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 | |
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386 | proc Sres(module M, int l) |
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387 | "USAGE: Sres(M, l) |
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388 | RETURN: ring, containing a list of modules RES and a module MRES |
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389 | PURPOSE: computes (at most l) syzygy modules of M wrt the classical Schreyer |
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390 | induced ordering with gen(i) > gen(j) if i > j, provided both gens |
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391 | are from the same syzygy level. |
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392 | NOTE: RES contains the images of maps subsituting the beginning of the |
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393 | Schreyer free resolution of baseRing^r/M, while MRES is a sum of |
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394 | these images in a big free sum, containing all the syzygy modules. |
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395 | The syzygy modules are shifted so that gen(i) correspons to MRES[i]. |
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396 | The leading zero module RES[0] indicates the fact that coker of the |
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397 | first map is zero. The number of zeroes inducates the rank of input. |
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398 | NOTE: If l == 0 then l is set to be nvars(basering) + 1 |
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399 | EXAMPLE: example Sres; shows an example |
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400 | " |
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401 | { |
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402 | def S = Sinit(M); setring S; |
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403 | |
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404 | if (l == 0) |
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405 | { |
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406 | l = nvars(basering) + 1; // not really an estimate...?! |
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407 | } |
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408 | |
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409 | Sstep(); l = l - 1; |
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410 | |
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411 | Scontinue(l); |
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412 | |
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413 | return (S); |
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414 | } |
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415 | example |
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416 | { "EXAMPLE:"; echo = 2; |
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417 | ring r; |
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418 | module M = maxideal(1); M; |
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419 | def S = Sres(M, 0); setring S; S; |
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420 | RES; |
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421 | MRES; |
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422 | kill S; |
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423 | setring r; kill M; |
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424 | |
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425 | def A = nc_algebra(-1,0); setring A; |
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426 | ideal Q = var(1)^2, var(2)^2, var(3)^2; |
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427 | qring SCA = twostd(Q); |
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428 | basering; |
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429 | |
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430 | module M = maxideal(1); |
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431 | def S = Sres(M, 2); setring S; S; |
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432 | RES; |
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433 | MRES; |
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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 | static proc loadme() |
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440 | { |
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441 | int @DEBUG = !system("with", "ndebug"); |
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442 | |
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443 | if( @DEBUG ) |
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444 | { |
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445 | |
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446 | "ndebug?: ", system("with", "ndebug"); |
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447 | "om_ndebug?: ", system("with", "om_ndebug"); |
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448 | |
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449 | listvar(Top); |
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450 | listvar(Schreyer); |
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451 | } |
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452 | // listvar(Syzextra); listvar(Syzextra_g); |
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453 | |
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454 | if( !defined(DetailedPrint) ) |
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455 | { |
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456 | if( !@DEBUG ) |
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457 | { |
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458 | |
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459 | if( @DEBUG ) |
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460 | { |
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461 | "Loading the Release version!"; |
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462 | } |
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463 | load("syzextra.so"); |
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464 | |
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465 | if( @DEBUG ) |
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466 | { |
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467 | listvar(Syzextra); |
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468 | } |
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469 | |
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470 | // export Syzextra; |
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471 | |
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472 | // exportto(Schreyer, Syzextra::noop); |
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473 | exportto(Schreyer, Syzextra::DetailedPrint); |
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474 | // exportto(Schreyer, Syzextra::leadmonom); |
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475 | exportto(Schreyer, Syzextra::leadcomp); |
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476 | // exportto(Schreyer, Syzextra::leadrawexp); |
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477 | // exportto(Schreyer, Syzextra::ISUpdateComponents); |
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478 | exportto(Schreyer, Syzextra::SetInducedReferrence); |
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479 | exportto(Schreyer, Syzextra::GetInducedData); |
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480 | // exportto(Schreyer, Syzextra::GetAMData); |
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481 | // exportto(Schreyer, Syzextra::SetSyzComp); |
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482 | exportto(Schreyer, Syzextra::MakeInducedSchreyerOrdering); |
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483 | // exportto(Schreyer, Syzextra::MakeSyzCompOrdering); |
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484 | exportto(Schreyer, Syzextra::idPrepare); |
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485 | // exportto(Schreyer, Syzextra::reduce_syz); |
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486 | // exportto(Schreyer, Syzextra::p_Content); |
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487 | |
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488 | } |
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489 | else |
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490 | { |
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491 | if( @DEBUG ) |
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492 | { |
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493 | "Loading the Debug version!"; |
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494 | } |
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495 | |
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496 | load("syzextra_g.so"); |
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497 | |
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498 | if( @DEBUG ) |
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499 | { |
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500 | listvar(Syzextra_g); |
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501 | } |
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502 | |
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503 | // export Syzextra_g; |
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504 | // exportto(Schreyer, Syzextra_g::noop); |
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505 | exportto(Schreyer, Syzextra_g::DetailedPrint); |
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506 | // exportto(Schreyer, Syzextra_g::leadmonom); |
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507 | exportto(Schreyer, Syzextra_g::leadcomp); |
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508 | // exportto(Schreyer, Syzextra_g::leadrawexp); |
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509 | // exportto(Schreyer, Syzextra_g::ISUpdateComponents); |
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510 | exportto(Schreyer, Syzextra_g::SetInducedReferrence); |
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511 | exportto(Schreyer, Syzextra_g::GetInducedData); |
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512 | // exportto(Schreyer, Syzextra_g::GetAMData); |
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513 | // exportto(Schreyer, Syzextra_g::SetSyzComp); |
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514 | exportto(Schreyer, Syzextra_g::MakeInducedSchreyerOrdering); |
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515 | // exportto(Schreyer, Syzextra_g::MakeSyzCompOrdering); |
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516 | exportto(Schreyer, Syzextra_g::idPrepare); |
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517 | // exportto(Schreyer, Syzextra_g::reduce_syz); |
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518 | // exportto(Schreyer, Syzextra_g::p_Content); |
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519 | |
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520 | |
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521 | } |
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522 | |
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523 | exportto(Top, DetailedPrint); |
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524 | exportto(Top, GetInducedData); |
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525 | |
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526 | if( @DEBUG ) |
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527 | { |
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528 | listvar(Top); |
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529 | listvar(Schreyer); |
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530 | } |
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531 | } |
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532 | |
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533 | if( !defined(GetInducedData) ) |
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534 | { |
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535 | ERROR("Sorry but we are missing the dynamic module (syzextra(_g)?.so)..."); |
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536 | } |
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537 | |
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538 | } |
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539 | |
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540 | static proc mod_init() |
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541 | { |
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542 | loadme(); |
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543 | } |
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544 | |
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545 | |
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546 | proc testallSexamples() |
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547 | { |
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548 | example Ssyz; |
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549 | example Scontinue; |
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550 | example Sres; |
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551 | } |
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552 | example |
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553 | { "EXAMPLE:"; echo = 2; |
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554 | testallSexamples(); |
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555 | } |
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