1 | ////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id$"; |
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3 | category="General purpose"; |
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4 | info=" |
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5 | LIBRARY: schreyer.lib Helpers for working with the Schreyer induced ordering |
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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,l) Schreyer resolution of module M of maximal length l |
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10 | Ssyz(M) Schreyer resolution of module M of length 1 |
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11 | Scontinue(l) continue the resolution computation by most l steps |
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12 | |
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13 | KEYWORDS: syzygy; Schreyer induced ordering; Schreyer free resolution |
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14 | NOTE: requires the dynamic 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 | attrib(M, "isHomog", @V); |
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147 | attrib(M, "isSB", 1); |
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148 | |
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149 | |
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150 | if( @DEBUG ) |
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151 | { |
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152 | "Sinit::SB_Input: "; |
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153 | type(M); |
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154 | attrib(M); |
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155 | attrib(M, "isHomog"); |
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156 | DetailedPrint(M); |
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157 | } |
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158 | |
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159 | // 0^th syz. property |
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160 | if( size(module(transpose( transpose(M) * transpose(MRES) ))) > 0 ) |
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161 | { |
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162 | transpose( transpose(M) * transpose(MRES) ); |
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163 | "transpose( transpose(M) * transpose(MRES) ) != 0!!!"; |
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164 | $ |
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165 | } |
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166 | |
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167 | RES[size(RES)+1] = M; // list of all syzygy modules |
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168 | MRES = MRES, M; |
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169 | |
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170 | attrib(MRES, "isHomog", @V); |
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171 | |
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172 | attrib(S, "InducionLeads", lead(M)); |
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173 | attrib(S, "InducionStart", @RANK); |
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174 | |
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175 | if( @DEBUG ) |
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176 | { |
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177 | "Sinit::MRES"; |
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178 | DetailedPrint(MRES); |
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179 | attrib(MRES, "isHomog"); |
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180 | attrib(S); |
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181 | } |
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182 | |
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183 | export RES; |
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184 | export MRES; |
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185 | return (S); |
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186 | } |
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187 | |
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188 | static proc Sstep() |
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189 | { |
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190 | int @DEBUG = !system("with", "ndebug"); |
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191 | |
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192 | if( @DEBUG ) |
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193 | { |
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194 | "Sstep::NextInducedRing"; |
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195 | DetailedPrint(basering); |
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196 | |
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197 | attrib(basering, "InducionLeads"); |
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198 | attrib(basering, "InducionStart"); |
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199 | |
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200 | GetInducedData(); |
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201 | } |
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202 | |
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203 | // syzygy step: |
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204 | |
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205 | /* |
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206 | // is initial weights are all zeroes! |
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207 | def L = lead(M); |
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208 | intvec @V = deg(M[1..ncols(M)]); @W; @V; @W = @V; attrib(L, "isHomog", @W); |
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209 | SetInducedReferrence(L, @RANK, 0); |
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210 | */ |
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211 | |
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212 | // def L = lead(MRES); |
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213 | // @W = @W, @V; |
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214 | // attrib(L, "isHomog", @W); |
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215 | |
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216 | |
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217 | // General setting: |
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218 | // SetInducedReferrence(MRES, 0, 0); // limit: 0! |
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219 | int @l = size(RES); |
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220 | |
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221 | module M = RES[@l]; |
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222 | |
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223 | module L = attrib(basering, "InducionLeads"); |
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224 | int limit = attrib(basering, "InducionStart"); |
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225 | |
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226 | // L; limit; |
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227 | |
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228 | int @RANK = ncols(MRES) - ncols(M); // nrows(M); // what if M is zero?! |
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229 | |
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230 | /* |
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231 | if( @RANK != nrows(M) ) |
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232 | { |
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233 | type(MRES); |
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234 | @RANK; |
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235 | type(M); |
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236 | pause(); |
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237 | } |
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238 | */ |
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239 | |
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240 | intvec @W = attrib(M, "isHomog"); |
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241 | intvec @V = deg(M[1..ncols(M)]); |
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242 | @V = @W, @V; |
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243 | |
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244 | if( @DEBUG ) |
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245 | { |
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246 | "Sstep::NextInput: "; |
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247 | M; |
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248 | @V; |
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249 | @RANK; |
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250 | DetailedPrint(MRES); |
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251 | attrib(MRES, "isHomog"); |
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252 | } |
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253 | |
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254 | |
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255 | |
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256 | SetInducedReferrence(L, limit, 0); |
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257 | |
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258 | def K = prepareSyz(M, @RANK); |
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259 | // K; |
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260 | |
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261 | // attrib(K, "isHomog", @V); DetailedPrint(K, 1000); |
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262 | |
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263 | // pause(); |
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264 | |
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265 | K = idPrepare(K, @RANK); // std(K); // ? |
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266 | K = simplify(K, 2); |
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267 | |
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268 | // K; |
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269 | |
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270 | module N = separateSyzGB(K, @RANK)[2]; // 1^st syz. module: vectors which start in lower part (comp >= @RANK) |
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271 | attrib(N, "isHomog", @V); |
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272 | |
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273 | // "N_0: "; N; DetailedPrint(N, 10); |
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274 | |
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275 | N = std(N); // TODO: fix "wrong weights"!!!? |
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276 | attrib(N, "isHomog", @V); |
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277 | |
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278 | // N; |
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279 | |
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280 | if( size(N) > 0 ) |
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281 | { |
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282 | // next syz. property |
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283 | if( size(module(transpose( transpose(N) * transpose(MRES) ))) > 0 ) |
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284 | { |
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285 | MRES; |
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286 | |
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287 | "N: "; N; DetailedPrint(N, 10); |
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288 | |
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289 | "K:"; K; DetailedPrint(K, 10); |
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290 | |
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291 | "RANKS: ", @RANK; |
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292 | |
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293 | "transpose( transpose(N) * transpose(MRES) ) != 0!!!"; |
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294 | transpose( transpose(N) * transpose(MRES) ); |
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295 | |
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296 | "transpose(N) * transpose(MRES): "; |
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297 | transpose(N) * transpose(MRES); |
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298 | DetailedPrint(module(_), 2); |
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299 | $ |
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300 | } |
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301 | } |
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302 | |
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303 | RES[@l + 1] = N; // list of all syzygy modules |
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304 | |
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305 | MRES = MRES, N; |
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306 | attrib(MRES, "isHomog", @V); |
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307 | |
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308 | |
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309 | L = L, lead(N); |
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310 | attrib(basering, "InducionLeads", L); |
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311 | |
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312 | if( @DEBUG ) |
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313 | { |
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314 | "Sstep::NextSyzOutput: "; |
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315 | DetailedPrint(N); |
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316 | attrib(N, "isHomog"); |
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317 | } |
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318 | |
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319 | } |
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320 | |
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321 | proc Scontinue(int l) |
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322 | "USAGE: Scontinue(l) |
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323 | RETURN: nothing, instead it changes RES and MRES variables in the current ring |
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324 | PURPOSE: computes further (at most l) syzygies |
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325 | NOTE: must be used within a ring returned by Sres or Ssyz. RES and MRES are |
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326 | explained in Sres |
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327 | EXAMPLE: example Scontinue; shows an example |
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328 | " |
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329 | { |
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330 | def data = GetInducedData(); |
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331 | |
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332 | if( (!defined(RES)) || (!defined(MRES)) || (typeof(data) != "list") || (size(data) != 2) ) |
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333 | { |
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334 | ERROR("Sorry, but basering does not seem to be returned by Sres or Ssyz"); |
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335 | } |
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336 | for (; (l != 0) && (size(RES[size(RES)]) > 0); l-- ) |
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337 | { |
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338 | Sstep(); |
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339 | } |
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340 | } |
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341 | example |
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342 | { "EXAMPLE:"; echo = 2; |
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343 | ring r; |
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344 | module M = maxideal(1); M; |
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345 | def S = Ssyz(M); setring S; S; |
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346 | "Only the first syzygy: "; |
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347 | RES; MRES; |
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348 | "More syzygies: "; |
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349 | Scontinue(10); |
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350 | RES; MRES; |
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351 | } |
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352 | |
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353 | proc Ssyz(module M) |
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354 | "USAGE: Ssyz(M) |
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355 | RETURN: ring, containing a list of modules RES and a module MRES |
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356 | PURPOSE: computes the first syzygy module of M (wrt some Schreyer ordering) |
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357 | NOTE: The output is explained in Sres |
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358 | EXAMPLE: example Ssyz; shows an example |
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359 | " |
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360 | { |
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361 | def S = Sinit(M); setring S; |
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362 | |
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363 | Sstep(); // NOTE: what if M is zero? |
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364 | |
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365 | return (S); |
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366 | } |
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367 | example |
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368 | { "EXAMPLE:"; echo = 2; |
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369 | ring r; |
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370 | module M = maxideal(1); M; |
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371 | def S = Ssyz(M); setring S; S; |
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372 | "Only the first syzygy: "; |
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373 | RES; |
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374 | MRES; // Note gen(i) |
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375 | kill S; |
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376 | setring r; kill M; |
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377 | |
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378 | module M = 0; |
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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; |
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383 | } |
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384 | |
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385 | proc Sres(module M, int l) |
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386 | "USAGE: Sres(M, l) |
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387 | RETURN: ring, containing a list of modules RES and a module MRES |
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388 | PURPOSE: computes (at most l) syzygy modules of M wrt the classical Schreyer |
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389 | induced ordering with gen(i) > gen(j) if i > j, provided both gens |
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390 | are from the same syzygy level. |
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391 | NOTE: RES contains the images of maps subsituting the beginning of the |
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392 | Schreyer free resolution of baseRing^r/M, while MRES is a sum of |
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393 | these images in a big free sum, containing all the syzygy modules. |
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394 | The syzygy modules are shifted so that gen(i) correspons to MRES[i]. |
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395 | The leading zero module RES[0] indicates the fact that coker of the |
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396 | first map is zero. The number of zeroes inducates the rank of input. |
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397 | NOTE: If l == 0 then l is set to be nvars(basering) + 1 |
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398 | EXAMPLE: example Sres; shows an example |
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399 | " |
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400 | { |
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401 | def S = Sinit(M); setring S; |
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402 | |
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403 | if (l == 0) |
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404 | { |
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405 | l = nvars(basering) + 1; // not really an estimate...?! |
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406 | } |
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407 | |
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408 | Sstep(); l = l - 1; |
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409 | |
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410 | Scontinue(l); |
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411 | |
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412 | return (S); |
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413 | } |
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414 | example |
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415 | { "EXAMPLE:"; echo = 2; |
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416 | ring r; |
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417 | module M = maxideal(1); M; |
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418 | def S = Sres(M, 0); setring S; S; |
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419 | RES; |
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420 | MRES; |
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421 | kill S; |
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422 | setring r; kill M; |
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423 | |
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424 | def A = nc_algebra(-1,0); setring A; |
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425 | ideal Q = var(1)^2, var(2)^2, var(3)^2; |
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426 | qring SCA = twostd(Q); |
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427 | basering; |
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428 | |
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429 | module M = maxideal(1); |
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430 | def S = Sres(M, 2); setring S; S; |
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431 | RES; |
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432 | MRES; |
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433 | } |
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434 | |
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435 | |
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436 | |
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437 | // ================================================================== // |
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438 | |
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439 | |
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440 | LIB "general.lib"; // for sort |
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441 | |
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442 | // TODO: in C++! |
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443 | static proc Tail(def M) |
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444 | { |
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445 | int i = ncols(M); def m; |
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446 | while (i > 0) |
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447 | { |
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448 | m = M[i]; |
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449 | |
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450 | m = m - lead(m); // m = tail(m) |
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451 | |
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452 | M[i] = m; |
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453 | |
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454 | i--; |
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455 | } |
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456 | return (M); |
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457 | } |
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458 | |
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459 | /* static */ proc SSinit(module M) |
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460 | { |
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461 | if( (typeof(M) != "module") && (typeof(M) != "ideal") ) |
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462 | { |
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463 | ERROR("Sorry: need an ideal or a module for input"); |
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464 | } |
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465 | |
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466 | // TODO! DONE? |
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467 | def @save = basering; |
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468 | |
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469 | int @DEBUG = !system("with", "ndebug"); |
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470 | if( @DEBUG ) |
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471 | { |
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472 | "SSinit::Input"; |
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473 | type(M); |
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474 | DetailedPrint(M); |
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475 | attrib(M); |
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476 | } |
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477 | |
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478 | int @RANK = nrows(M); int @SIZE = ncols(M); |
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479 | |
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480 | int @IS_A_SB = attrib(M, "isSB"); // ??? only if all weights were zero?! |
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481 | |
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482 | if( !@IS_A_SB ) |
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483 | { |
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484 | def opts = option(get); |
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485 | option(redSB); option(redTail); |
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486 | M = std(M); |
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487 | option(set, opts); |
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488 | kill opts; |
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489 | } else |
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490 | { |
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491 | M = simplify(M, 2 + 4 + 32); |
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492 | } |
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493 | |
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494 | def LEAD = lead(M); |
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495 | |
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496 | // sort wrt neg.deg.rev.lex! |
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497 | intvec iv_ds = sort(LEAD, "ds", 1)[2]; // ,1 => reversed! |
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498 | |
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499 | M = M[iv_ds]; // sort M wrt ds on current leading terms |
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500 | LEAD = LEAD[iv_ds]; |
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501 | |
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502 | def TAIL = Tail(M); |
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503 | |
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504 | intvec @DEGS = deg(M[1..@SIZE]); // store actuall degrees of input elements |
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505 | |
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506 | // TODO: what about real modules? weighted ones? |
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507 | |
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508 | list @l = ringlist(@save); |
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509 | |
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510 | int @z = 0; ideal @m = maxideal(1); intvec @wdeg = deg(@m[1..ncols(@m)]); |
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511 | |
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512 | // NOTE: @wdeg will be ignored anyway :( |
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513 | @l[3] = list(list("C", @z), list("lp", @wdeg)); |
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514 | |
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515 | kill @z, @wdeg; // since these vars are ring independent! |
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516 | |
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517 | def S = ring(@l); // --MakeInducedSchreyerOrdering(1); |
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518 | |
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519 | module F = freemodule(@RANK); |
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520 | intvec @V = deg(F[1..@RANK]); |
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521 | |
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522 | setring S; // ring with an easy divisibility test ("C, lex") |
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523 | |
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524 | if( @DEBUG ) |
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525 | { |
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526 | "SSinit::StartingISRing"; |
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527 | basering; |
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528 | // DetailedPrint(basering); |
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529 | } |
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530 | |
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531 | // Setup the leading syzygy^{-1} module to zero: |
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532 | module Z = 0; Z[@RANK] = 0; attrib(Z, "isHomog", intvec(0)); |
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533 | |
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534 | module MRES = Z; |
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535 | |
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536 | list RES; RES[1] = Z; |
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537 | list LRES; LRES[1] = Z; |
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538 | list TRES; TRES[1] = Z; |
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539 | |
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540 | def M = imap(@save, M); |
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541 | attrib(M, "isHomog", @V); |
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542 | attrib(M, "isSB", 1); |
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543 | |
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544 | attrib(M, "degrees", @DEGS); |
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545 | |
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546 | def LEAD = imap(@save, LEAD); |
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547 | attrib(LEAD, "isHomog", @V); |
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548 | attrib(LEAD, "isSB", 1); |
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549 | |
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550 | def TAIL = imap(@save, TAIL); |
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551 | |
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552 | if( @DEBUG ) |
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553 | { |
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554 | "SSinit::(sorted) SB_Input: "; |
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555 | type(M); |
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556 | attrib(M); |
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557 | attrib(M, "isHomog"); |
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558 | DetailedPrint(M); |
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559 | } |
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560 | |
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561 | // 0^th syz. property |
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562 | if( size(module(transpose( transpose(M) * transpose(MRES) ))) > 0 ) |
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563 | { |
---|
564 | transpose( transpose(M) * transpose(MRES) ); |
---|
565 | "ERROR: transpose( transpose(M) * transpose(MRES) ) != 0!!!"; |
---|
566 | $ |
---|
567 | } |
---|
568 | |
---|
569 | RES[size(RES)+1] = M; // list of all syzygy modules |
---|
570 | LRES[size(LRES)+1] = LEAD; // list of all syzygy modules |
---|
571 | TRES[size(TRES)+1] = TAIL; // list of all syzygy modules |
---|
572 | |
---|
573 | MRES = MRES, M; //? |
---|
574 | |
---|
575 | attrib(MRES, "isHomog", @V); |
---|
576 | |
---|
577 | attrib(S, "InducionStart", @RANK); |
---|
578 | |
---|
579 | if( @DEBUG ) |
---|
580 | { |
---|
581 | "SSinit::MRES"; |
---|
582 | DetailedPrint(MRES); |
---|
583 | attrib(MRES, "isHomog"); |
---|
584 | attrib(S); |
---|
585 | } |
---|
586 | |
---|
587 | export RES; |
---|
588 | export MRES; |
---|
589 | export LRES; |
---|
590 | export TRES; |
---|
591 | return (S); |
---|
592 | } |
---|
593 | example |
---|
594 | { "EXAMPLE:"; echo = 2; |
---|
595 | ring R = 0, (w, x, y, z), dp; |
---|
596 | |
---|
597 | def M = maxideal(1); |
---|
598 | def S = SSinit(M); setring S; S; |
---|
599 | |
---|
600 | "Only the first initialization: "; |
---|
601 | RES; LRES; TRES; |
---|
602 | MRES; |
---|
603 | |
---|
604 | kill S; setring R; kill M; |
---|
605 | |
---|
606 | ideal M = w^2 - x*z, w*x - y*z, x^2 - w*y, x*y - z^2, y^2 - w*z; |
---|
607 | def S = SSinit(M); setring S; S; |
---|
608 | |
---|
609 | "Only the first initialization: "; |
---|
610 | RES; LRES; TRES; |
---|
611 | MRES; |
---|
612 | |
---|
613 | kill S; setring R; kill M; |
---|
614 | } |
---|
615 | |
---|
616 | |
---|
617 | LIB "poly.lib"; // for lcm |
---|
618 | |
---|
619 | |
---|
620 | proc SSComputeLeadingSyzygyTerms(def L, int iCompShift) |
---|
621 | { |
---|
622 | int @DEBUG = !system("with", "ndebug"); |
---|
623 | |
---|
624 | if( @DEBUG ) |
---|
625 | { |
---|
626 | "SSComputeLeadingSyzygyTerms::Input: "; |
---|
627 | L; |
---|
628 | "iCompShift: ", iCompShift; |
---|
629 | |
---|
630 | } |
---|
631 | |
---|
632 | int i, j, r; intvec iv_ds; |
---|
633 | int N = ncols(L); |
---|
634 | def a, b; |
---|
635 | poly aa, bb; |
---|
636 | |
---|
637 | bigint c; |
---|
638 | |
---|
639 | ideal M; |
---|
640 | |
---|
641 | module S = 0; |
---|
642 | |
---|
643 | for(i = 1; i <= N; i++) |
---|
644 | { |
---|
645 | a = L[i]; |
---|
646 | "a: ", a; |
---|
647 | c = leadcomp(a); |
---|
648 | r = int(c); |
---|
649 | aa = a[r]; |
---|
650 | |
---|
651 | M = 0; |
---|
652 | |
---|
653 | for(j = i-1; j > 0; j--) |
---|
654 | { |
---|
655 | b = L[j]; |
---|
656 | "b: ", b; |
---|
657 | |
---|
658 | if( leadcomp(b) == c ) |
---|
659 | { |
---|
660 | bb = b[r]; |
---|
661 | |
---|
662 | M[j] = (lcm(aa, bb) / aa); |
---|
663 | } |
---|
664 | } |
---|
665 | |
---|
666 | M = simplify(M, 1 + 2 + 32); |
---|
667 | |
---|
668 | iv_ds = sort(M, "ds", 1)[2]; // ,1 => reversed! |
---|
669 | |
---|
670 | M = M[iv_ds]; |
---|
671 | |
---|
672 | S = S, M * gen(i + iCompShift); |
---|
673 | } |
---|
674 | |
---|
675 | S = simplify(S, 2); |
---|
676 | |
---|
677 | |
---|
678 | if( @DEBUG ) |
---|
679 | { |
---|
680 | "SSComputeLeadingSyzygyTerms::Output: "; |
---|
681 | S; |
---|
682 | } |
---|
683 | |
---|
684 | return (S); |
---|
685 | |
---|
686 | } |
---|
687 | |
---|
688 | proc Traverse(mult, index, Leads, Tails, LSyz) |
---|
689 | { |
---|
690 | |
---|
691 | |
---|
692 | } |
---|
693 | |
---|
694 | |
---|
695 | |
---|
696 | // module (N, LL, TT) = SSComputeSyzygy(L, T, @RANK); // shift syz.comp by @RANK! |
---|
697 | proc SSComputeSyzygy(def M, def L, def T, int iCompShift) |
---|
698 | { |
---|
699 | int @DEBUG = !system("with", "ndebug"); |
---|
700 | |
---|
701 | if( @DEBUG ) |
---|
702 | { |
---|
703 | "SSComputeSyzygy::Input"; |
---|
704 | "basering: ", basering; attrib(basering); |
---|
705 | // DetailedPrint(basering); |
---|
706 | |
---|
707 | "iCompShift: ", iCompShift; |
---|
708 | |
---|
709 | "M: "; M; |
---|
710 | "L: "; L; |
---|
711 | "T: "; T; |
---|
712 | } |
---|
713 | |
---|
714 | |
---|
715 | // !!!!!!!!!!!!!!!!!!!!!!!!!!!! TODO !!!!!!!!!!!!!!!!!!!!!!!!!!!! // |
---|
716 | // !!!!!!!!!!!!!!!!!!!!!!!!!!!! TODO !!!!!!!!!!!!!!!!!!!!!!!!!!!! // |
---|
717 | // !!!!!!!!!!!!!!!!!!!!!!!!!!!! TODO !!!!!!!!!!!!!!!!!!!!!!!!!!!! // |
---|
718 | // !!!!!!!!!!!!!!!!!!!!!!!!!!!! TODO !!!!!!!!!!!!!!!!!!!!!!!!!!!! // |
---|
719 | // !!!!!!!!!!!!!!!!!!!!!!!!!!!! TODO !!!!!!!!!!!!!!!!!!!!!!!!!!!! // |
---|
720 | // !!!!!!!!!!!!!!!!!!!!!!!!!!!! TODO !!!!!!!!!!!!!!!!!!!!!!!!!!!! // |
---|
721 | // !!!!!!!!!!!!!!!!!!!!!!!!!!!! TODO !!!!!!!!!!!!!!!!!!!!!!!!!!!! // |
---|
722 | |
---|
723 | def a; bigint c; int r, k; poly aa; |
---|
724 | |
---|
725 | def LS = SSComputeLeadingSyzygyTerms(L, 0); // iCompShift // 0? |
---|
726 | for(k = ncols(LS); k >= 0; k-- ) |
---|
727 | { |
---|
728 | a = LS[k]; |
---|
729 | "A: ", a; |
---|
730 | c = leadcomp(a); |
---|
731 | r = int(c); |
---|
732 | aa = a[r]; |
---|
733 | |
---|
734 | Traverse(aa, r, L, T, LS); |
---|
735 | } |
---|
736 | |
---|
737 | |
---|
738 | def opts = option(get); option(redSB); option(redTail); |
---|
739 | module SYZ = std(syz(M)); // TODO: !!!!!!!!!!! |
---|
740 | option(set, opts); kill opts; |
---|
741 | |
---|
742 | "SYZ: "; SYZ; print(SYZ); |
---|
743 | |
---|
744 | module Z; Z[iCompShift] = 0; Z = Z, transpose(SYZ); SYZ = transpose(Z); |
---|
745 | |
---|
746 | "shifted SYZ: "; SYZ; print(SYZ); |
---|
747 | |
---|
748 | module LL, TT; |
---|
749 | |
---|
750 | LL = lead(SYZ); // TODO: WRONG ORDERING!!!!!!!! |
---|
751 | TT = Tail(SYZ); |
---|
752 | |
---|
753 | |
---|
754 | // !!!!!!!!!!!!!!!!!!!!!!!!!!!! TODO !!!!!!!!!!!!!!!!!!!!!!!!!!!! // |
---|
755 | // !!!!!!!!!!!!!!!!!!!!!!!!!!!! TODO !!!!!!!!!!!!!!!!!!!!!!!!!!!! // |
---|
756 | // !!!!!!!!!!!!!!!!!!!!!!!!!!!! TODO !!!!!!!!!!!!!!!!!!!!!!!!!!!! // |
---|
757 | // !!!!!!!!!!!!!!!!!!!!!!!!!!!! TODO !!!!!!!!!!!!!!!!!!!!!!!!!!!! // |
---|
758 | // !!!!!!!!!!!!!!!!!!!!!!!!!!!! TODO !!!!!!!!!!!!!!!!!!!!!!!!!!!! // |
---|
759 | // !!!!!!!!!!!!!!!!!!!!!!!!!!!! TODO !!!!!!!!!!!!!!!!!!!!!!!!!!!! // |
---|
760 | // !!!!!!!!!!!!!!!!!!!!!!!!!!!! TODO !!!!!!!!!!!!!!!!!!!!!!!!!!!! // |
---|
761 | |
---|
762 | intvec iv_ds = sort(LL, "ds", 1)[2]; // ,1 => reversed! |
---|
763 | LL = LL[iv_ds]; |
---|
764 | TT = TT[iv_ds]; |
---|
765 | |
---|
766 | if( @DEBUG ) |
---|
767 | { |
---|
768 | "SSComputeSyzygy::Output"; |
---|
769 | |
---|
770 | "SYZ: "; SYZ; |
---|
771 | "LL: "; LL; |
---|
772 | "TT: "; TT; |
---|
773 | } |
---|
774 | |
---|
775 | return (SYZ, LL, TT); |
---|
776 | } |
---|
777 | |
---|
778 | // resolution/syzygy step: |
---|
779 | static proc SSstep() |
---|
780 | { |
---|
781 | /// TODO!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! |
---|
782 | int @DEBUG = !system("with", "ndebug"); |
---|
783 | |
---|
784 | if( @DEBUG ) |
---|
785 | { |
---|
786 | "SSstep::NextInducedRing"; |
---|
787 | "basering: ", basering; attrib(basering); |
---|
788 | // DetailedPrint(basering); |
---|
789 | } |
---|
790 | |
---|
791 | /* |
---|
792 | // is initial weights are all zeroes! |
---|
793 | def L = lead(M); |
---|
794 | intvec @V = deg(M[1..ncols(M)]); @W; @V; @W = @V; attrib(L, "isHomog", @W); |
---|
795 | SetInducedReferrence(L, @RANK, 0); |
---|
796 | */ |
---|
797 | |
---|
798 | // def L = lead(MRES); |
---|
799 | // @W = @W, @V; |
---|
800 | // attrib(L, "isHomog", @W); |
---|
801 | |
---|
802 | |
---|
803 | // General setting: |
---|
804 | // SetInducedReferrence(MRES, 0, 0); // limit: 0! |
---|
805 | int @l = size(RES); |
---|
806 | |
---|
807 | def M = RES[@l]; |
---|
808 | |
---|
809 | def L = LRES[@l]; |
---|
810 | def T = TRES[@l]; |
---|
811 | |
---|
812 | |
---|
813 | //// TODO: wrong !!!!! |
---|
814 | int @RANK = ncols(MRES) - ncols(M); // nrows(M); // what if M is zero?! |
---|
815 | |
---|
816 | |
---|
817 | |
---|
818 | /* |
---|
819 | if( @RANK != nrows(M) ) |
---|
820 | { |
---|
821 | type(MRES); |
---|
822 | @RANK; |
---|
823 | type(M); |
---|
824 | pause(); |
---|
825 | } |
---|
826 | */ |
---|
827 | |
---|
828 | intvec @W = attrib(M, "isHomog"); intvec @V = attrib(M, "degrees"); @V = @W, @V; |
---|
829 | |
---|
830 | if( @DEBUG ) |
---|
831 | { |
---|
832 | "Sstep::NextInput: "; |
---|
833 | M; |
---|
834 | L; |
---|
835 | @V; |
---|
836 | @RANK; |
---|
837 | DetailedPrint(MRES); |
---|
838 | attrib(MRES, "isHomog"); |
---|
839 | } |
---|
840 | |
---|
841 | |
---|
842 | // TODO: N = SYZ( M )!!! |
---|
843 | module N, LL, TT; (N, LL, TT) = SSComputeSyzygy(M, L, T, @RANK); // shift syz.comp by @RANK! |
---|
844 | |
---|
845 | attrib(N, "isHomog", @V); |
---|
846 | |
---|
847 | // TODO: correct the following: |
---|
848 | intvec @DEGS = deg(N[1..ncols(N)]); // no mod. comp. weights :( |
---|
849 | attrib(N, "degrees", @DEGS); |
---|
850 | |
---|
851 | if( size(N) > 0 ) |
---|
852 | { |
---|
853 | N; |
---|
854 | MRES; |
---|
855 | // next syz. property |
---|
856 | if( size(module(transpose( transpose(N) * transpose(MRES) ))) > 0 ) |
---|
857 | { |
---|
858 | MRES; |
---|
859 | |
---|
860 | "N: "; N; // DetailedPrint(N, 2); |
---|
861 | |
---|
862 | "LL:"; LL; // DetailedPrint(LL, 1); |
---|
863 | "TT:"; TT; // DetailedPrint(TT, 10); |
---|
864 | |
---|
865 | "RANKS: ", @RANK; |
---|
866 | |
---|
867 | "transpose( transpose(N) * transpose(MRES) ) != 0!!!"; |
---|
868 | transpose( transpose(N) * transpose(MRES) ); |
---|
869 | |
---|
870 | "transpose(N) * transpose(MRES): "; |
---|
871 | transpose(N) * transpose(MRES); |
---|
872 | // DetailedPrint(module(_), 2); |
---|
873 | $ |
---|
874 | } |
---|
875 | } |
---|
876 | |
---|
877 | RES[@l + 1] = N; // list of all syzygy modules |
---|
878 | LRES[@l + 1] = LL; // list of all syzygy modules |
---|
879 | TRES[@l + 1] = TT; // list of all syzygy modules |
---|
880 | |
---|
881 | MRES = MRES, N; |
---|
882 | attrib(MRES, "isHomog", @V); |
---|
883 | |
---|
884 | // L = L, lead(N); attrib(basering, "InducionLeads", L); |
---|
885 | |
---|
886 | if( @DEBUG ) |
---|
887 | { |
---|
888 | "SSstep::NextSyzOutput: "; |
---|
889 | DetailedPrint(N); |
---|
890 | attrib(N); |
---|
891 | } |
---|
892 | |
---|
893 | } |
---|
894 | |
---|
895 | proc SScontinue(int l) |
---|
896 | "USAGE: SScontinue(l) |
---|
897 | RETURN: nothing, instead it changes RES and MRES variables in the current ring |
---|
898 | PURPOSE: computes further (at most l) syzygies |
---|
899 | NOTE: must be used within a ring returned by Sres or Ssyz. RES and MRES are |
---|
900 | explained in Sres |
---|
901 | EXAMPLE: example Scontinue; shows an example |
---|
902 | " |
---|
903 | { |
---|
904 | |
---|
905 | /// TODO! |
---|
906 | // def data = GetInducedData(); |
---|
907 | |
---|
908 | if( (!defined(RES)) || (!defined(MRES)) ) /* || (typeof(data) != "list") || (size(data) != 2) */ |
---|
909 | { |
---|
910 | ERROR("Sorry, but basering does not seem to be returned by Sres or Ssyz"); |
---|
911 | } |
---|
912 | for (; (l != 0) && (size(RES[size(RES)]) > 0); l-- ) |
---|
913 | { |
---|
914 | SSstep(); |
---|
915 | } |
---|
916 | } |
---|
917 | example |
---|
918 | { "EXAMPLE:"; echo = 2; |
---|
919 | ring r; |
---|
920 | module M = maxideal(1); M; |
---|
921 | def S = SSsyz(M); setring S; S; |
---|
922 | "Only the first syzygy: "; |
---|
923 | RES; MRES; |
---|
924 | "More syzygies: "; |
---|
925 | SScontinue(10); |
---|
926 | RES; MRES; |
---|
927 | } |
---|
928 | |
---|
929 | proc SSsyz(def M) |
---|
930 | "USAGE: SSsyz(M) |
---|
931 | RETURN: ring, containing a list of modules RES and a module MRES |
---|
932 | PURPOSE: computes the first syzygy module of M (wrt some Schreyer ordering)? |
---|
933 | NOTE: The output is explained in Sres |
---|
934 | EXAMPLE: example Ssyz; shows an example |
---|
935 | " |
---|
936 | { |
---|
937 | if( (typeof(M) != "module") && (typeof(M) != "ideal") ) |
---|
938 | { |
---|
939 | ERROR("Sorry: need an ideal or a module for input"); |
---|
940 | } |
---|
941 | |
---|
942 | def SS = SSinit(M); setring SS; |
---|
943 | |
---|
944 | SSstep(); // NOTE: what if M is zero? |
---|
945 | |
---|
946 | return (SS); |
---|
947 | } |
---|
948 | example |
---|
949 | { "EXAMPLE:"; echo = 2; |
---|
950 | ring r; |
---|
951 | |
---|
952 | /* ideal M = 0; |
---|
953 | def S = SSsyz(M); setring S; S; |
---|
954 | "Only the first syzygy: "; |
---|
955 | RES; LRES; TRES; |
---|
956 | MRES; |
---|
957 | |
---|
958 | kill S; setring r; kill M; |
---|
959 | */ |
---|
960 | |
---|
961 | module M = maxideal(1); M; |
---|
962 | def S = SSsyz(M); setring S; S; |
---|
963 | "Only the first syzygy: "; |
---|
964 | RES; LRES; TRES; |
---|
965 | MRES; |
---|
966 | |
---|
967 | kill S; setring r; kill M; |
---|
968 | |
---|
969 | kill r; |
---|
970 | |
---|
971 | ring R = 0, (w, x, y, z), dp; |
---|
972 | ideal M = w^2 - x*z, w*x - y*z, x^2 - w*y, x*y - z^2, y^2 - w*z; |
---|
973 | |
---|
974 | def S = SSsyz(M); setring S; S; |
---|
975 | "Only the first syzygy: "; |
---|
976 | RES; LRES; TRES; |
---|
977 | MRES; |
---|
978 | } |
---|
979 | |
---|
980 | proc SSres(def M, int l) |
---|
981 | "USAGE: SSres(I, l) |
---|
982 | RETURN: ring, containing a list of modules RES and a module MRES |
---|
983 | PURPOSE: computes (at most l) syzygy modules of M wrt the classical Schreyer |
---|
984 | induced ordering with gen(i) > gen(j) if i > j, provided both gens |
---|
985 | are from the same syzygy level.??? |
---|
986 | NOTE: RES contains the images of maps subsituting the beginning of the |
---|
987 | Schreyer free resolution of baseRing^r/M, while MRES is a sum of |
---|
988 | these images in a big free sum, containing all the syzygy modules. |
---|
989 | The syzygy modules are shifted so that gen(i) correspons to MRES[i]. |
---|
990 | The leading zero module RES[0] indicates the fact that coker of the |
---|
991 | first map is zero. The number of zeroes inducates the rank of input. |
---|
992 | NOTE: If l == 0 then l is set to be nvars(basering) + 1 |
---|
993 | EXAMPLE: example SSres; shows an example |
---|
994 | " |
---|
995 | { |
---|
996 | if( (typeof(M) != "module") && (typeof(M) != "ideal") ) |
---|
997 | { |
---|
998 | ERROR("Sorry: need an ideal or a module for input"); |
---|
999 | } |
---|
1000 | |
---|
1001 | def SS = SSinit(M); setring SS; |
---|
1002 | |
---|
1003 | if (l == 0) |
---|
1004 | { |
---|
1005 | l = nvars(basering) + 1; // not really an estimate...?! |
---|
1006 | } |
---|
1007 | |
---|
1008 | SSstep(); l = l - 1; |
---|
1009 | |
---|
1010 | SScontinue(l); |
---|
1011 | |
---|
1012 | return (SS); |
---|
1013 | } |
---|
1014 | example |
---|
1015 | { "EXAMPLE:"; echo = 2; |
---|
1016 | ring r; |
---|
1017 | module M = maxideal(1); M; |
---|
1018 | def S = SSres(M, 0); setring S; S; |
---|
1019 | RES; |
---|
1020 | MRES; |
---|
1021 | kill S; |
---|
1022 | setring r; kill M; |
---|
1023 | |
---|
1024 | def A = nc_algebra(-1,0); setring A; |
---|
1025 | ideal Q = var(1)^2, var(2)^2, var(3)^2; |
---|
1026 | qring SCA = twostd(Q); |
---|
1027 | basering; |
---|
1028 | |
---|
1029 | module M = maxideal(1); |
---|
1030 | def S = SSres(M, 2); setring S; S; |
---|
1031 | RES; |
---|
1032 | MRES; |
---|
1033 | } |
---|
1034 | |
---|
1035 | |
---|
1036 | |
---|
1037 | static proc loadme() |
---|
1038 | { |
---|
1039 | int @DEBUG = !system("with", "ndebug"); |
---|
1040 | |
---|
1041 | if( @DEBUG ) |
---|
1042 | { |
---|
1043 | |
---|
1044 | "ndebug?: ", system("with", "ndebug"); |
---|
1045 | "om_ndebug?: ", system("with", "om_ndebug"); |
---|
1046 | |
---|
1047 | listvar(Top); |
---|
1048 | listvar(Schreyer); |
---|
1049 | } |
---|
1050 | // listvar(Syzextra); listvar(Syzextra_g); |
---|
1051 | |
---|
1052 | if( !defined(DetailedPrint) ) |
---|
1053 | { |
---|
1054 | if( !@DEBUG ) |
---|
1055 | { |
---|
1056 | |
---|
1057 | if( @DEBUG ) |
---|
1058 | { |
---|
1059 | "Loading the Release version!"; |
---|
1060 | } |
---|
1061 | load("syzextra.so"); |
---|
1062 | |
---|
1063 | if( @DEBUG ) |
---|
1064 | { |
---|
1065 | listvar(Syzextra); |
---|
1066 | } |
---|
1067 | |
---|
1068 | // export Syzextra; |
---|
1069 | |
---|
1070 | // exportto(Schreyer, Syzextra::noop); |
---|
1071 | exportto(Schreyer, Syzextra::DetailedPrint); |
---|
1072 | // exportto(Schreyer, Syzextra::leadmonom); |
---|
1073 | exportto(Schreyer, Syzextra::leadcomp); |
---|
1074 | // exportto(Schreyer, Syzextra::leadrawexp); |
---|
1075 | // exportto(Schreyer, Syzextra::ISUpdateComponents); |
---|
1076 | exportto(Schreyer, Syzextra::SetInducedReferrence); |
---|
1077 | exportto(Schreyer, Syzextra::GetInducedData); |
---|
1078 | // exportto(Schreyer, Syzextra::GetAMData); |
---|
1079 | // exportto(Schreyer, Syzextra::SetSyzComp); |
---|
1080 | exportto(Schreyer, Syzextra::MakeInducedSchreyerOrdering); |
---|
1081 | // exportto(Schreyer, Syzextra::MakeSyzCompOrdering); |
---|
1082 | exportto(Schreyer, Syzextra::idPrepare); |
---|
1083 | // exportto(Schreyer, Syzextra::reduce_syz); |
---|
1084 | // exportto(Schreyer, Syzextra::p_Content); |
---|
1085 | |
---|
1086 | } |
---|
1087 | else |
---|
1088 | { |
---|
1089 | if( @DEBUG ) |
---|
1090 | { |
---|
1091 | "Loading the Debug version!"; |
---|
1092 | } |
---|
1093 | |
---|
1094 | load("syzextra_g.so"); |
---|
1095 | |
---|
1096 | if( @DEBUG ) |
---|
1097 | { |
---|
1098 | listvar(Syzextra_g); |
---|
1099 | } |
---|
1100 | |
---|
1101 | // export Syzextra_g; |
---|
1102 | // exportto(Schreyer, Syzextra_g::noop); |
---|
1103 | exportto(Schreyer, Syzextra_g::DetailedPrint); |
---|
1104 | // exportto(Schreyer, Syzextra_g::leadmonom); |
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1105 | exportto(Schreyer, Syzextra_g::leadcomp); |
---|
1106 | // exportto(Schreyer, Syzextra_g::leadrawexp); |
---|
1107 | // exportto(Schreyer, Syzextra_g::ISUpdateComponents); |
---|
1108 | exportto(Schreyer, Syzextra_g::SetInducedReferrence); |
---|
1109 | exportto(Schreyer, Syzextra_g::GetInducedData); |
---|
1110 | // exportto(Schreyer, Syzextra_g::GetAMData); |
---|
1111 | // exportto(Schreyer, Syzextra_g::SetSyzComp); |
---|
1112 | exportto(Schreyer, Syzextra_g::MakeInducedSchreyerOrdering); |
---|
1113 | // exportto(Schreyer, Syzextra_g::MakeSyzCompOrdering); |
---|
1114 | exportto(Schreyer, Syzextra_g::idPrepare); |
---|
1115 | // exportto(Schreyer, Syzextra_g::reduce_syz); |
---|
1116 | // exportto(Schreyer, Syzextra_g::p_Content); |
---|
1117 | |
---|
1118 | |
---|
1119 | } |
---|
1120 | |
---|
1121 | exportto(Top, DetailedPrint); |
---|
1122 | exportto(Top, GetInducedData); |
---|
1123 | |
---|
1124 | if( @DEBUG ) |
---|
1125 | { |
---|
1126 | listvar(Top); |
---|
1127 | listvar(Schreyer); |
---|
1128 | } |
---|
1129 | } |
---|
1130 | |
---|
1131 | if( !defined(GetInducedData) ) |
---|
1132 | { |
---|
1133 | ERROR("Sorry but we are missing the dynamic module (syzextra(_g)?.so)..."); |
---|
1134 | } |
---|
1135 | |
---|
1136 | } |
---|
1137 | |
---|
1138 | static proc mod_init() |
---|
1139 | { |
---|
1140 | loadme(); |
---|
1141 | } |
---|
1142 | |
---|
1143 | |
---|
1144 | proc testallSexamples() |
---|
1145 | { |
---|
1146 | example Ssyz; |
---|
1147 | example Scontinue; |
---|
1148 | example Sres; |
---|
1149 | } |
---|
1150 | |
---|
1151 | proc testallSSexamples() |
---|
1152 | { |
---|
1153 | example SSsyz; |
---|
1154 | example SScontinue; |
---|
1155 | example SSres; |
---|
1156 | } |
---|
1157 | |
---|
1158 | example |
---|
1159 | { "EXAMPLE:"; echo = 2; |
---|
1160 | testallSexamples(); |
---|
1161 | testallSSexamples(); |
---|
1162 | } |
---|