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 resoltion (cf. [SFO]), |
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10 | originally meant to be used by @code{derham.lib} (which requires resolutions over the homogenized Weyl algebra). |
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11 | The library works both in the commutative and non-commutative setting (cf. [MO]). |
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12 | Here, we call a free resolution a Schreyer resolution if each syzygy module is given by a Groebner basis |
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13 | with respect to the corresponding Schreyer ordering. |
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14 | A Schreyer resolution can be much bigger than a minimal resolution of the same module, but may be easier to construct. |
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15 | @* The input for the resolution computations is a set of vectors @code{M} in form of a module over some basering @code{R}. |
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16 | The ring @code{R} may be non-commutative, in which case the ring ordering should be global. |
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17 | @* These procedures produce/work with partial Schreyer resolutions of @code{(R^rank(M))/M} in form of |
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18 | a ring (endowed with a special ring ordering that will be extended in the course of a resolution computation) |
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19 | containing a list of modules @code{RES} and a module @code{MRES}: |
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20 | @* The list of modules @code{RES} contains the images of maps (also called syzygy modules) substituting the |
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21 | computed beginning of a Schreyer resolution, that is, each syzygy module is given by a Groebner basis |
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22 | with respect to the corresponding Schreyer ordering. |
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23 | @* 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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24 | the first differential map is zero. The second map @code{RES[2]} is given by @code{M}, which indicates that |
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25 | the resolution of @code{(R^rank(M))/M} is being computed. |
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26 | @* The module @code{MRES} is a direct sum of modules from @code{RES} and thus comprises all computed differentials. |
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27 | @* Syzygies are shifted so that @code{gen(i)} is mapped to @code{MRES[i]} under the differential map. |
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28 | @* The Schreyer ordering succesively extends the starting module ordering on @code{M} (defined in Singular by the basering @code{R}) |
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29 | and is extended to higher syzygies using the following definition: |
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30 | @* a < b if and only if (d(a) < d(b)) OR ( (d(a) = d(b) AND (comp(a) < comp(b)) ), |
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31 | @* where @code{d(a)} is the image of a under the differential (given by @code{MRES}), |
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32 | and @code{comp(a)} is the module component, for any module terms @code{a} and @code{b} from the same higher syzygy module. |
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33 | REFERENCES: |
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34 | [SFO] Schreyer, F.O.: Die Berechnung von Syzygien mit dem verallgemeinerten Weierstrassschen Divisionssatz, |
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35 | Master's thesis, Univ. Hamburg, 1980. |
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36 | [MO] Motsak, O.: Non-commutative Computer Algebra with applications: Graded commutative algebra and related |
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37 | structures in Singular with applications, Ph.D. thesis, TU Kaiserslautern, 2010 |
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38 | |
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39 | NOTE: requires the dynamic or built-in module @code{syzextra} |
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40 | |
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41 | PROCEDURES: |
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42 | Sres(M,len) compute Schreyer resolution of module M of maximal length len |
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43 | Ssyz(M) compute Schreyer resolution of module M of length 1 |
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44 | Scontinue(len) extend currently active resolution by (at most) len syszygies |
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45 | "; |
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46 | |
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47 | static proc prepareSyz( module I, list # ) |
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48 | { |
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49 | int i; |
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50 | int k = 0; |
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51 | int r = nrows(I); |
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52 | int c = ncols(I); |
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53 | |
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54 | |
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55 | if( size(#) > 0 ) |
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56 | { |
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57 | if( typeof(#[1]) == "int" || typeof(#[1]) == "bigint" ) |
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58 | { |
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59 | k = #[1]; |
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60 | } |
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61 | } |
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62 | |
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63 | if( k < r ) |
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64 | { |
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65 | "// *** Wrong k: ", k, " < nrows: ", r, " => setting k = r = ", r; |
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66 | k = r; |
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67 | } |
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68 | |
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69 | // "k: ", k; "c: ", c; "I: ", I; |
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70 | |
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71 | for( i = c; i > 0; i-- ) |
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72 | { |
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73 | I[i] = I[i] + gen(k + i); |
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74 | } |
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75 | |
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76 | // DetailedPrint(I); |
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77 | |
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78 | return(I); |
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79 | } |
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80 | |
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81 | static proc separateSyzGB( module J, int c ) |
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82 | { |
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83 | module II, G; vector v; int i; |
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84 | |
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85 | J = simplify(J, 2); |
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86 | |
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87 | for( i = ncols(J); i > 0; i-- ) |
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88 | { |
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89 | v = J[i]; |
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90 | if( leadcomp(v) > c ) |
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91 | { |
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92 | II[i] = v; |
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93 | } else |
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94 | { |
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95 | G[i] = v; // leave only gen(i): i <= c |
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96 | } |
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97 | } |
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98 | |
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99 | II = simplify(II, 2); |
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100 | G = simplify(G, 2); |
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101 | |
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102 | return (list(G, II)); |
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103 | } |
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104 | |
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105 | static proc splitSyzGB( module J, int c ) |
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106 | { |
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107 | module JJ; vector v, vv; int i; |
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108 | |
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109 | for( i = ncols(J); i > 0; i-- ) |
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110 | { |
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111 | v = J[i]; |
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112 | |
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113 | vv = 0; |
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114 | |
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115 | while( leadcomp(v) <= c ) |
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116 | { |
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117 | vv = vv + lead(v); |
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118 | v = v - lead(v); |
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119 | } |
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120 | |
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121 | J[i] = vv; |
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122 | JJ[i] = v; |
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123 | } |
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124 | |
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125 | J = simplify(J, 2); |
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126 | JJ = simplify(JJ, 2); |
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127 | |
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128 | return (list(J, JJ)); |
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129 | } |
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130 | |
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131 | |
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132 | static proc Sinit(module M) |
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133 | { |
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134 | def @save = basering; |
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135 | |
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136 | int @DEBUG = !system("with", "ndebug"); |
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137 | if( @DEBUG ) |
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138 | { |
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139 | "Sinit::Input"; |
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140 | type(M); |
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141 | DetailedPrint(M); |
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142 | attrib(M); |
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143 | } |
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144 | |
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145 | int @RANK = nrows(M); int @SIZE = ncols(M); |
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146 | |
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147 | int @IS_A_SB = attrib(M, "isSB"); // ??? only if all weights were zero?! |
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148 | |
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149 | if( !@IS_A_SB ) |
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150 | { |
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151 | M = std(M); // this should be faster than computing std in S (later on) |
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152 | } |
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153 | |
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154 | def S = MakeInducedSchreyerOrdering(1); // 1 puts history terms to the back |
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155 | // TODO: NOTE: +1 causes trouble to Singular interpreter!!!??? |
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156 | setring S; // a new ring with a Schreyer ordering |
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157 | |
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158 | if( @DEBUG ) |
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159 | { |
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160 | "Sinit::StartingISRing"; |
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161 | basering; |
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162 | // DetailedPrint(basering); |
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163 | } |
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164 | |
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165 | // Setup the leading syzygy^{-1} module to zero: |
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166 | module Z = 0; Z[@RANK] = 0; attrib(Z, "isHomog", intvec(0)); |
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167 | |
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168 | module MRES = Z; |
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169 | |
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170 | list RES; RES[1] = Z; |
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171 | |
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172 | module F = freemodule(@RANK); |
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173 | intvec @V = deg(F[1..@RANK]); |
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174 | |
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175 | module M = imap(@save, M); |
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176 | |
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177 | attrib(M, "isHomog", @V); |
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178 | attrib(M, "isSB", 1); |
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179 | |
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180 | |
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181 | if( @DEBUG ) |
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182 | { |
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183 | "Sinit::SB_Input: "; |
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184 | type(M); |
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185 | attrib(M); |
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186 | attrib(M, "isHomog"); |
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187 | DetailedPrint(M); |
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188 | } |
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189 | |
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190 | if( @DEBUG ) |
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191 | { |
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192 | // 0^th syz. property |
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193 | if( size(module(transpose( transpose(M) * transpose(MRES) ))) > 0 ) |
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194 | { |
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195 | transpose( transpose(M) * transpose(MRES) ); |
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196 | "transpose( transpose(M) * transpose(MRES) ) != 0!!!"; |
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197 | $ |
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198 | } |
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199 | } |
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200 | |
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201 | RES[size(RES)+1] = M; // list of all syzygy modules |
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202 | MRES = MRES, M; |
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203 | |
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204 | attrib(MRES, "isHomog", @V); |
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205 | |
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206 | attrib(S, "InducionLeads", lead(M)); |
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207 | attrib(S, "InducionStart", @RANK); |
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208 | |
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209 | if( @DEBUG ) |
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210 | { |
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211 | "Sinit::MRES"; |
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212 | DetailedPrint(MRES); |
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213 | attrib(MRES, "isHomog"); |
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214 | attrib(S); |
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215 | } |
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216 | |
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217 | export RES; |
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218 | export MRES; |
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219 | return (S); |
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220 | } |
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221 | |
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222 | static proc Sstep() |
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223 | { |
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224 | int @DEBUG = !system("with", "ndebug"); |
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225 | |
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226 | if( @DEBUG ) |
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227 | { |
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228 | "Sstep::NextInducedRing"; |
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229 | DetailedPrint(basering); |
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230 | |
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231 | attrib(basering, "InducionLeads"); |
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232 | attrib(basering, "InducionStart"); |
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233 | |
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234 | GetInducedData(); |
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235 | } |
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236 | |
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237 | // syzygy step: |
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238 | |
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239 | /* |
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240 | // is initial weights are all zeroes! |
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241 | def L = lead(M); |
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242 | intvec @V = deg(M[1..ncols(M)]); @W; @V; @W = @V; attrib(L, "isHomog", @W); |
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243 | SetInducedReferrence(L, @RANK, 0); |
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244 | */ |
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245 | |
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246 | // def L = lead(MRES); |
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247 | // @W = @W, @V; |
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248 | // attrib(L, "isHomog", @W); |
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249 | |
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250 | |
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251 | // General setting: |
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252 | // SetInducedReferrence(MRES, 0, 0); // limit: 0! |
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253 | int @l = size(RES); |
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254 | |
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255 | module M = RES[@l]; |
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256 | |
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257 | module L = attrib(basering, "InducionLeads"); |
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258 | int limit = attrib(basering, "InducionStart"); |
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259 | |
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260 | // L; limit; |
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261 | |
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262 | int @RANK = ncols(MRES) - ncols(M); // nrows(M); // what if M is zero?! |
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263 | |
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264 | /* |
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265 | if( @RANK != nrows(M) ) |
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266 | { |
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267 | type(MRES); |
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268 | @RANK; |
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269 | type(M); |
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270 | pause(); |
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271 | } |
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272 | */ |
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273 | |
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274 | intvec @W = attrib(M, "isHomog"); |
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275 | intvec @V = deg(M[1..ncols(M)]); |
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276 | @V = @W, @V; |
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277 | |
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278 | if( @DEBUG ) |
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279 | { |
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280 | "Sstep::NextInput: "; |
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281 | M; |
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282 | deg(M[1..ncols(M)]); // no use of @W :(? |
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283 | @RANK; |
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284 | DetailedPrint(MRES); |
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285 | attrib(MRES, "isHomog"); @W; |
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286 | deg(MRES[1..ncols(MRES)]); |
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287 | } |
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288 | |
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289 | |
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290 | |
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291 | SetInducedReferrence(L, limit, 0); |
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292 | |
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293 | def K = prepareSyz(M, @RANK); |
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294 | // K; |
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295 | |
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296 | // attrib(K, "isHomog", @V); DetailedPrint(K, 1000); |
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297 | |
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298 | // pause(); |
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299 | |
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300 | K = idPrepare(K, @RANK); // std(K); // ? |
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301 | K = simplify(K, 2); |
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302 | |
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303 | // K; |
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304 | |
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305 | module N = separateSyzGB(K, @RANK)[2]; // 1^st syz. module: vectors which start in lower part (comp >= @RANK) |
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306 | |
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307 | // "N_0: "; N; DetailedPrint(N, 10); |
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308 | |
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309 | // basering; print(@V); type(N); |
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310 | // attrib(N, "isHomog", @V); // TODO: fix "wrong weights"!!!? deg is wrong :((( |
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311 | N = std(N); |
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312 | attrib(N, "isHomog", @V); |
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313 | |
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314 | // N; |
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315 | |
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316 | if( @DEBUG ) |
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317 | { |
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318 | if( size(N) > 0 ) |
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319 | { |
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320 | // next syz. property |
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321 | if( size(module(transpose( transpose(N) * transpose(MRES) ))) > 0 ) |
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322 | { |
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323 | MRES; |
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324 | |
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325 | "N: "; N; DetailedPrint(N, 10); |
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326 | |
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327 | "K:"; K; DetailedPrint(K, 10); |
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328 | |
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329 | "RANKS: ", @RANK; |
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330 | |
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331 | "transpose( transpose(N) * transpose(MRES) ) != 0!!!"; |
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332 | transpose( transpose(N) * transpose(MRES) ); |
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333 | |
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334 | "transpose(N) * transpose(MRES): "; |
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335 | transpose(N) * transpose(MRES); |
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336 | DetailedPrint(module(_), 2); |
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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 | |
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342 | RES[@l + 1] = N; // list of all syzygy modules |
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343 | |
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344 | MRES = MRES, N; |
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345 | attrib(MRES, "isHomog", @V); |
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346 | |
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347 | |
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348 | L = L, lead(N); |
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349 | attrib(basering, "InducionLeads", L); |
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350 | |
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351 | if( @DEBUG ) |
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352 | { |
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353 | "Sstep::NextSyzOutput: "; |
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354 | DetailedPrint(N); |
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355 | attrib(N, "isHomog"); |
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356 | } |
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357 | |
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358 | } |
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359 | |
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360 | proc Scontinue(int l) |
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361 | "USAGE: Scontinue(int len) |
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362 | RETURN: nothing, instead it changes the currently active resolution |
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363 | PURPOSE: extends the currently active resolution by at most len syzygies |
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364 | ASSUME: must be used within a ring returned by Sres or Ssyz |
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365 | EXAMPLE: example Scontinue; shows an example |
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366 | " |
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367 | { |
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368 | def data = GetInducedData(); |
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369 | |
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370 | if( (!defined(RES)) || (!defined(MRES)) || (typeof(data) != "list") || (size(data) != 2) ) |
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371 | { |
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372 | ERROR("Sorry, but basering does not seem to be returned by Sres or Ssyz"); |
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373 | } |
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374 | for (; (l != 0) && (size(RES[size(RES)]) > 0); l-- ) |
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375 | { |
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376 | Sstep(); |
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377 | } |
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378 | } |
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379 | example |
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380 | { "EXAMPLE:"; echo = 2; |
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381 | ring r; |
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382 | module M = maxideal(1); M; |
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383 | def S = Ssyz(M); setring S; S; |
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384 | "Only the first syzygy: "; |
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385 | RES; MRES; |
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386 | "More syzygies: "; |
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387 | Scontinue(10); |
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388 | RES; MRES; |
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389 | } |
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390 | |
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391 | proc Ssyz(module M) |
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392 | "USAGE: Ssyz(module M) |
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393 | RETURN: ring, containing a Schreyer resolution |
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394 | PURPOSE: computes a Schreyer resolution of M of length 1 (see the library overview) |
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395 | SEE ALSO: Sres |
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396 | EXAMPLE: example Ssyz; shows an example |
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397 | " |
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398 | { |
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399 | def S = Sinit(M); setring S; |
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400 | |
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401 | Sstep(); // NOTE: what if M is zero? |
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402 | |
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403 | return (S); |
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404 | } |
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405 | example |
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406 | { "EXAMPLE:"; echo = 2; |
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407 | ring r; |
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408 | module M = maxideal(1); M; |
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409 | def S = Ssyz(M); setring S; S; |
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410 | "Only the first syzygy: "; |
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411 | RES; |
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412 | MRES; // Note gen(i) |
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413 | kill S; |
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414 | setring r; kill M; |
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415 | |
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416 | module M = 0; |
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417 | def S = Ssyz(M); setring S; S; |
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418 | "Only the first syzygy: "; |
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419 | RES; |
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420 | MRES; |
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421 | } |
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422 | |
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423 | proc Sres(module M, int l) |
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424 | "USAGE: Sres(module M, int len) |
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425 | RETURN: ring, containing a Schreyer resolution |
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426 | PURPOSE: computes a Schreyer resolution of M of length at most len (see the library overview) |
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427 | NOTE: If given len is zero then nvars(basering) + 1 is used instead. |
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428 | SEE ALSO: Ssyz |
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429 | EXAMPLE: example Sres; shows an example |
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430 | " |
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431 | { |
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432 | def S = Sinit(M); setring S; |
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433 | |
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434 | if (l == 0) |
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435 | { |
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436 | l = nvars(basering) + 1; // not really an estimate...?! |
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437 | } |
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438 | |
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439 | Sstep(); l = l - 1; |
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440 | |
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441 | Scontinue(l); |
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442 | |
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443 | return (S); |
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444 | } |
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445 | example |
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446 | { "EXAMPLE:"; echo = 2; |
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447 | ring r; |
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448 | module M = maxideal(1); M; |
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449 | def S = Sres(M, 0); setring S; S; |
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450 | RES; |
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451 | MRES; |
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452 | kill S; |
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453 | setring r; kill M; |
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454 | |
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455 | def A = nc_algebra(-1,0); setring A; |
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456 | ideal Q = var(1)^2, var(2)^2, var(3)^2; |
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457 | qring SCA = twostd(Q); |
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458 | basering; |
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459 | |
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460 | module M = maxideal(1); |
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461 | def S = Sres(M, 2); setring S; S; |
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462 | RES; |
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463 | MRES; |
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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 | |
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471 | LIB "general.lib"; // for sort |
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472 | |
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473 | /* static proc Tail(def M) // DONE: in C++ (dyn. module: syzextra)! |
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474 | { |
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475 | int i = ncols(M); def m; |
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476 | while (i > 0) |
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477 | { |
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478 | m = M[i]; |
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479 | m = m - lead(m); // m = tail(m) |
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480 | M[i] = m; |
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481 | i--; |
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482 | } |
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483 | return (M); |
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484 | }*/ |
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485 | |
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486 | /* static */ |
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487 | proc MySort(def M) |
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488 | { |
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489 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
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490 | { |
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491 | int @DEBUG = attrib(basering, "DEBUG"); |
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492 | } else |
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493 | { |
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494 | int @DEBUG = !system("with", "ndebug"); |
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495 | } |
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496 | |
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497 | if( typeof( attrib(basering, "SYZCHECK") ) == "int" ) |
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498 | { |
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499 | int @SYZCHECK = attrib(basering, "SYZCHECK"); |
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500 | } else |
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501 | { |
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502 | int @SYZCHECK = @DEBUG; |
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503 | } |
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504 | |
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505 | |
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506 | if( @DEBUG ) |
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507 | { |
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508 | "MySort:: Input: "; M; |
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509 | } |
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510 | |
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511 | def @N = M; Sort_c_ds(@N); |
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512 | |
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513 | if( @SYZCHECK ) |
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514 | { |
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515 | def iv = sort(lead(M), "c,ds", 1)[2]; // ,1 => reversed! // TODO: not needed? |
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516 | def @M = M; |
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517 | @M = M[iv]; |
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518 | |
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519 | // 0^th syz. property |
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520 | if( size(module( matrix(@N) - matrix(@M) )) > 0 ) |
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521 | { |
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522 | "@M:"; @M; |
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523 | "@N:"; @N; |
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524 | |
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525 | "module( matrix(@N) - matrix(@M) ): "; |
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526 | module( matrix(@N) - matrix(@M) ); |
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527 | |
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528 | "ERROR: wrong sorting in 'MySort': @N != @M!!!"; |
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529 | $ |
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530 | } |
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531 | } |
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532 | |
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533 | if( @DEBUG ) |
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534 | { |
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535 | "MySort:: Ouput: "; @N; |
---|
536 | } |
---|
537 | |
---|
538 | return (@N); |
---|
539 | } |
---|
540 | |
---|
541 | |
---|
542 | /* static */ proc SSinit(def M) |
---|
543 | { |
---|
544 | if( (typeof(M) != "module") && (typeof(M) != "ideal") ) |
---|
545 | { |
---|
546 | ERROR("Sorry: need an ideal or a module for input"); |
---|
547 | } |
---|
548 | |
---|
549 | // TODO! DONE? |
---|
550 | def @save = basering; |
---|
551 | |
---|
552 | int @DEBUG = !system("with", "ndebug"); |
---|
553 | |
---|
554 | if( typeof( attrib(SSinit, "DEBUG") ) == "int" ) |
---|
555 | { |
---|
556 | @DEBUG = attrib(SSinit, "DEBUG"); |
---|
557 | } |
---|
558 | |
---|
559 | int @SYZCHECK = @DEBUG; |
---|
560 | |
---|
561 | if( typeof( attrib(SSinit, "SYZCHECK") ) == "int" ) |
---|
562 | { |
---|
563 | @SYZCHECK = attrib(SSinit, "SYZCHECK"); |
---|
564 | } |
---|
565 | |
---|
566 | if( @DEBUG ) |
---|
567 | { |
---|
568 | "SSinit::Input"; |
---|
569 | type(M); |
---|
570 | // DetailedPrint(M); |
---|
571 | attrib(M); |
---|
572 | } |
---|
573 | |
---|
574 | int @RANK = nrows(M); int @SIZE = ncols(M); |
---|
575 | |
---|
576 | int @IS_A_SB = attrib(M, "isSB"); // ??? only if all weights were zero?! |
---|
577 | |
---|
578 | if( !@IS_A_SB ) |
---|
579 | { |
---|
580 | def opts = option(get); |
---|
581 | option(redSB); option(redTail); |
---|
582 | M = std(M); |
---|
583 | option(set, opts); |
---|
584 | kill opts; |
---|
585 | } else |
---|
586 | { |
---|
587 | M = simplify(M, 2 + 4 + 32); |
---|
588 | } |
---|
589 | |
---|
590 | def @N = MySort(M); // TODO: replace with inplace sorting!!! |
---|
591 | def LEAD = lead(@N); |
---|
592 | |
---|
593 | if( @SYZCHECK ) |
---|
594 | { |
---|
595 | def @LEAD = lead(M); |
---|
596 | |
---|
597 | // sort wrt neg.deg.rev.lex! |
---|
598 | intvec iv_ds = sort(@LEAD, "c,ds", 1)[2]; // ,1 => reversed! |
---|
599 | |
---|
600 | M = M[iv_ds]; // sort M wrt ds on current leading terms |
---|
601 | @LEAD = @LEAD[iv_ds]; |
---|
602 | |
---|
603 | // 0^th syz. property |
---|
604 | if( size(module( matrix(@N) - matrix(M) )) > 0 ) |
---|
605 | { |
---|
606 | "M:"; M; |
---|
607 | "@N:"; @N; |
---|
608 | |
---|
609 | "module( matrix(@N) - matrix(M) ): "; |
---|
610 | module( matrix(@N) - matrix(M) ); |
---|
611 | |
---|
612 | "ERROR: wrong sorting (in SSnit): @N != M!!!"; |
---|
613 | $ |
---|
614 | } |
---|
615 | |
---|
616 | // 0^th syz. property |
---|
617 | if( size(module( matrix(@LEAD) - matrix(LEAD) )) > 0 ) |
---|
618 | { |
---|
619 | "LEAD:"; LEAD; |
---|
620 | "@LEAD:"; @LEAD; |
---|
621 | |
---|
622 | "module( matrix(@LEAD) - matrix(LEAD) ): "; |
---|
623 | module( matrix(@LEAD) - matrix(LEAD) ); |
---|
624 | |
---|
625 | "ERROR: wrong sorting (in SSnit): @LEAD != LEAD!!!"; |
---|
626 | $ |
---|
627 | } |
---|
628 | |
---|
629 | } |
---|
630 | |
---|
631 | M = @N; |
---|
632 | |
---|
633 | |
---|
634 | |
---|
635 | |
---|
636 | def TAIL = Tail(M); |
---|
637 | |
---|
638 | intvec @DEGS = deg(M[1..@SIZE]); // store actuall degrees of input elements |
---|
639 | |
---|
640 | // TODO: what about real modules? weighted ones? |
---|
641 | |
---|
642 | list @l = ringlist(@save); |
---|
643 | |
---|
644 | int @z = 0; ideal @m = maxideal(1); intvec @wdeg = deg(@m[1..ncols(@m)]); |
---|
645 | |
---|
646 | // NOTE: @wdeg will be ignored anyway :( |
---|
647 | @l[3] = list(list("C", @z), list("lp", @wdeg)); |
---|
648 | |
---|
649 | kill @z, @wdeg; // since these vars are ring independent! |
---|
650 | |
---|
651 | def S = ring(@l); // --MakeInducedSchreyerOrdering(1); |
---|
652 | |
---|
653 | module F = freemodule(@RANK); |
---|
654 | intvec @V = deg(F[1..@RANK]); |
---|
655 | |
---|
656 | setring S; // ring with an easy divisibility test ("C, lex") |
---|
657 | |
---|
658 | if( @DEBUG ) |
---|
659 | { |
---|
660 | "SSinit::NewRing(C, lex)"; |
---|
661 | basering; |
---|
662 | DetailedPrint(basering); |
---|
663 | } |
---|
664 | |
---|
665 | // Setup the leading syzygy^{-1} module to zero: |
---|
666 | module Z = 0; Z[@RANK] = 0; attrib(Z, "isHomog", intvec(0)); |
---|
667 | |
---|
668 | module MRES = Z; |
---|
669 | |
---|
670 | list RES; RES[1] = Z; |
---|
671 | list LRES; LRES[1] = Z; |
---|
672 | list TRES; TRES[1] = Z; |
---|
673 | |
---|
674 | def M = imap(@save, M); |
---|
675 | |
---|
676 | attrib(M, "isHomog", @V); |
---|
677 | attrib(M, "isSB", 1); |
---|
678 | attrib(M, "degrees", @DEGS); |
---|
679 | |
---|
680 | def LEAD = imap(@save, LEAD); |
---|
681 | |
---|
682 | attrib(LEAD, "isHomog", @V); |
---|
683 | attrib(LEAD, "isSB", 1); |
---|
684 | |
---|
685 | def TAIL = imap(@save, TAIL); |
---|
686 | |
---|
687 | if( @DEBUG ) |
---|
688 | { |
---|
689 | "SSinit::(sorted) SB_Input: "; |
---|
690 | type(M); |
---|
691 | attrib(M); |
---|
692 | attrib(M, "isHomog"); |
---|
693 | // DetailedPrint(M); |
---|
694 | } |
---|
695 | |
---|
696 | if( @SYZCHECK ) |
---|
697 | { |
---|
698 | // 0^th syz. property |
---|
699 | if( size(module(transpose( transpose(M) * transpose(MRES) ))) > 0 ) |
---|
700 | { |
---|
701 | transpose( transpose(M) * transpose(MRES) ); |
---|
702 | "ERROR: transpose( transpose(M) * transpose(MRES) ) != 0!!!"; |
---|
703 | $ |
---|
704 | } |
---|
705 | } |
---|
706 | |
---|
707 | RES [size(RES)+1] = M; // list of all syzygy modules |
---|
708 | LRES[size(LRES)+1] = LEAD; // list of all syzygy modules |
---|
709 | TRES[size(TRES)+1] = TAIL; // list of all syzygy modules |
---|
710 | |
---|
711 | MRES = MRES, M; //? |
---|
712 | |
---|
713 | attrib(MRES, "isHomog", @V); |
---|
714 | |
---|
715 | // attrib(S, "InducionStart", @RANK); |
---|
716 | |
---|
717 | |
---|
718 | if( typeof( attrib(SSinit, "LEAD2SYZ") ) == "int" ) |
---|
719 | { |
---|
720 | attrib(S, "LEAD2SYZ", attrib(SSinit, "LEAD2SYZ") ); |
---|
721 | } else |
---|
722 | { |
---|
723 | attrib(S, "LEAD2SYZ", 1); |
---|
724 | } |
---|
725 | |
---|
726 | if( typeof( attrib(SSinit, "TAILREDSYZ") ) == "int" ) |
---|
727 | { |
---|
728 | attrib(S, "TAILREDSYZ", attrib(SSinit, "TAILREDSYZ") ); |
---|
729 | } else |
---|
730 | { |
---|
731 | attrib(S, "TAILREDSYZ", 1); |
---|
732 | } |
---|
733 | |
---|
734 | |
---|
735 | if( typeof( attrib(SSinit, "HYBRIDNF") ) == "int" ) |
---|
736 | { |
---|
737 | attrib(S, "HYBRIDNF", attrib(SSinit, "HYBRIDNF") ); |
---|
738 | } else |
---|
739 | { |
---|
740 | attrib(S, "HYBRIDNF", 0); |
---|
741 | } |
---|
742 | |
---|
743 | attrib(S, "DEBUG", @DEBUG); |
---|
744 | attrib(S, "SYZCHECK", @SYZCHECK); |
---|
745 | |
---|
746 | if( @DEBUG ) |
---|
747 | { |
---|
748 | "SSinit::MRES"; |
---|
749 | MRES; |
---|
750 | // DetailedPrint(MRES); |
---|
751 | attrib(MRES, "isHomog"); |
---|
752 | attrib(S); |
---|
753 | } |
---|
754 | |
---|
755 | export RES; |
---|
756 | export MRES; |
---|
757 | export LRES; |
---|
758 | export TRES; |
---|
759 | return (S); |
---|
760 | } |
---|
761 | example |
---|
762 | { "EXAMPLE:"; echo = 2; |
---|
763 | ring R = 0, (w, x, y, z), dp; |
---|
764 | |
---|
765 | def M = maxideal(1); |
---|
766 | def S = SSinit(M); setring S; S; |
---|
767 | |
---|
768 | "Only the first initialization: "; |
---|
769 | RES; LRES; TRES; |
---|
770 | MRES; |
---|
771 | |
---|
772 | kill S; setring R; kill M; |
---|
773 | |
---|
774 | ideal M = w^2 - x*z, w*x - y*z, x^2 - w*y, x*y - z^2, y^2 - w*z; |
---|
775 | def S = SSinit(M); setring S; S; |
---|
776 | |
---|
777 | "Only the first initialization: "; |
---|
778 | RES; LRES; TRES; |
---|
779 | MRES; |
---|
780 | |
---|
781 | kill S; setring R; kill M; |
---|
782 | } |
---|
783 | |
---|
784 | |
---|
785 | LIB "poly.lib"; // for lcm |
---|
786 | |
---|
787 | |
---|
788 | |
---|
789 | /// Compute L(Syz(L)) |
---|
790 | proc SSComputeLeadingSyzygyTerms(def L) |
---|
791 | { |
---|
792 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
793 | { |
---|
794 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
795 | } else |
---|
796 | { |
---|
797 | int @DEBUG = !system("with", "ndebug"); |
---|
798 | } |
---|
799 | |
---|
800 | if( typeof( attrib(basering, "SYZCHECK") ) == "int" ) |
---|
801 | { |
---|
802 | int @SYZCHECK = attrib(basering, "SYZCHECK"); |
---|
803 | } else |
---|
804 | { |
---|
805 | int @SYZCHECK = @DEBUG; |
---|
806 | } |
---|
807 | |
---|
808 | if( @DEBUG ) |
---|
809 | { |
---|
810 | "SSComputeLeadingSyzygyTerms::Input: "; |
---|
811 | L; |
---|
812 | } |
---|
813 | |
---|
814 | module SS = ComputeLeadingSyzygyTerms(L); |
---|
815 | |
---|
816 | if( @SYZCHECK ) |
---|
817 | { |
---|
818 | int i, j, r; |
---|
819 | int N = ncols(L); |
---|
820 | def a, b; |
---|
821 | poly aa, bb; |
---|
822 | |
---|
823 | bigint c; |
---|
824 | |
---|
825 | ideal M; |
---|
826 | |
---|
827 | module S = 0; |
---|
828 | |
---|
829 | for(i = 1; i <= N; i++) |
---|
830 | { |
---|
831 | a = L[i]; |
---|
832 | c = leadcomp(a); |
---|
833 | r = int(c); |
---|
834 | |
---|
835 | aa = leadmonomial(a); |
---|
836 | |
---|
837 | M = 0; |
---|
838 | |
---|
839 | for(j = i-1; j > 0; j--) |
---|
840 | { |
---|
841 | b = L[j]; |
---|
842 | |
---|
843 | if( leadcomp(b) == c ) |
---|
844 | { |
---|
845 | bb = leadmonomial(b); |
---|
846 | |
---|
847 | M[j] = (lcm(aa, bb) / aa); |
---|
848 | } |
---|
849 | } |
---|
850 | |
---|
851 | // TODO: add quotient relations here... |
---|
852 | |
---|
853 | M = simplify(M, 1 + 2 + 32); |
---|
854 | |
---|
855 | M = MySort(M); |
---|
856 | |
---|
857 | S = S, M * gen(i); |
---|
858 | } |
---|
859 | |
---|
860 | S = MySort(simplify(S, 2)); |
---|
861 | |
---|
862 | if( size(module(matrix(S) - matrix(SS))) > 0 ) |
---|
863 | { |
---|
864 | "ERROR: S != SS "; |
---|
865 | |
---|
866 | "basering: "; |
---|
867 | DetailedPrint(basering); |
---|
868 | |
---|
869 | "S: "; S; |
---|
870 | DetailedPrint(S, 1); |
---|
871 | "SS: "; SS; |
---|
872 | DetailedPrint(SS, 1); |
---|
873 | |
---|
874 | "DIFF: "; |
---|
875 | print(matrix(S) - matrix(SS)); |
---|
876 | DetailedPrint(module(matrix(S) - matrix(SS)), 2); |
---|
877 | $ |
---|
878 | } |
---|
879 | } |
---|
880 | |
---|
881 | |
---|
882 | if( @DEBUG ) |
---|
883 | { |
---|
884 | "SSComputeLeadingSyzygyTerms::Output: "; |
---|
885 | "SS: "; |
---|
886 | SS; |
---|
887 | } |
---|
888 | |
---|
889 | attrib(SS, "isSB", 1); |
---|
890 | |
---|
891 | return (SS); |
---|
892 | } |
---|
893 | |
---|
894 | /// Compute Syz(L), where L is a monomial (leading) module |
---|
895 | proc SSCompute2LeadingSyzygyTerms(def L) |
---|
896 | { |
---|
897 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
898 | { |
---|
899 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
900 | } else |
---|
901 | { |
---|
902 | int @DEBUG = !system("with", "ndebug"); |
---|
903 | } |
---|
904 | |
---|
905 | if( typeof( attrib(basering, "SYZCHECK") ) == "int" ) |
---|
906 | { |
---|
907 | int @SYZCHECK = attrib(basering, "SYZCHECK"); |
---|
908 | } else |
---|
909 | { |
---|
910 | int @SYZCHECK = @DEBUG; |
---|
911 | } |
---|
912 | |
---|
913 | int @TAILREDSYZ = 1; |
---|
914 | if( typeof( attrib(basering, "TAILREDSYZ") ) == "int" ) |
---|
915 | { |
---|
916 | @TAILREDSYZ = attrib(basering, "TAILREDSYZ"); |
---|
917 | } |
---|
918 | |
---|
919 | |
---|
920 | if( @DEBUG ) |
---|
921 | { |
---|
922 | "SSCompute2LeadingSyzygyTerms::Input: "; |
---|
923 | L; |
---|
924 | } |
---|
925 | |
---|
926 | int i, j, r; |
---|
927 | int N = ncols(L); |
---|
928 | def a, b; |
---|
929 | |
---|
930 | poly aa, bb, @lcm; |
---|
931 | |
---|
932 | bigint c; |
---|
933 | |
---|
934 | module M; |
---|
935 | |
---|
936 | module S = 0; |
---|
937 | |
---|
938 | for(i = 1; i <= N; i++) |
---|
939 | { |
---|
940 | a = L[i]; |
---|
941 | // "a: ", a; |
---|
942 | c = leadcomp(a); |
---|
943 | r = int(c); |
---|
944 | |
---|
945 | aa = leadmonomial(a); |
---|
946 | |
---|
947 | M = 0; |
---|
948 | |
---|
949 | for(j = i-1; j > 0; j--) |
---|
950 | { |
---|
951 | b = L[j]; |
---|
952 | // "b: ", b; |
---|
953 | |
---|
954 | if( leadcomp(b) == c ) |
---|
955 | { |
---|
956 | bb = leadmonomial(b); |
---|
957 | @lcm = lcm(aa, bb); |
---|
958 | |
---|
959 | M[j] = (@lcm / aa)* gen(i) - (@lcm / bb)* gen(j); |
---|
960 | } |
---|
961 | } |
---|
962 | |
---|
963 | M = simplify(M, 2); |
---|
964 | |
---|
965 | // TODO: add quotient relations here... |
---|
966 | S = S, M; |
---|
967 | } |
---|
968 | |
---|
969 | if( @TAILREDSYZ ) |
---|
970 | { |
---|
971 | // Make sure that 2nd syzygy terms are not reducible by 1st |
---|
972 | def opts = option(get); |
---|
973 | option(redSB); option(redTail); |
---|
974 | S = std(S); // binomial module |
---|
975 | option(set, opts); |
---|
976 | // kill opts; |
---|
977 | } else |
---|
978 | { |
---|
979 | S = simplify(S, 2 + 32); |
---|
980 | } |
---|
981 | |
---|
982 | S = MySort(S); |
---|
983 | |
---|
984 | if( @DEBUG ) |
---|
985 | { |
---|
986 | "SSCompute2LeadingSyzygyTerms::Syz(LEAD): "; S; |
---|
987 | } |
---|
988 | |
---|
989 | if( @SYZCHECK ) |
---|
990 | { |
---|
991 | if( size(S) > 0 and size(L) > 0 ) |
---|
992 | { |
---|
993 | if( size(module(transpose( transpose(S) * transpose(L) ))) > 0 ) |
---|
994 | { |
---|
995 | transpose( transpose(S) * transpose(L) ); |
---|
996 | "ERROR: transpose( transpose(S) * transpose(L) ) != 0!!!"; |
---|
997 | $ |
---|
998 | } |
---|
999 | } |
---|
1000 | |
---|
1001 | module SS = Compute2LeadingSyzygyTerms(L); |
---|
1002 | |
---|
1003 | "S: "; DetailedPrint(S); |
---|
1004 | "SS: "; DetailedPrint(SS); |
---|
1005 | |
---|
1006 | if( size(module(matrix(S) - matrix(SS))) > 0 ) |
---|
1007 | { |
---|
1008 | "ERROR: S != SS "; |
---|
1009 | |
---|
1010 | "basering: "; |
---|
1011 | DetailedPrint(basering); |
---|
1012 | |
---|
1013 | "S: "; S; |
---|
1014 | DetailedPrint(S, 2); |
---|
1015 | "SS: "; SS; |
---|
1016 | DetailedPrint(SS, 2); |
---|
1017 | |
---|
1018 | "DIFF: "; |
---|
1019 | print(matrix(S) - matrix(SS)); |
---|
1020 | DetailedPrint(module(matrix(S) - matrix(SS)), 4); |
---|
1021 | $ |
---|
1022 | } |
---|
1023 | |
---|
1024 | } |
---|
1025 | |
---|
1026 | module S2 = Tail(S); |
---|
1027 | S = lead(S); // (C,lp) on base ring! |
---|
1028 | |
---|
1029 | if( @DEBUG ) |
---|
1030 | { |
---|
1031 | "SSCompute2LeadingSyzygyTerms::Output: "; S; S2; |
---|
1032 | } |
---|
1033 | |
---|
1034 | attrib(S, "isSB", 1); |
---|
1035 | |
---|
1036 | return (S, S2); |
---|
1037 | } |
---|
1038 | |
---|
1039 | // -------------------------------------------------------- // |
---|
1040 | |
---|
1041 | /// TODO: save shortcut LM(m) * "t" -> ? |
---|
1042 | proc SSReduceTerm(poly m, def t, def L, def T, list #) |
---|
1043 | { |
---|
1044 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
1045 | { |
---|
1046 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
1047 | } else |
---|
1048 | { |
---|
1049 | int @DEBUG = !system("with", "ndebug"); |
---|
1050 | } |
---|
1051 | |
---|
1052 | if( @DEBUG ) |
---|
1053 | { |
---|
1054 | "SSReduce::Input: "; |
---|
1055 | |
---|
1056 | "mult: ", m; |
---|
1057 | "term: ", t; |
---|
1058 | "L: ", L; |
---|
1059 | "T: ", T; |
---|
1060 | if( size(#) > 0 ) |
---|
1061 | { |
---|
1062 | "LSyz: ", #; |
---|
1063 | } |
---|
1064 | // "attrib(LS, 'isSB')", attrib(LS, "isSB"); |
---|
1065 | } |
---|
1066 | |
---|
1067 | vector s = 0; |
---|
1068 | |
---|
1069 | if( t == 0 ) |
---|
1070 | { |
---|
1071 | return (s); |
---|
1072 | } |
---|
1073 | |
---|
1074 | def product = m * t; |
---|
1075 | |
---|
1076 | bigint c = leadcomp(t); |
---|
1077 | int r = int(c); |
---|
1078 | |
---|
1079 | def a, b, nf, bb; |
---|
1080 | |
---|
1081 | // looking for an appropriate reducer |
---|
1082 | for( int k = ncols(L); k > 0; k-- ) |
---|
1083 | { |
---|
1084 | a = L[k]; |
---|
1085 | // with the same mod. component |
---|
1086 | if( leadcomp(a) == c ) |
---|
1087 | { |
---|
1088 | b = - (leadmonomial(product) / leadmonomial(L[k])); |
---|
1089 | |
---|
1090 | // which divides the product |
---|
1091 | if( b != 0 ) |
---|
1092 | { |
---|
1093 | // "b: ", b; |
---|
1094 | bb = b * gen(k); |
---|
1095 | nf = bb; |
---|
1096 | |
---|
1097 | if( size(#) > 0 ) |
---|
1098 | { |
---|
1099 | if( typeof(#[1]) == "module" ) |
---|
1100 | { |
---|
1101 | nf = NF(bb, #[1]); |
---|
1102 | // "NF: ", nf; |
---|
1103 | } |
---|
1104 | } |
---|
1105 | |
---|
1106 | // while the complement (the fraction) is not reducible by leading syzygies |
---|
1107 | if( nf != 0 ) |
---|
1108 | { |
---|
1109 | /// TODO: save shortcut LM(m) * T[i] -> ? |
---|
1110 | |
---|
1111 | // choose ANY such reduction... (with the biggest index?) |
---|
1112 | s = bb + SSTraverseTail(b, T[k], L, T, #); |
---|
1113 | break; |
---|
1114 | } |
---|
1115 | } |
---|
1116 | } |
---|
1117 | } |
---|
1118 | if( @DEBUG ) |
---|
1119 | { |
---|
1120 | "SSReduceTerm::Output: ", s; |
---|
1121 | } |
---|
1122 | return (s); |
---|
1123 | } |
---|
1124 | |
---|
1125 | // TODO: store m * @tail -.-^-.-^-.--> ? |
---|
1126 | proc SSTraverseTail(poly m, def @tail, def L, def T, list #) |
---|
1127 | { |
---|
1128 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
1129 | { |
---|
1130 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
1131 | } else |
---|
1132 | { |
---|
1133 | int @DEBUG = !system("with", "ndebug"); |
---|
1134 | } |
---|
1135 | |
---|
1136 | if( @DEBUG ) |
---|
1137 | { |
---|
1138 | "SSTraverse::Input: "; |
---|
1139 | |
---|
1140 | "mult: ", m; |
---|
1141 | "tail: ", @tail; // T[i]; |
---|
1142 | |
---|
1143 | if( size(#) > 0 ) |
---|
1144 | { |
---|
1145 | "LSyz: "; #[1]; |
---|
1146 | } |
---|
1147 | } |
---|
1148 | |
---|
1149 | vector s = 0; |
---|
1150 | |
---|
1151 | def @l; |
---|
1152 | |
---|
1153 | // iterate tail-terms in ANY order! |
---|
1154 | while( size(@tail) > 0 ) |
---|
1155 | { |
---|
1156 | @l = lead(@tail); |
---|
1157 | s = s + SSReduceTerm(m, @l, L, T, #); |
---|
1158 | @tail = @tail - @l; |
---|
1159 | } |
---|
1160 | |
---|
1161 | if( @DEBUG ) |
---|
1162 | { |
---|
1163 | "SSTraverseTail::Output: ", s; |
---|
1164 | } |
---|
1165 | return (s); |
---|
1166 | } |
---|
1167 | |
---|
1168 | // -------------------------------------------------------- // |
---|
1169 | |
---|
1170 | // module (N, LL, TT) = SSComputeSyzygy(L, T); |
---|
1171 | // Compute Syz(L ++ T) = N = LL ++ TT |
---|
1172 | proc SSComputeSyzygy(def L, def T) |
---|
1173 | { |
---|
1174 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
1175 | { |
---|
1176 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
1177 | } else |
---|
1178 | { |
---|
1179 | int @DEBUG = !system("with", "ndebug"); |
---|
1180 | } |
---|
1181 | |
---|
1182 | if( typeof( attrib(basering, "SYZCHECK") ) == "int" ) |
---|
1183 | { |
---|
1184 | int @SYZCHECK = attrib(basering, "SYZCHECK"); |
---|
1185 | } else |
---|
1186 | { |
---|
1187 | int @SYZCHECK = @DEBUG; |
---|
1188 | } |
---|
1189 | |
---|
1190 | |
---|
1191 | if( @DEBUG ) |
---|
1192 | { |
---|
1193 | "SSComputeSyzygy::Input"; |
---|
1194 | "basering: ", basering; attrib(basering); |
---|
1195 | // DetailedPrint(basering); |
---|
1196 | |
---|
1197 | // "iCompShift: ", iCompShift; |
---|
1198 | |
---|
1199 | "L: "; L; |
---|
1200 | "T: "; T; |
---|
1201 | } |
---|
1202 | |
---|
1203 | def a; bigint c; int r, k; poly aa; |
---|
1204 | |
---|
1205 | int @LEAD2SYZ = 0; |
---|
1206 | if( typeof( attrib(basering, "LEAD2SYZ") ) == "int" ) |
---|
1207 | { |
---|
1208 | @LEAD2SYZ = attrib(basering, "LEAD2SYZ"); |
---|
1209 | } |
---|
1210 | |
---|
1211 | int @TAILREDSYZ = 1; |
---|
1212 | if( typeof( attrib(basering, "TAILREDSYZ") ) == "int" ) |
---|
1213 | { |
---|
1214 | @TAILREDSYZ = attrib(basering, "TAILREDSYZ"); |
---|
1215 | // @TAILREDSYZ; |
---|
1216 | } |
---|
1217 | |
---|
1218 | /// Get the critical leading syzygy terms |
---|
1219 | if( @LEAD2SYZ ) // & 2nd syz. term |
---|
1220 | { |
---|
1221 | def a2; int r2; poly aa2; |
---|
1222 | module LL, LL2; |
---|
1223 | (LL, LL2) = SSCompute2LeadingSyzygyTerms(L); // ++ |
---|
1224 | } else |
---|
1225 | { |
---|
1226 | module LL = SSComputeLeadingSyzygyTerms(L); |
---|
1227 | } |
---|
1228 | |
---|
1229 | module TT, SYZ; |
---|
1230 | |
---|
1231 | if( size(LL) > 0 ) |
---|
1232 | { |
---|
1233 | list LS; |
---|
1234 | |
---|
1235 | if( @TAILREDSYZ ) |
---|
1236 | { |
---|
1237 | LS = list(LL); |
---|
1238 | } |
---|
1239 | |
---|
1240 | vector @tail; |
---|
1241 | |
---|
1242 | for(k = ncols(LL); k > 0; k-- ) |
---|
1243 | { |
---|
1244 | // leading syz. term: |
---|
1245 | a = LL[k]; c = leadcomp(a); r = int(c); aa = leadmonomial(a); |
---|
1246 | // "A: ", a, " --->>>> ", aa, " **** [", r, "]: "; |
---|
1247 | |
---|
1248 | /// TODO: save shortcut (aa) * T[r] -> ? |
---|
1249 | @tail = SSTraverseTail(aa, T[r], L, T, LS); |
---|
1250 | |
---|
1251 | // get the 2nd syzygy term... |
---|
1252 | |
---|
1253 | if( @LEAD2SYZ ) // with the 2nd syz. term: |
---|
1254 | { |
---|
1255 | a2 = LL2[k]; c = leadcomp(a2); r2 = int(c); aa2 = leadmonomial(a2); |
---|
1256 | @tail = @tail + |
---|
1257 | /// TODO: save shortcut (aa2) * T[r2] -> ? |
---|
1258 | a2 + SSTraverseTail(aa2, T[r2], L, T, LS); |
---|
1259 | } else |
---|
1260 | { |
---|
1261 | @tail = @tail + SSReduceTerm(aa, L[r], L, T, LS); |
---|
1262 | } |
---|
1263 | |
---|
1264 | TT[k] = @tail; |
---|
1265 | SYZ[k] = a + @tail; |
---|
1266 | } |
---|
1267 | } |
---|
1268 | |
---|
1269 | /* |
---|
1270 | def opts = option(get); option(redSB); option(redTail); |
---|
1271 | module SYZ = std(syz(M)); |
---|
1272 | option(set, opts); kill opts; |
---|
1273 | |
---|
1274 | module LL = lead(SYZ); // TODO: WRONG ORDERING!!!!!!!! |
---|
1275 | module TT = Tail(SYZ); |
---|
1276 | */ |
---|
1277 | |
---|
1278 | if( @DEBUG ) |
---|
1279 | { |
---|
1280 | "SSComputeSyzygy::Output"; |
---|
1281 | |
---|
1282 | "SYZ: "; SYZ; |
---|
1283 | "LL: "; LL; |
---|
1284 | "TT: "; TT; |
---|
1285 | } |
---|
1286 | |
---|
1287 | return (SYZ, LL, TT); |
---|
1288 | } |
---|
1289 | |
---|
1290 | // resolution/syzygy step: |
---|
1291 | static proc SSstep() |
---|
1292 | { |
---|
1293 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
1294 | { |
---|
1295 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
1296 | } else |
---|
1297 | { |
---|
1298 | int @DEBUG = !system("with", "ndebug"); |
---|
1299 | } |
---|
1300 | |
---|
1301 | |
---|
1302 | if( typeof( attrib(basering, "SYZCHECK") ) == "int" ) |
---|
1303 | { |
---|
1304 | int @SYZCHECK = attrib(basering, "SYZCHECK"); |
---|
1305 | } else |
---|
1306 | { |
---|
1307 | int @SYZCHECK = @DEBUG; |
---|
1308 | } |
---|
1309 | |
---|
1310 | if( @DEBUG ) |
---|
1311 | { |
---|
1312 | "SSstep::NextInducedRing"; |
---|
1313 | "basering: ", basering; attrib(basering); |
---|
1314 | } |
---|
1315 | |
---|
1316 | /* |
---|
1317 | // is initial weights are all zeroes! |
---|
1318 | def L = lead(M); |
---|
1319 | intvec @V = deg(M[1..ncols(M)]); @W; @V; @W = @V; attrib(L, "isHomog", @W); |
---|
1320 | SetInducedReferrence(L, @RANK, 0); |
---|
1321 | */ |
---|
1322 | |
---|
1323 | // def L = lead(MRES); |
---|
1324 | // @W = @W, @V; |
---|
1325 | // attrib(L, "isHomog", @W); |
---|
1326 | |
---|
1327 | |
---|
1328 | // General setting: |
---|
1329 | // SetInducedReferrence(MRES, 0, 0); // limit: 0! |
---|
1330 | int @l = size(RES); |
---|
1331 | |
---|
1332 | def M = RES[@l]; |
---|
1333 | |
---|
1334 | def L = LRES[@l]; |
---|
1335 | def T = TRES[@l]; |
---|
1336 | |
---|
1337 | |
---|
1338 | //// TODO: wrong !!!!! |
---|
1339 | int @RANK = ncols(MRES) - ncols(M); // nrows(M); // what if M is zero?! |
---|
1340 | |
---|
1341 | |
---|
1342 | |
---|
1343 | /* |
---|
1344 | if( @RANK != nrows(M) ) |
---|
1345 | { |
---|
1346 | type(MRES); |
---|
1347 | @RANK; |
---|
1348 | type(M); |
---|
1349 | pause(); |
---|
1350 | } |
---|
1351 | */ |
---|
1352 | |
---|
1353 | intvec @W = attrib(M, "isHomog"); intvec @V = attrib(M, "degrees"); @V = @W, @V; |
---|
1354 | |
---|
1355 | if( @DEBUG ) |
---|
1356 | { |
---|
1357 | "Sstep::NextInput: "; |
---|
1358 | M; |
---|
1359 | L; |
---|
1360 | @V; |
---|
1361 | @RANK; |
---|
1362 | // DetailedPrint(MRES); |
---|
1363 | attrib(MRES, "isHomog"); |
---|
1364 | } |
---|
1365 | |
---|
1366 | |
---|
1367 | // TODO: N = SYZ( M )!!! |
---|
1368 | module N, LL, TT; |
---|
1369 | (N, LL, TT) = SSComputeSyzygy(/*M, */L, T/*, @RANK*/); |
---|
1370 | |
---|
1371 | // shift syz.comp by @RANK: |
---|
1372 | module Z; |
---|
1373 | Z = 0; Z[@RANK] = 0; Z = Z, transpose(LL); LL = transpose(Z); |
---|
1374 | Z = 0; Z[@RANK] = 0; Z = Z, transpose(TT); TT = transpose(Z); |
---|
1375 | Z = 0; Z[@RANK] = 0; Z = Z, transpose(N); N = transpose(Z); |
---|
1376 | |
---|
1377 | |
---|
1378 | if( @SYZCHECK ) |
---|
1379 | { |
---|
1380 | if( size(N) > 0 ) |
---|
1381 | { |
---|
1382 | // next syz. property |
---|
1383 | if( size(module(transpose( transpose(N) * transpose(MRES) ))) > 0 ) |
---|
1384 | { |
---|
1385 | "MRES", MRES; |
---|
1386 | |
---|
1387 | "N: "; N; // DetailedPrint(N, 2); |
---|
1388 | |
---|
1389 | "LL:"; LL; // DetailedPrint(LL, 1); |
---|
1390 | "TT:"; TT; // DetailedPrint(TT, 10); |
---|
1391 | |
---|
1392 | "RANKS: ", @RANK; |
---|
1393 | |
---|
1394 | "transpose( transpose(N) * transpose(MRES) ) != 0!!!"; |
---|
1395 | transpose( transpose(N) * transpose(MRES) ); |
---|
1396 | |
---|
1397 | "transpose(N) * transpose(MRES): "; |
---|
1398 | transpose(N) * transpose(MRES); |
---|
1399 | // DetailedPrint(module(_), 2); |
---|
1400 | $ |
---|
1401 | } |
---|
1402 | } |
---|
1403 | } |
---|
1404 | |
---|
1405 | attrib(N, "isHomog", @V); |
---|
1406 | |
---|
1407 | // TODO: correct the following: |
---|
1408 | intvec @DEGS = deg(N[1..ncols(N)]); // no mod. comp. weights :( |
---|
1409 | |
---|
1410 | |
---|
1411 | attrib(N, "degrees", @DEGS); |
---|
1412 | |
---|
1413 | RES[@l + 1] = N; // list of all syzygy modules |
---|
1414 | LRES[@l + 1] = LL; // list of all syzygy modules |
---|
1415 | TRES[@l + 1] = TT; // list of all syzygy modules |
---|
1416 | |
---|
1417 | MRES = MRES, N; |
---|
1418 | |
---|
1419 | attrib(MRES, "isHomog", @V); |
---|
1420 | |
---|
1421 | // L = L, lead(N); attrib(basering, "InducionLeads", L); |
---|
1422 | |
---|
1423 | if( @DEBUG ) |
---|
1424 | { |
---|
1425 | "SSstep::NextSyzOutput: "; |
---|
1426 | N; |
---|
1427 | // DetailedPrint(N); |
---|
1428 | attrib(N); |
---|
1429 | } |
---|
1430 | |
---|
1431 | } |
---|
1432 | |
---|
1433 | proc SScontinue(int l) |
---|
1434 | "USAGE: SScontinue(l) |
---|
1435 | RETURN: nothing, instead it changes RES and MRES variables in the current ring |
---|
1436 | PURPOSE: computes further (at most l) syzygies |
---|
1437 | NOTE: must be used within a ring returned by Sres or Ssyz. RES and MRES are |
---|
1438 | explained in Sres |
---|
1439 | EXAMPLE: example Scontinue; shows an example |
---|
1440 | " |
---|
1441 | { |
---|
1442 | |
---|
1443 | /// TODO! |
---|
1444 | // def data = GetInducedData(); |
---|
1445 | |
---|
1446 | if( (!defined(RES)) || (!defined(MRES)) ) /* || (typeof(data) != "list") || (size(data) != 2) */ |
---|
1447 | { |
---|
1448 | ERROR("Sorry, but basering does not seem to be returned by Sres or Ssyz"); |
---|
1449 | } |
---|
1450 | for (; (l != 0) && (size(RES[size(RES)]) > 0); l-- ) |
---|
1451 | { |
---|
1452 | SSstep(); |
---|
1453 | } |
---|
1454 | } |
---|
1455 | example |
---|
1456 | { "EXAMPLE:"; echo = 2; |
---|
1457 | ring r; |
---|
1458 | module M = maxideal(1); M; |
---|
1459 | def S = SSsyz(M); setring S; S; |
---|
1460 | "Only the first syzygy: "; |
---|
1461 | RES; MRES; |
---|
1462 | "More syzygies: "; |
---|
1463 | SScontinue(10); |
---|
1464 | RES; MRES; |
---|
1465 | } |
---|
1466 | |
---|
1467 | proc SSsyz(def M) |
---|
1468 | "USAGE: SSsyz(M) |
---|
1469 | RETURN: ring, containing a list of modules RES and a module MRES |
---|
1470 | PURPOSE: computes the first syzygy module of M (wrt some Schreyer ordering)? |
---|
1471 | NOTE: The output is explained in Sres |
---|
1472 | EXAMPLE: example Ssyz; shows an example |
---|
1473 | " |
---|
1474 | { |
---|
1475 | if( (typeof(M) != "module") && (typeof(M) != "ideal") ) |
---|
1476 | { |
---|
1477 | ERROR("Sorry: need an ideal or a module for input"); |
---|
1478 | } |
---|
1479 | |
---|
1480 | def SS = SSinit(M); setring SS; |
---|
1481 | |
---|
1482 | SSstep(); // NOTE: what if M is zero? |
---|
1483 | |
---|
1484 | return (SS); |
---|
1485 | } |
---|
1486 | example |
---|
1487 | { "EXAMPLE:"; echo = 2; |
---|
1488 | ring r; |
---|
1489 | |
---|
1490 | /* ideal M = 0; |
---|
1491 | def S = SSsyz(M); setring S; S; |
---|
1492 | "Only the first syzygy: "; |
---|
1493 | RES; LRES; TRES; |
---|
1494 | MRES; |
---|
1495 | |
---|
1496 | kill S; setring r; kill M; |
---|
1497 | */ |
---|
1498 | |
---|
1499 | ideal M = maxideal(1); M; |
---|
1500 | def S = SSres(M, 0); setring S; S; |
---|
1501 | MRES; |
---|
1502 | RES; |
---|
1503 | ""; |
---|
1504 | LRES; |
---|
1505 | ""; |
---|
1506 | TRES; |
---|
1507 | |
---|
1508 | kill S; setring r; kill M; |
---|
1509 | |
---|
1510 | kill r; |
---|
1511 | |
---|
1512 | ring R = 0, (w, x, y, z), dp; |
---|
1513 | ideal M = w^2 - x*z, w*x - y*z, x^2 - w*y, x*y - z^2, y^2 - w*z; |
---|
1514 | |
---|
1515 | def S = SSres(M, 0); setring S; S; |
---|
1516 | MRES; |
---|
1517 | RES; |
---|
1518 | ""; |
---|
1519 | LRES; |
---|
1520 | ""; |
---|
1521 | TRES; |
---|
1522 | } |
---|
1523 | |
---|
1524 | proc SSres(def M, int l) |
---|
1525 | "USAGE: SSres(I, l) |
---|
1526 | RETURN: ring, containing a list of modules RES and a module MRES |
---|
1527 | PURPOSE: computes (at most l) syzygy modules of M wrt the classical Schreyer |
---|
1528 | induced ordering with gen(i) > gen(j) if i > j, provided both gens |
---|
1529 | are from the same syzygy level.??? |
---|
1530 | NOTE: RES contains the images of maps subsituting the beginning of the |
---|
1531 | Schreyer free resolution of baseRing^r/M, while MRES is a sum of |
---|
1532 | these images in a big free sum, containing all the syzygy modules. |
---|
1533 | The syzygy modules are shifted so that gen(i) correspons to MRES[i]. |
---|
1534 | The leading zero module RES[0] indicates the fact that coker of the |
---|
1535 | first map is zero. The number of zeroes inducates the rank of input. |
---|
1536 | NOTE: If l == 0 then l is set to be nvars(basering) + 1 |
---|
1537 | EXAMPLE: example SSres; shows an example |
---|
1538 | " |
---|
1539 | { |
---|
1540 | if( (typeof(M) != "module") && (typeof(M) != "ideal") ) |
---|
1541 | { |
---|
1542 | ERROR("Sorry: need an ideal or a module for input"); |
---|
1543 | } |
---|
1544 | |
---|
1545 | def SS = SSinit(M); setring SS; |
---|
1546 | |
---|
1547 | if (l == 0) |
---|
1548 | { |
---|
1549 | l = nvars(basering) + 1; // not really an estimate...?! |
---|
1550 | } |
---|
1551 | |
---|
1552 | SSstep(); l = l - 1; |
---|
1553 | |
---|
1554 | SScontinue(l); |
---|
1555 | |
---|
1556 | return (SS); |
---|
1557 | } |
---|
1558 | example |
---|
1559 | { "EXAMPLE:"; echo = 2; |
---|
1560 | ring r; |
---|
1561 | module M = maxideal(1); M; |
---|
1562 | def S = SSres(M, 0); setring S; S; |
---|
1563 | RES; |
---|
1564 | MRES; |
---|
1565 | kill S; |
---|
1566 | setring r; kill M; |
---|
1567 | |
---|
1568 | def A = nc_algebra(-1,0); setring A; |
---|
1569 | ideal Q = var(1)^2, var(2)^2, var(3)^2; |
---|
1570 | qring SCA = twostd(Q); |
---|
1571 | basering; |
---|
1572 | |
---|
1573 | module M = maxideal(1); |
---|
1574 | def S = SSres(M, 2); setring S; S; |
---|
1575 | RES; |
---|
1576 | MRES; |
---|
1577 | } |
---|
1578 | |
---|
1579 | |
---|
1580 | |
---|
1581 | static proc loadme() |
---|
1582 | { |
---|
1583 | int @DEBUG = !system("with", "ndebug"); |
---|
1584 | |
---|
1585 | if( @DEBUG ) |
---|
1586 | { |
---|
1587 | |
---|
1588 | "ndebug?: ", system("with", "ndebug"); |
---|
1589 | "om_ndebug?: ", system("with", "om_ndebug"); |
---|
1590 | |
---|
1591 | listvar(Top); |
---|
1592 | listvar(Schreyer); |
---|
1593 | } |
---|
1594 | // listvar(Syzextra); listvar(Syzextra_g); |
---|
1595 | |
---|
1596 | if( !defined(DetailedPrint) ) |
---|
1597 | { |
---|
1598 | if( 1 ) |
---|
1599 | { |
---|
1600 | |
---|
1601 | if( @DEBUG ) |
---|
1602 | { |
---|
1603 | "Loading the Release version!"; |
---|
1604 | } |
---|
1605 | load("syzextra.so"); |
---|
1606 | |
---|
1607 | if( @DEBUG ) |
---|
1608 | { |
---|
1609 | listvar(Syzextra); |
---|
1610 | } |
---|
1611 | |
---|
1612 | exportto(Top, Syzextra::ClearContent); |
---|
1613 | exportto(Top, Syzextra::ClearDenominators); |
---|
1614 | |
---|
1615 | exportto(Schreyer, Syzextra::m2_end); |
---|
1616 | |
---|
1617 | // export Syzextra; |
---|
1618 | |
---|
1619 | // exportto(Schreyer, Syzextra::noop); |
---|
1620 | exportto(Schreyer, Syzextra::DetailedPrint); |
---|
1621 | exportto(Schreyer, Syzextra::leadmonomial); |
---|
1622 | exportto(Schreyer, Syzextra::leadcomp); |
---|
1623 | // exportto(Schreyer, Syzextra::leadrawexp); |
---|
1624 | // exportto(Schreyer, Syzextra::ISUpdateComponents); |
---|
1625 | exportto(Schreyer, Syzextra::SetInducedReferrence); |
---|
1626 | exportto(Schreyer, Syzextra::GetInducedData); |
---|
1627 | // exportto(Schreyer, Syzextra::GetAMData); |
---|
1628 | // exportto(Schreyer, Syzextra::SetSyzComp); |
---|
1629 | exportto(Schreyer, Syzextra::MakeInducedSchreyerOrdering); |
---|
1630 | // exportto(Schreyer, Syzextra::MakeSyzCompOrdering); |
---|
1631 | exportto(Schreyer, Syzextra::idPrepare); |
---|
1632 | // exportto(Schreyer, Syzextra::reduce_syz); |
---|
1633 | // exportto(Schreyer, Syzextra::p_Content); |
---|
1634 | |
---|
1635 | exportto(Schreyer, Syzextra::ProfilerStart); exportto(Schreyer, Syzextra::ProfilerStop); |
---|
1636 | |
---|
1637 | exportto(Schreyer, Syzextra::Tail); |
---|
1638 | exportto(Schreyer, Syzextra::ComputeLeadingSyzygyTerms); |
---|
1639 | exportto(Schreyer, Syzextra::Compute2LeadingSyzygyTerms); |
---|
1640 | exportto(Schreyer, Syzextra::Sort_c_ds); |
---|
1641 | } |
---|
1642 | /* |
---|
1643 | else |
---|
1644 | { |
---|
1645 | if( @DEBUG ) |
---|
1646 | { |
---|
1647 | "Loading the Debug version!"; |
---|
1648 | } |
---|
1649 | |
---|
1650 | load("syzextra.so"); |
---|
1651 | |
---|
1652 | if( @DEBUG ) |
---|
1653 | { |
---|
1654 | listvar(Syzextra_g); |
---|
1655 | } |
---|
1656 | |
---|
1657 | exportto(Top, Syzextra_g::ClearContent); |
---|
1658 | exportto(Top, Syzextra_g::ClearDenominators); |
---|
1659 | |
---|
1660 | exportto(Schreyer, Syzextra_g::m2_end); |
---|
1661 | |
---|
1662 | // export Syzextra_g; |
---|
1663 | // exportto(Schreyer, Syzextra_g::noop); |
---|
1664 | exportto(Schreyer, Syzextra_g::DetailedPrint); |
---|
1665 | exportto(Schreyer, Syzextra_g::leadmonomial); |
---|
1666 | exportto(Schreyer, Syzextra_g::leadcomp); |
---|
1667 | // exportto(Schreyer, Syzextra_g::leadrawexp); |
---|
1668 | // exportto(Schreyer, Syzextra_g::ISUpdateComponents); |
---|
1669 | exportto(Schreyer, Syzextra_g::SetInducedReferrence); |
---|
1670 | exportto(Schreyer, Syzextra_g::GetInducedData); |
---|
1671 | // exportto(Schreyer, Syzextra_g::GetAMData); |
---|
1672 | // exportto(Schreyer, Syzextra_g::SetSyzComp); |
---|
1673 | exportto(Schreyer, Syzextra_g::MakeInducedSchreyerOrdering); |
---|
1674 | // exportto(Schreyer, Syzextra_g::MakeSyzCompOrdering); |
---|
1675 | exportto(Schreyer, Syzextra_g::idPrepare); |
---|
1676 | // exportto(Schreyer, Syzextra_g::reduce_syz); |
---|
1677 | // exportto(Schreyer, Syzextra_g::p_Content); |
---|
1678 | |
---|
1679 | exportto(Schreyer, Syzextra_g::ProfilerStart); exportto(Schreyer, Syzextra_g::ProfilerStop); |
---|
1680 | |
---|
1681 | exportto(Schreyer, Syzextra_g::Tail); |
---|
1682 | exportto(Schreyer, Syzextra_g::ComputeLeadingSyzygyTerms); |
---|
1683 | exportto(Schreyer, Syzextra_g::Compute2LeadingSyzygyTerms); |
---|
1684 | exportto(Schreyer, Syzextra_g::Sort_c_ds); |
---|
1685 | |
---|
1686 | } |
---|
1687 | */ |
---|
1688 | |
---|
1689 | exportto(Top, DetailedPrint); |
---|
1690 | exportto(Top, GetInducedData); |
---|
1691 | |
---|
1692 | if( @DEBUG ) |
---|
1693 | { |
---|
1694 | listvar(Top); |
---|
1695 | listvar(Schreyer); |
---|
1696 | } |
---|
1697 | } |
---|
1698 | |
---|
1699 | if( !defined(GetInducedData) ) |
---|
1700 | { |
---|
1701 | ERROR("Sorry but we are missing the dynamic module (syzextra.so)..."); |
---|
1702 | } |
---|
1703 | |
---|
1704 | } |
---|
1705 | |
---|
1706 | static proc mod_init() |
---|
1707 | { |
---|
1708 | loadme(); |
---|
1709 | } |
---|
1710 | |
---|
1711 | |
---|
1712 | proc testallSexamples() |
---|
1713 | { |
---|
1714 | example Ssyz; |
---|
1715 | example Scontinue; |
---|
1716 | example Sres; |
---|
1717 | } |
---|
1718 | |
---|
1719 | proc testallSSexamples() |
---|
1720 | { |
---|
1721 | example SSsyz; |
---|
1722 | example SScontinue; |
---|
1723 | example SSres; |
---|
1724 | } |
---|
1725 | |
---|
1726 | example |
---|
1727 | { "EXAMPLE:"; echo = 2; |
---|
1728 | testallSexamples(); |
---|
1729 | testallSSexamples(); |
---|
1730 | } |
---|
1731 | |
---|
1732 | proc TestSSres(def M) |
---|
1733 | { |
---|
1734 | "-------------------------------------"; |
---|
1735 | "options: ", attrib(SSinit, "LEAD2SYZ"), attrib(SSinit, "TAILREDSYZ"), attrib(SSinit, "HYBRIDNF"), ": "; |
---|
1736 | int t = timer; |
---|
1737 | def S = SSres(M, 0); |
---|
1738 | int tt = timer; |
---|
1739 | /* |
---|
1740 | setring S; |
---|
1741 | |
---|
1742 | MRES; |
---|
1743 | RES; |
---|
1744 | ""; |
---|
1745 | LRES; |
---|
1746 | ""; |
---|
1747 | TRES; |
---|
1748 | */ |
---|
1749 | kill S; |
---|
1750 | "0-----------------------------------0 => ", tt - t; |
---|
1751 | } |
---|
1752 | |
---|
1753 | |
---|
1754 | proc TestSSresAttribs(def M) |
---|
1755 | { |
---|
1756 | |
---|
1757 | // the following 2 setups are bad for AGR@101n3d002s004%1:((( |
---|
1758 | // attrib(SSinit, "LEAD2SYZ", 0); attrib(SSinit, "TAILREDSYZ", 0); attrib(SSinit, "HYBRIDNF", 0); TestSSres(M); |
---|
1759 | // attrib(SSinit, "LEAD2SYZ", 0); attrib(SSinit, "TAILREDSYZ", 0); attrib(SSinit, "HYBRIDNF", 1); TestSSres(M); |
---|
1760 | |
---|
1761 | attrib(SSinit, "LEAD2SYZ", 0); attrib(SSinit, "TAILREDSYZ", 1); attrib(SSinit, "HYBRIDNF", 0); TestSSres(M); |
---|
1762 | attrib(SSinit, "LEAD2SYZ", 0); attrib(SSinit, "TAILREDSYZ", 1); attrib(SSinit, "HYBRIDNF", 1); TestSSres(M); |
---|
1763 | |
---|
1764 | attrib(SSinit, "LEAD2SYZ", 1); attrib(SSinit, "TAILREDSYZ", 0); attrib(SSinit, "HYBRIDNF", 0); TestSSres(M); |
---|
1765 | attrib(SSinit, "LEAD2SYZ", 1); attrib(SSinit, "TAILREDSYZ", 0); attrib(SSinit, "HYBRIDNF", 1); TestSSres(M); |
---|
1766 | |
---|
1767 | attrib(SSinit, "LEAD2SYZ", 1); attrib(SSinit, "TAILREDSYZ", 1); attrib(SSinit, "HYBRIDNF", 0); TestSSres(M); |
---|
1768 | attrib(SSinit, "LEAD2SYZ", 1); attrib(SSinit, "TAILREDSYZ", 1); attrib(SSinit, "HYBRIDNF", 1); TestSSres(M); |
---|
1769 | } |
---|
1770 | |
---|
1771 | |
---|
1772 | proc testALLA() |
---|
1773 | { |
---|
1774 | attrib(SSinit, "SYZCHECK", 1); // TODO: only for now!! |
---|
1775 | |
---|
1776 | ring r; r; ideal M = maxideal(1); M; |
---|
1777 | TestSSresAttribs(M); |
---|
1778 | kill r; |
---|
1779 | |
---|
1780 | ring r = 0, (a, b, c, d), lp; r; ideal M = maxideal(1); M; |
---|
1781 | TestSSresAttribs(M); |
---|
1782 | kill r; |
---|
1783 | |
---|
1784 | ring R = 0, (w, x, y, z), dp; R; |
---|
1785 | ideal M = w^2 - x*z, w*x - y*z, x^2 - w*y, x*y - z^2, y^2 - w*z; M; |
---|
1786 | TestSSresAttribs(M); |
---|
1787 | kill R; |
---|
1788 | |
---|
1789 | |
---|
1790 | ring AGR = (101), (a, b, c, d), dp; AGR; |
---|
1791 | // simple: AGR@101n3d002s004%1: |
---|
1792 | ideal M = c*d, b*d, a*d, c^2-d^2, b*c, a*c, b^2-d^2, a*b, a^2-d^2; |
---|
1793 | M; |
---|
1794 | TestSSresAttribs(M); |
---|
1795 | |
---|
1796 | |
---|
1797 | // medium: AGR@101n3d004s009%1; |
---|
1798 | ideal M = a*b+7*a*c-16*b*c-27*a*d+37*b*d-2*c*d, d^3, c*d^2, b*d^2, a*d^2, c^2*d, b*c*d, a*c*d, b^2*d, a^2*d, c^3, b*c^2, a*c^2, b^2*c, a^2*c, b^3, a^3; |
---|
1799 | M; |
---|
1800 | TestSSresAttribs(M); |
---|
1801 | |
---|
1802 | /* |
---|
1803 | // lengthy: AGR@101n3d008s058%3, toooo long!!! :(((( |
---|
1804 | ideal M = 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|
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1805 | M; |
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1806 | TestSSresAttribs(M); |
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1807 | */ |
---|
1808 | kill AGR; |
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1809 | |
---|
1810 | ring r = 0, (a, b, c, d, e, f), dp; r; ideal M = maxideal(1); M; |
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1811 | TestSSresAttribs(M); |
---|
1812 | kill r; |
---|
1813 | } |
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1814 | |
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1815 | // TODO: betti!!! |
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1816 | |
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