1 | /////////////////////////////////////////////////////////////////////////// |
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2 | version="version schreyer.lib 4.0.0.0 Jun_2013 "; // $Id$ |
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3 | category="General purpose"; |
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4 | info=" |
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5 | LIBRARY: schreyer.lib Helpers for computing a Schreyer resolution in @code{derham.lib} |
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6 | AUTHOR: Oleksandr Motsak <U@D>, where U={motsak}, D={mathematik.uni-kl.de} |
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7 | KEYWORDS: Schreyer ordering; Schreyer resolution; syzygy |
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8 | OVERVIEW: |
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9 | @* The library contains helper procedures for computing a Schreyer resultion (cf. [SFO]) |
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10 | for @code{derham.lib} also in non-commutative setting (cf. [MO]). |
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11 | We call a free resulution a Schreyer resolution if any higher syzygy is a Goebner base of a previous one |
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12 | with respect to the corresponding Schreyer ordering. |
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13 | Schreyer resolution can be much bigger than a minimal resolution of the same module one but can be easier to construct. |
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14 | @* Input for resolution computations is a set of vectors @code{M} in form of a module over some basering @code{R}. |
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15 | The ring @code{R} may be non-commutative, in which case the ring ordering should be global. |
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16 | @* These procedures produce/work with partial Schreyer resolutions of @code{(R^rank(M))/M} in form of |
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17 | a ring (endowed with a special ring ordering that will be extended in the course of a resolution computation) |
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18 | containing a list of modules @code{RES} and a module @code{MRES}: |
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19 | @* The list of modules @code{RES} contains the images of maps (also called syzygies) substituting the |
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20 | computed beginning of a Schreyer resolution, that is, each syzygy module is a Groebner Basis |
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21 | of the previous one with respect to corresponding Schreyer induced ordering. |
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22 | @* The list @code{RES} starts with a zero map given by @code{rank(M)} zero generators indicating that the image of |
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23 | the first differential map is zero. The second map @code{RES[2]} is given by @code{M}, which indicates that |
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24 | the resolution is of @code{(R^rank(M))/M} is being computed. |
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25 | @* The module @code{MRES} is a direct sum of modules from @code{RES} and thus comprises all computed differentials. |
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26 | @* Syzygies are shifted so that @code{gen(i)} is mapped to @code{MRES[i]} under the differential map. |
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27 | @* Schreyer ordering extends the starting module ordering on @code{M} (defined in Singular by the basering @code{R}) |
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28 | and is extended to higher syzygies using the following definition: |
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29 | @* a < b if and only if (d(a) < d(b)) OR ( (d(a) = d(b) AND (comp(a) < comp(b)) ), |
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30 | @* where @code{d(a)} is the image of a under the differential (given by @code{MRES}), |
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31 | and @code{comp(a)} is the mod. component, for any module terms @code{a} and @code{b} from the same free module. |
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32 | REFERENCES: |
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33 | [SFO] Schreyer, F.O.: Die Berechnung von Syzygien mit dem verallgemeinerten Weierstrassschen Divisionssatz, |
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34 | Master's thesis, Univ. Hamburg, 1980. |
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35 | [MO] Motsak, O.: Non-commutative Computer Algebra with applications: Graded commutative algebra and related |
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36 | structures in Singular with applications, Ph.D. thesis, TU Kaiserslautern, 2010 |
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37 | |
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38 | NOTE: requires the dynamic or built-in module @code{syzextra} |
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39 | |
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40 | PROCEDURES: |
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41 | Sres(M,len) compute Schreyer resolution of module M of maximal length len |
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42 | Ssyz(M) compute Schreyer resolution of module M of length 1 |
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43 | Scontinue(len) extend currently active resolution by (at most) len syszygies |
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44 | "; |
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45 | |
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46 | static proc prepareSyz( module I, list # ) |
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47 | { |
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48 | int i; |
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49 | int k = 0; |
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50 | int r = nrows(I); |
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51 | int c = ncols(I); |
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52 | |
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53 | |
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54 | if( size(#) > 0 ) |
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55 | { |
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56 | if( typeof(#[1]) == "int" || typeof(#[1]) == "bigint" ) |
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57 | { |
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58 | k = #[1]; |
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59 | } |
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60 | } |
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61 | |
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62 | if( k < r ) |
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63 | { |
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64 | "// *** Wrong k: ", k, " < nrows: ", r, " => setting k = r = ", r; |
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65 | k = r; |
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66 | } |
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67 | |
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68 | // "k: ", k; "c: ", c; "I: ", I; |
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69 | |
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70 | for( i = c; i > 0; i-- ) |
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71 | { |
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72 | I[i] = I[i] + gen(k + i); |
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73 | } |
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74 | |
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75 | // DetailedPrint(I); |
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76 | |
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77 | return(I); |
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78 | } |
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79 | |
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80 | static proc separateSyzGB( module J, int c ) |
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81 | { |
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82 | module II, G; vector v; int i; |
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83 | |
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84 | J = simplify(J, 2); |
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85 | |
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86 | for( i = ncols(J); i > 0; i-- ) |
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87 | { |
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88 | v = J[i]; |
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89 | if( leadcomp(v) > c ) |
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90 | { |
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91 | II[i] = v; |
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92 | } else |
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93 | { |
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94 | G[i] = v; // leave only gen(i): i <= c |
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95 | } |
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96 | } |
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97 | |
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98 | II = simplify(II, 2); |
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99 | G = simplify(G, 2); |
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100 | |
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101 | return (list(G, II)); |
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102 | } |
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103 | |
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104 | static proc splitSyzGB( module J, int c ) |
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105 | { |
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106 | module JJ; vector v, vv; int i; |
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107 | |
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108 | for( i = ncols(J); i > 0; i-- ) |
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109 | { |
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110 | v = J[i]; |
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111 | |
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112 | vv = 0; |
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113 | |
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114 | while( leadcomp(v) <= c ) |
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115 | { |
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116 | vv = vv + lead(v); |
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117 | v = v - lead(v); |
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118 | } |
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119 | |
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120 | J[i] = vv; |
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121 | JJ[i] = v; |
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122 | } |
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123 | |
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124 | J = simplify(J, 2); |
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125 | JJ = simplify(JJ, 2); |
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126 | |
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127 | return (list(J, JJ)); |
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128 | } |
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129 | |
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130 | |
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131 | static proc Sinit(module M) |
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132 | { |
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133 | def @save = basering; |
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134 | |
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135 | int @DEBUG = !system("with", "ndebug"); |
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136 | if( @DEBUG ) |
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137 | { |
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138 | "Sinit::Input"; |
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139 | type(M); |
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140 | DetailedPrint(M); |
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141 | attrib(M); |
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142 | } |
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143 | |
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144 | int @RANK = nrows(M); int @SIZE = ncols(M); |
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145 | |
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146 | int @IS_A_SB = attrib(M, "isSB"); // ??? only if all weights were zero?! |
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147 | |
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148 | if( !@IS_A_SB ) |
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149 | { |
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150 | M = std(M); // this should be faster than computing std in S (later on) |
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151 | } |
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152 | |
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153 | def S = MakeInducedSchreyerOrdering(1); // 1 puts history terms to the back |
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154 | // TODO: NOTE: +1 causes trouble to Singular interpreter!!!??? |
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155 | setring S; // a new ring with a Schreyer ordering |
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156 | |
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157 | if( @DEBUG ) |
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158 | { |
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159 | "Sinit::StartingISRing"; |
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160 | basering; |
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161 | // DetailedPrint(basering); |
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162 | } |
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163 | |
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164 | // Setup the leading syzygy^{-1} module to zero: |
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165 | module Z = 0; Z[@RANK] = 0; attrib(Z, "isHomog", intvec(0)); |
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166 | |
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167 | module MRES = Z; |
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168 | |
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169 | list RES; RES[1] = Z; |
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170 | |
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171 | module F = freemodule(@RANK); |
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172 | intvec @V = deg(F[1..@RANK]); |
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173 | |
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174 | module M = imap(@save, M); |
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175 | |
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176 | attrib(M, "isHomog", @V); |
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177 | attrib(M, "isSB", 1); |
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178 | |
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179 | |
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180 | if( @DEBUG ) |
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181 | { |
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182 | "Sinit::SB_Input: "; |
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183 | type(M); |
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184 | attrib(M); |
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185 | attrib(M, "isHomog"); |
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186 | DetailedPrint(M); |
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187 | } |
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188 | |
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189 | if( @DEBUG ) |
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190 | { |
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191 | // 0^th syz. property |
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192 | if( size(module(transpose( transpose(M) * transpose(MRES) ))) > 0 ) |
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193 | { |
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194 | transpose( transpose(M) * transpose(MRES) ); |
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195 | "transpose( transpose(M) * transpose(MRES) ) != 0!!!"; |
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196 | $ |
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197 | } |
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198 | } |
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199 | |
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200 | RES[size(RES)+1] = M; // list of all syzygy modules |
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201 | MRES = MRES, M; |
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202 | |
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203 | attrib(MRES, "isHomog", @V); |
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204 | |
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205 | attrib(S, "InducionLeads", lead(M)); |
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206 | attrib(S, "InducionStart", @RANK); |
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207 | |
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208 | if( @DEBUG ) |
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209 | { |
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210 | "Sinit::MRES"; |
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211 | DetailedPrint(MRES); |
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212 | attrib(MRES, "isHomog"); |
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213 | attrib(S); |
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214 | } |
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215 | |
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216 | export RES; |
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217 | export MRES; |
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218 | return (S); |
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219 | } |
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220 | |
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221 | static proc Sstep() |
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222 | { |
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223 | int @DEBUG = !system("with", "ndebug"); |
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224 | |
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225 | if( @DEBUG ) |
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226 | { |
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227 | "Sstep::NextInducedRing"; |
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228 | DetailedPrint(basering); |
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229 | |
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230 | attrib(basering, "InducionLeads"); |
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231 | attrib(basering, "InducionStart"); |
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232 | |
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233 | GetInducedData(); |
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234 | } |
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235 | |
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236 | // syzygy step: |
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237 | |
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238 | /* |
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239 | // is initial weights are all zeroes! |
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240 | def L = lead(M); |
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241 | intvec @V = deg(M[1..ncols(M)]); @W; @V; @W = @V; attrib(L, "isHomog", @W); |
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242 | SetInducedReferrence(L, @RANK, 0); |
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243 | */ |
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244 | |
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245 | // def L = lead(MRES); |
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246 | // @W = @W, @V; |
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247 | // attrib(L, "isHomog", @W); |
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248 | |
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249 | |
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250 | // General setting: |
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251 | // SetInducedReferrence(MRES, 0, 0); // limit: 0! |
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252 | int @l = size(RES); |
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253 | |
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254 | module M = RES[@l]; |
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255 | |
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256 | module L = attrib(basering, "InducionLeads"); |
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257 | int limit = attrib(basering, "InducionStart"); |
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258 | |
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259 | // L; limit; |
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260 | |
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261 | int @RANK = ncols(MRES) - ncols(M); // nrows(M); // what if M is zero?! |
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262 | |
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263 | /* |
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264 | if( @RANK != nrows(M) ) |
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265 | { |
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266 | type(MRES); |
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267 | @RANK; |
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268 | type(M); |
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269 | pause(); |
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270 | } |
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271 | */ |
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272 | |
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273 | intvec @W = attrib(M, "isHomog"); |
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274 | intvec @V = deg(M[1..ncols(M)]); |
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275 | @V = @W, @V; |
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276 | |
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277 | if( @DEBUG ) |
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278 | { |
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279 | "Sstep::NextInput: "; |
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280 | M; |
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281 | deg(M[1..ncols(M)]); // no use of @W :(? |
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282 | @RANK; |
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283 | DetailedPrint(MRES); |
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284 | attrib(MRES, "isHomog"); @W; |
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285 | deg(MRES[1..ncols(MRES)]); |
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286 | } |
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287 | |
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288 | |
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289 | |
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290 | SetInducedReferrence(L, limit, 0); |
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291 | |
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292 | def K = prepareSyz(M, @RANK); |
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293 | // K; |
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294 | |
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295 | // attrib(K, "isHomog", @V); DetailedPrint(K, 1000); |
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296 | |
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297 | // pause(); |
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298 | |
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299 | K = idPrepare(K, @RANK); // std(K); // ? |
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300 | K = simplify(K, 2); |
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301 | |
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302 | // K; |
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303 | |
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304 | module N = separateSyzGB(K, @RANK)[2]; // 1^st syz. module: vectors which start in lower part (comp >= @RANK) |
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305 | |
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306 | // "N_0: "; N; DetailedPrint(N, 10); |
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307 | |
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308 | // basering; print(@V); type(N); |
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309 | // attrib(N, "isHomog", @V); // TODO: fix "wrong weights"!!!? deg is wrong :((( |
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310 | N = std(N); |
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311 | attrib(N, "isHomog", @V); |
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312 | |
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313 | // N; |
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314 | |
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315 | if( @DEBUG ) |
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316 | { |
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317 | if( size(N) > 0 ) |
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318 | { |
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319 | // next syz. property |
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320 | if( size(module(transpose( transpose(N) * transpose(MRES) ))) > 0 ) |
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321 | { |
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322 | MRES; |
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323 | |
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324 | "N: "; N; DetailedPrint(N, 10); |
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325 | |
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326 | "K:"; K; DetailedPrint(K, 10); |
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327 | |
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328 | "RANKS: ", @RANK; |
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329 | |
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330 | "transpose( transpose(N) * transpose(MRES) ) != 0!!!"; |
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331 | transpose( transpose(N) * transpose(MRES) ); |
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332 | |
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333 | "transpose(N) * transpose(MRES): "; |
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334 | transpose(N) * transpose(MRES); |
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335 | DetailedPrint(module(_), 2); |
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336 | $ |
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337 | } |
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338 | } |
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339 | } |
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340 | |
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341 | RES[@l + 1] = N; // list of all syzygy modules |
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342 | |
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343 | MRES = MRES, N; |
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344 | attrib(MRES, "isHomog", @V); |
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345 | |
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346 | |
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347 | L = L, lead(N); |
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348 | attrib(basering, "InducionLeads", L); |
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349 | |
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350 | if( @DEBUG ) |
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351 | { |
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352 | "Sstep::NextSyzOutput: "; |
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353 | DetailedPrint(N); |
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354 | attrib(N, "isHomog"); |
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355 | } |
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356 | |
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357 | } |
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358 | |
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359 | proc Scontinue(int l) |
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360 | "USAGE: Scontinue(int len) |
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361 | RETURN: nothing, instead it changes the currently active resolution |
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362 | PURPOSE: extends the currently active resolution by at most len syzygies |
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363 | ASSUME: must be used within a ring returned by Sres or Ssyz |
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364 | EXAMPLE: example Scontinue; shows an example |
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365 | " |
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366 | { |
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367 | def data = GetInducedData(); |
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368 | |
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369 | if( (!defined(RES)) || (!defined(MRES)) || (typeof(data) != "list") || (size(data) != 2) ) |
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370 | { |
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371 | ERROR("Sorry, but basering does not seem to be returned by Sres or Ssyz"); |
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372 | } |
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373 | for (; (l != 0) && (size(RES[size(RES)]) > 0); l-- ) |
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374 | { |
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375 | Sstep(); |
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376 | } |
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377 | } |
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378 | example |
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379 | { "EXAMPLE:"; echo = 2; |
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380 | ring r; |
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381 | module M = maxideal(1); M; |
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382 | def S = Ssyz(M); setring S; S; |
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383 | "Only the first syzygy: "; |
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384 | RES; MRES; |
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385 | "More syzygies: "; |
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386 | Scontinue(10); |
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387 | RES; MRES; |
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388 | } |
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389 | |
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390 | proc Ssyz(module M) |
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391 | "USAGE: Ssyz(module M) |
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392 | RETURN: ring, containing a Schreyer resolution |
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393 | PURPOSE: computes a Schreyer resolution of M of length 1 (see the library overview) |
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394 | SEE ALSO: Sres |
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395 | EXAMPLE: example Ssyz; shows an example |
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396 | " |
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397 | { |
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398 | def S = Sinit(M); setring S; |
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399 | |
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400 | Sstep(); // NOTE: what if M is zero? |
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401 | |
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402 | return (S); |
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403 | } |
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404 | example |
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405 | { "EXAMPLE:"; echo = 2; |
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406 | ring r; |
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407 | module M = maxideal(1); M; |
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408 | def S = Ssyz(M); setring S; S; |
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409 | "Only the first syzygy: "; |
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410 | RES; |
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411 | MRES; // Note gen(i) |
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412 | kill S; |
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413 | setring r; kill M; |
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414 | |
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415 | module M = 0; |
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416 | def S = Ssyz(M); setring S; S; |
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417 | "Only the first syzygy: "; |
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418 | RES; |
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419 | MRES; |
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420 | } |
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421 | |
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422 | proc Sres(module M, int l) |
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423 | "USAGE: Sres(module M, int len) |
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424 | RETURN: ring, containing a Schreyer resolution |
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425 | PURPOSE: computes a Schreyer resolution of M of length at most len (see the library overview) |
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426 | NOTE: If given len is zero then nvars(basering) + 1 is used instead. |
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427 | SEE ALSO: Ssyz |
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428 | EXAMPLE: example Sres; shows an example |
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429 | " |
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430 | { |
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431 | def S = Sinit(M); setring S; |
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432 | |
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433 | if (l == 0) |
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434 | { |
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435 | l = nvars(basering) + 1; // not really an estimate...?! |
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436 | } |
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437 | |
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438 | Sstep(); l = l - 1; |
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439 | |
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440 | Scontinue(l); |
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441 | |
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442 | return (S); |
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443 | } |
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444 | example |
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445 | { "EXAMPLE:"; echo = 2; |
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446 | ring r; |
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447 | module M = maxideal(1); M; |
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448 | def S = Sres(M, 0); setring S; S; |
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449 | RES; |
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450 | MRES; |
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451 | kill S; |
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452 | setring r; kill M; |
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453 | |
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454 | def A = nc_algebra(-1,0); setring A; |
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455 | ideal Q = var(1)^2, var(2)^2, var(3)^2; |
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456 | qring SCA = twostd(Q); |
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457 | basering; |
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458 | |
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459 | module M = maxideal(1); |
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460 | def S = Sres(M, 2); setring S; S; |
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461 | RES; |
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462 | MRES; |
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463 | } |
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464 | |
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465 | |
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466 | |
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467 | // ================================================================== // |
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468 | |
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469 | |
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470 | LIB "general.lib"; // for sort |
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471 | |
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472 | /* static proc Tail(def M) // DONE: in C++ (dyn. module: syzextra)! |
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473 | { |
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474 | int i = ncols(M); def m; |
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475 | while (i > 0) |
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476 | { |
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477 | m = M[i]; |
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478 | m = m - lead(m); // m = tail(m) |
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479 | M[i] = m; |
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480 | i--; |
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481 | } |
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482 | return (M); |
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483 | }*/ |
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484 | |
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485 | |
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486 | /* static */ proc SSinit(def M) |
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487 | { |
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488 | if( (typeof(M) != "module") && (typeof(M) != "ideal") ) |
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489 | { |
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490 | ERROR("Sorry: need an ideal or a module for input"); |
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491 | } |
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492 | |
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493 | // TODO! DONE? |
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494 | def @save = basering; |
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495 | |
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496 | int @DEBUG = !system("with", "ndebug"); |
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497 | int @SYZCHECK = 1 || @DEBUG; // TODO: only for now!! |
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498 | |
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499 | if( @DEBUG ) |
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500 | { |
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501 | "SSinit::Input"; |
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502 | type(M); |
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503 | // DetailedPrint(M); |
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504 | attrib(M); |
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505 | } |
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506 | |
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507 | int @RANK = nrows(M); int @SIZE = ncols(M); |
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508 | |
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509 | int @IS_A_SB = attrib(M, "isSB"); // ??? only if all weights were zero?! |
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510 | |
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511 | if( !@IS_A_SB ) |
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512 | { |
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513 | def opts = option(get); |
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514 | option(redSB); option(redTail); |
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515 | M = std(M); |
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516 | option(set, opts); |
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517 | kill opts; |
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518 | } else |
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519 | { |
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520 | M = simplify(M, 2 + 4 + 32); |
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521 | } |
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522 | |
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523 | def LEAD = lead(M); |
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524 | |
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525 | // sort wrt neg.deg.rev.lex! |
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526 | intvec iv_ds = sort(LEAD, "ds", 1)[2]; // ,1 => reversed! |
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527 | |
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528 | M = M[iv_ds]; // sort M wrt ds on current leading terms |
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529 | LEAD = LEAD[iv_ds]; |
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530 | |
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531 | def TAIL = Tail(M); |
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532 | |
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533 | intvec @DEGS = deg(M[1..@SIZE]); // store actuall degrees of input elements |
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534 | |
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535 | // TODO: what about real modules? weighted ones? |
---|
536 | |
---|
537 | list @l = ringlist(@save); |
---|
538 | |
---|
539 | int @z = 0; ideal @m = maxideal(1); intvec @wdeg = deg(@m[1..ncols(@m)]); |
---|
540 | |
---|
541 | // NOTE: @wdeg will be ignored anyway :( |
---|
542 | @l[3] = list(list("C", @z), list("lp", @wdeg)); |
---|
543 | |
---|
544 | kill @z, @wdeg; // since these vars are ring independent! |
---|
545 | |
---|
546 | def S = ring(@l); // --MakeInducedSchreyerOrdering(1); |
---|
547 | |
---|
548 | module F = freemodule(@RANK); |
---|
549 | intvec @V = deg(F[1..@RANK]); |
---|
550 | |
---|
551 | setring S; // ring with an easy divisibility test ("C, lex") |
---|
552 | |
---|
553 | if( @DEBUG ) |
---|
554 | { |
---|
555 | "SSinit::NewRing(C, lex)"; |
---|
556 | basering; |
---|
557 | // DetailedPrint(basering); |
---|
558 | } |
---|
559 | |
---|
560 | // Setup the leading syzygy^{-1} module to zero: |
---|
561 | module Z = 0; Z[@RANK] = 0; attrib(Z, "isHomog", intvec(0)); |
---|
562 | |
---|
563 | module MRES = Z; |
---|
564 | |
---|
565 | list RES; RES[1] = Z; |
---|
566 | list LRES; LRES[1] = Z; |
---|
567 | list TRES; TRES[1] = Z; |
---|
568 | |
---|
569 | def M = imap(@save, M); |
---|
570 | |
---|
571 | attrib(M, "isHomog", @V); |
---|
572 | attrib(M, "isSB", 1); |
---|
573 | attrib(M, "degrees", @DEGS); |
---|
574 | |
---|
575 | def LEAD = imap(@save, LEAD); |
---|
576 | |
---|
577 | attrib(LEAD, "isHomog", @V); |
---|
578 | attrib(LEAD, "isSB", 1); |
---|
579 | |
---|
580 | def TAIL = imap(@save, TAIL); |
---|
581 | |
---|
582 | if( @DEBUG ) |
---|
583 | { |
---|
584 | "SSinit::(sorted) SB_Input: "; |
---|
585 | type(M); |
---|
586 | attrib(M); |
---|
587 | attrib(M, "isHomog"); |
---|
588 | // DetailedPrint(M); |
---|
589 | } |
---|
590 | |
---|
591 | if( @SYZCHECK ) |
---|
592 | { |
---|
593 | // 0^th syz. property |
---|
594 | if( size(module(transpose( transpose(M) * transpose(MRES) ))) > 0 ) |
---|
595 | { |
---|
596 | transpose( transpose(M) * transpose(MRES) ); |
---|
597 | "ERROR: transpose( transpose(M) * transpose(MRES) ) != 0!!!"; |
---|
598 | $ |
---|
599 | } |
---|
600 | } |
---|
601 | |
---|
602 | RES[size(RES)+1] = M; // list of all syzygy modules |
---|
603 | LRES[size(LRES)+1] = LEAD; // list of all syzygy modules |
---|
604 | TRES[size(TRES)+1] = TAIL; // list of all syzygy modules |
---|
605 | |
---|
606 | MRES = MRES, M; //? |
---|
607 | |
---|
608 | attrib(MRES, "isHomog", @V); |
---|
609 | |
---|
610 | // attrib(S, "InducionStart", @RANK); |
---|
611 | attrib(S, "LEAD2SYZ", 1); |
---|
612 | attrib(S, "TAILREDSYZ", 0); |
---|
613 | attrib(S, "DEBUG", @DEBUG); |
---|
614 | attrib(S, "SYZCHECK", @SYZCHECK); |
---|
615 | |
---|
616 | if( @DEBUG ) |
---|
617 | { |
---|
618 | "SSinit::MRES"; |
---|
619 | MRES; |
---|
620 | // DetailedPrint(MRES); |
---|
621 | attrib(MRES, "isHomog"); |
---|
622 | attrib(S); |
---|
623 | } |
---|
624 | |
---|
625 | export RES; |
---|
626 | export MRES; |
---|
627 | export LRES; |
---|
628 | export TRES; |
---|
629 | return (S); |
---|
630 | } |
---|
631 | example |
---|
632 | { "EXAMPLE:"; echo = 2; |
---|
633 | ring R = 0, (w, x, y, z), dp; |
---|
634 | |
---|
635 | def M = maxideal(1); |
---|
636 | def S = SSinit(M); setring S; S; |
---|
637 | |
---|
638 | "Only the first initialization: "; |
---|
639 | RES; LRES; TRES; |
---|
640 | MRES; |
---|
641 | |
---|
642 | kill S; setring R; kill M; |
---|
643 | |
---|
644 | ideal M = w^2 - x*z, w*x - y*z, x^2 - w*y, x*y - z^2, y^2 - w*z; |
---|
645 | def S = SSinit(M); setring S; S; |
---|
646 | |
---|
647 | "Only the first initialization: "; |
---|
648 | RES; LRES; TRES; |
---|
649 | MRES; |
---|
650 | |
---|
651 | kill S; setring R; kill M; |
---|
652 | } |
---|
653 | |
---|
654 | |
---|
655 | LIB "poly.lib"; // for lcm |
---|
656 | |
---|
657 | |
---|
658 | |
---|
659 | /// Compute L(Syz(L)) |
---|
660 | proc SSComputeLeadingSyzygyTerms(def L) |
---|
661 | { |
---|
662 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
663 | { |
---|
664 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
665 | } else |
---|
666 | { |
---|
667 | int @DEBUG = !system("with", "ndebug"); |
---|
668 | } |
---|
669 | |
---|
670 | if( @DEBUG ) |
---|
671 | { |
---|
672 | "SSComputeLeadingSyzygyTerms::Input: "; |
---|
673 | L; |
---|
674 | } |
---|
675 | |
---|
676 | int i, j, r; intvec iv_ds; |
---|
677 | int N = ncols(L); |
---|
678 | def a, b; |
---|
679 | poly aa, bb; |
---|
680 | |
---|
681 | bigint c; |
---|
682 | |
---|
683 | ideal M; |
---|
684 | |
---|
685 | module S = 0; |
---|
686 | |
---|
687 | for(i = 1; i <= N; i++) |
---|
688 | { |
---|
689 | a = L[i]; |
---|
690 | // "a: ", a; |
---|
691 | c = leadcomp(a); |
---|
692 | r = int(c); |
---|
693 | |
---|
694 | if(r > 0) |
---|
695 | { |
---|
696 | aa = a[r]; |
---|
697 | } else |
---|
698 | { |
---|
699 | aa = a; |
---|
700 | } |
---|
701 | |
---|
702 | M = 0; |
---|
703 | |
---|
704 | for(j = i-1; j > 0; j--) |
---|
705 | { |
---|
706 | b = L[j]; |
---|
707 | // "b: ", b; |
---|
708 | |
---|
709 | if( leadcomp(b) == c ) |
---|
710 | { |
---|
711 | if(r > 0) |
---|
712 | { |
---|
713 | bb = b[r]; |
---|
714 | } else |
---|
715 | { |
---|
716 | bb = b; |
---|
717 | } |
---|
718 | |
---|
719 | M[j] = (lcm(aa, bb) / aa); |
---|
720 | } |
---|
721 | } |
---|
722 | |
---|
723 | // TODO: add quotient relations here... |
---|
724 | |
---|
725 | M = simplify(M, 1 + 2 + 32); |
---|
726 | |
---|
727 | iv_ds = sort(M, "ds", 1)[2]; // ,1 => reversed! |
---|
728 | |
---|
729 | M = M[iv_ds]; |
---|
730 | |
---|
731 | S = S, M * gen(i); |
---|
732 | } |
---|
733 | |
---|
734 | S = simplify(S, 2); |
---|
735 | |
---|
736 | S = sort(S, "ds", 1)[1]; // ,1 => reversed! // TODO: not needed? |
---|
737 | |
---|
738 | if( @DEBUG ) |
---|
739 | { |
---|
740 | "SSComputeLeadingSyzygyTerms::Output: "; |
---|
741 | S; |
---|
742 | } |
---|
743 | |
---|
744 | attrib(S, "isSB", 1); |
---|
745 | |
---|
746 | return (S); |
---|
747 | } |
---|
748 | |
---|
749 | /// Compute Syz(L), where L is a monomial (leading) module |
---|
750 | proc SSCompute2LeadingSyzygyTerms(def L, int @TAILREDSYZ) |
---|
751 | { |
---|
752 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
753 | { |
---|
754 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
755 | } else |
---|
756 | { |
---|
757 | int @DEBUG = !system("with", "ndebug"); |
---|
758 | } |
---|
759 | |
---|
760 | if( typeof( attrib(basering, "SYZCHECK") ) == "int" ) |
---|
761 | { |
---|
762 | int @SYZCHECK = attrib(basering, "SYZCHECK"); |
---|
763 | } else |
---|
764 | { |
---|
765 | int @SYZCHECK = @DEBUG; |
---|
766 | } |
---|
767 | |
---|
768 | |
---|
769 | if( @DEBUG ) |
---|
770 | { |
---|
771 | "SSCompute2LeadingSyzygyTerms::Input: "; |
---|
772 | L; |
---|
773 | "@TAILREDSYZ: ", @TAILREDSYZ; |
---|
774 | } |
---|
775 | |
---|
776 | int i, j, r; |
---|
777 | int N = ncols(L); |
---|
778 | def a, b; |
---|
779 | |
---|
780 | poly aa, bb, @lcm; |
---|
781 | |
---|
782 | bigint c; |
---|
783 | |
---|
784 | module M; |
---|
785 | |
---|
786 | module S = 0; |
---|
787 | |
---|
788 | for(i = 1; i <= N; i++) |
---|
789 | { |
---|
790 | a = L[i]; |
---|
791 | // "a: ", a; |
---|
792 | c = leadcomp(a); |
---|
793 | r = int(c); |
---|
794 | |
---|
795 | aa = leadmonomial(a); |
---|
796 | |
---|
797 | M = 0; |
---|
798 | |
---|
799 | for(j = i-1; j > 0; j--) |
---|
800 | { |
---|
801 | b = L[j]; |
---|
802 | // "b: ", b; |
---|
803 | |
---|
804 | if( leadcomp(b) == c ) |
---|
805 | { |
---|
806 | bb = leadmonomial(b); |
---|
807 | @lcm = lcm(aa, bb); |
---|
808 | |
---|
809 | M[j] = (@lcm / aa)* gen(i) - (@lcm / bb)* gen(j); |
---|
810 | } |
---|
811 | } |
---|
812 | |
---|
813 | M = simplify(M, 2); |
---|
814 | |
---|
815 | // TODO: add quotient relations here... |
---|
816 | S = S, M; |
---|
817 | } |
---|
818 | |
---|
819 | if( @TAILREDSYZ ) |
---|
820 | { |
---|
821 | // Make sure that 2nd syzygy terms are not reducible by 1st |
---|
822 | def opts = option(get); |
---|
823 | option(redSB); option(redTail); |
---|
824 | S = std(S); // binomial module |
---|
825 | option(set, opts); |
---|
826 | // kill opts; |
---|
827 | } else |
---|
828 | { |
---|
829 | S = simplify(S, 2 + 32); |
---|
830 | } |
---|
831 | |
---|
832 | S = sort(S, "ds", 1)[1]; // ,1 => reversed! |
---|
833 | |
---|
834 | if( @DEBUG ) |
---|
835 | { |
---|
836 | "SSCompute2LeadingSyzygyTerms::Syz(LEAD): "; S; |
---|
837 | } |
---|
838 | |
---|
839 | if( @SYZCHECK ) |
---|
840 | { |
---|
841 | if( size(S) > 0 and size(L) > 0 ) |
---|
842 | { |
---|
843 | if( size(module(transpose( transpose(S) * transpose(L) ))) > 0 ) |
---|
844 | { |
---|
845 | transpose( transpose(S) * transpose(L) ); |
---|
846 | "ERROR: transpose( transpose(S) * transpose(L) ) != 0!!!"; |
---|
847 | $ |
---|
848 | } |
---|
849 | } |
---|
850 | } |
---|
851 | |
---|
852 | module S2 = Tail(S); |
---|
853 | S = lead(S); // (C,lp) on base ring! |
---|
854 | |
---|
855 | if( @DEBUG ) |
---|
856 | { |
---|
857 | "SSCompute2LeadingSyzygyTerms::Output: "; S; S2; |
---|
858 | } |
---|
859 | |
---|
860 | attrib(S, "isSB", 1); |
---|
861 | |
---|
862 | return (S, S2); |
---|
863 | } |
---|
864 | |
---|
865 | // -------------------------------------------------------- // |
---|
866 | |
---|
867 | /// TODO: save shortcut LM(m) * "t" -> ? |
---|
868 | proc SSReduceTerm(poly m, def t, def L, def T, list #) |
---|
869 | { |
---|
870 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
871 | { |
---|
872 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
873 | } else |
---|
874 | { |
---|
875 | int @DEBUG = !system("with", "ndebug"); |
---|
876 | } |
---|
877 | |
---|
878 | if( @DEBUG ) |
---|
879 | { |
---|
880 | "SSReduce::Input: "; |
---|
881 | |
---|
882 | "mult: ", m; |
---|
883 | "term: ", t; |
---|
884 | "L: ", L; |
---|
885 | "T: ", T; |
---|
886 | if( size(#) > 0 ) |
---|
887 | { |
---|
888 | "LSyz: ", #; |
---|
889 | } |
---|
890 | // "attrib(LS, 'isSB')", attrib(LS, "isSB"); |
---|
891 | } |
---|
892 | |
---|
893 | vector s = 0; |
---|
894 | |
---|
895 | if( t == 0 ) |
---|
896 | { |
---|
897 | return (s); |
---|
898 | } |
---|
899 | |
---|
900 | def product = m * t; |
---|
901 | |
---|
902 | bigint c = leadcomp(t); |
---|
903 | int r = int(c); |
---|
904 | |
---|
905 | def a, b, nf, bb; |
---|
906 | |
---|
907 | // looking for an appropriate reducer |
---|
908 | for( int k = ncols(L); k > 0; k-- ) |
---|
909 | { |
---|
910 | a = L[k]; |
---|
911 | // with the same mod. component |
---|
912 | if( leadcomp(a) == c ) |
---|
913 | { |
---|
914 | b = - (leadmonomial(product) / leadmonomial(L[k])); |
---|
915 | |
---|
916 | // which divides the product |
---|
917 | if( b != 0 ) |
---|
918 | { |
---|
919 | // "b: ", b; |
---|
920 | bb = b * gen(k); |
---|
921 | nf = bb; |
---|
922 | |
---|
923 | if( size(#) > 0 ) |
---|
924 | { |
---|
925 | if( typeof(#[1]) == "module" ) |
---|
926 | { |
---|
927 | nf = NF(bb, #[1]); |
---|
928 | // "NF: ", nf; |
---|
929 | } |
---|
930 | } |
---|
931 | |
---|
932 | // while the complement (the fraction) is not reducible by leading syzygies |
---|
933 | if( nf != 0 ) |
---|
934 | { |
---|
935 | /// TODO: save shortcut LM(m) * T[i] -> ? |
---|
936 | |
---|
937 | // choose ANY such reduction... (with the biggest index?) |
---|
938 | s = bb + SSTraverseTail(b, T[k], L, T, #); |
---|
939 | break; |
---|
940 | } |
---|
941 | } |
---|
942 | } |
---|
943 | } |
---|
944 | if( @DEBUG ) |
---|
945 | { |
---|
946 | "SSReduceTerm::Output: ", s; |
---|
947 | } |
---|
948 | return (s); |
---|
949 | } |
---|
950 | |
---|
951 | // TODO: store m * @tail -.-^-.-^-.--> ? |
---|
952 | proc SSTraverseTail(poly m, def @tail, def L, def T, list #) |
---|
953 | { |
---|
954 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
955 | { |
---|
956 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
957 | } else |
---|
958 | { |
---|
959 | int @DEBUG = !system("with", "ndebug"); |
---|
960 | } |
---|
961 | |
---|
962 | if( @DEBUG ) |
---|
963 | { |
---|
964 | "SSTraverse::Input: "; |
---|
965 | |
---|
966 | "mult: ", m; |
---|
967 | "tail: ", @tail; // T[i]; |
---|
968 | |
---|
969 | if( size(#) > 0 ) |
---|
970 | { |
---|
971 | "LSyz: "; #[1]; |
---|
972 | } |
---|
973 | } |
---|
974 | |
---|
975 | vector s = 0; |
---|
976 | |
---|
977 | def @l; |
---|
978 | |
---|
979 | // iterate tail-terms in ANY order! |
---|
980 | while( size(@tail) > 0 ) |
---|
981 | { |
---|
982 | @l = lead(@tail); |
---|
983 | s = s + SSReduceTerm(m, @l, L, T, #); |
---|
984 | @tail = @tail - @l; |
---|
985 | } |
---|
986 | |
---|
987 | if( @DEBUG ) |
---|
988 | { |
---|
989 | "SSTraverseTail::Output: ", s; |
---|
990 | } |
---|
991 | return (s); |
---|
992 | } |
---|
993 | |
---|
994 | // -------------------------------------------------------- // |
---|
995 | |
---|
996 | // module (N, LL, TT) = SSComputeSyzygy(L, T); |
---|
997 | // Compute Syz(L ++ T) = N = LL ++ TT |
---|
998 | proc SSComputeSyzygy(def L, def T) |
---|
999 | { |
---|
1000 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
1001 | { |
---|
1002 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
1003 | } else |
---|
1004 | { |
---|
1005 | int @DEBUG = !system("with", "ndebug"); |
---|
1006 | } |
---|
1007 | |
---|
1008 | if( @DEBUG ) |
---|
1009 | { |
---|
1010 | "SSComputeSyzygy::Input"; |
---|
1011 | "basering: ", basering; attrib(basering); |
---|
1012 | // DetailedPrint(basering); |
---|
1013 | |
---|
1014 | // "iCompShift: ", iCompShift; |
---|
1015 | |
---|
1016 | "L: "; L; |
---|
1017 | "T: "; T; |
---|
1018 | } |
---|
1019 | |
---|
1020 | def a; bigint c; int r, k; poly aa; |
---|
1021 | |
---|
1022 | int @LEAD2SYZ = 0; |
---|
1023 | if( typeof( attrib(basering, "LEAD2SYZ") ) == "int" ) |
---|
1024 | { |
---|
1025 | @LEAD2SYZ = attrib(basering, "LEAD2SYZ"); |
---|
1026 | } |
---|
1027 | |
---|
1028 | int @TAILREDSYZ = 1; |
---|
1029 | if( typeof( attrib(basering, "TAILREDSYZ") ) == "int" ) |
---|
1030 | { |
---|
1031 | @TAILREDSYZ = attrib(basering, "TAILREDSYZ"); |
---|
1032 | // @TAILREDSYZ; |
---|
1033 | } |
---|
1034 | |
---|
1035 | /// Get the critical leading syzygy terms |
---|
1036 | if( @LEAD2SYZ ) // & 2nd syz. term |
---|
1037 | { |
---|
1038 | def a2; int r2; poly aa2; |
---|
1039 | module LL, LL2; |
---|
1040 | (LL, LL2) = SSCompute2LeadingSyzygyTerms(L, @TAILREDSYZ); // ++ |
---|
1041 | } else |
---|
1042 | { |
---|
1043 | module LL = SSComputeLeadingSyzygyTerms(L); |
---|
1044 | } |
---|
1045 | |
---|
1046 | module TT, SYZ; |
---|
1047 | |
---|
1048 | if( size(LL) > 0 ) |
---|
1049 | { |
---|
1050 | list LS; |
---|
1051 | |
---|
1052 | if( @TAILREDSYZ ) |
---|
1053 | { |
---|
1054 | LS = list(LL); |
---|
1055 | } |
---|
1056 | |
---|
1057 | vector @tail; |
---|
1058 | |
---|
1059 | for(k = ncols(LL); k > 0; k-- ) |
---|
1060 | { |
---|
1061 | // leading syz. term: |
---|
1062 | a = LL[k]; c = leadcomp(a); r = int(c); aa = leadmonomial(a); |
---|
1063 | // "A: ", a, " --->>>> ", aa, " **** [", r, "]: "; |
---|
1064 | |
---|
1065 | /// TODO: save shortcut (aa) * T[r] -> ? |
---|
1066 | @tail = SSTraverseTail(aa, T[r], L, T, LS); |
---|
1067 | |
---|
1068 | // get the 2nd syzygy term... |
---|
1069 | |
---|
1070 | if( @LEAD2SYZ ) // with the 2nd syz. term: |
---|
1071 | { |
---|
1072 | a2 = LL2[k]; c = leadcomp(a2); r2 = int(c); aa2 = leadmonomial(a2); |
---|
1073 | @tail = @tail + |
---|
1074 | /// TODO: save shortcut (aa2) * T[r2] -> ? |
---|
1075 | a2 + SSTraverseTail(aa2, T[r2], L, T, LS); |
---|
1076 | } else |
---|
1077 | { |
---|
1078 | @tail = @tail + SSReduceTerm(aa, L[r], L, T, LS); |
---|
1079 | } |
---|
1080 | |
---|
1081 | |
---|
1082 | TT[k] = @tail; |
---|
1083 | SYZ[k] = a + @tail; |
---|
1084 | } |
---|
1085 | } |
---|
1086 | |
---|
1087 | /* |
---|
1088 | def opts = option(get); option(redSB); option(redTail); |
---|
1089 | module SYZ = std(syz(M)); |
---|
1090 | option(set, opts); kill opts; |
---|
1091 | |
---|
1092 | module LL = lead(SYZ); // TODO: WRONG ORDERING!!!!!!!! |
---|
1093 | module TT = Tail(SYZ); |
---|
1094 | */ |
---|
1095 | |
---|
1096 | if( @DEBUG ) |
---|
1097 | { |
---|
1098 | "SSComputeSyzygy::Output"; |
---|
1099 | |
---|
1100 | "SYZ: "; SYZ; |
---|
1101 | "LL: "; LL; |
---|
1102 | "TT: "; TT; |
---|
1103 | } |
---|
1104 | |
---|
1105 | return (SYZ, LL, TT); |
---|
1106 | } |
---|
1107 | |
---|
1108 | // resolution/syzygy step: |
---|
1109 | static proc SSstep() |
---|
1110 | { |
---|
1111 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
1112 | { |
---|
1113 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
1114 | } else |
---|
1115 | { |
---|
1116 | int @DEBUG = !system("with", "ndebug"); |
---|
1117 | } |
---|
1118 | |
---|
1119 | |
---|
1120 | if( typeof( attrib(basering, "SYZCHECK") ) == "int" ) |
---|
1121 | { |
---|
1122 | int @SYZCHECK = attrib(basering, "SYZCHECK"); |
---|
1123 | } else |
---|
1124 | { |
---|
1125 | int @SYZCHECK = @DEBUG; |
---|
1126 | } |
---|
1127 | |
---|
1128 | if( @DEBUG ) |
---|
1129 | { |
---|
1130 | "SSstep::NextInducedRing"; |
---|
1131 | "basering: ", basering; attrib(basering); |
---|
1132 | } |
---|
1133 | |
---|
1134 | /* |
---|
1135 | // is initial weights are all zeroes! |
---|
1136 | def L = lead(M); |
---|
1137 | intvec @V = deg(M[1..ncols(M)]); @W; @V; @W = @V; attrib(L, "isHomog", @W); |
---|
1138 | SetInducedReferrence(L, @RANK, 0); |
---|
1139 | */ |
---|
1140 | |
---|
1141 | // def L = lead(MRES); |
---|
1142 | // @W = @W, @V; |
---|
1143 | // attrib(L, "isHomog", @W); |
---|
1144 | |
---|
1145 | |
---|
1146 | // General setting: |
---|
1147 | // SetInducedReferrence(MRES, 0, 0); // limit: 0! |
---|
1148 | int @l = size(RES); |
---|
1149 | |
---|
1150 | def M = RES[@l]; |
---|
1151 | |
---|
1152 | def L = LRES[@l]; |
---|
1153 | def T = TRES[@l]; |
---|
1154 | |
---|
1155 | |
---|
1156 | //// TODO: wrong !!!!! |
---|
1157 | int @RANK = ncols(MRES) - ncols(M); // nrows(M); // what if M is zero?! |
---|
1158 | |
---|
1159 | |
---|
1160 | |
---|
1161 | /* |
---|
1162 | if( @RANK != nrows(M) ) |
---|
1163 | { |
---|
1164 | type(MRES); |
---|
1165 | @RANK; |
---|
1166 | type(M); |
---|
1167 | pause(); |
---|
1168 | } |
---|
1169 | */ |
---|
1170 | |
---|
1171 | intvec @W = attrib(M, "isHomog"); intvec @V = attrib(M, "degrees"); @V = @W, @V; |
---|
1172 | |
---|
1173 | if( @DEBUG ) |
---|
1174 | { |
---|
1175 | "Sstep::NextInput: "; |
---|
1176 | M; |
---|
1177 | L; |
---|
1178 | @V; |
---|
1179 | @RANK; |
---|
1180 | // DetailedPrint(MRES); |
---|
1181 | attrib(MRES, "isHomog"); |
---|
1182 | } |
---|
1183 | |
---|
1184 | |
---|
1185 | // TODO: N = SYZ( M )!!! |
---|
1186 | module N, LL, TT; |
---|
1187 | (N, LL, TT) = SSComputeSyzygy(/*M, */L, T/*, @RANK*/); |
---|
1188 | |
---|
1189 | // shift syz.comp by @RANK: |
---|
1190 | module Z; |
---|
1191 | Z = 0; Z[@RANK] = 0; Z = Z, transpose(LL); LL = transpose(Z); |
---|
1192 | Z = 0; Z[@RANK] = 0; Z = Z, transpose(TT); TT = transpose(Z); |
---|
1193 | Z = 0; Z[@RANK] = 0; Z = Z, transpose(N); N = transpose(Z); |
---|
1194 | |
---|
1195 | |
---|
1196 | if( @SYZCHECK ) |
---|
1197 | { |
---|
1198 | if( size(N) > 0 ) |
---|
1199 | { |
---|
1200 | // next syz. property |
---|
1201 | if( size(module(transpose( transpose(N) * transpose(MRES) ))) > 0 ) |
---|
1202 | { |
---|
1203 | "MRES", MRES; |
---|
1204 | |
---|
1205 | "N: "; N; // DetailedPrint(N, 2); |
---|
1206 | |
---|
1207 | "LL:"; LL; // DetailedPrint(LL, 1); |
---|
1208 | "TT:"; TT; // DetailedPrint(TT, 10); |
---|
1209 | |
---|
1210 | "RANKS: ", @RANK; |
---|
1211 | |
---|
1212 | "transpose( transpose(N) * transpose(MRES) ) != 0!!!"; |
---|
1213 | transpose( transpose(N) * transpose(MRES) ); |
---|
1214 | |
---|
1215 | "transpose(N) * transpose(MRES): "; |
---|
1216 | transpose(N) * transpose(MRES); |
---|
1217 | // DetailedPrint(module(_), 2); |
---|
1218 | $ |
---|
1219 | } |
---|
1220 | } |
---|
1221 | } |
---|
1222 | |
---|
1223 | attrib(N, "isHomog", @V); |
---|
1224 | |
---|
1225 | // TODO: correct the following: |
---|
1226 | intvec @DEGS = deg(N[1..ncols(N)]); // no mod. comp. weights :( |
---|
1227 | |
---|
1228 | |
---|
1229 | attrib(N, "degrees", @DEGS); |
---|
1230 | |
---|
1231 | RES[@l + 1] = N; // list of all syzygy modules |
---|
1232 | LRES[@l + 1] = LL; // list of all syzygy modules |
---|
1233 | TRES[@l + 1] = TT; // list of all syzygy modules |
---|
1234 | |
---|
1235 | MRES = MRES, N; |
---|
1236 | |
---|
1237 | attrib(MRES, "isHomog", @V); |
---|
1238 | |
---|
1239 | // L = L, lead(N); attrib(basering, "InducionLeads", L); |
---|
1240 | |
---|
1241 | if( @DEBUG ) |
---|
1242 | { |
---|
1243 | "SSstep::NextSyzOutput: "; |
---|
1244 | N; |
---|
1245 | // DetailedPrint(N); |
---|
1246 | attrib(N); |
---|
1247 | } |
---|
1248 | |
---|
1249 | } |
---|
1250 | |
---|
1251 | proc SScontinue(int l) |
---|
1252 | "USAGE: SScontinue(l) |
---|
1253 | RETURN: nothing, instead it changes RES and MRES variables in the current ring |
---|
1254 | PURPOSE: computes further (at most l) syzygies |
---|
1255 | NOTE: must be used within a ring returned by Sres or Ssyz. RES and MRES are |
---|
1256 | explained in Sres |
---|
1257 | EXAMPLE: example Scontinue; shows an example |
---|
1258 | " |
---|
1259 | { |
---|
1260 | |
---|
1261 | /// TODO! |
---|
1262 | // def data = GetInducedData(); |
---|
1263 | |
---|
1264 | if( (!defined(RES)) || (!defined(MRES)) ) /* || (typeof(data) != "list") || (size(data) != 2) */ |
---|
1265 | { |
---|
1266 | ERROR("Sorry, but basering does not seem to be returned by Sres or Ssyz"); |
---|
1267 | } |
---|
1268 | for (; (l != 0) && (size(RES[size(RES)]) > 0); l-- ) |
---|
1269 | { |
---|
1270 | SSstep(); |
---|
1271 | } |
---|
1272 | } |
---|
1273 | example |
---|
1274 | { "EXAMPLE:"; echo = 2; |
---|
1275 | ring r; |
---|
1276 | module M = maxideal(1); M; |
---|
1277 | def S = SSsyz(M); setring S; S; |
---|
1278 | "Only the first syzygy: "; |
---|
1279 | RES; MRES; |
---|
1280 | "More syzygies: "; |
---|
1281 | SScontinue(10); |
---|
1282 | RES; MRES; |
---|
1283 | } |
---|
1284 | |
---|
1285 | proc SSsyz(def M) |
---|
1286 | "USAGE: SSsyz(M) |
---|
1287 | RETURN: ring, containing a list of modules RES and a module MRES |
---|
1288 | PURPOSE: computes the first syzygy module of M (wrt some Schreyer ordering)? |
---|
1289 | NOTE: The output is explained in Sres |
---|
1290 | EXAMPLE: example Ssyz; shows an example |
---|
1291 | " |
---|
1292 | { |
---|
1293 | if( (typeof(M) != "module") && (typeof(M) != "ideal") ) |
---|
1294 | { |
---|
1295 | ERROR("Sorry: need an ideal or a module for input"); |
---|
1296 | } |
---|
1297 | |
---|
1298 | def SS = SSinit(M); setring SS; |
---|
1299 | |
---|
1300 | SSstep(); // NOTE: what if M is zero? |
---|
1301 | |
---|
1302 | return (SS); |
---|
1303 | } |
---|
1304 | example |
---|
1305 | { "EXAMPLE:"; echo = 2; |
---|
1306 | ring r; |
---|
1307 | |
---|
1308 | /* ideal M = 0; |
---|
1309 | def S = SSsyz(M); setring S; S; |
---|
1310 | "Only the first syzygy: "; |
---|
1311 | RES; LRES; TRES; |
---|
1312 | MRES; |
---|
1313 | |
---|
1314 | kill S; setring r; kill M; |
---|
1315 | */ |
---|
1316 | |
---|
1317 | module M = maxideal(1); M; |
---|
1318 | def S = SSres(M, 0); setring S; S; |
---|
1319 | MRES; |
---|
1320 | RES; |
---|
1321 | ""; |
---|
1322 | LRES; |
---|
1323 | ""; |
---|
1324 | TRES; |
---|
1325 | |
---|
1326 | kill S; setring r; kill M; |
---|
1327 | |
---|
1328 | kill r; |
---|
1329 | |
---|
1330 | ring R = 0, (w, x, y, z), dp; |
---|
1331 | ideal M = w^2 - x*z, w*x - y*z, x^2 - w*y, x*y - z^2, y^2 - w*z; |
---|
1332 | |
---|
1333 | def S = SSres(M, 0); setring S; S; |
---|
1334 | MRES; |
---|
1335 | RES; |
---|
1336 | ""; |
---|
1337 | LRES; |
---|
1338 | ""; |
---|
1339 | TRES; |
---|
1340 | } |
---|
1341 | |
---|
1342 | proc SSres(def M, int l) |
---|
1343 | "USAGE: SSres(I, l) |
---|
1344 | RETURN: ring, containing a list of modules RES and a module MRES |
---|
1345 | PURPOSE: computes (at most l) syzygy modules of M wrt the classical Schreyer |
---|
1346 | induced ordering with gen(i) > gen(j) if i > j, provided both gens |
---|
1347 | are from the same syzygy level.??? |
---|
1348 | NOTE: RES contains the images of maps subsituting the beginning of the |
---|
1349 | Schreyer free resolution of baseRing^r/M, while MRES is a sum of |
---|
1350 | these images in a big free sum, containing all the syzygy modules. |
---|
1351 | The syzygy modules are shifted so that gen(i) correspons to MRES[i]. |
---|
1352 | The leading zero module RES[0] indicates the fact that coker of the |
---|
1353 | first map is zero. The number of zeroes inducates the rank of input. |
---|
1354 | NOTE: If l == 0 then l is set to be nvars(basering) + 1 |
---|
1355 | EXAMPLE: example SSres; shows an example |
---|
1356 | " |
---|
1357 | { |
---|
1358 | if( (typeof(M) != "module") && (typeof(M) != "ideal") ) |
---|
1359 | { |
---|
1360 | ERROR("Sorry: need an ideal or a module for input"); |
---|
1361 | } |
---|
1362 | |
---|
1363 | def SS = SSinit(M); setring SS; |
---|
1364 | |
---|
1365 | if (l == 0) |
---|
1366 | { |
---|
1367 | l = nvars(basering) + 1; // not really an estimate...?! |
---|
1368 | } |
---|
1369 | |
---|
1370 | SSstep(); l = l - 1; |
---|
1371 | |
---|
1372 | SScontinue(l); |
---|
1373 | |
---|
1374 | return (SS); |
---|
1375 | } |
---|
1376 | example |
---|
1377 | { "EXAMPLE:"; echo = 2; |
---|
1378 | ring r; |
---|
1379 | module M = maxideal(1); M; |
---|
1380 | def S = SSres(M, 0); setring S; S; |
---|
1381 | RES; |
---|
1382 | MRES; |
---|
1383 | kill S; |
---|
1384 | setring r; kill M; |
---|
1385 | |
---|
1386 | def A = nc_algebra(-1,0); setring A; |
---|
1387 | ideal Q = var(1)^2, var(2)^2, var(3)^2; |
---|
1388 | qring SCA = twostd(Q); |
---|
1389 | basering; |
---|
1390 | |
---|
1391 | module M = maxideal(1); |
---|
1392 | def S = SSres(M, 2); setring S; S; |
---|
1393 | RES; |
---|
1394 | MRES; |
---|
1395 | } |
---|
1396 | |
---|
1397 | |
---|
1398 | |
---|
1399 | static proc loadme() |
---|
1400 | { |
---|
1401 | int @DEBUG = !system("with", "ndebug"); |
---|
1402 | |
---|
1403 | if( @DEBUG ) |
---|
1404 | { |
---|
1405 | |
---|
1406 | "ndebug?: ", system("with", "ndebug"); |
---|
1407 | "om_ndebug?: ", system("with", "om_ndebug"); |
---|
1408 | |
---|
1409 | listvar(Top); |
---|
1410 | listvar(Schreyer); |
---|
1411 | } |
---|
1412 | // listvar(Syzextra); listvar(Syzextra_g); |
---|
1413 | |
---|
1414 | if( !defined(DetailedPrint) ) |
---|
1415 | { |
---|
1416 | if( 1 ) |
---|
1417 | { |
---|
1418 | |
---|
1419 | if( @DEBUG ) |
---|
1420 | { |
---|
1421 | "Loading the Release version!"; |
---|
1422 | } |
---|
1423 | load("syzextra.so"); |
---|
1424 | |
---|
1425 | if( @DEBUG ) |
---|
1426 | { |
---|
1427 | listvar(Syzextra); |
---|
1428 | } |
---|
1429 | |
---|
1430 | exportto(Top, Syzextra::ClearContent); |
---|
1431 | exportto(Top, Syzextra::ClearDenominators); |
---|
1432 | |
---|
1433 | // export Syzextra; |
---|
1434 | |
---|
1435 | // exportto(Schreyer, Syzextra::noop); |
---|
1436 | exportto(Schreyer, Syzextra::DetailedPrint); |
---|
1437 | exportto(Schreyer, Syzextra::leadmonomial); |
---|
1438 | exportto(Schreyer, Syzextra::leadcomp); |
---|
1439 | // exportto(Schreyer, Syzextra::leadrawexp); |
---|
1440 | // exportto(Schreyer, Syzextra::ISUpdateComponents); |
---|
1441 | exportto(Schreyer, Syzextra::SetInducedReferrence); |
---|
1442 | exportto(Schreyer, Syzextra::GetInducedData); |
---|
1443 | // exportto(Schreyer, Syzextra::GetAMData); |
---|
1444 | // exportto(Schreyer, Syzextra::SetSyzComp); |
---|
1445 | exportto(Schreyer, Syzextra::MakeInducedSchreyerOrdering); |
---|
1446 | // exportto(Schreyer, Syzextra::MakeSyzCompOrdering); |
---|
1447 | exportto(Schreyer, Syzextra::idPrepare); |
---|
1448 | // exportto(Schreyer, Syzextra::reduce_syz); |
---|
1449 | // exportto(Schreyer, Syzextra::p_Content); |
---|
1450 | |
---|
1451 | exportto(Schreyer, Syzextra::ProfilerStart); exportto(Schreyer, Syzextra::ProfilerStop); |
---|
1452 | |
---|
1453 | exportto(Schreyer, Syzextra::Tail); |
---|
1454 | |
---|
1455 | exportto(Schreyer, Syzextra::m2_end); |
---|
1456 | } |
---|
1457 | /* |
---|
1458 | else |
---|
1459 | { |
---|
1460 | if( @DEBUG ) |
---|
1461 | { |
---|
1462 | "Loading the Debug version!"; |
---|
1463 | } |
---|
1464 | |
---|
1465 | load("syzextra.so"); |
---|
1466 | |
---|
1467 | if( @DEBUG ) |
---|
1468 | { |
---|
1469 | listvar(Syzextra_g); |
---|
1470 | } |
---|
1471 | |
---|
1472 | exportto(Top, Syzextra_g::ClearContent); |
---|
1473 | exportto(Top, Syzextra_g::ClearDenominators); |
---|
1474 | |
---|
1475 | // export Syzextra_g; |
---|
1476 | // exportto(Schreyer, Syzextra_g::noop); |
---|
1477 | exportto(Schreyer, Syzextra_g::DetailedPrint); |
---|
1478 | exportto(Schreyer, Syzextra_g::leadmonomial); |
---|
1479 | exportto(Schreyer, Syzextra_g::leadcomp); |
---|
1480 | // exportto(Schreyer, Syzextra_g::leadrawexp); |
---|
1481 | // exportto(Schreyer, Syzextra_g::ISUpdateComponents); |
---|
1482 | exportto(Schreyer, Syzextra_g::SetInducedReferrence); |
---|
1483 | exportto(Schreyer, Syzextra_g::GetInducedData); |
---|
1484 | // exportto(Schreyer, Syzextra_g::GetAMData); |
---|
1485 | // exportto(Schreyer, Syzextra_g::SetSyzComp); |
---|
1486 | exportto(Schreyer, Syzextra_g::MakeInducedSchreyerOrdering); |
---|
1487 | // exportto(Schreyer, Syzextra_g::MakeSyzCompOrdering); |
---|
1488 | exportto(Schreyer, Syzextra_g::idPrepare); |
---|
1489 | // exportto(Schreyer, Syzextra_g::reduce_syz); |
---|
1490 | // exportto(Schreyer, Syzextra_g::p_Content); |
---|
1491 | |
---|
1492 | exportto(Schreyer, Syzextra_g::ProfilerStart); exportto(Schreyer, Syzextra_g::ProfilerStop); |
---|
1493 | |
---|
1494 | exportto(Schreyer, Syzextra_g::Tail); |
---|
1495 | |
---|
1496 | |
---|
1497 | exportto(Schreyer, Syzextra_g::m2_end); |
---|
1498 | } |
---|
1499 | */ |
---|
1500 | |
---|
1501 | exportto(Top, DetailedPrint); |
---|
1502 | exportto(Top, GetInducedData); |
---|
1503 | |
---|
1504 | if( @DEBUG ) |
---|
1505 | { |
---|
1506 | listvar(Top); |
---|
1507 | listvar(Schreyer); |
---|
1508 | } |
---|
1509 | } |
---|
1510 | |
---|
1511 | if( !defined(GetInducedData) ) |
---|
1512 | { |
---|
1513 | ERROR("Sorry but we are missing the dynamic module (syzextra.so)..."); |
---|
1514 | } |
---|
1515 | |
---|
1516 | } |
---|
1517 | |
---|
1518 | static proc mod_init() |
---|
1519 | { |
---|
1520 | loadme(); |
---|
1521 | } |
---|
1522 | |
---|
1523 | |
---|
1524 | proc testallSexamples() |
---|
1525 | { |
---|
1526 | example Ssyz; |
---|
1527 | example Scontinue; |
---|
1528 | example Sres; |
---|
1529 | } |
---|
1530 | |
---|
1531 | proc testallSSexamples() |
---|
1532 | { |
---|
1533 | example SSsyz; |
---|
1534 | example SScontinue; |
---|
1535 | example SSres; |
---|
1536 | } |
---|
1537 | |
---|
1538 | example |
---|
1539 | { "EXAMPLE:"; echo = 2; |
---|
1540 | testallSexamples(); |
---|
1541 | testallSSexamples(); |
---|
1542 | } |
---|