSE: computes a Schreyer resolution of M of length 1 |
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365 | NOTE: the output is explained in Sres |
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366 | EXAMPLE: example Ssyz; shows an example |
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367 | " |
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368 | { |
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369 | def S = Sinit(M); setring S; |
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370 | |
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371 | Sstep(); // NOTE: what if M is zero? |
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372 | |
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373 | return (S); |
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374 | } |
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375 | example |
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376 | { "EXAMPLE:"; echo = 2; |
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377 | ring r; |
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378 | module M = maxideal(1); M; |
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379 | def S = Ssyz(M); setring S; S; |
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380 | "Only the first syzygy: "; |
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381 | RES; |
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382 | MRES; // Note gen(i) |
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383 | kill S; |
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384 | setring r; kill M; |
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385 | |
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386 | module M = 0; |
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387 | def S = Ssyz(M); setring S; S; |
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388 | "Only the first syzygy: "; |
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389 | RES; |
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390 | MRES; |
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391 | } |
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392 | |
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393 | proc Sres(module M, int l) |
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394 | "USAGE: Sres(module M, int len) |
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395 | RETURN: ring, containing a Schreyer resolution |
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396 | PURPOSE: computes a Schreyer resolution of (basering^rank(M))/M with at most len syzygy modules, |
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397 | computed with respect to a Schreyer (induced) ordering. |
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398 | NOTE: Input is a set of vectors M over a basering. basering may be non-commutative. |
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399 | NOTE: Schreyer resolution is represented by a list of modules RES and a module MRES |
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400 | belonging to a specially constructed ring, which is endowed with a Schreyer ordering. |
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401 | The list of modules RES contains the images of maps (also called syzygies) subsituting the |
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402 | computed beginning of a Schreyer free resolution of (baseRing^rank(M))/M. |
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403 | The leading zero map RES[1] with rank(M) zero generators indicates that the image of |
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404 | the first differential map is zero. The second map RES[2] is given by M, which indicates that |
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405 | the resolution is of (baseRing^rank(M))/M is being computed. |
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406 | The module MRES is a direct sum of modules from RES and comprises all computed differential maps. |
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407 | Syzygies are shifted so that gen(i) is mapped to MRES[i] under the differential. |
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408 | Syzygies are given by Groebner bases with respect to corresponding Schreyer orderings. |
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409 | NOTE: Schreyer ordering extends an arbitrary starting module ordeing (defined by basering) |
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410 | and is extended to higher syzygt modules using the following definition: |
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411 | a < b if and only if (d(a) < d(b)) OR ( (d(a) = d(b) AND (comp(a) < comp(b)) ), |
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412 | where d(a) is the image of a under the differential (given by MRES), |
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413 | and comp(a) is the mod. component, for any module terms a and b. |
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414 | NOTE: If len == 0 then len is set to be nvars(basering) + 1 |
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415 | EXAMPLE: example Sres; shows an example |
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416 | " |
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417 | { |
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418 | def S = Sinit(M); setring S; |
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419 | |
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420 | if (l == 0) |
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421 | { |
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422 | l = nvars(basering) + 1; // not really an estimate...?! |
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423 | } |
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424 | |
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425 | Sstep(); l = l - 1; |
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426 | |
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427 | Scontinue(l); |
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428 | |
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429 | return (S); |
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430 | } |
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431 | example |
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432 | { "EXAMPLE:"; echo = 2; |
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433 | ring r; |
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434 | module M = maxideal(1); M; |
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435 | def S = Sres(M, 0); setring S; S; |
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436 | RES; |
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437 | MRES; |
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438 | kill S; |
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439 | setring r; kill M; |
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440 | |
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441 | def A = nc_algebra(-1,0); setring A; |
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442 | ideal Q = var(1)^2, var(2)^2, var(3)^2; |
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443 | qring SCA = twostd(Q); |
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444 | basering; |
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445 | |
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446 | module M = maxideal(1); |
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447 | def S = Sres(M, 2); setring S; S; |
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448 | RES; |
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449 | MRES; |
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450 | } |
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451 | |
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452 | |
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453 | |
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454 | // ================================================================== // |
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455 | |
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456 | |
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457 | LIB "general.lib"; // for sort |
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458 | |
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459 | /* static proc Tail(def M) // DONE: in C++ (dyn. module: syzextra)! |
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460 | { |
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461 | int i = ncols(M); def m; |
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462 | while (i > 0) |
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463 | { |
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464 | m = M[i]; |
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465 | m = m - lead(m); // m = tail(m) |
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466 | M[i] = m; |
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467 | i--; |
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468 | } |
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469 | return (M); |
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470 | }*/ |
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471 | |
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472 | |
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473 | /* static */ proc SSinit(def M) |
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474 | { |
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475 | if( (typeof(M) != "module") && (typeof(M) != "ideal") ) |
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476 | { |
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477 | ERROR("Sorry: need an ideal or a module for input"); |
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478 | } |
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479 | |
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480 | // TODO! DONE? |
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481 | def @save = basering; |
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482 | |
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483 | int @DEBUG = !system("with", "ndebug"); |
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484 | int @SYZCHECK = 1 || @DEBUG; // TODO: only for now!! |
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485 | |
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486 | if( @DEBUG ) |
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487 | { |
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488 | "SSinit::Input"; |
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489 | type(M); |
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490 | // DetailedPrint(M); |
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491 | attrib(M); |
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492 | } |
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493 | |
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494 | int @RANK = nrows(M); int @SIZE = ncols(M); |
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495 | |
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496 | int @IS_A_SB = attrib(M, "isSB"); // ??? only if all weights were zero?! |
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497 | |
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498 | if( !@IS_A_SB ) |
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499 | { |
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500 | def opts = option(get); |
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501 | option(redSB); option(redTail); |
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502 | M = std(M); |
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503 | option(set, opts); |
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504 | kill opts; |
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505 | } else |
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506 | { |
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507 | M = simplify(M, 2 + 4 + 32); |
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508 | } |
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509 | |
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510 | def LEAD = lead(M); |
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511 | |
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512 | // sort wrt neg.deg.rev.lex! |
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513 | intvec iv_ds = sort(LEAD, "ds", 1)[2]; // ,1 => reversed! |
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514 | |
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515 | M = M[iv_ds]; // sort M wrt ds on current leading terms |
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516 | LEAD = LEAD[iv_ds]; |
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517 | |
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518 | def TAIL = Tail(M); |
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519 | |
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520 | intvec @DEGS = deg(M[1..@SIZE]); // store actuall degrees of input elements |
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521 | |
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522 | // TODO: what about real modules? weighted ones? |
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523 | |
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524 | list @l = ringlist(@save); |
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525 | |
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526 | int @z = 0; ideal @m = maxideal(1); intvec @wdeg = deg(@m[1..ncols(@m)]); |
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527 | |
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528 | // NOTE: @wdeg will be ignored anyway :( |
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529 | @l[3] = list(list("C", @z), list("lp", @wdeg)); |
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530 | |
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531 | kill @z, @wdeg; // since these vars are ring independent! |
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532 | |
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533 | def S = ring(@l); // --MakeInducedSchreyerOrdering(1); |
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534 | |
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535 | module F = freemodule(@RANK); |
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536 | intvec @V = deg(F[1..@RANK]); |
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537 | |
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538 | setring S; // ring with an easy divisibility test ("C, lex") |
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539 | |
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540 | if( @DEBUG ) |
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541 | { |
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542 | "SSinit::NewRing(C, lex)"; |
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543 | basering; |
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544 | // DetailedPrint(basering); |
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545 | } |
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546 | |
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547 | // Setup the leading syzygy^{-1} module to zero: |
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548 | module Z = 0; Z[@RANK] = 0; attrib(Z, "isHomog", intvec(0)); |
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549 | |
---|
550 | module MRES = Z; |
---|
551 | |
---|
552 | list RES; RES[1] = Z; |
---|
553 | list LRES; LRES[1] = Z; |
---|
554 | list TRES; TRES[1] = Z; |
---|
555 | |
---|
556 | def M = imap(@save, M); |
---|
557 | |
---|
558 | attrib(M, "isHomog", @V); |
---|
559 | attrib(M, "isSB", 1); |
---|
560 | attrib(M, "degrees", @DEGS); |
---|
561 | |
---|
562 | def LEAD = imap(@save, LEAD); |
---|
563 | |
---|
564 | attrib(LEAD, "isHomog", @V); |
---|
565 | attrib(LEAD, "isSB", 1); |
---|
566 | |
---|
567 | def TAIL = imap(@save, TAIL); |
---|
568 | |
---|
569 | if( @DEBUG ) |
---|
570 | { |
---|
571 | "SSinit::(sorted) SB_Input: "; |
---|
572 | type(M); |
---|
573 | attrib(M); |
---|
574 | attrib(M, "isHomog"); |
---|
575 | // DetailedPrint(M); |
---|
576 | } |
---|
577 | |
---|
578 | if( @SYZCHECK ) |
---|
579 | { |
---|
580 | // 0^th syz. property |
---|
581 | if( size(module(transpose( transpose(M) * transpose(MRES) ))) > 0 ) |
---|
582 | { |
---|
583 | transpose( transpose(M) * transpose(MRES) ); |
---|
584 | "ERROR: transpose( transpose(M) * transpose(MRES) ) != 0!!!"; |
---|
585 | $ |
---|
586 | } |
---|
587 | } |
---|
588 | |
---|
589 | RES[size(RES)+1] = M; // list of all syzygy modules |
---|
590 | LRES[size(LRES)+1] = LEAD; // list of all syzygy modules |
---|
591 | TRES[size(TRES)+1] = TAIL; // list of all syzygy modules |
---|
592 | |
---|
593 | MRES = MRES, M; //? |
---|
594 | |
---|
595 | attrib(MRES, "isHomog", @V); |
---|
596 | |
---|
597 | // attrib(S, "InducionStart", @RANK); |
---|
598 | attrib(S, "LEAD2SYZ", 1); |
---|
599 | attrib(S, "TAILREDSYZ", 0); |
---|
600 | attrib(S, "DEBUG", @DEBUG); |
---|
601 | attrib(S, "SYZCHECK", @SYZCHECK); |
---|
602 | |
---|
603 | if( @DEBUG ) |
---|
604 | { |
---|
605 | "SSinit::MRES"; |
---|
606 | MRES; |
---|
607 | // DetailedPrint(MRES); |
---|
608 | attrib(MRES, "isHomog"); |
---|
609 | attrib(S); |
---|
610 | } |
---|
611 | |
---|
612 | export RES; |
---|
613 | export MRES; |
---|
614 | export LRES; |
---|
615 | export TRES; |
---|
616 | return (S); |
---|
617 | } |
---|
618 | example |
---|
619 | { "EXAMPLE:"; echo = 2; |
---|
620 | ring R = 0, (w, x, y, z), dp; |
---|
621 | |
---|
622 | def M = maxideal(1); |
---|
623 | def S = SSinit(M); setring S; S; |
---|
624 | |
---|
625 | "Only the first initialization: "; |
---|
626 | RES; LRES; TRES; |
---|
627 | MRES; |
---|
628 | |
---|
629 | kill S; setring R; kill M; |
---|
630 | |
---|
631 | ideal M = w^2 - x*z, w*x - y*z, x^2 - w*y, x*y - z^2, y^2 - w*z; |
---|
632 | def S = SSinit(M); setring S; S; |
---|
633 | |
---|
634 | "Only the first initialization: "; |
---|
635 | RES; LRES; TRES; |
---|
636 | MRES; |
---|
637 | |
---|
638 | kill S; setring R; kill M; |
---|
639 | } |
---|
640 | |
---|
641 | |
---|
642 | LIB "poly.lib"; // for lcm |
---|
643 | |
---|
644 | |
---|
645 | |
---|
646 | /// Compute L(Syz(L)) |
---|
647 | proc SSComputeLeadingSyzygyTerms(def L) |
---|
648 | { |
---|
649 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
650 | { |
---|
651 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
652 | } else |
---|
653 | { |
---|
654 | int @DEBUG = !system("with", "ndebug"); |
---|
655 | } |
---|
656 | |
---|
657 | if( @DEBUG ) |
---|
658 | { |
---|
659 | "SSComputeLeadingSyzygyTerms::Input: "; |
---|
660 | L; |
---|
661 | } |
---|
662 | |
---|
663 | int i, j, r; intvec iv_ds; |
---|
664 | int N = ncols(L); |
---|
665 | def a, b; |
---|
666 | poly aa, bb; |
---|
667 | |
---|
668 | bigint c; |
---|
669 | |
---|
670 | ideal M; |
---|
671 | |
---|
672 | module S = 0; |
---|
673 | |
---|
674 | for(i = 1; i <= N; i++) |
---|
675 | { |
---|
676 | a = L[i]; |
---|
677 | // "a: ", a; |
---|
678 | c = leadcomp(a); |
---|
679 | r = int(c); |
---|
680 | |
---|
681 | if(r > 0) |
---|
682 | { |
---|
683 | aa = a[r]; |
---|
684 | } else |
---|
685 | { |
---|
686 | aa = a; |
---|
687 | } |
---|
688 | |
---|
689 | M = 0; |
---|
690 | |
---|
691 | for(j = i-1; j > 0; j--) |
---|
692 | { |
---|
693 | b = L[j]; |
---|
694 | // "b: ", b; |
---|
695 | |
---|
696 | if( leadcomp(b) == c ) |
---|
697 | { |
---|
698 | if(r > 0) |
---|
699 | { |
---|
700 | bb = b[r]; |
---|
701 | } else |
---|
702 | { |
---|
703 | bb = b; |
---|
704 | } |
---|
705 | |
---|
706 | M[j] = (lcm(aa, bb) / aa); |
---|
707 | } |
---|
708 | } |
---|
709 | |
---|
710 | // TODO: add quotient relations here... |
---|
711 | |
---|
712 | M = simplify(M, 1 + 2 + 32); |
---|
713 | |
---|
714 | iv_ds = sort(M, "ds", 1)[2]; // ,1 => reversed! |
---|
715 | |
---|
716 | M = M[iv_ds]; |
---|
717 | |
---|
718 | S = S, M * gen(i); |
---|
719 | } |
---|
720 | |
---|
721 | S = simplify(S, 2); |
---|
722 | |
---|
723 | S = sort(S, "ds", 1)[1]; // ,1 => reversed! // TODO: not needed? |
---|
724 | |
---|
725 | if( @DEBUG ) |
---|
726 | { |
---|
727 | "SSComputeLeadingSyzygyTerms::Output: "; |
---|
728 | S; |
---|
729 | } |
---|
730 | |
---|
731 | attrib(S, "isSB", 1); |
---|
732 | |
---|
733 | return (S); |
---|
734 | } |
---|
735 | |
---|
736 | /// Compute Syz(L), where L is a monomial (leading) module |
---|
737 | proc SSCompute2LeadingSyzygyTerms(def L, int @TAILREDSYZ) |
---|
738 | { |
---|
739 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
740 | { |
---|
741 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
742 | } else |
---|
743 | { |
---|
744 | int @DEBUG = !system("with", "ndebug"); |
---|
745 | } |
---|
746 | |
---|
747 | if( typeof( attrib(basering, "SYZCHECK") ) == "int" ) |
---|
748 | { |
---|
749 | int @SYZCHECK = attrib(basering, "SYZCHECK"); |
---|
750 | } else |
---|
751 | { |
---|
752 | int @SYZCHECK = @DEBUG; |
---|
753 | } |
---|
754 | |
---|
755 | |
---|
756 | if( @DEBUG ) |
---|
757 | { |
---|
758 | "SSCompute2LeadingSyzygyTerms::Input: "; |
---|
759 | L; |
---|
760 | "@TAILREDSYZ: ", @TAILREDSYZ; |
---|
761 | } |
---|
762 | |
---|
763 | int i, j, r; |
---|
764 | int N = ncols(L); |
---|
765 | def a, b; |
---|
766 | |
---|
767 | poly aa, bb, @lcm; |
---|
768 | |
---|
769 | bigint c; |
---|
770 | |
---|
771 | module M; |
---|
772 | |
---|
773 | module S = 0; |
---|
774 | |
---|
775 | for(i = 1; i <= N; i++) |
---|
776 | { |
---|
777 | a = L[i]; |
---|
778 | // "a: ", a; |
---|
779 | c = leadcomp(a); |
---|
780 | r = int(c); |
---|
781 | |
---|
782 | aa = leadmonomial(a); |
---|
783 | |
---|
784 | M = 0; |
---|
785 | |
---|
786 | for(j = i-1; j > 0; j--) |
---|
787 | { |
---|
788 | b = L[j]; |
---|
789 | // "b: ", b; |
---|
790 | |
---|
791 | if( leadcomp(b) == c ) |
---|
792 | { |
---|
793 | bb = leadmonomial(b); |
---|
794 | @lcm = lcm(aa, bb); |
---|
795 | |
---|
796 | M[j] = (@lcm / aa)* gen(i) - (@lcm / bb)* gen(j); |
---|
797 | } |
---|
798 | } |
---|
799 | |
---|
800 | M = simplify(M, 2); |
---|
801 | |
---|
802 | // TODO: add quotient relations here... |
---|
803 | S = S, M; |
---|
804 | } |
---|
805 | |
---|
806 | if( @TAILREDSYZ ) |
---|
807 | { |
---|
808 | // Make sure that 2nd syzygy terms are not reducible by 1st |
---|
809 | def opts = option(get); |
---|
810 | option(redSB); option(redTail); |
---|
811 | S = std(S); // binomial module |
---|
812 | option(set, opts); |
---|
813 | // kill opts; |
---|
814 | } else |
---|
815 | { |
---|
816 | S = simplify(S, 2 + 32); |
---|
817 | } |
---|
818 | |
---|
819 | S = sort(S, "ds", 1)[1]; // ,1 => reversed! |
---|
820 | |
---|
821 | if( @DEBUG ) |
---|
822 | { |
---|
823 | "SSCompute2LeadingSyzygyTerms::Syz(LEAD): "; S; |
---|
824 | } |
---|
825 | |
---|
826 | if( @SYZCHECK ) |
---|
827 | { |
---|
828 | if( size(S) > 0 and size(L) > 0 ) |
---|
829 | { |
---|
830 | if( size(module(transpose( transpose(S) * transpose(L) ))) > 0 ) |
---|
831 | { |
---|
832 | transpose( transpose(S) * transpose(L) ); |
---|
833 | "ERROR: transpose( transpose(S) * transpose(L) ) != 0!!!"; |
---|
834 | $ |
---|
835 | } |
---|
836 | } |
---|
837 | } |
---|
838 | |
---|
839 | module S2 = Tail(S); |
---|
840 | S = lead(S); // (C,lp) on base ring! |
---|
841 | |
---|
842 | if( @DEBUG ) |
---|
843 | { |
---|
844 | "SSCompute2LeadingSyzygyTerms::Output: "; S; S2; |
---|
845 | } |
---|
846 | |
---|
847 | attrib(S, "isSB", 1); |
---|
848 | |
---|
849 | return (S, S2); |
---|
850 | } |
---|
851 | |
---|
852 | // -------------------------------------------------------- // |
---|
853 | |
---|
854 | /// TODO: save shortcut LM(m) * "t" -> ? |
---|
855 | proc SSReduceTerm(poly m, def t, def L, def T, list #) |
---|
856 | { |
---|
857 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
858 | { |
---|
859 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
860 | } else |
---|
861 | { |
---|
862 | int @DEBUG = !system("with", "ndebug"); |
---|
863 | } |
---|
864 | |
---|
865 | if( @DEBUG ) |
---|
866 | { |
---|
867 | "SSReduce::Input: "; |
---|
868 | |
---|
869 | "mult: ", m; |
---|
870 | "term: ", t; |
---|
871 | "L: ", L; |
---|
872 | "T: ", T; |
---|
873 | if( size(#) > 0 ) |
---|
874 | { |
---|
875 | "LSyz: ", #; |
---|
876 | } |
---|
877 | // "attrib(LS, 'isSB')", attrib(LS, "isSB"); |
---|
878 | } |
---|
879 | |
---|
880 | vector s = 0; |
---|
881 | |
---|
882 | if( t == 0 ) |
---|
883 | { |
---|
884 | return (s); |
---|
885 | } |
---|
886 | |
---|
887 | def product = m * t; |
---|
888 | |
---|
889 | bigint c = leadcomp(t); |
---|
890 | int r = int(c); |
---|
891 | |
---|
892 | def a, b, nf, bb; |
---|
893 | |
---|
894 | // looking for an appropriate reducer |
---|
895 | for( int k = ncols(L); k > 0; k-- ) |
---|
896 | { |
---|
897 | a = L[k]; |
---|
898 | // with the same mod. component |
---|
899 | if( leadcomp(a) == c ) |
---|
900 | { |
---|
901 | b = - (leadmonomial(product) / leadmonomial(L[k])); |
---|
902 | |
---|
903 | // which divides the product |
---|
904 | if( b != 0 ) |
---|
905 | { |
---|
906 | // "b: ", b; |
---|
907 | bb = b * gen(k); |
---|
908 | nf = bb; |
---|
909 | |
---|
910 | if( size(#) > 0 ) |
---|
911 | { |
---|
912 | if( typeof(#[1]) == "module" ) |
---|
913 | { |
---|
914 | nf = NF(bb, #[1]); |
---|
915 | // "NF: ", nf; |
---|
916 | } |
---|
917 | } |
---|
918 | |
---|
919 | // while the complement (the fraction) is not reducible by leading syzygies |
---|
920 | if( nf != 0 ) |
---|
921 | { |
---|
922 | /// TODO: save shortcut LM(m) * T[i] -> ? |
---|
923 | |
---|
924 | // choose ANY such reduction... (with the biggest index?) |
---|
925 | s = bb + SSTraverseTail(b, T[k], L, T, #); |
---|
926 | break; |
---|
927 | } |
---|
928 | } |
---|
929 | } |
---|
930 | } |
---|
931 | if( @DEBUG ) |
---|
932 | { |
---|
933 | "SSReduceTerm::Output: ", s; |
---|
934 | } |
---|
935 | return (s); |
---|
936 | } |
---|
937 | |
---|
938 | // TODO: store m * @tail -.-^-.-^-.--> ? |
---|
939 | proc SSTraverseTail(poly m, def @tail, def L, def T, list #) |
---|
940 | { |
---|
941 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
942 | { |
---|
943 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
944 | } else |
---|
945 | { |
---|
946 | int @DEBUG = !system("with", "ndebug"); |
---|
947 | } |
---|
948 | |
---|
949 | if( @DEBUG ) |
---|
950 | { |
---|
951 | "SSTraverse::Input: "; |
---|
952 | |
---|
953 | "mult: ", m; |
---|
954 | "tail: ", @tail; // T[i]; |
---|
955 | |
---|
956 | if( size(#) > 0 ) |
---|
957 | { |
---|
958 | "LSyz: "; #[1]; |
---|
959 | } |
---|
960 | } |
---|
961 | |
---|
962 | vector s = 0; |
---|
963 | |
---|
964 | def @l; |
---|
965 | |
---|
966 | // iterate tail-terms in ANY order! |
---|
967 | while( size(@tail) > 0 ) |
---|
968 | { |
---|
969 | @l = lead(@tail); |
---|
970 | s = s + SSReduceTerm(m, @l, L, T, #); |
---|
971 | @tail = @tail - @l; |
---|
972 | } |
---|
973 | |
---|
974 | if( @DEBUG ) |
---|
975 | { |
---|
976 | "SSTraverseTail::Output: ", s; |
---|
977 | } |
---|
978 | return (s); |
---|
979 | } |
---|
980 | |
---|
981 | // -------------------------------------------------------- // |
---|
982 | |
---|
983 | // module (N, LL, TT) = SSComputeSyzygy(L, T); |
---|
984 | // Compute Syz(L ++ T) = N = LL ++ TT |
---|
985 | proc SSComputeSyzygy(def L, def T) |
---|
986 | { |
---|
987 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
988 | { |
---|
989 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
990 | } else |
---|
991 | { |
---|
992 | int @DEBUG = !system("with", "ndebug"); |
---|
993 | } |
---|
994 | |
---|
995 | if( @DEBUG ) |
---|
996 | { |
---|
997 | "SSComputeSyzygy::Input"; |
---|
998 | "basering: ", basering; attrib(basering); |
---|
999 | // DetailedPrint(basering); |
---|
1000 | |
---|
1001 | // "iCompShift: ", iCompShift; |
---|
1002 | |
---|
1003 | "L: "; L; |
---|
1004 | "T: "; T; |
---|
1005 | } |
---|
1006 | |
---|
1007 | def a; bigint c; int r, k; poly aa; |
---|
1008 | |
---|
1009 | int @LEAD2SYZ = 0; |
---|
1010 | if( typeof( attrib(basering, "LEAD2SYZ") ) == "int" ) |
---|
1011 | { |
---|
1012 | @LEAD2SYZ = attrib(basering, "LEAD2SYZ"); |
---|
1013 | } |
---|
1014 | |
---|
1015 | int @TAILREDSYZ = 1; |
---|
1016 | if( typeof( attrib(basering, "TAILREDSYZ") ) == "int" ) |
---|
1017 | { |
---|
1018 | @TAILREDSYZ = attrib(basering, "TAILREDSYZ"); |
---|
1019 | // @TAILREDSYZ; |
---|
1020 | } |
---|
1021 | |
---|
1022 | /// Get the critical leading syzygy terms |
---|
1023 | if( @LEAD2SYZ ) // & 2nd syz. term |
---|
1024 | { |
---|
1025 | def a2; int r2; poly aa2; |
---|
1026 | module LL, LL2; |
---|
1027 | (LL, LL2) = SSCompute2LeadingSyzygyTerms(L, @TAILREDSYZ); // ++ |
---|
1028 | } else |
---|
1029 | { |
---|
1030 | module LL = SSComputeLeadingSyzygyTerms(L); |
---|
1031 | } |
---|
1032 | |
---|
1033 | module TT, SYZ; |
---|
1034 | |
---|
1035 | if( size(LL) > 0 ) |
---|
1036 | { |
---|
1037 | list LS; |
---|
1038 | |
---|
1039 | if( @TAILREDSYZ ) |
---|
1040 | { |
---|
1041 | LS = list(LL); |
---|
1042 | } |
---|
1043 | |
---|
1044 | vector @tail; |
---|
1045 | |
---|
1046 | for(k = ncols(LL); k > 0; k-- ) |
---|
1047 | { |
---|
1048 | // leading syz. term: |
---|
1049 | a = LL[k]; c = leadcomp(a); r = int(c); aa = leadmonomial(a); |
---|
1050 | // "A: ", a, " --->>>> ", aa, " **** [", r, "]: "; |
---|
1051 | |
---|
1052 | /// TODO: save shortcut (aa) * T[r] -> ? |
---|
1053 | @tail = SSTraverseTail(aa, T[r], L, T, LS); |
---|
1054 | |
---|
1055 | // get the 2nd syzygy term... |
---|
1056 | |
---|
1057 | if( @LEAD2SYZ ) // with the 2nd syz. term: |
---|
1058 | { |
---|
1059 | a2 = LL2[k]; c = leadcomp(a2); r2 = int(c); aa2 = leadmonomial(a2); |
---|
1060 | @tail = @tail + |
---|
1061 | /// TODO: save shortcut (aa2) * T[r2] -> ? |
---|
1062 | a2 + SSTraverseTail(aa2, T[r2], L, T, LS); |
---|
1063 | } else |
---|
1064 | { |
---|
1065 | @tail = @tail + SSReduceTerm(aa, L[r], L, T, LS); |
---|
1066 | } |
---|
1067 | |
---|
1068 | |
---|
1069 | TT[k] = @tail; |
---|
1070 | SYZ[k] = a + @tail; |
---|
1071 | } |
---|
1072 | } |
---|
1073 | |
---|
1074 | /* |
---|
1075 | def opts = option(get); option(redSB); option(redTail); |
---|
1076 | module SYZ = std(syz(M)); |
---|
1077 | option(set, opts); kill opts; |
---|
1078 | |
---|
1079 | module LL = lead(SYZ); // TODO: WRONG ORDERING!!!!!!!! |
---|
1080 | module TT = Tail(SYZ); |
---|
1081 | */ |
---|
1082 | |
---|
1083 | if( @DEBUG ) |
---|
1084 | { |
---|
1085 | "SSComputeSyzygy::Output"; |
---|
1086 | |
---|
1087 | "SYZ: "; SYZ; |
---|
1088 | "LL: "; LL; |
---|
1089 | "TT: "; TT; |
---|
1090 | } |
---|
1091 | |
---|
1092 | return (SYZ, LL, TT); |
---|
1093 | } |
---|
1094 | |
---|
1095 | // resolution/syzygy step: |
---|
1096 | static proc SSstep() |
---|
1097 | { |
---|
1098 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
1099 | { |
---|
1100 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
1101 | } else |
---|
1102 | { |
---|
1103 | int @DEBUG = !system("with", "ndebug"); |
---|
1104 | } |
---|
1105 | |
---|
1106 | |
---|
1107 | if( typeof( attrib(basering, "SYZCHECK") ) == "int" ) |
---|
1108 | { |
---|
1109 | int @SYZCHECK = attrib(basering, "SYZCHECK"); |
---|
1110 | } else |
---|
1111 | { |
---|
1112 | int @SYZCHECK = @DEBUG; |
---|
1113 | } |
---|
1114 | |
---|
1115 | if( @DEBUG ) |
---|
1116 | { |
---|
1117 | "SSstep::NextInducedRing"; |
---|
1118 | "basering: ", basering; attrib(basering); |
---|
1119 | } |
---|
1120 | |
---|
1121 | /* |
---|
1122 | // is initial weights are all zeroes! |
---|
1123 | def L = lead(M); |
---|
1124 | intvec @V = deg(M[1..ncols(M)]); @W; @V; @W = @V; attrib(L, "isHomog", @W); |
---|
1125 | SetInducedReferrence(L, @RANK, 0); |
---|
1126 | */ |
---|
1127 | |
---|
1128 | // def L = lead(MRES); |
---|
1129 | // @W = @W, @V; |
---|
1130 | // attrib(L, "isHomog", @W); |
---|
1131 | |
---|
1132 | |
---|
1133 | // General setting: |
---|
1134 | // SetInducedReferrence(MRES, 0, 0); // limit: 0! |
---|
1135 | int @l = size(RES); |
---|
1136 | |
---|
1137 | def M = RES[@l]; |
---|
1138 | |
---|
1139 | def L = LRES[@l]; |
---|
1140 | def T = TRES[@l]; |
---|
1141 | |
---|
1142 | |
---|
1143 | //// TODO: wrong !!!!! |
---|
1144 | int @RANK = ncols(MRES) - ncols(M); // nrows(M); // what if M is zero?! |
---|
1145 | |
---|
1146 | |
---|
1147 | |
---|
1148 | /* |
---|
1149 | if( @RANK != nrows(M) ) |
---|
1150 | { |
---|
1151 | type(MRES); |
---|
1152 | @RANK; |
---|
1153 | type(M); |
---|
1154 | pause(); |
---|
1155 | } |
---|
1156 | */ |
---|
1157 | |
---|
1158 | intvec @W = attrib(M, "isHomog"); intvec @V = attrib(M, "degrees"); @V = @W, @V; |
---|
1159 | |
---|
1160 | if( @DEBUG ) |
---|
1161 | { |
---|
1162 | "Sstep::NextInput: "; |
---|
1163 | M; |
---|
1164 | L; |
---|
1165 | @V; |
---|
1166 | @RANK; |
---|
1167 | // DetailedPrint(MRES); |
---|
1168 | attrib(MRES, "isHomog"); |
---|
1169 | } |
---|
1170 | |
---|
1171 | |
---|
1172 | // TODO: N = SYZ( M )!!! |
---|
1173 | module N, LL, TT; |
---|
1174 | (N, LL, TT) = SSComputeSyzygy(/*M, */L, T/*, @RANK*/); |
---|
1175 | |
---|
1176 | // shift syz.comp by @RANK: |
---|
1177 | module Z; |
---|
1178 | Z = 0; Z[@RANK] = 0; Z = Z, transpose(LL); LL = transpose(Z); |
---|
1179 | Z = 0; Z[@RANK] = 0; Z = Z, transpose(TT); TT = transpose(Z); |
---|
1180 | Z = 0; Z[@RANK] = 0; Z = Z, transpose(N); N = transpose(Z); |
---|
1181 | |
---|
1182 | |
---|
1183 | if( @SYZCHECK ) |
---|
1184 | { |
---|
1185 | if( size(N) > 0 ) |
---|
1186 | { |
---|
1187 | // next syz. property |
---|
1188 | if( size(module(transpose( transpose(N) * transpose(MRES) ))) > 0 ) |
---|
1189 | { |
---|
1190 | "MRES", MRES; |
---|
1191 | |
---|
1192 | "N: "; N; // DetailedPrint(N, 2); |
---|
1193 | |
---|
1194 | "LL:"; LL; // DetailedPrint(LL, 1); |
---|
1195 | "TT:"; TT; // DetailedPrint(TT, 10); |
---|
1196 | |
---|
1197 | "RANKS: ", @RANK; |
---|
1198 | |
---|
1199 | "transpose( transpose(N) * transpose(MRES) ) != 0!!!"; |
---|
1200 | transpose( transpose(N) * transpose(MRES) ); |
---|
1201 | |
---|
1202 | "transpose(N) * transpose(MRES): "; |
---|
1203 | transpose(N) * transpose(MRES); |
---|
1204 | // DetailedPrint(module(_), 2); |
---|
1205 | $ |
---|
1206 | } |
---|
1207 | } |
---|
1208 | } |
---|
1209 | |
---|
1210 | attrib(N, "isHomog", @V); |
---|
1211 | |
---|
1212 | // TODO: correct the following: |
---|
1213 | intvec @DEGS = deg(N[1..ncols(N)]); // no mod. comp. weights :( |
---|
1214 | |
---|
1215 | |
---|
1216 | attrib(N, "degrees", @DEGS); |
---|
1217 | |
---|
1218 | RES[@l + 1] = N; // list of all syzygy modules |
---|
1219 | LRES[@l + 1] = LL; // list of all syzygy modules |
---|
1220 | TRES[@l + 1] = TT; // list of all syzygy modules |
---|
1221 | |
---|
1222 | MRES = MRES, N; |
---|
1223 | |
---|
1224 | attrib(MRES, "isHomog", @V); |
---|
1225 | |
---|
1226 | // L = L, lead(N); attrib(basering, "InducionLeads", L); |
---|
1227 | |
---|
1228 | if( @DEBUG ) |
---|
1229 | { |
---|
1230 | "SSstep::NextSyzOutput: "; |
---|
1231 | N; |
---|
1232 | // DetailedPrint(N); |
---|
1233 | attrib(N); |
---|
1234 | } |
---|
1235 | |
---|
1236 | } |
---|
1237 | |
---|
1238 | proc SScontinue(int l) |
---|
1239 | "USAGE: SScontinue(l) |
---|
1240 | RETURN: nothing, instead it changes RES and MRES variables in the current ring |
---|
1241 | PURPOSE: computes further (at most l) syzygies |
---|
1242 | NOTE: must be used within a ring returned by Sres or Ssyz. RES and MRES are |
---|
1243 | explained in Sres |
---|
1244 | EXAMPLE: example Scontinue; shows an example |
---|
1245 | " |
---|
1246 | { |
---|
1247 | |
---|
1248 | /// TODO! |
---|
1249 | // def data = GetInducedData(); |
---|
1250 | |
---|
1251 | if( (!defined(RES)) || (!defined(MRES)) ) /* || (typeof(data) != "list") || (size(data) != 2) */ |
---|
1252 | { |
---|
1253 | ERROR("Sorry, but basering does not seem to be returned by Sres or Ssyz"); |
---|
1254 | } |
---|
1255 | for (; (l != 0) && (size(RES[size(RES)]) > 0); l-- ) |
---|
1256 | { |
---|
1257 | SSstep(); |
---|
1258 | } |
---|
1259 | } |
---|
1260 | example |
---|
1261 | { "EXAMPLE:"; echo = 2; |
---|
1262 | ring r; |
---|
1263 | module M = maxideal(1); M; |
---|
1264 | def S = SSsyz(M); setring S; S; |
---|
1265 | "Only the first syzygy: "; |
---|
1266 | RES; MRES; |
---|
1267 | "More syzygies: "; |
---|
1268 | SScontinue(10); |
---|
1269 | RES; MRES; |
---|
1270 | } |
---|
1271 | |
---|
1272 | proc SSsyz(def M) |
---|
1273 | "USAGE: SSsyz(M) |
---|
1274 | RETURN: ring, containing a list of modules RES and a module MRES |
---|
1275 | PURPOSE: computes the first syzygy module of M (wrt some Schreyer ordering)? |
---|
1276 | NOTE: The output is explained in Sres |
---|
1277 | EXAMPLE: example Ssyz; shows an example |
---|
1278 | " |
---|
1279 | { |
---|
1280 | if( (typeof(M) != "module") && (typeof(M) != "ideal") ) |
---|
1281 | { |
---|
1282 | ERROR("Sorry: need an ideal or a module for input"); |
---|
1283 | } |
---|
1284 | |
---|
1285 | def SS = SSinit(M); setring SS; |
---|
1286 | |
---|
1287 | SSstep(); // NOTE: what if M is zero? |
---|
1288 | |
---|
1289 | return (SS); |
---|
1290 | } |
---|
1291 | example |
---|
1292 | { "EXAMPLE:"; echo = 2; |
---|
1293 | ring r; |
---|
1294 | |
---|
1295 | /* ideal M = 0; |
---|
1296 | def S = SSsyz(M); setring S; S; |
---|
1297 | "Only the first syzygy: "; |
---|
1298 | RES; LRES; TRES; |
---|
1299 | MRES; |
---|
1300 | |
---|
1301 | kill S; setring r; kill M; |
---|
1302 | */ |
---|
1303 | |
---|
1304 | module M = maxideal(1); M; |
---|
1305 | def S = SSres(M, 0); setring S; S; |
---|
1306 | MRES; |
---|
1307 | RES; |
---|
1308 | ""; |
---|
1309 | LRES; |
---|
1310 | ""; |
---|
1311 | TRES; |
---|
1312 | |
---|
1313 | kill S; setring r; kill M; |
---|
1314 | |
---|
1315 | kill r; |
---|
1316 | |
---|
1317 | ring R = 0, (w, x, y, z), dp; |
---|
1318 | ideal M = w^2 - x*z, w*x - y*z, x^2 - w*y, x*y - z^2, y^2 - w*z; |
---|
1319 | |
---|
1320 | def S = SSres(M, 0); setring S; S; |
---|
1321 | MRES; |
---|
1322 | RES; |
---|
1323 | ""; |
---|
1324 | LRES; |
---|
1325 | ""; |
---|
1326 | TRES; |
---|
1327 | } |
---|
1328 | |
---|
1329 | proc SSres(def M, int l) |
---|
1330 | "USAGE: SSres(I, l) |
---|
1331 | RETURN: ring, containing a list of modules RES and a module MRES |
---|
1332 | PURPOSE: computes (at most l) syzygy modules of M wrt the classical Schreyer |
---|
1333 | induced ordering with gen(i) > gen(j) if i > j, provided both gens |
---|
1334 | are from the same syzygy level.??? |
---|
1335 | NOTE: RES contains the images of maps subsituting the beginning of the |
---|
1336 | Schreyer free resolution of baseRing^r/M, while MRES is a sum of |
---|
1337 | these images in a big free sum, containing all the syzygy modules. |
---|
1338 | The syzygy modules are shifted so that gen(i) correspons to MRES[i]. |
---|
1339 | The leading zero module RES[0] indicates the fact that coker of the |
---|
1340 | first map is zero. The number of zeroes inducates the rank of input. |
---|
1341 | NOTE: If l == 0 then l is set to be nvars(basering) + 1 |
---|
1342 | EXAMPLE: example SSres; shows an example |
---|
1343 | " |
---|
1344 | { |
---|
1345 | if( (typeof(M) != "module") && (typeof(M) != "ideal") ) |
---|
1346 | { |
---|
1347 | ERROR("Sorry: need an ideal or a module for input"); |
---|
1348 | } |
---|
1349 | |
---|
1350 | def SS = SSinit(M); setring SS; |
---|
1351 | |
---|
1352 | if (l == 0) |
---|
1353 | { |
---|
1354 | l = nvars(basering) + 1; // not really an estimate...?! |
---|
1355 | } |
---|
1356 | |
---|
1357 | SSstep(); l = l - 1; |
---|
1358 | |
---|
1359 | SScontinue(l); |
---|
1360 | |
---|
1361 | return (SS); |
---|
1362 | } |
---|
1363 | example |
---|
1364 | { "EXAMPLE:"; echo = 2; |
---|
1365 | ring r; |
---|
1366 | module M = maxideal(1); M; |
---|
1367 | def S = SSres(M, 0); setring S; S; |
---|
1368 | RES; |
---|
1369 | MRES; |
---|
1370 | kill S; |
---|
1371 | setring r; kill M; |
---|
1372 | |
---|
1373 | def A = nc_algebra(-1,0); setring A; |
---|
1374 | ideal Q = var(1)^2, var(2)^2, var(3)^2; |
---|
1375 | qring SCA = twostd(Q); |
---|
1376 | basering; |
---|
1377 | |
---|
1378 | module M = maxideal(1); |
---|
1379 | def S = SSres(M, 2); setring S; S; |
---|
1380 | RES; |
---|
1381 | MRES; |
---|
1382 | } |
---|
1383 | |
---|
1384 | |
---|
1385 | |
---|
1386 | static proc loadme() |
---|
1387 | { |
---|
1388 | int @DEBUG = !system("with", "ndebug"); |
---|
1389 | |
---|
1390 | if( @DEBUG ) |
---|
1391 | { |
---|
1392 | |
---|
1393 | "ndebug?: ", system("with", "ndebug"); |
---|
1394 | "om_ndebug?: ", system("with", "om_ndebug"); |
---|
1395 | |
---|
1396 | listvar(Top); |
---|
1397 | listvar(Schreyer); |
---|
1398 | } |
---|
1399 | // listvar(Syzextra); listvar(Syzextra_g); |
---|
1400 | |
---|
1401 | if( !defined(DetailedPrint) ) |
---|
1402 | { |
---|
1403 | if( 1 ) |
---|
1404 | { |
---|
1405 | |
---|
1406 | if( @DEBUG ) |
---|
1407 | { |
---|
1408 | "Loading the Release version!"; |
---|
1409 | } |
---|
1410 | load("syzextra.so"); |
---|
1411 | |
---|
1412 | if( @DEBUG ) |
---|
1413 | { |
---|
1414 | listvar(Syzextra); |
---|
1415 | } |
---|
1416 | |
---|
1417 | exportto(Top, Syzextra::ClearContent); |
---|
1418 | exportto(Top, Syzextra::ClearDenominators); |
---|
1419 | |
---|
1420 | // export Syzextra; |
---|
1421 | |
---|
1422 | // exportto(Schreyer, Syzextra::noop); |
---|
1423 | exportto(Schreyer, Syzextra::DetailedPrint); |
---|
1424 | exportto(Schreyer, Syzextra::leadmonomial); |
---|
1425 | exportto(Schreyer, Syzextra::leadcomp); |
---|
1426 | // exportto(Schreyer, Syzextra::leadrawexp); |
---|
1427 | // exportto(Schreyer, Syzextra::ISUpdateComponents); |
---|
1428 | exportto(Schreyer, Syzextra::SetInducedReferrence); |
---|
1429 | exportto(Schreyer, Syzextra::GetInducedData); |
---|
1430 | // exportto(Schreyer, Syzextra::GetAMData); |
---|
1431 | // exportto(Schreyer, Syzextra::SetSyzComp); |
---|
1432 | exportto(Schreyer, Syzextra::MakeInducedSchreyerOrdering); |
---|
1433 | // exportto(Schreyer, Syzextra::MakeSyzCompOrdering); |
---|
1434 | exportto(Schreyer, Syzextra::idPrepare); |
---|
1435 | // exportto(Schreyer, Syzextra::reduce_syz); |
---|
1436 | // exportto(Schreyer, Syzextra::p_Content); |
---|
1437 | |
---|
1438 | exportto(Schreyer, Syzextra::ProfilerStart); exportto(Schreyer, Syzextra::ProfilerStop); |
---|
1439 | |
---|
1440 | exportto(Schreyer, Syzextra::Tail); |
---|
1441 | |
---|
1442 | exportto(Schreyer, Syzextra::m2_end); |
---|
1443 | } |
---|
1444 | /* |
---|
1445 | else |
---|
1446 | { |
---|
1447 | if( @DEBUG ) |
---|
1448 | { |
---|
1449 | "Loading the Debug version!"; |
---|
1450 | } |
---|
1451 | |
---|
1452 | load("syzextra.so"); |
---|
1453 | |
---|
1454 | if( @DEBUG ) |
---|
1455 | { |
---|
1456 | listvar(Syzextra_g); |
---|
1457 | } |
---|
1458 | |
---|
1459 | exportto(Top, Syzextra_g::ClearContent); |
---|
1460 | exportto(Top, Syzextra_g::ClearDenominators); |
---|
1461 | |
---|
1462 | // export Syzextra_g; |
---|
1463 | // exportto(Schreyer, Syzextra_g::noop); |
---|
1464 | exportto(Schreyer, Syzextra_g::DetailedPrint); |
---|
1465 | exportto(Schreyer, Syzextra_g::leadmonomial); |
---|
1466 | exportto(Schreyer, Syzextra_g::leadcomp); |
---|
1467 | // exportto(Schreyer, Syzextra_g::leadrawexp); |
---|
1468 | // exportto(Schreyer, Syzextra_g::ISUpdateComponents); |
---|
1469 | exportto(Schreyer, Syzextra_g::SetInducedReferrence); |
---|
1470 | exportto(Schreyer, Syzextra_g::GetInducedData); |
---|
1471 | // exportto(Schreyer, Syzextra_g::GetAMData); |
---|
1472 | // exportto(Schreyer, Syzextra_g::SetSyzComp); |
---|
1473 | exportto(Schreyer, Syzextra_g::MakeInducedSchreyerOrdering); |
---|
1474 | // exportto(Schreyer, Syzextra_g::MakeSyzCompOrdering); |
---|
1475 | exportto(Schreyer, Syzextra_g::idPrepare); |
---|
1476 | // exportto(Schreyer, Syzextra_g::reduce_syz); |
---|
1477 | // exportto(Schreyer, Syzextra_g::p_Content); |
---|
1478 | |
---|
1479 | exportto(Schreyer, Syzextra_g::ProfilerStart); exportto(Schreyer, Syzextra_g::ProfilerStop); |
---|
1480 | |
---|
1481 | exportto(Schreyer, Syzextra_g::Tail); |
---|
1482 | |
---|
1483 | |
---|
1484 | exportto(Schreyer, Syzextra_g::m2_end); |
---|
1485 | } |
---|
1486 | */ |
---|
1487 | |
---|
1488 | exportto(Top, DetailedPrint); |
---|
1489 | exportto(Top, GetInducedData); |
---|
1490 | |
---|
1491 | if( @DEBUG ) |
---|
1492 | { |
---|
1493 | listvar(Top); |
---|
1494 | listvar(Schreyer); |
---|
1495 | } |
---|
1496 | } |
---|
1497 | |
---|
1498 | if( !defined(GetInducedData) ) |
---|
1499 | { |
---|
1500 | ERROR("Sorry but we are missing the dynamic module (syzextra.so)..."); |
---|
1501 | } |
---|
1502 | |
---|
1503 | } |
---|
1504 | |
---|
1505 | static proc mod_init() |
---|
1506 | { |
---|
1507 | loadme(); |
---|
1508 | } |
---|
1509 | |
---|
1510 | |
---|
1511 | proc testallSexamples() |
---|
1512 | { |
---|
1513 | example Ssyz; |
---|
1514 | example Scontinue; |
---|
1515 | example Sres; |
---|
1516 | } |
---|
1517 | |
---|
1518 | proc testallSSexamples() |
---|
1519 | { |
---|
1520 | example SSsyz; |
---|
1521 | example SScontinue; |
---|
1522 | example SSres; |
---|
1523 | } |
---|
1524 | |
---|
1525 | example |
---|
1526 | { "EXAMPLE:"; echo = 2; |
---|
1527 | testallSexamples(); |
---|
1528 | testallSSexamples(); |
---|
1529 | } |
---|