1 | /////////////////////////////////////////////////////////////////////////// |
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2 | version="version standard.lib 4.0.0.0 Jun_2013 "; // $Id$ |
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3 | category="Miscellaneous"; |
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4 | info=" |
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5 | LIBRARY: standard.lib Procedures which are always loaded at Start-up |
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6 | |
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7 | PROCEDURES: |
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8 | stdfglm(ideal[,ord]) standard basis of ideal via fglm [and ordering ord] |
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9 | stdhilb(ideal[,h]) Hilbert driven Groebner basis of ideal |
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10 | groebner(ideal,...) standard basis using a heuristically chosen method |
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11 | res(ideal/module,[i]) free resolution of ideal or module |
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12 | sprintf(fmt,...) returns fomatted string |
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13 | fprintf(link,fmt,..) writes formatted string to link |
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14 | printf(fmt,...) displays formatted string |
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15 | weightKB(stc,dd,vl) degree dd part of a kbase w.r.t. some weigths |
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16 | qslimgb(i) computes a standard basis with slimgb in a qring |
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17 | par2varRing([i]) create a ring making pars to vars, together with i |
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18 | datetime() return date and time as a string |
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19 | max(i,j) maximum of i and j |
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20 | min(i,j) minimum of i and j |
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21 | |
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22 | "; |
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23 | //AUXILIARY PROCEDURES: |
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24 | // hilbRing([i]) ring for computing the (weighted) hilbert series |
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25 | // quotientList(L,...) ringlist for creating a correct quotient ring |
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26 | |
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27 | ////////////////////////////////////////////////////////////////////////////// |
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28 | |
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29 | proc stdfglm (ideal i, list #) |
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30 | "SYNTAX: @code{stdfglm (} ideal_expression @code{)} @* |
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31 | @code{stdfglm (} ideal_expression@code{,} string_expression @code{)} |
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32 | TYPE: ideal |
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33 | PURPOSE: computes the standard basis of the ideal in the basering |
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34 | via @code{fglm} from the ordering given as the second argument |
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35 | to the ordering of the basering. If no second argument is given, |
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36 | \"dp\" is used. The standard basis for the given ordering (resp. for |
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37 | \"dp\") is computed via the command groebner except if a further |
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38 | argument \"std\" or \"slimgb\" is given in which case std resp. |
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39 | slimgb is used. |
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40 | SEE ALSO: fglm, groebner, std, slimgb, stdhilb |
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41 | KEYWORDS: fglm |
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42 | EXAMPLE: example stdfglm; shows an example" |
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43 | { |
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44 | string os; |
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45 | int s = size(#); |
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46 | def P= basering; |
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47 | string algorithm; |
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48 | int ii; |
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49 | for( ii=1; ii<=s; ii++) |
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50 | { |
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51 | if ( typeof(#[ii])== "string" ) |
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52 | { |
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53 | if ( #[ii]=="std" || #[ii]=="slimgb" ) |
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54 | { |
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55 | algorithm = #[ii]; |
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56 | # = delete(#,ii); |
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57 | s--; |
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58 | ii--; |
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59 | } |
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60 | } |
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61 | } |
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62 | |
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63 | if((s > 0) && (typeof(#[1]) == "string")) |
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64 | { |
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65 | os = #[1]; |
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66 | ideal Qideal = ideal(P); |
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67 | int sQ = size(Qideal); |
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68 | int sM = size(minpoly); |
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69 | if ( sM!=0 ) |
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70 | { |
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71 | string mpoly = string(minpoly); |
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72 | } |
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73 | if (sQ!=0 ) |
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74 | { |
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75 | execute("ring Rfglm=("+charstr(P)+"),("+varstr(P)+"),"+os+";"); |
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76 | ideal Qideal = fetch(P,Qideal); |
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77 | qring Pfglm = groebner(Qideal,"std","slimgb"); |
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78 | } |
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79 | else |
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80 | { |
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81 | execute("ring Pfglm=("+charstr(P)+"),("+varstr(P)+"),"+os+";"); |
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82 | } |
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83 | if ( sM!=0 ) |
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84 | { |
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85 | execute("minpoly="+mpoly+";"); |
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86 | } |
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87 | } |
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88 | else |
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89 | { |
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90 | list BRlist = ringlist(P); |
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91 | int nvarP = nvars(P); |
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92 | intvec w; //for ringweights of basering P |
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93 | int k; |
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94 | for(k=1; k <= nvarP; k++) |
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95 | { |
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96 | w[k]=deg(var(k)); |
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97 | } |
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98 | |
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99 | BRlist[3] = list(); |
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100 | if( s==0 or (typeof(#[1]) != "string") ) |
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101 | { |
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102 | if( w==1 ) |
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103 | { |
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104 | BRlist[3][1]=list("dp",w); |
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105 | } |
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106 | else |
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107 | { |
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108 | BRlist[3][1]=list("wp",w); |
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109 | } |
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110 | BRlist[3][2]=list("C",intvec(0)); |
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111 | def Pfglm = ring(quotientList(BRlist)); |
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112 | setring Pfglm; |
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113 | } |
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114 | } |
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115 | ideal i = fetch(P,i); |
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116 | |
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117 | intvec opt = option(get); //save options |
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118 | option(redSB); |
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119 | if (size(algorithm) > 0) |
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120 | { |
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121 | i = groebner(i,algorithm); |
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122 | } |
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123 | else |
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124 | { |
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125 | i = groebner(i); |
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126 | } |
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127 | option(set,opt); |
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128 | setring P; |
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129 | return (fglm(Pfglm,i)); |
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130 | } |
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131 | example |
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132 | { "EXAMPLE:"; echo = 2; |
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133 | ring r = 0,(x,y,z),lp; |
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134 | ideal i = y3+x2,x2y+x2,x3-x2,z4-x2-y; |
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135 | stdfglm(i); //uses fglm from "dp" (with groebner) to "lp" |
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136 | stdfglm(i,"std"); //uses fglm from "dp" (with std) to "lp" |
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137 | |
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138 | ring s = (0,x),(y,z,u,v),lp; |
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139 | minpoly = x2+1; |
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140 | ideal i = u5-v4,zv-u2,zu3-v3,z2u-v2,z3-uv,yv-zu,yu-z2,yz-v,y2-u,u-xy2; |
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141 | weight(i); |
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142 | stdfglm(i,"(a(2,3,4,5),dp)"); //uses fglm from "(a(2,3,4,5),dp)" to "lp" |
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143 | } |
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144 | |
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145 | ///////////////////////////////////////////////////////////////////////////// |
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146 | |
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147 | proc stdhilb(def i,list #) |
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148 | "SYNTAX: @code{stdhilb (} ideal_expression @code{)} @* |
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149 | @code{stdhilb (} module_expression @code{)} @* |
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150 | @code{stdhilb (} ideal_expression, intvec_expression @code{)}@* |
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151 | @code{stdhilb (} module_expression, intvec_expression @code{)}@* |
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152 | @code{stdhilb (} ideal_expression@code{,} list of string_expressions, |
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153 | and intvec_expression @code{)} @* |
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154 | TYPE: type of the first argument |
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155 | PURPOSE: Compute a Groebner basis of the ideal/module in the basering by |
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156 | using the Hilbert driven Groebner basis algorithm. |
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157 | If an argument of type string, stating @code{\"std\"} resp. @code{\"slimgb\"}, |
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158 | is given, the standard basis computation uses @code{std} or |
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159 | @code{slimgb}, otherwise a heuristically chosen method (default)@* |
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160 | If an optional second argument w of type intvec is given, w is used |
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161 | as variable weights. If w is not given, it is computed as w[i] = |
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162 | deg(var(i)). If the ideal is homogeneous w.r.t. w then the |
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163 | Hilbert series is computed w.r.t. to these weights. |
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164 | THEORY: If the ideal is not homogeneous compute first a Groebner basis |
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165 | of the homogenization [w.r.t. the weights w] of the ideal/module, |
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166 | then the Hilbert function and, finally, a Groebner basis in the |
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167 | original ring by using the computed Hilbert function. If the given |
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168 | w does not coincide with the variable weights of the basering, the |
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169 | result may not be a groebner basis in the original ring. |
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170 | NOTE: 'Homogeneous' means weighted homogeneous with respect to the weights |
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171 | w[i] of the variables var(i) of the basering. Parameters are not |
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172 | converted to variables. |
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173 | SEE ALSO: stdfglm, std, slimgb, groebner |
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174 | KEYWORDS: Hilbert function |
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175 | EXAMPLE: example stdhilb; shows an example" |
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176 | { |
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177 | |
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178 | //--------------------- save data from basering -------------------------- |
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179 | def P=basering; |
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180 | int nr; |
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181 | if (typeof(i)=="ideal") { nr=1;} |
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182 | else { nr= nrows(i); } //nr=1 if i is an ideal |
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183 | ideal Qideal = ideal(P); //defining the quotient ideal if P is a qring |
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184 | int was_qring; //remembers if basering was a qring |
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185 | int is_homog =homog(i); //check for homogeneity of i and Qideal |
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186 | if (size(Qideal) > 0) |
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187 | { |
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188 | was_qring = 1; |
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189 | } |
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190 | |
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191 | // save ordering of basering P for later use |
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192 | list ord_P = ringlist(P)[3]; //ordering of basering in ringlist |
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193 | string ordstr_P = ordstr(P); //ordering of basering as string |
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194 | int nvarP = nvars(P); |
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195 | |
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196 | //save options: |
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197 | intvec gopt = option(get); |
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198 | int p_opt; |
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199 | string s_opt = option(); |
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200 | if (find(s_opt, "prot")) { p_opt = 1; } |
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201 | |
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202 | //-------------------- check the given method and weights --------------------- |
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203 | //Note: stdhilb is used in elim where it is applied to an elimination ordering |
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204 | //a(1..1,0..0),wp(w). In such a ring deg(var(k)=0 for all vars corresponding to |
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205 | //0 in a(1..1,0..0), hence we cannot identify w via w[k] = deg(var(k)); |
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206 | //Therefore hilbstd has the option to give ringweights. |
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207 | |
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208 | int k; |
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209 | string method; |
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210 | for (k=1; k<=size(#); k++) |
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211 | { |
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212 | if (typeof(#[k]) == "intvec") |
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213 | { |
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214 | intvec w = #[k]; //given ringweights of basering P |
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215 | } |
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216 | if (typeof(#[k]) == "string") |
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217 | { |
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218 | method = method + "," + #[k]; |
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219 | } |
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220 | } |
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221 | |
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222 | if ( defined(w)!=voice ) |
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223 | { |
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224 | intvec w; |
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225 | for(k=nvarP; k>=1; k--) |
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226 | { |
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227 | w[k] = deg(var(k)); //compute ring weights |
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228 | } |
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229 | } |
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230 | |
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231 | |
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232 | if (npars(P) > 0) //clear denominators of parameters |
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233 | { |
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234 | for( k=ncols(i); k>0; k-- ) |
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235 | { |
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236 | i[k]=cleardenom(i[k]); |
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237 | } |
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238 | } |
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239 | |
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240 | //---------- exclude cases to which stdhilb should no be applied ---------- |
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241 | //Note that quotient ideal of qring must be homogeneous too |
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242 | |
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243 | int neg=1-attrib (P,"global"); |
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244 | |
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245 | if( //find(ordstr_P,"s") ||// covered by neg |
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246 | find(ordstr_P,"M") || neg ) |
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247 | { |
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248 | // if( defined(hi) && is_homog ) |
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249 | // { |
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250 | // if (p_opt){"std with given Hilbert function in basering";} |
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251 | // return( std(i,hi,w) ); |
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252 | //### here we would need Hibert-Samuel function |
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253 | // } |
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254 | |
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255 | if (p_opt) |
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256 | {"//-- stdhilb not implemented, we use std in ring:"; string(P);} |
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257 | return( std(i) ); |
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258 | } |
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259 | |
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260 | //------------------------ change to hilbRing ---------------------------- |
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261 | //The ground field of P and Philb coincide, Philb has an extra variable |
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262 | //@ or @(k). Philb is no qring and the predefined ideal/module Id(1) in |
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263 | //Philb is homogeneous (it is the homogenized i w.r.t. @ or @(k)) |
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264 | //Parameters of P are not converted in Philb |
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265 | //Philb has only 1 block dp or wp(w) |
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266 | |
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267 | list hiRi = hilbRing(i,w); |
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268 | intvec W = hiRi[2]; |
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269 | def Philb = hiRi[1]; |
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270 | setring Philb; |
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271 | |
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272 | //-------- compute Hilbert series of homogenized ideal in Philb --------- |
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273 | //There are three cases |
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274 | |
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275 | string algorithm; //possibilities: std, slimgb, stdorslimgb |
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276 | //define algorithm: |
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277 | if( find(method,"std") && !find(method,"slimgb") ) |
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278 | { |
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279 | algorithm = "std"; |
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280 | } |
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281 | if( find(method,"slimgb") && !find(method,"std") ) |
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282 | { |
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283 | algorithm = "slimgb"; |
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284 | } |
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285 | if( find(method,"std") && find(method,"slimgb") || |
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286 | (!find(method,"std") && !find(method,"slimgb")) ) |
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287 | { |
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288 | algorithm = "stdorslimgb"; |
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289 | } |
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290 | |
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291 | //### geaendert Dez08: es wird std(Id(1)) statt Id(1) aus Philb nach Phelp |
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292 | // weitergegeben fuer hilbertgetriebenen std |
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293 | |
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294 | if (( algorithm=="std" || ( algorithm=="stdorslimgb" && char(P)>0 ) ) |
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295 | && (defined(hi)!=voice)) |
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296 | { |
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297 | if (p_opt) {"compute hilbert series with std in ring " + string(Philb); |
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298 | "weights used for hilbert series:",W;} |
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299 | Id(1) = std(Id(1)); |
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300 | intvec hi = hilb( Id(1),1,W ); |
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301 | } |
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302 | if (( algorithm=="slimgb" || ( algorithm=="stdorslimgb" && char(P)==0 ) ) |
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303 | && (defined(hi)!=voice)) |
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304 | { |
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305 | if (p_opt) {"compute hilbert series with slimgb in ring " + string(Philb); |
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306 | "weights used for hilbert series:",W;} |
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307 | Id(1) = qslimgb(Id(1)); |
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308 | intvec hi = hilb( Id(1),1,W ); |
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309 | } |
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310 | |
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311 | //-------------- we need another intermediate ring Phelp ---------------- |
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312 | //In Phelp we change only the ordering from Philb (otherwise it coincides |
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313 | //with Philb). Phelp has in addition to P an extra homogenizing variable |
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314 | //with name @ (resp. @(i) if @ and @(1), ..., @(i-1) are defined) with |
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315 | //ordering an extra last block dp(1). |
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316 | //Phelp has the same ordering as P on common variables. In Phelp |
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317 | //a quotient ideal from P is added to the input |
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318 | |
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319 | list BRlist = ringlist(Philb); |
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320 | BRlist[3] = list(); |
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321 | int so = size(ord_P); |
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322 | if( ord_P[so][1] =="c" || ord_P[so][1] =="C" ) |
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323 | { |
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324 | list moduleord = ord_P[so]; |
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325 | so = so-1; |
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326 | } |
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327 | for (k=1; k<=so; k++) |
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328 | { |
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329 | BRlist[3][k] = ord_P[k]; |
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330 | } |
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331 | |
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332 | BRlist[3][so+1] = list("dp",1); |
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333 | w = w,1; |
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334 | |
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335 | if( defined(moduleord)==voice ) |
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336 | { |
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337 | BRlist[3][so+2] = moduleord; |
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338 | } |
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339 | |
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340 | //--- change to extended ring Phelp and compute std with hilbert series ---- |
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341 | def Phelp = ring(quotientList(BRlist)); |
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342 | setring Phelp; |
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343 | def i = imap(Philb, Id(1)); |
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344 | kill Philb; |
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345 | |
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346 | // compute std with Hilbert series |
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347 | option(redThrough); |
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348 | if (w == 1) |
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349 | { |
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350 | if (p_opt){ "std with hilb in " + string(Phelp);} |
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351 | i = std(i, hi); |
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352 | } |
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353 | else |
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354 | { |
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355 | if(p_opt){"std with weighted hilb in "+string(Phelp);} |
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356 | i = std(i, hi, w); |
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357 | } |
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358 | |
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359 | //-------------------- go back to original ring --------------------------- |
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360 | //The main computation is done. Do not forget to simplfy before maping. |
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361 | |
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362 | // subst 1 for homogenizing var |
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363 | if ( p_opt ) { "dehomogenization"; } |
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364 | i = subst(i, var(nvars(basering)), 1); |
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365 | |
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366 | if (p_opt) { "simplification"; } |
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367 | i= simplify(i,34); |
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368 | |
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369 | setring P; |
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370 | if (p_opt) { "imap to ring "+string(P); } |
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371 | i = imap(Phelp,i); |
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372 | kill Phelp; |
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373 | if( was_qring ) |
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374 | { |
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375 | i = NF(i,std(0)); |
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376 | } |
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377 | i = simplify(i,34); |
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378 | // compute reduced SB |
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379 | if (find(s_opt, "redSB") > 0) |
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380 | { |
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381 | if (p_opt) { "//interreduction"; } |
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382 | i=interred(i); |
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383 | } |
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384 | attrib(i, "isSB", 1); |
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385 | option(set,gopt); |
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386 | return (i); |
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387 | } |
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388 | example |
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389 | { "EXAMPLE:"; echo = 2; |
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390 | ring r = 0,(x,y,z),lp; |
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391 | ideal i = y3+x2,x2y+x2z2,x3-z9,z4-y2-xz; |
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392 | ideal j = stdhilb(i); j; |
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393 | |
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394 | ring r1 = 0,(x,y,z),wp(3,2,1); |
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395 | ideal i = y3+x2,x2y+x2z2,x3-z9,z4-y2-xz; //ideal is homogeneous |
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396 | ideal j = stdhilb(i,"std"); j; |
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397 | //this is equivalent to: |
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398 | intvec v = hilb(std(i),1); |
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399 | ideal j1 = std(i,v,intvec(3,2,1)); j1; |
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400 | |
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401 | size(NF(j,j1))+size(NF(j1,j)); //j and j1 define the same ideal |
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402 | } |
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403 | |
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404 | /////////////////////////////////////////////////////////////////////////////// |
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405 | proc quotientList (list RL, list #) |
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406 | "SYNTAX: @code{quotientList (} list_expression @code{)} @* |
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407 | @code{quotientList (} list_expression @code{,} string_expression@code{)} |
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408 | TYPE: list |
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409 | PURPOSE: define a ringlist, say QL, of the first argument, say RL, which is |
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410 | assumed to be the ringlist of a qring, but where the quotient ideal |
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411 | RL[4] is not a standard basis with respect to the given monomial |
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412 | order in RL[3]. Then QL will be obtained from RL just by replacing |
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413 | RL[4] by a standard of it with respect to this order. RL itself |
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414 | will be returnd if size(RL[4]) <= 1 (in which case it is known to be |
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415 | a standard basis w.r.t. any ordering) or if a second argument |
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416 | \"isSB\" of type string is given. |
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417 | NOTE: the command ring(quotientList(RL)) defines a quotient ring correctly |
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418 | and should be used instead of ring(RL) if the quotient ideal RL[4] |
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419 | is not (or not known to be) a standard basis with respect to the |
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420 | monomial ordering specified in RL[3]. |
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421 | SEE ALSO: ringlist, ring |
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422 | EXAMPLE: example quotientList; shows an example" |
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423 | { |
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424 | def P = basering; |
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425 | if( size(#) > 0 ) |
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426 | { |
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427 | if ( #[1] == "isSB") |
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428 | { |
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429 | return (RL); |
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430 | } |
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431 | } |
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432 | ideal Qideal = RL[4]; //##Achtung: falls basering Nullteiler hat, kann |
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433 | //die SB eines Elements mehrere Elemente enthalten |
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434 | if( size(Qideal) <= 0) |
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435 | { |
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436 | return (RL); |
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437 | } |
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438 | |
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439 | RL[4] = ideal(0); |
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440 | def Phelp = ring(RL); |
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441 | setring Phelp; |
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442 | ideal Qideal = groebner(fetch(P,Qideal)); |
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443 | setring P; |
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444 | RL[4]=fetch(Phelp,Qideal); |
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445 | return (RL); |
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446 | } |
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447 | example |
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448 | { "EXAMPLE:"; echo = 2; |
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449 | ring P = 0,(y,z,u,v),lp; |
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450 | ideal i = y+u2+uv3, z+uv3; //i is an lp-SB but not a dp_SB |
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451 | qring Q = std(i); |
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452 | list LQ = ringlist(Q); |
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453 | LQ[3][1][1]="dp"; |
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454 | def Q1 = ring(quotientList(LQ)); |
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455 | setring Q1; |
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456 | Q1; |
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457 | |
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458 | setring Q; |
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459 | ideal q1 = uv3+z, u2+y-z, yv3-zv3-zu; //q1 is a dp-standard basis |
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460 | LQ[4] = q1; |
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461 | def Q2 = ring(quotientList(LQ,"isSB")); |
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462 | setring Q2; |
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463 | Q2; |
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464 | } |
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465 | |
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466 | /////////////////////////////////////////////////////////////////////////////// |
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467 | proc par2varRing (list #) |
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468 | "USAGE: par2varRing([l]); l list of ideals/modules [default:l=empty list] |
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469 | RETURN: list, say L, with L[1] a ring where the parameters of the |
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470 | basering have been converted to an additional last block of |
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471 | variables, all of weight 1, and ordering dp. |
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472 | If a list l with l[i] an ideal/module is given, then |
---|
473 | l[i] + minpoly*freemodule(nrows(l[i])) is mapped to an ideal/module |
---|
474 | in L[1] with name Id(i). |
---|
475 | If the basering has no parameters then L[1] is the basering. |
---|
476 | EXAMPLE: example par2varRing; shows an example" |
---|
477 | { |
---|
478 | def P = basering; |
---|
479 | int npar = npars(P); //number of parameters |
---|
480 | int s = size(#); |
---|
481 | int ii; |
---|
482 | if ( npar == 0) |
---|
483 | { |
---|
484 | dbprint(printlevel-voice+3,"// ** no parameters, ring was not changed"); |
---|
485 | for( ii = 1; ii <= s; ii++) |
---|
486 | { |
---|
487 | def Id(ii) = #[ii]; |
---|
488 | export (Id(ii)); |
---|
489 | } |
---|
490 | return(list(P)); |
---|
491 | } |
---|
492 | |
---|
493 | list rlist = ringlist(P); |
---|
494 | list parlist = rlist[1]; |
---|
495 | rlist[1] = parlist[1]; |
---|
496 | |
---|
497 | string @Minpoly = string(minpoly); //check for minpoly: |
---|
498 | int sm = size(minpoly); |
---|
499 | //now create new ring |
---|
500 | for( ii = 1; ii <= s; ii++) |
---|
501 | { |
---|
502 | def Id(ii) = #[ii]; |
---|
503 | } |
---|
504 | int nvar = size(rlist[2]); |
---|
505 | int nblock = size(rlist[3]); |
---|
506 | int k; |
---|
507 | for (k=1; k<=npar; k++) |
---|
508 | { |
---|
509 | rlist[2][nvar+k] = parlist[2][k]; //change variable list |
---|
510 | } |
---|
511 | |
---|
512 | //converted parameters get one block dp. If module ordering was in front |
---|
513 | //it stays in front, otherwise it will be moved to the end |
---|
514 | intvec OW = 1:npar; |
---|
515 | if( rlist[3][nblock][1] =="c" || rlist[3][nblock][1] =="C" ) |
---|
516 | { |
---|
517 | rlist[3][nblock+1] = rlist[3][nblock]; |
---|
518 | rlist[3][nblock] = list("dp",OW); |
---|
519 | } |
---|
520 | else |
---|
521 | { |
---|
522 | rlist[3][nblock+1] = list("dp",OW); |
---|
523 | } |
---|
524 | |
---|
525 | def Ppar2var = ring(quotientList(rlist)); |
---|
526 | setring Ppar2var; |
---|
527 | if ( sm == 0 ) |
---|
528 | { |
---|
529 | for( ii = 1; ii <= s; ii++) |
---|
530 | { |
---|
531 | def Id(ii) = imap(P,Id(ii)); |
---|
532 | export (Id(ii)); |
---|
533 | } |
---|
534 | } |
---|
535 | else |
---|
536 | { |
---|
537 | if( find(option(),"prot") ){"//add minpoly to input";} |
---|
538 | execute("poly Minpoly = " + @Minpoly + " ;"); |
---|
539 | for( ii = 1; ii <= s; ii++) |
---|
540 | { |
---|
541 | def Id(ii) = imap(P,Id(ii)); |
---|
542 | if (typeof(Id(ii))=="module") |
---|
543 | { |
---|
544 | Id(ii) = Id(ii),Minpoly*freemodule(nrows(Id(ii))); |
---|
545 | } |
---|
546 | else |
---|
547 | { |
---|
548 | Id(ii) = Id(ii),Minpoly; |
---|
549 | } |
---|
550 | export (Id(ii)); |
---|
551 | } |
---|
552 | } |
---|
553 | list Lpar2var = Ppar2var; |
---|
554 | return(Lpar2var); |
---|
555 | } |
---|
556 | example |
---|
557 | { "EXAMPLE:"; echo = 2; |
---|
558 | ring R = (0,x),(y,z,u,v),lp; |
---|
559 | minpoly = x2+1; |
---|
560 | ideal i = x3,x2+y+z+u+v,xyzuv-1; i; |
---|
561 | def P = par2varRing(i)[1]; P; |
---|
562 | setring(P); |
---|
563 | Id(1); |
---|
564 | |
---|
565 | setring R; |
---|
566 | module m = x3*[1,1,1], (xyzuv-1)*[1,0,1]; |
---|
567 | def Q = par2varRing(m)[1]; Q; |
---|
568 | setring(Q); |
---|
569 | print(Id(1)); |
---|
570 | } |
---|
571 | |
---|
572 | ////////////////////////////////////////////////////////////////////////////// |
---|
573 | proc hilbRing ( list # ) |
---|
574 | "USAGE: hilbRing([w,l]); w = intvec, l = list of ideals/modules |
---|
575 | RETURN: list, say L: L[1] is a ring and L[2] an intvec |
---|
576 | L[1] is a ring whith an extra homogenizing variable with name @, |
---|
577 | resp. @(i) if @ and @(1), ..., @(i-1) are defined. |
---|
578 | The monomial ordering of L[1] is consists of 1 block: dp if the |
---|
579 | weights of the variables of the basering, say R, are all 1, resp. |
---|
580 | wp(w,1) wehre w is either given or the intvec of weights of the |
---|
581 | variables of R, i.e. w[k]=deg(var(k)). |
---|
582 | If R is a quotient ring P/Q, then L[1] is not a quotient ring but |
---|
583 | contains the ideal @Qidealhilb@, the homogenized ideal Q of P. |
---|
584 | (Parameters of R are not touched). |
---|
585 | If a list l is given with l[i] an ideal/module, then l[i] is mapped |
---|
586 | to Id(i), the homogenized l[i]+Q*freemodule(nrows(l[i]) in L[1] |
---|
587 | (Id(i) = l[i] if l[i] is already homogeneous). |
---|
588 | L[2] is the intvec (w,1). |
---|
589 | PURPOSE: Prepare a ring for computing the (weighted) hilbert series of |
---|
590 | an ideal/module with an easy monomial ordering. |
---|
591 | NOTE: For this purpose we need w[k]=deg(var(k)). However, if the ordering |
---|
592 | contains an extra weight vector a(v,0..0)) deg(var(k)) returns 0 for |
---|
593 | k being an index which is 0 in a. Therefore we must compute w |
---|
594 | beforehand and give it to hilbRing. |
---|
595 | EXAMPLE: example hilbRing; shows an example |
---|
596 | " |
---|
597 | { |
---|
598 | def P = basering; |
---|
599 | ideal Qideal = ideal(P); //defining the quotient ideal if P is a qring |
---|
600 | if( size(Qideal) != 0 ) |
---|
601 | { |
---|
602 | int is_qring =1; |
---|
603 | } |
---|
604 | list BRlist = ringlist(P); |
---|
605 | BRlist[4] = ideal(0); //kill quotient ideal in BRlist |
---|
606 | |
---|
607 | int nvarP = nvars(P); |
---|
608 | int s = size(#); |
---|
609 | int k; |
---|
610 | |
---|
611 | for(k = 1; k <= s; k++) |
---|
612 | { |
---|
613 | if ( typeof(#[k]) == "intvec" ) |
---|
614 | { |
---|
615 | intvec w = #[k]; //given weights for the variables |
---|
616 | # = delete (#,k); |
---|
617 | } |
---|
618 | } |
---|
619 | |
---|
620 | s = size(#); |
---|
621 | for(k = 1; k <= s; k++) |
---|
622 | { |
---|
623 | def Id(k) = #[k]; |
---|
624 | int nr(k) = nrows(Id(k)); |
---|
625 | } |
---|
626 | |
---|
627 | if ( defined(w)!=voice ) |
---|
628 | { |
---|
629 | intvec w; //for ringweights of basering P |
---|
630 | for(k=1; k<=nvarP; k++) |
---|
631 | { |
---|
632 | w[k]=deg(var(k)); //degree of kth variable |
---|
633 | } |
---|
634 | } |
---|
635 | //--------------------- a homogenizing variable is added ------------------ |
---|
636 | // call it @, resp. @(k) if @(1),...,@(k-1) are defined |
---|
637 | string homvar; |
---|
638 | if ( defined(@)==0 ) |
---|
639 | { |
---|
640 | homvar = "@"; |
---|
641 | } |
---|
642 | else |
---|
643 | { |
---|
644 | k=1; |
---|
645 | while( defined(@(k)) != 0 ) |
---|
646 | { |
---|
647 | k++; |
---|
648 | } |
---|
649 | homvar = "@("+string(k)+")"; |
---|
650 | } |
---|
651 | BRlist[2][nvarP+1] = homvar; |
---|
652 | w[nvarP +1]=1; |
---|
653 | |
---|
654 | //ordering is set to (dp,C) if weights of all variables are 1 |
---|
655 | //resp. to (wp(w,1),C) where w are the ringweights of basering P |
---|
656 | //homogenizing var gets weight 1: |
---|
657 | |
---|
658 | BRlist[3] = list(); |
---|
659 | BRlist[3][2]=list("C",intvec(0)); //put module ordering always last |
---|
660 | if(w==1) |
---|
661 | { |
---|
662 | BRlist[3][1]=list("dp",w); |
---|
663 | } |
---|
664 | else |
---|
665 | { |
---|
666 | BRlist[3][1]=list("wp",w); |
---|
667 | } |
---|
668 | |
---|
669 | //-------------- change ring and get ideal from previous ring --------------- |
---|
670 | def Philb = ring(quotientList(BRlist)); |
---|
671 | kill BRlist; |
---|
672 | setring Philb; |
---|
673 | if( defined(is_qring)==voice ) |
---|
674 | { |
---|
675 | ideal @Qidealhilb@ = imap(P,Qideal); |
---|
676 | if ( ! homog(@Qidealhilb@) ) |
---|
677 | { |
---|
678 | @Qidealhilb@ = homog( @Qidealhilb@, `homvar` ); |
---|
679 | } |
---|
680 | export(@Qidealhilb@); |
---|
681 | |
---|
682 | if( find(option(),"prot") ){"add quotient ideal to input";} |
---|
683 | |
---|
684 | for(k = 1; k <= s; k++) |
---|
685 | { //homogenize if necessary |
---|
686 | def Id(k) = imap(P,Id(k)); |
---|
687 | if ( ! homog(Id(k)) ) |
---|
688 | { |
---|
689 | Id(k) = homog( imap(P,Id(k)), `homvar` ); |
---|
690 | } |
---|
691 | if (typeof(Id(k))=="module") |
---|
692 | { |
---|
693 | Id(k) = Id(k),@Qidealhilb@*freemodule(nr(k)) ; |
---|
694 | } |
---|
695 | else |
---|
696 | { |
---|
697 | Id(k) = Id(k),@Qidealhilb@ ; |
---|
698 | } |
---|
699 | export(Id(k)); |
---|
700 | } |
---|
701 | } |
---|
702 | else |
---|
703 | { |
---|
704 | for(k = 1; k <= s; k++) |
---|
705 | { //homogenize if necessary |
---|
706 | def Id(k) = imap(P,Id(k)); |
---|
707 | if ( ! homog(Id(k)) ) |
---|
708 | { |
---|
709 | Id(k) = homog( imap(P,Id(k)), `homvar` ); |
---|
710 | } |
---|
711 | export(Id(k)); |
---|
712 | } |
---|
713 | } |
---|
714 | list Lhilb = Philb,w; |
---|
715 | setring(P); return(Lhilb); |
---|
716 | } |
---|
717 | example |
---|
718 | { "EXAMPLE:"; echo = 2; |
---|
719 | ring R = 0,(x,y,z,u,v),lp; |
---|
720 | ideal i = x+y2+z3,xy+xv+yz+zu+uv,xyzuv-1; |
---|
721 | intvec w = 6,3,2,1,1; |
---|
722 | hilbRing(i,w); |
---|
723 | def P = hilbRing(w,i)[1]; |
---|
724 | setring P; |
---|
725 | Id(1); |
---|
726 | hilb(std(Id(1)),1); |
---|
727 | |
---|
728 | ring S = 0,(x,y,z,u,v),lp; |
---|
729 | qring T = std(x+y2+z3); |
---|
730 | ideal i = xy+xv+yz+zu+uv,xyzuv-v5; |
---|
731 | module m = i*[0,1,1] + (xyzuv-v5)*[1,1,0]; |
---|
732 | def Q = hilbRing(m)[1]; Q; |
---|
733 | setring Q; |
---|
734 | print(Id(1)); |
---|
735 | } |
---|
736 | |
---|
737 | ////////////////////////////////////////////////////////////////////////////// |
---|
738 | proc qslimgb (def i) |
---|
739 | "USAGE: qslimgb(i); i ideal or module |
---|
740 | RETURN: same type as input, a standard basis of i computed with slimgb |
---|
741 | NOTE: Only as long as slimgb does not know qrings qslimgb should be used |
---|
742 | in case the basering is (possibly) a quotient ring. |
---|
743 | The quotient ideal is added to the input and slimgb is applied. |
---|
744 | EXAMPLE: example qslimgb; shows an example" |
---|
745 | { |
---|
746 | def P = basering; |
---|
747 | ideal Qideal = ideal(P); //defining the quotient ideal if P is a qring |
---|
748 | int p_opt; |
---|
749 | if( find(option(),"prot") ) |
---|
750 | { |
---|
751 | p_opt=1; |
---|
752 | } |
---|
753 | if (size(Qideal) == 0) |
---|
754 | { |
---|
755 | if (p_opt) { "slimgb in ring " + string(P); } |
---|
756 | return(slimgb(i)); |
---|
757 | } |
---|
758 | |
---|
759 | //case of a qring; since slimgb does not know qrings we |
---|
760 | //delete the quotient ideal and add it to i |
---|
761 | |
---|
762 | list BRlist = ringlist(P); |
---|
763 | BRlist[4] = ideal(0); |
---|
764 | def Phelp = ring(BRlist); |
---|
765 | kill BRlist; |
---|
766 | setring Phelp; |
---|
767 | // module case: |
---|
768 | def iq = imap(P,i); |
---|
769 | iq = iq, imap(P,Qideal)*freemodule(nrows(iq)); |
---|
770 | if (p_opt) |
---|
771 | { |
---|
772 | "slimgb in ring " + string(Phelp); |
---|
773 | "(with quotient ideal added to input)"; |
---|
774 | } |
---|
775 | iq = slimgb(iq); |
---|
776 | |
---|
777 | setring P; |
---|
778 | if (p_opt) { "//imap to original ring"; } |
---|
779 | i = imap(Phelp,iq); |
---|
780 | kill Phelp; |
---|
781 | |
---|
782 | if (find(option(),"redSB") > 0) |
---|
783 | { |
---|
784 | if (p_opt) { "//interreduction"; } |
---|
785 | i=reduce(i,std(0)); |
---|
786 | i=interred(i); |
---|
787 | } |
---|
788 | attrib(i, "isSB", 1); |
---|
789 | return (i); |
---|
790 | } |
---|
791 | example |
---|
792 | { "EXAMPLE:"; echo = 2; |
---|
793 | ring R = (0,v),(x,y,z,u),dp; |
---|
794 | qring Q = std(x2-y3); |
---|
795 | ideal i = x+y2,xy+yz+zu+u*v,xyzu*v-1; |
---|
796 | ideal j = qslimgb(i); j; |
---|
797 | |
---|
798 | module m = [x+y2,1,0], [1,1,x2+y2+xyz]; |
---|
799 | print(qslimgb(m)); |
---|
800 | } |
---|
801 | |
---|
802 | ////////////////////////////////////////////////////////////////////////////// |
---|
803 | proc groebner(def i_par, list #) |
---|
804 | "SYNTAX: @code{groebner (} ideal_expression @code{)} @* |
---|
805 | @code{groebner (} module_expression @code{)} @* |
---|
806 | @code{groebner (} ideal_expression@code{,} int_expression @code{)} @* |
---|
807 | @code{groebner (} module_expression@code{,} int_expression @code{)} @* |
---|
808 | @code{groebner (} ideal_expression@code{,} list of string_expressions |
---|
809 | @code{)} @* |
---|
810 | @code{groebner (} ideal_expression@code{,} list of string_expressions |
---|
811 | and int_expression @code{)} @* |
---|
812 | @code{groebner (} ideal_expression@code{,} int_expression @code{)} |
---|
813 | TYPE: type of the first argument |
---|
814 | PURPOSE: computes a standard basis of the first argument @code{I} |
---|
815 | (ideal or module) by a heuristically chosen method (default) |
---|
816 | or by a method specified by further arguments of type string. |
---|
817 | Possible methods are: @* |
---|
818 | - the direct methods @code{\"std\"} or @code{\"slimgb\"} without |
---|
819 | conversion, @* |
---|
820 | - conversion methods @code{\"hilb\"} or @code{\"fglm\"} where |
---|
821 | a Groebner basis is first computed with an \"easy\" ordering |
---|
822 | and then converted to the ordering of the basering by the |
---|
823 | Hilbert driven Groebner basis computation or by linear algebra. |
---|
824 | The actual computation of the Groebner basis can be |
---|
825 | specified by @code{\"std\"} or by @code{\"slimgb\"} |
---|
826 | (not for all orderings implemented). @* |
---|
827 | A further string @code{\"par2var\"} converts parameters to an extra |
---|
828 | block of variables before a Groebner basis computation (and |
---|
829 | afterwards back). |
---|
830 | @code{option(prot)} informs about the chosen method. |
---|
831 | NOTE: If an additional argument, say @code{wait}, of type int is given, |
---|
832 | then the computation runs for at most @code{wait} seconds. |
---|
833 | That is, if no result could be computed in @code{wait} seconds, |
---|
834 | then the computation is interrupted, 0 is returned, a warning |
---|
835 | message is displayed, and the global variable |
---|
836 | @code{Standard::groebner_error} is defined. |
---|
837 | This feature uses MP and hence it is available on UNIX platforms, only. |
---|
838 | HINT: Since there exists no uniform best method for computing standard |
---|
839 | bases, and since the difference in performance of a method on |
---|
840 | different examples can be huge, it is recommended to test, for hard |
---|
841 | examples, first various methods on a simplified example (e.g. use |
---|
842 | characteristic 32003 instead of 0 or substitute a subset of |
---|
843 | parameters/variables by integers, etc.). @* |
---|
844 | SEE ALSO: stdhilb, stdfglm, std, slimgb |
---|
845 | KEYWORDS: time limit on computations; MP, groebner basis computations |
---|
846 | EXAMPLE: example groebner; shows an example" |
---|
847 | |
---|
848 | { |
---|
849 | //Vorgabe einer Teilmenge aus {hilb,fglm,par2var,std,slimgb} |
---|
850 | //V1: Erste Einstellungen (Jan 2007) |
---|
851 | //V2: Aktuelle Aenderungen (Juni 2008) |
---|
852 | //--------------------------------- |
---|
853 | //0. Immer Aufruf von std unabhaengig von der Vorgabe: |
---|
854 | // gemischte Ordnungen, extra Gewichtsvektor, Matrix Ordnungen |
---|
855 | // ### Todo: extra Gewichtsvektor sollte nicht immer mit std wirken, |
---|
856 | // sondern z.B. mit "hilb" arbeiten koennen |
---|
857 | // ### Todo: es sollte ein Gewichtsvektor mitgegeben werden koennen (oder |
---|
858 | // berechnet werden), z.B. groebner(I,"hilb",w) oder groebner(I,"withWeights") |
---|
859 | // wie bei elim in elim.lib |
---|
860 | |
---|
861 | //1. Keine Vorgabe: es wirkt die aktuelle Heuristk: |
---|
862 | // - Char = p: std |
---|
863 | //V1 - Char = 0: slimgb (im qring wird Quotientenideal zum Input addiert) |
---|
864 | //V2 - Char = 0: std |
---|
865 | // - 1-Block-Ordnungen/non-commutative: direkt Aufruf von std oder slimgb |
---|
866 | // - Komplizierte Ordnungen (lp oder > 1 Block): hilb |
---|
867 | //V1 - Parameter werden grundsaetzlich nicht in Variable umgewandelt |
---|
868 | //V2 - Mehr als ein Parmeter wird zu Variable konvertiert |
---|
869 | // - fglm is keine Heuristik, da sonst vorher dim==0 peprueft werden muss |
---|
870 | |
---|
871 | //2. Vorgabe aus {std,slimgb}: es wird wo immer moeglich das Angegebene |
---|
872 | // gewaehlt (da slimgb keine Hilbertfunktion kennt, wird std verwendet). |
---|
873 | // Bei slimgb im qring, wird das Quotientenideal zum Ideal addiert. |
---|
874 | // Bei Angabe von std zusammen mit slimgb (aequivalent zur Angabe von |
---|
875 | // keinem von beidem) wirkt obige Heuristik. |
---|
876 | |
---|
877 | //3. Nichtleere Vorgabe aus {hilb,fglm,std,slimgb}: |
---|
878 | // es wird nur das Angegebene und Moegliche sowie das Notwendige verwendet |
---|
879 | // und bei Wahlmoeglickeit je nach Heuristik. |
---|
880 | // Z.B. Vorgabe von {hilb} ist aequivalent zu {hilb,std,slimgb} und es wird |
---|
881 | // hilb und nach Heuristik std oder slimgb verwendet, |
---|
882 | // (V1: aber nicht par2var) |
---|
883 | // bei Vorgabe von {hilb,slimgb} wird hilb und wo moeglich slimgb verwendet. |
---|
884 | |
---|
885 | //4. Bei Vorgabe von {par2var} wird par2var immer mit hilb und nach Heuristik |
---|
886 | // std oder slimgb verwendet. Zu Variablen konvertierte Parameter haben |
---|
887 | // extra letzten Block und Gewichte 1. |
---|
888 | |
---|
889 | def P=basering; |
---|
890 | if ((typeof(i_par)=="vector")||(typeof(i_par)=="module")||(typeof(i_par)=="matrix")) {module i=i_par;} |
---|
891 | else {ideal i=i_par; } // int, poly, number, ideal |
---|
892 | kill i_par; |
---|
893 | // check for integer etc coefficients |
---|
894 | if (charstr(basering)[1]=="i") // either integer or integer,q |
---|
895 | { |
---|
896 | if (find(option(),"prot")) { "calling std for ideals in ring with ring coefficients"; } |
---|
897 | return (std(i)); |
---|
898 | } |
---|
899 | |
---|
900 | //----------------------- save the given method --------------------------- |
---|
901 | string method; //all given methods as a coma separated string |
---|
902 | list Method; //all given methods as a list |
---|
903 | int k; |
---|
904 | for (k=1; k<=size(#); k++) |
---|
905 | { |
---|
906 | if (typeof(#[k]) == "int") |
---|
907 | { |
---|
908 | int wait = #[k]; |
---|
909 | } |
---|
910 | if (typeof(#[k]) == "string") |
---|
911 | { |
---|
912 | method = method + "," + #[k]; |
---|
913 | Method = Method + list(#[k]); |
---|
914 | } |
---|
915 | } |
---|
916 | |
---|
917 | //======= we have an argument of type int -- try to use MPfork links ======= |
---|
918 | if ( defined(wait) == voice ) |
---|
919 | { |
---|
920 | if ( system("with", "MP") ) |
---|
921 | { |
---|
922 | int j = 10; |
---|
923 | string bs = nameof(basering); |
---|
924 | link l_fork = "MPtcp:fork"; |
---|
925 | open(l_fork); |
---|
926 | write(l_fork, quote(system("pid"))); |
---|
927 | int pid = read(l_fork); |
---|
928 | // write(l_fork, quote(groebner(eval(i)))); |
---|
929 | write(l_fork, quote(groebner(eval(i),eval(Method)))); |
---|
930 | //###Fehlermeldung: |
---|
931 | // ***dError: undef. ringorder used |
---|
932 | // occured at: |
---|
933 | |
---|
934 | // sleep in small intervalls for appr. one second |
---|
935 | if (wait > 0) |
---|
936 | { |
---|
937 | while(j < 1000000) |
---|
938 | { |
---|
939 | if (status(l_fork, "read", "ready", j)) {break;} |
---|
940 | j = j + j; |
---|
941 | } |
---|
942 | } |
---|
943 | |
---|
944 | // sleep in intervalls of one second from now on |
---|
945 | j = 1; |
---|
946 | while (j < wait) |
---|
947 | { |
---|
948 | if (status(l_fork, "read", "ready", 1000000)) {break;} |
---|
949 | j = j + 1; |
---|
950 | } |
---|
951 | |
---|
952 | if (status(l_fork, "read", "ready")) |
---|
953 | { |
---|
954 | def result = read(l_fork); |
---|
955 | if (bs != nameof(basering)) |
---|
956 | { |
---|
957 | def PP = basering; |
---|
958 | setring P; |
---|
959 | def result = imap(PP, result); |
---|
960 | kill PP; |
---|
961 | } |
---|
962 | if (defined(groebner_error)==1) |
---|
963 | { |
---|
964 | kill groebner_error; |
---|
965 | } |
---|
966 | kill l_fork; |
---|
967 | } |
---|
968 | else |
---|
969 | { |
---|
970 | ideal result; |
---|
971 | if (! defined(groebner_error)) |
---|
972 | { |
---|
973 | int groebner_error = 1; |
---|
974 | export groebner_error; |
---|
975 | } |
---|
976 | "** groebner did not finish"; |
---|
977 | j = system("sh", "kill " + string(pid)); |
---|
978 | } |
---|
979 | return (result); |
---|
980 | } |
---|
981 | else |
---|
982 | { |
---|
983 | "** groebner with a time limit on computation is not supported |
---|
984 | in this configuration"; |
---|
985 | } |
---|
986 | } |
---|
987 | |
---|
988 | //=========== we are still here -- do the actual computation ============= |
---|
989 | |
---|
990 | //--------------------- save data from basering --------------------------- |
---|
991 | string @Minpoly = string(minpoly); //minimal polynomial |
---|
992 | int was_minpoly; //remembers if there was a minpoly in P |
---|
993 | if (size(minpoly) > 0) |
---|
994 | { |
---|
995 | was_minpoly = 1; |
---|
996 | } |
---|
997 | |
---|
998 | ideal Qideal = ideal(P); //defining the quotient ideal if P is a qring |
---|
999 | int was_qring; //remembers if basering was a qring |
---|
1000 | //int is_homog = 1; |
---|
1001 | if (size(Qideal) > 0) |
---|
1002 | { |
---|
1003 | was_qring = 1; |
---|
1004 | //is_homog = homog(Qideal); //remembers if Qideal was homog (homog(0)=1) |
---|
1005 | } |
---|
1006 | list BRlist = ringlist(P); //ringlist of basering |
---|
1007 | |
---|
1008 | // save ordering of basering P for later use |
---|
1009 | list ord_P = BRlist[3]; //should be available in all rings |
---|
1010 | string ordstr_P = ordstr(P); |
---|
1011 | int nvars_P = nvars(P); |
---|
1012 | int npars_P = npars(P); |
---|
1013 | intvec w; //for ringweights of basering P |
---|
1014 | for(k=1; k<=nvars_P; k++) |
---|
1015 | { |
---|
1016 | w[k]=deg(var(k)); |
---|
1017 | } |
---|
1018 | int neg=1-attrib (P,"global"); |
---|
1019 | |
---|
1020 | //save options: |
---|
1021 | intvec opt=option(get); |
---|
1022 | string s_opt = option(); |
---|
1023 | int p_opt; |
---|
1024 | if (find(s_opt, "prot")) { p_opt = 1; } |
---|
1025 | |
---|
1026 | //------------------ cases where std is always used ------------------------ |
---|
1027 | //If other methods are not implemented or do not make sense, i.e. for |
---|
1028 | //local or mixed orderings, matrix orderings, extra weight vector |
---|
1029 | //### Todo: extra weight vector should be allowed for e.g. with "hilb" |
---|
1030 | |
---|
1031 | if( //( find(ordstr_P,"s") > 0 ) || // covered by neg |
---|
1032 | ( find(ordstr_P,"M") > 0 ) || ( find(ordstr_P,"a") > 0 ) || neg ) |
---|
1033 | { |
---|
1034 | if (p_opt) { "std in basering"; } |
---|
1035 | return(std(i)); |
---|
1036 | } |
---|
1037 | |
---|
1038 | //now we have: |
---|
1039 | //ideal or module, global ordering, no matrix ordering, no extra weight vector |
---|
1040 | //The interesting cases start now. |
---|
1041 | |
---|
1042 | //------------------ classify the possible settings --------------------- |
---|
1043 | string algorithm; //possibilities: std, slimgb, stdorslimgb |
---|
1044 | string conversion; //possibilities: hilb, fglm, hilborfglm, no |
---|
1045 | string partovar; //possibilities: yes, no |
---|
1046 | string order; //possibilities: simple, !simple |
---|
1047 | string direct; //possibilities: yes, no |
---|
1048 | |
---|
1049 | //define algorithm: |
---|
1050 | if( find(method,"std") && !find(method,"slimgb") ) |
---|
1051 | { |
---|
1052 | algorithm = "std"; |
---|
1053 | } |
---|
1054 | if( find(method,"slimgb") && !find(method,"std") ) |
---|
1055 | { |
---|
1056 | algorithm = "slimgb"; |
---|
1057 | } |
---|
1058 | if( find(method,"std") && find(method,"slimgb") || |
---|
1059 | (!find(method,"std") && !find(method,"slimgb")) ) |
---|
1060 | { |
---|
1061 | algorithm = "stdorslimgb"; |
---|
1062 | } |
---|
1063 | |
---|
1064 | //define conversion: |
---|
1065 | if( find(method,"hilb") && !find(method,"fglm") ) |
---|
1066 | { |
---|
1067 | conversion = "hilb"; // $Id$ |
---|
1068 | } |
---|
1069 | if( find(method,"fglm") && !find(method,"hilb") ) |
---|
1070 | { |
---|
1071 | conversion = "fglm"; // $Id$ |
---|
1072 | } |
---|
1073 | if( find(method,"fglm") && find(method,"hilb") ) |
---|
1074 | { |
---|
1075 | conversion = "hilborfglm"; // $Id$ |
---|
1076 | } |
---|
1077 | if( !find(method,"fglm") && !find(method,"hilb") ) |
---|
1078 | { |
---|
1079 | conversion = "no"; // $Id$ |
---|
1080 | } |
---|
1081 | |
---|
1082 | //define partovar: |
---|
1083 | //if( find(method,"par2var") && npars_P > 0 ) //V1 |
---|
1084 | if( find(method,"par2var") || npars_P > 1 ) //V2 |
---|
1085 | { |
---|
1086 | partovar = "yes"; |
---|
1087 | } |
---|
1088 | else |
---|
1089 | { |
---|
1090 | partovar = "no"; |
---|
1091 | } |
---|
1092 | |
---|
1093 | //define order: |
---|
1094 | if (system("nblocks") <= 2) |
---|
1095 | { |
---|
1096 | if ( find(ordstr_P,"M")+find(ordstr_P,"lp")+find(ordstr_P,"rp") <= 0 ) |
---|
1097 | { |
---|
1098 | order = "simple"; |
---|
1099 | } |
---|
1100 | } |
---|
1101 | |
---|
1102 | //define direct: |
---|
1103 | if ( (order=="simple" && (size(method)==0)) || |
---|
1104 | (size(BRlist)>4) || |
---|
1105 | (order=="simple" && (method==",par2var" && npars_P==0 )) || |
---|
1106 | (conversion=="no" && partovar=="no" && |
---|
1107 | (algorithm=="std" || algorithm=="slimgb" || |
---|
1108 | (find(method,"std") && find(method,"slimgb")) ) ) ) |
---|
1109 | { |
---|
1110 | direct = "yes"; |
---|
1111 | } |
---|
1112 | else |
---|
1113 | { |
---|
1114 | direct = "no"; |
---|
1115 | } |
---|
1116 | |
---|
1117 | //order=="simple" means that the ordering of the variables consists of one |
---|
1118 | //block which is not a matrix ordering and not a lexicographical ordering. |
---|
1119 | //(Note:Singular counts always least 2 blocks, one is for module component): |
---|
1120 | //Call a method "direct" if conversion=="no" && partovar="no" which means |
---|
1121 | //that we apply std or slimgb dircet in the basering (exception |
---|
1122 | //as long as slimgb does not know qrings: in a qring of a ring P |
---|
1123 | //the ideal Qideal is added to the ideal and slimgb is applied in P). |
---|
1124 | //We apply a direct method if we have a simple monomial ordering, if no |
---|
1125 | //conversion (fglm or hilb) is specified and if the parameters shall |
---|
1126 | //not be made to variables |
---|
1127 | //BRlist (=ringlist of basering) > 4 if the basering is non-commutative |
---|
1128 | //---------------------------- direct methods ----------------------------- |
---|
1129 | if ( direct == "yes" ) |
---|
1130 | { |
---|
1131 | //if ( algorithm=="std" || (algorithm=="stdorslimgb" && char(P)>0) ) //V1 |
---|
1132 | if ( algorithm=="std" || (algorithm=="stdorslimgb") ) //V2 |
---|
1133 | { |
---|
1134 | if (p_opt) { "std in " + string(P); } |
---|
1135 | return(std(i)); |
---|
1136 | } |
---|
1137 | //if( algorithm=="slimgb" || (algorithm=="stdorslimgb" && char(P)==0)) //V1 |
---|
1138 | if ( algorithm=="slimgb" ) //V2 |
---|
1139 | { |
---|
1140 | return(qslimgb(i)); |
---|
1141 | } |
---|
1142 | } |
---|
1143 | |
---|
1144 | //--------------------------- indirect methods ----------------------------- |
---|
1145 | //indirect methods are methods where a conversion is used with a ring change |
---|
1146 | //We are in the following situation: |
---|
1147 | //direct=="no" (i.e. "hilb" or "fglm" or "par2var" is given) |
---|
1148 | //or no method is given and we have a complicated monomial ordering |
---|
1149 | //V1: "par2var" is not a default strategy, it must be explicitely |
---|
1150 | //given in order to be performed. |
---|
1151 | //V2: "par2var" is a default strategy if there are more than 1 parameters |
---|
1152 | |
---|
1153 | //------------ case where no parameters are made to variables ------------- |
---|
1154 | if ( partovar == "no" && conversion == "hilb" |
---|
1155 | || (partovar == "no" && conversion == "fglm" ) |
---|
1156 | || (partovar == "no" && conversion == "hilborfglm" ) |
---|
1157 | || (partovar == "no" && conversion == "no" && direct == "no") ) |
---|
1158 | //last case: heuristic |
---|
1159 | { |
---|
1160 | if ( conversion=="fglm" ) |
---|
1161 | { |
---|
1162 | //if ( algorithm=="std" || (algorithm=="stdorslimgb" && char(P)>0) ) //V1 |
---|
1163 | if ( algorithm=="std" || (algorithm=="stdorslimgb") ) //V2 |
---|
1164 | { |
---|
1165 | return (stdfglm(i,"std")); |
---|
1166 | } |
---|
1167 | //if(algorithm=="slimgb" || (algorithm=="stdorslimgb" && char(P)==0))//V1 |
---|
1168 | if( algorithm=="slimgb" ) //V2 |
---|
1169 | { |
---|
1170 | return (stdfglm(i,"slimgb")); |
---|
1171 | } |
---|
1172 | } |
---|
1173 | else |
---|
1174 | { |
---|
1175 | //if ( algorithm=="std" || (algorithm=="stdorslimgb" && char(P)>0) )//V1 |
---|
1176 | if ( algorithm=="std" || (algorithm=="stdorslimgb" ) ) //V2 |
---|
1177 | { |
---|
1178 | return (stdhilb(i,"std")); |
---|
1179 | } |
---|
1180 | //if(algorithm=="slimgb" || (algorithm=="stdorslimgb" && char(P)==0))//V1 |
---|
1181 | if ( algorithm=="slimgb" ) //V2 |
---|
1182 | { |
---|
1183 | return (stdhilb(i,"slimgb")); |
---|
1184 | } |
---|
1185 | } |
---|
1186 | } |
---|
1187 | |
---|
1188 | //------------ case where parameters are made to variables ---------------- |
---|
1189 | //define a ring Phelp via par2varRing in which the parameters are variables |
---|
1190 | |
---|
1191 | else |
---|
1192 | { |
---|
1193 | // reset options |
---|
1194 | option(none); |
---|
1195 | // turn on options prot, mem, redSB, intStrategy if previously set |
---|
1196 | if ( find(s_opt, "prot") ) |
---|
1197 | { option(prot); } |
---|
1198 | if ( find(s_opt, "mem") ) |
---|
1199 | { option(mem); } |
---|
1200 | if ( find(s_opt, "redSB") ) |
---|
1201 | { option(redSB); } |
---|
1202 | if ( find(s_opt, "intStrategy") ) |
---|
1203 | { option(intStrategy); } |
---|
1204 | |
---|
1205 | //first clear denominators of parameters |
---|
1206 | if (npars_P > 0) |
---|
1207 | { |
---|
1208 | for( k=ncols(i); k>0; k-- ) |
---|
1209 | { i[k]=cleardenom(i[k]); } |
---|
1210 | } |
---|
1211 | |
---|
1212 | def Phelp = par2varRing(i)[1]; //minpoly is mapped with i |
---|
1213 | setring Phelp; |
---|
1214 | def i = Id(1); |
---|
1215 | //is_homog = homog(i); |
---|
1216 | |
---|
1217 | //If parameters are converted to ring variables, they appear in an extra |
---|
1218 | //block. Therefore we use always hilb for this block ordering: |
---|
1219 | if ( conversion=="fglm" ) |
---|
1220 | { |
---|
1221 | i = (stdfglm(i)); //only uesful for 1 parameter with minpoly |
---|
1222 | } |
---|
1223 | else |
---|
1224 | { |
---|
1225 | //if ( algorithm=="std" || (algorithm=="stdorslimgb" && char(P)>0) )//V1 |
---|
1226 | if ( algorithm=="std" || (algorithm=="stdorslimgb" )) //V2 |
---|
1227 | { |
---|
1228 | i = stdhilb(i,"std"); |
---|
1229 | } |
---|
1230 | //if(algorithm=="slimgb" || (algorithm=="stdorslimgb" && char(P)==0))//V1 |
---|
1231 | if ( algorithm=="slimgb" ) //V2 |
---|
1232 | { |
---|
1233 | i = stdhilb(i,"slimgb"); |
---|
1234 | } |
---|
1235 | } |
---|
1236 | } |
---|
1237 | |
---|
1238 | //-------------------- go back to original ring --------------------------- |
---|
1239 | //The main computation is done. However, the SB coming from a ring with |
---|
1240 | //extra variables is in general too big. We simplify it before mapping it |
---|
1241 | //to the basering. |
---|
1242 | |
---|
1243 | if (p_opt) { "//simplification"; } |
---|
1244 | |
---|
1245 | if (was_minpoly) |
---|
1246 | { |
---|
1247 | execute("ideal Minpoly = " + @Minpoly + ";"); |
---|
1248 | attrib(Minpoly,"isSB",1); |
---|
1249 | i = simplify(NF(i,Minpoly),2); |
---|
1250 | } |
---|
1251 | |
---|
1252 | def Li = lead(i); |
---|
1253 | setring P; |
---|
1254 | def Li = imap(Phelp,Li); |
---|
1255 | Li = simplify(Li,32); |
---|
1256 | intvec vi; |
---|
1257 | for (k=1; k<=ncols(Li); k++) |
---|
1258 | { |
---|
1259 | vi[k] = Li[k]==0; |
---|
1260 | } |
---|
1261 | |
---|
1262 | setring Phelp; |
---|
1263 | for (k=1; k<=size(i) ;k++) |
---|
1264 | { |
---|
1265 | if(vi[k]==1) |
---|
1266 | { |
---|
1267 | i[k]=0; |
---|
1268 | } |
---|
1269 | } |
---|
1270 | i = simplify(i,2); |
---|
1271 | |
---|
1272 | setring P; |
---|
1273 | if (p_opt) { "//imap to original ring"; } |
---|
1274 | i = imap(Phelp,i); |
---|
1275 | kill Phelp; |
---|
1276 | i = simplify(i,34); |
---|
1277 | |
---|
1278 | // clean-up time |
---|
1279 | option(set, opt); |
---|
1280 | if (find(s_opt, "redSB") > 0) |
---|
1281 | { |
---|
1282 | if (p_opt) { "//interreduction"; } |
---|
1283 | i=interred(i); |
---|
1284 | } |
---|
1285 | attrib(i, "isSB", 1); |
---|
1286 | return (i); |
---|
1287 | } |
---|
1288 | example |
---|
1289 | { "EXAMPLE: "; echo=2; |
---|
1290 | intvec opt = option(get); |
---|
1291 | option(prot); |
---|
1292 | ring r = 0,(a,b,c,d),dp; |
---|
1293 | ideal i = a+b+c+d,ab+ad+bc+cd,abc+abd+acd+bcd,abcd-1; |
---|
1294 | groebner(i); |
---|
1295 | |
---|
1296 | ring s = 0,(a,b,c,d),lp; |
---|
1297 | ideal i = imap(r,i); |
---|
1298 | groebner(i,"hilb"); |
---|
1299 | |
---|
1300 | ring R = (0,a),(b,c,d),lp; |
---|
1301 | minpoly = a2+1; |
---|
1302 | ideal i = a+b+c+d,ab+ad+bc+cd,abc+abd+acd+bcd,d2-c2b2; |
---|
1303 | groebner(i,"par2var","slimgb"); |
---|
1304 | |
---|
1305 | groebner(i,"fglm"); //computes a reduced standard basis |
---|
1306 | |
---|
1307 | option(set,opt); |
---|
1308 | } |
---|
1309 | |
---|
1310 | ////////////////////////////////////////////////////////////////////////// |
---|
1311 | |
---|
1312 | proc res(list #) |
---|
1313 | "@c we do texinfo here: |
---|
1314 | @cindex resolution, computation of |
---|
1315 | @table @code |
---|
1316 | @item @strong{Syntax:} |
---|
1317 | @code{res (} ideal_expression@code{,} int_expression @code{[,} any_expression @code{])} |
---|
1318 | @*@code{res (} module_expression@code{,} int_expression @code{[,} any_expression @code{])} |
---|
1319 | @item @strong{Type:} |
---|
1320 | resolution |
---|
1321 | @item @strong{Purpose:} |
---|
1322 | computes a (possibly minimal) free resolution of an ideal or module using |
---|
1323 | a heuristically chosen method. |
---|
1324 | @* The second (int) argument (say @code{k}) specifies the length of |
---|
1325 | the resolution. If it is not positive then @code{k} is assumed to be the |
---|
1326 | number of variables of the basering. |
---|
1327 | @* If a third argument is given, the returned resolution is minimized. |
---|
1328 | |
---|
1329 | Depending on the input, the returned resolution is computed using the |
---|
1330 | following methods: |
---|
1331 | @table @asis |
---|
1332 | @item @strong{quotient rings:} |
---|
1333 | @code{nres} (classical method using syzygies) , see @ref{nres}. |
---|
1334 | |
---|
1335 | @item @strong{homogeneous ideals and k=0:} |
---|
1336 | @code{lres} (La'Scala's method), see @ref{lres}. |
---|
1337 | |
---|
1338 | @item @strong{not minimized resolution and (homogeneous input with k not 0, or local rings):} |
---|
1339 | @code{sres} (Schreyer's method), see @ref{sres}. |
---|
1340 | |
---|
1341 | @item @strong{all other inputs:} |
---|
1342 | @code{mres} (classical method), see @ref{mres}. |
---|
1343 | @end table |
---|
1344 | @item @strong{Note:} |
---|
1345 | Accessing single elements of a resolution may require some partial |
---|
1346 | computations to be finished and may therefore take some time. |
---|
1347 | @end table |
---|
1348 | @c ref |
---|
1349 | See also |
---|
1350 | @ref{betti}; |
---|
1351 | @ref{ideal}; |
---|
1352 | @ref{minres}; |
---|
1353 | @ref{module}; |
---|
1354 | @ref{mres}; |
---|
1355 | @ref{nres}; |
---|
1356 | @ref{lres}; |
---|
1357 | @ref{hres}; |
---|
1358 | @ref{sres}; |
---|
1359 | @ref{resolution}. |
---|
1360 | @c ref |
---|
1361 | " |
---|
1362 | { |
---|
1363 | def P=basering; |
---|
1364 | if (size(#) < 2) |
---|
1365 | { |
---|
1366 | ERROR("res: need at least two arguments: ideal/module, int"); |
---|
1367 | } |
---|
1368 | |
---|
1369 | def m=#[1]; //the ideal or module |
---|
1370 | int i=#[2]; //the length of the resolution |
---|
1371 | if (i< 0) { i=0;} |
---|
1372 | |
---|
1373 | string varstr_P = varstr(P); |
---|
1374 | |
---|
1375 | int p_opt; |
---|
1376 | string s_opt = option(); |
---|
1377 | // set p_opt, if option(prot) is set |
---|
1378 | if (find(s_opt, "prot")) |
---|
1379 | { |
---|
1380 | p_opt = 1; |
---|
1381 | } |
---|
1382 | |
---|
1383 | if( (size(ideal(basering)) > 0) || (size(ringlist(P)) > 4) ) |
---|
1384 | { |
---|
1385 | // the quick hack for qrings - seems to fit most needs |
---|
1386 | // (lres is not implemented for qrings, sres is not so efficient) |
---|
1387 | // || non-commutative, since only n/m-res are implemented for NC rings |
---|
1388 | if (p_opt) { "using nres";} |
---|
1389 | return(nres(m,i)); |
---|
1390 | } |
---|
1391 | |
---|
1392 | if(homog(m)==1) |
---|
1393 | { |
---|
1394 | resolution re; |
---|
1395 | if (((i==0) or (i>=nvars(basering))) && (typeof(m) != "module") && (nvars(basering)>1)) |
---|
1396 | { |
---|
1397 | //LaScala for the homogeneous case and i == 0 |
---|
1398 | if (p_opt) { "using lres";} |
---|
1399 | re=lres(m,i); |
---|
1400 | if(size(#)>2) |
---|
1401 | { |
---|
1402 | re=minres(re); |
---|
1403 | } |
---|
1404 | } |
---|
1405 | else |
---|
1406 | { |
---|
1407 | if(size(#)>2) |
---|
1408 | { |
---|
1409 | if (p_opt) { "using mres";} |
---|
1410 | re=mres(m,i); |
---|
1411 | } |
---|
1412 | else |
---|
1413 | { |
---|
1414 | if (p_opt) { "using sres";} |
---|
1415 | re=sres(std(m),i); |
---|
1416 | } |
---|
1417 | } |
---|
1418 | return(re); |
---|
1419 | } |
---|
1420 | |
---|
1421 | //mres for the global non homogeneous case |
---|
1422 | if(find(ordstr(P),"s")==0) |
---|
1423 | { |
---|
1424 | string ri= "ring Phelp =" |
---|
1425 | +string(char(P))+",("+varstr_P+"),(dp,C);"; |
---|
1426 | ri = ri + "minpoly = "+string(minpoly) + ";"; |
---|
1427 | execute(ri); |
---|
1428 | def m=imap(P,m); |
---|
1429 | if (p_opt) { "using mres in another ring";} |
---|
1430 | list re=mres(m,i); |
---|
1431 | setring P; |
---|
1432 | resolution result=imap(Phelp,re); |
---|
1433 | if (size(#) > 2) {result = minres(result);} |
---|
1434 | return(result); |
---|
1435 | } |
---|
1436 | |
---|
1437 | //sres for the local case and not minimal resolution |
---|
1438 | if(size(#)<=2) |
---|
1439 | { |
---|
1440 | string ri= "ring Phelp =" |
---|
1441 | +string(char(P))+",("+varstr_P+"),(ls,c);"; |
---|
1442 | ri = ri + "minpoly = "+string(minpoly) + ";"; |
---|
1443 | execute(ri); |
---|
1444 | def m=imap(P,m); |
---|
1445 | m=std(m); |
---|
1446 | if (p_opt) { "using sres in another ring";} |
---|
1447 | list re=sres(m,i); |
---|
1448 | setring P; |
---|
1449 | resolution result=imap(Phelp,re); |
---|
1450 | return(result); |
---|
1451 | } |
---|
1452 | |
---|
1453 | //mres for the local case and minimal resolution |
---|
1454 | string ri= "ring Phelp =" |
---|
1455 | +string(char(P))+",("+varstr_P+"),(ls,C);"; |
---|
1456 | ri = ri + "minpoly = "+string(minpoly) + ";"; |
---|
1457 | execute(ri); |
---|
1458 | def m=imap(P,m); |
---|
1459 | if (p_opt) { "using mres in another ring";} |
---|
1460 | list re=mres(m,i); |
---|
1461 | setring P; |
---|
1462 | resolution result=imap(Phelp,re); |
---|
1463 | result = minres(result); |
---|
1464 | return(result); |
---|
1465 | } |
---|
1466 | example |
---|
1467 | {"EXAMPLE:"; echo = 2; |
---|
1468 | ring r=0,(x,y,z),dp; |
---|
1469 | ideal i=xz,yz,x3-y3; |
---|
1470 | def l=res(i,0); // homogeneous ideal: uses lres |
---|
1471 | l; |
---|
1472 | print(betti(l), "betti"); // input to betti may be of type resolution |
---|
1473 | l[2]; // element access may take some time |
---|
1474 | i=i,x+1; |
---|
1475 | l=res(i,0); // inhomogeneous ideal: uses mres |
---|
1476 | l; |
---|
1477 | ring rs=0,(x,y,z),ds; |
---|
1478 | ideal i=imap(r,i); |
---|
1479 | def l=res(i,0); // local ring not minimized: uses sres |
---|
1480 | l; |
---|
1481 | res(i,0,0); // local ring and minimized: uses mres |
---|
1482 | } |
---|
1483 | ///////////////////////////////////////////////////////////////////////// |
---|
1484 | |
---|
1485 | proc quot (def m1,def m2,list #) |
---|
1486 | "SYNTAX: @code{quot (} module_expression@code{,} module_expression @code{)} |
---|
1487 | @*@code{quot (} module_expression@code{,} module_expression@code{,} |
---|
1488 | int_expression @code{)} |
---|
1489 | @*@code{quot (} ideal_expression@code{,} ideal_expression @code{)} |
---|
1490 | @*@code{quot (} ideal_expression@code{,} ideal_expression@code{,} |
---|
1491 | int_expression @code{)} |
---|
1492 | TYPE: ideal |
---|
1493 | SYNTAX: @code{quot (} module_expression@code{,} ideal_expression @code{)} |
---|
1494 | TYPE: module |
---|
1495 | PURPOSE: computes the quotient of the 1st and the 2nd argument. |
---|
1496 | If a 3rd argument @code{n} is given the @code{n}-th method is used |
---|
1497 | (@code{n}=1...5). |
---|
1498 | SEE ALSO: quotient |
---|
1499 | EXAMPLE: example quot; shows an example" |
---|
1500 | { |
---|
1501 | if (((typeof(m1)!="ideal") and (typeof(m1)!="module")) |
---|
1502 | or ((typeof(m2)!="ideal") and (typeof(m2)!="module"))) |
---|
1503 | { |
---|
1504 | "USAGE: quot(m1, m2[, n]); m1, m2 two submodules of k^s,"; |
---|
1505 | " n (optional) integer (1<= n <=5)"; |
---|
1506 | "RETURN: the quotient of m1 and m2"; |
---|
1507 | "EXAMPLE: example quot; shows an example"; |
---|
1508 | return(); |
---|
1509 | } |
---|
1510 | if (typeof(m1)!=typeof(m2)) |
---|
1511 | { |
---|
1512 | return(quotient(m1,m2)); |
---|
1513 | } |
---|
1514 | if (size(#)>0) |
---|
1515 | { |
---|
1516 | if (typeof(#[1])=="int" ) |
---|
1517 | { |
---|
1518 | return(quot1(m1,m2,#[1])); |
---|
1519 | } |
---|
1520 | } |
---|
1521 | else |
---|
1522 | { |
---|
1523 | return(quot1(m1,m2,2)); |
---|
1524 | } |
---|
1525 | } |
---|
1526 | example |
---|
1527 | { "EXAMPLE:"; echo = 2; |
---|
1528 | ring r=181,(x,y,z),(c,ls); |
---|
1529 | ideal id1=maxideal(4); |
---|
1530 | ideal id2=x2+xyz,y2-z3y,z3+y5xz; |
---|
1531 | option(prot); |
---|
1532 | ideal id3=quotient(id1,id2); |
---|
1533 | id3; |
---|
1534 | ideal id4=quot(id1,id2,1); |
---|
1535 | id4; |
---|
1536 | ideal id5=quot(id1,id2,2); |
---|
1537 | id5; |
---|
1538 | } |
---|
1539 | |
---|
1540 | static proc quot1 (module m1, module m2,int n) |
---|
1541 | "USAGE: quot1(m1, m2, n); m1, m2 two submodules of k^s, |
---|
1542 | n integer (1<= n <=5) |
---|
1543 | RETURN: the quotient of m1 and m2 |
---|
1544 | EXAMPLE: example quot1; shows an example" |
---|
1545 | { |
---|
1546 | if (n==1) |
---|
1547 | { |
---|
1548 | return(quotient1(m1,m2)); |
---|
1549 | } |
---|
1550 | else |
---|
1551 | { |
---|
1552 | if (n==2) |
---|
1553 | { |
---|
1554 | return(quotient2(m1,m2)); |
---|
1555 | } |
---|
1556 | else |
---|
1557 | { |
---|
1558 | if (n==3) |
---|
1559 | { |
---|
1560 | return(quotient3(m1,m2)); |
---|
1561 | } |
---|
1562 | else |
---|
1563 | { |
---|
1564 | if (n==4) |
---|
1565 | { |
---|
1566 | return(quotient4(m1,m2)); |
---|
1567 | } |
---|
1568 | else |
---|
1569 | { |
---|
1570 | if (n==5) |
---|
1571 | { |
---|
1572 | return(quotient5(m1,m2)); |
---|
1573 | } |
---|
1574 | else |
---|
1575 | { |
---|
1576 | return(quotient(m1,m2)); |
---|
1577 | } |
---|
1578 | } |
---|
1579 | } |
---|
1580 | } |
---|
1581 | } |
---|
1582 | } |
---|
1583 | example |
---|
1584 | { "EXAMPLE:"; echo = 2; |
---|
1585 | ring r=181,(x,y,z),(c,ls); |
---|
1586 | ideal id1=maxideal(4); |
---|
1587 | ideal id2=x2+xyz,y2-z3y,z3+y5xz; |
---|
1588 | option(prot); |
---|
1589 | ideal id6=quotient(id1,id2); |
---|
1590 | id6; |
---|
1591 | ideal id7=quot1(id1,id2,1); |
---|
1592 | id7; |
---|
1593 | ideal id8=quot1(id1,id2,2); |
---|
1594 | id8; |
---|
1595 | } |
---|
1596 | |
---|
1597 | static proc quotient0(module a,module b) |
---|
1598 | { |
---|
1599 | module mm=b+a; |
---|
1600 | resolution rs=lres(mm,0); |
---|
1601 | list I=list(rs); |
---|
1602 | matrix M=I[2]; |
---|
1603 | matrix A[1][nrows(M)]=M[1..nrows(M),1]; |
---|
1604 | ideal i=A; |
---|
1605 | return (i); |
---|
1606 | } |
---|
1607 | proc quotient1(module a,module b) //17sec |
---|
1608 | "USAGE: quotient1(m1, m2); m1, m2 two submodules of k^s, |
---|
1609 | RETURN: the quotient of m1 and m2" |
---|
1610 | { |
---|
1611 | int i; |
---|
1612 | a=std(a); |
---|
1613 | module dummy; |
---|
1614 | module B=NF(b,a)+dummy; |
---|
1615 | ideal re=quotient(a,module(B[1])); |
---|
1616 | for(i=2;i<=ncols(B);i++) |
---|
1617 | { |
---|
1618 | re=intersect1(re,quotient(a,module(B[i]))); |
---|
1619 | } |
---|
1620 | return(re); |
---|
1621 | } |
---|
1622 | proc quotient2(module a,module b) //13sec |
---|
1623 | "USAGE: quotient2(m1, m2); m1, m2 two submodules of k^s, |
---|
1624 | RETURN: the quotient of m1 and m2" |
---|
1625 | { |
---|
1626 | a=std(a); |
---|
1627 | module dummy; |
---|
1628 | module bb=NF(b,a)+dummy; |
---|
1629 | int i=ncols(bb); |
---|
1630 | ideal re=quotient(a,module(bb[i])); |
---|
1631 | bb[i]=0; |
---|
1632 | module temp; |
---|
1633 | module temp1; |
---|
1634 | module bbb; |
---|
1635 | int mx; |
---|
1636 | i=i-1; |
---|
1637 | while (1) |
---|
1638 | { |
---|
1639 | if (i==0) break; |
---|
1640 | temp = a+bb*re; |
---|
1641 | temp1 = lead(interred(temp)); |
---|
1642 | mx=ncols(a); |
---|
1643 | if (ncols(temp1)>ncols(a)) |
---|
1644 | { |
---|
1645 | mx=ncols(temp1); |
---|
1646 | } |
---|
1647 | temp1 = matrix(temp1,1,mx)-matrix(lead(a),1,mx); |
---|
1648 | temp1 = dummy+temp1; |
---|
1649 | if (deg(temp1[1])<0) break; |
---|
1650 | re=intersect1(re,quotient(a,module(bb[i]))); |
---|
1651 | bb[i]=0; |
---|
1652 | i = i-1; |
---|
1653 | } |
---|
1654 | return(re); |
---|
1655 | } |
---|
1656 | proc quotient3(module a,module b) //89sec |
---|
1657 | "USAGE: quotient3(m1, m2); m1, m2 two submodules of k^s, |
---|
1658 | only for global rings |
---|
1659 | RETURN: the quotient of m1 and m2" |
---|
1660 | { |
---|
1661 | string s="ring @newr=("+charstr(basering)+ |
---|
1662 | "),("+varstr(basering)+",@t,@w),dp;"; |
---|
1663 | def @newP=basering; |
---|
1664 | execute(s); |
---|
1665 | module b=imap(@newP,b); |
---|
1666 | module a=imap(@newP,a); |
---|
1667 | int i; |
---|
1668 | int j=ncols(b); |
---|
1669 | vector @b; |
---|
1670 | for(i=1;i<=j;i++) |
---|
1671 | { |
---|
1672 | @b=@b+@t^(i-1)*@w^(j-i+1)*b[i]; |
---|
1673 | } |
---|
1674 | ideal re=quotient(a,module(@b)); |
---|
1675 | setring @newP; |
---|
1676 | ideal re=imap(@newr,re); |
---|
1677 | return(re); |
---|
1678 | } |
---|
1679 | proc quotient5(module a,module b) //89sec |
---|
1680 | "USAGE: quotient5(m1, m2); m1, m2 two submodules of k^s, |
---|
1681 | only for global rings |
---|
1682 | RETURN: the quotient of m1 and m2" |
---|
1683 | { |
---|
1684 | string s="ring @newr=("+charstr(basering)+ |
---|
1685 | "),("+varstr(basering)+",@t),dp;"; |
---|
1686 | def @newP=basering; |
---|
1687 | execute(s); |
---|
1688 | module b=imap(@newP,b); |
---|
1689 | module a=imap(@newP,a); |
---|
1690 | int i; |
---|
1691 | int j=ncols(b); |
---|
1692 | vector @b; |
---|
1693 | for(i=1;i<=j;i++) |
---|
1694 | { |
---|
1695 | @b=@b+@t^(i-1)*b[i]; |
---|
1696 | } |
---|
1697 | @b=homog(@b,@w); |
---|
1698 | ideal re=quotient(a,module(@b)); |
---|
1699 | setring @newP; |
---|
1700 | ideal re=imap(@newr,re); |
---|
1701 | return(re); |
---|
1702 | } |
---|
1703 | proc quotient4(module a,module b) //95sec |
---|
1704 | "USAGE: quotient4(m1, m2); m1, m2 two submodules of k^s, |
---|
1705 | only for global rings |
---|
1706 | RETURN: the quotient of m1 and m2" |
---|
1707 | { |
---|
1708 | string s="ring @newr=("+charstr(basering)+ |
---|
1709 | "),("+varstr(basering)+",@t),dp;"; |
---|
1710 | def @newP=basering; |
---|
1711 | execute(s); |
---|
1712 | module b=imap(@newP,b); |
---|
1713 | module a=imap(@newP,a); |
---|
1714 | int i; |
---|
1715 | vector @b=b[1]; |
---|
1716 | for(i=2;i<=ncols(b);i++) |
---|
1717 | { |
---|
1718 | @b=@b+@t^(i-1)*b[i]; |
---|
1719 | } |
---|
1720 | matrix sy=modulo(@b,a); |
---|
1721 | ideal re=sy; |
---|
1722 | setring @newP; |
---|
1723 | ideal re=imap(@newr,re); |
---|
1724 | return(re); |
---|
1725 | } |
---|
1726 | static proc intersect1(ideal i,ideal j) |
---|
1727 | { |
---|
1728 | def R=basering; |
---|
1729 | execute("ring gnir = ("+charstr(basering)+"), |
---|
1730 | ("+varstr(basering)+",@t),(C,dp);"); |
---|
1731 | ideal i=var(nvars(basering))*imap(R,i)+(var(nvars(basering))-1)*imap(R,j); |
---|
1732 | ideal j=eliminate(i,var(nvars(basering))); |
---|
1733 | setring R; |
---|
1734 | map phi=gnir,maxideal(1); |
---|
1735 | return(phi(j)); |
---|
1736 | } |
---|
1737 | |
---|
1738 | ////////////////////////////////////////////////////////////////// |
---|
1739 | /// |
---|
1740 | /// sprintf, fprintf printf |
---|
1741 | /// |
---|
1742 | proc sprintf(string fmt, list #) |
---|
1743 | "SYNTAX: @code{sprintf (} string_expression @code{[,} any_expressions |
---|
1744 | @code{] )} |
---|
1745 | RETURN: string |
---|
1746 | PURPOSE: @code{sprintf(fmt,...);} performs output formatting. The first |
---|
1747 | argument is a format control string. Additional arguments may be |
---|
1748 | required, depending on the content of the control string. A series |
---|
1749 | of output characters is generated as directed by the control string; |
---|
1750 | these characters are returned as a string. @* |
---|
1751 | The control string @code{fmt} is simply text to be copied, |
---|
1752 | except that the string may contain conversion specifications.@* |
---|
1753 | Type @code{help print;} for a listing of valid conversion |
---|
1754 | specifications. As an addition to the conversions of @code{print}, |
---|
1755 | the @code{%n} and @code{%2} conversion specification does not |
---|
1756 | consume an additional argument, but simply generates a newline |
---|
1757 | character. |
---|
1758 | NOTE: If one of the additional arguments is a list, then it should be |
---|
1759 | wrapped in an additional @code{list()} command, since passing a list |
---|
1760 | as an argument flattens the list by one level. |
---|
1761 | SEE ALSO: fprintf, printf, print, string |
---|
1762 | EXAMPLE : example sprintf; shows an example |
---|
1763 | " |
---|
1764 | { |
---|
1765 | int sfmt = size(fmt); |
---|
1766 | if (sfmt <= 1) |
---|
1767 | { |
---|
1768 | return (fmt); |
---|
1769 | } |
---|
1770 | int next, l, nnext; |
---|
1771 | string ret; |
---|
1772 | list formats = "%l", "%s", "%2l", "%2s", "%t", "%;", "%p", "%b", "%n", "%2"; |
---|
1773 | while (1) |
---|
1774 | { |
---|
1775 | if (size(#) <= 0) |
---|
1776 | { |
---|
1777 | return (ret + fmt); |
---|
1778 | } |
---|
1779 | nnext = 0; |
---|
1780 | while (nnext < sfmt) |
---|
1781 | { |
---|
1782 | nnext = find(fmt, "%", nnext + 1); |
---|
1783 | if (nnext == 0) |
---|
1784 | { |
---|
1785 | next = 0; |
---|
1786 | break; |
---|
1787 | } |
---|
1788 | l = 1; |
---|
1789 | while (l <= size(formats)) |
---|
1790 | { |
---|
1791 | next = find(fmt, formats[l], nnext); |
---|
1792 | if (next == nnext) break; |
---|
1793 | l++; |
---|
1794 | } |
---|
1795 | if (next == nnext) break; |
---|
1796 | } |
---|
1797 | if (next == 0) |
---|
1798 | { |
---|
1799 | return (ret + fmt); |
---|
1800 | } |
---|
1801 | if (formats[l] != "%2" && formats[l] != "%n") |
---|
1802 | { |
---|
1803 | ret = ret + fmt[1, next - 1] + print(#[1], formats[l]); |
---|
1804 | # = delete(#, 1); |
---|
1805 | } |
---|
1806 | else |
---|
1807 | { |
---|
1808 | ret = ret + fmt[1, next - 1] + print("", "%2s"); |
---|
1809 | } |
---|
1810 | if (size(fmt) <= (next + size(formats[l]) - 1)) |
---|
1811 | { |
---|
1812 | return (ret); |
---|
1813 | } |
---|
1814 | fmt = fmt[next + size(formats[l]), size(fmt)-next-size(formats[l]) + 1]; |
---|
1815 | } |
---|
1816 | } |
---|
1817 | example |
---|
1818 | { "EXAMPLE:"; echo=2; |
---|
1819 | ring r=0,(x,y,z),dp; |
---|
1820 | module m=[1,y],[0,x+z]; |
---|
1821 | intmat M=betti(mres(m,0)); |
---|
1822 | list l = r, m, M; |
---|
1823 | string s = sprintf("s:%s,%n l:%l", 1, 2); s; |
---|
1824 | s = sprintf("s:%n%s", l); s; |
---|
1825 | s = sprintf("s:%2%s", list(l)); s; |
---|
1826 | s = sprintf("2l:%n%2l", list(l)); s; |
---|
1827 | s = sprintf("%p", list(l)); s; |
---|
1828 | s = sprintf("%;", list(l)); s; |
---|
1829 | s = sprintf("%b", M); s; |
---|
1830 | } |
---|
1831 | |
---|
1832 | proc printf(string fmt, list #) |
---|
1833 | "SYNTAX: @code{printf (} string_expression @code{[,} any_expressions@code{] )} |
---|
1834 | RETURN: none |
---|
1835 | PURPOSE: @code{printf(fmt,...);} performs output formatting. The first |
---|
1836 | argument is a format control string. Additional arguments may be |
---|
1837 | required, depending on the content of the control string. A series |
---|
1838 | of output characters is generated as directed by the control string; |
---|
1839 | these characters are displayed (i.e., printed to standard out). @* |
---|
1840 | The control string @code{fmt} is simply text to be copied, except |
---|
1841 | that the string may contain conversion specifications. @* |
---|
1842 | Type @code{help print;} for a listing of valid conversion |
---|
1843 | specifications. As an addition to the conversions of @code{print}, |
---|
1844 | the @code{%n} and @code{%2} conversion specification does not |
---|
1845 | consume an additional argument, but simply generates a newline |
---|
1846 | character. |
---|
1847 | NOTE: If one of the additional arguments is a list, then it should be |
---|
1848 | enclosed once more into a @code{list()} command, since passing a |
---|
1849 | list as an argument flattens the list by one level. |
---|
1850 | SEE ALSO: sprintf, fprintf, print, string |
---|
1851 | EXAMPLE : example printf; shows an example |
---|
1852 | " |
---|
1853 | { |
---|
1854 | write("", sprintf(fmt, #)); |
---|
1855 | } |
---|
1856 | example |
---|
1857 | { "EXAMPLE:"; echo=2; |
---|
1858 | ring r=0,(x,y,z),dp; |
---|
1859 | module m=[1,y],[0,x+z]; |
---|
1860 | intmat M=betti(mres(m,0)); |
---|
1861 | list l=r,m,matrix(M); |
---|
1862 | printf("s:%s,l:%l",1,2); |
---|
1863 | printf("s:%s",l); |
---|
1864 | printf("s:%s",list(l)); |
---|
1865 | printf("2l:%2l",list(l)); |
---|
1866 | printf("%p",matrix(M)); |
---|
1867 | printf("%;",matrix(M)); |
---|
1868 | printf("%b",M); |
---|
1869 | } |
---|
1870 | |
---|
1871 | |
---|
1872 | proc fprintf(link l, string fmt, list #) |
---|
1873 | "SYNTAX: @code{fprintf (} link_expression@code{,} string_expression @code{[,} |
---|
1874 | any_expressions@code{] )} |
---|
1875 | RETURN: none |
---|
1876 | PURPOSE: @code{fprintf(l,fmt,...);} performs output formatting. |
---|
1877 | The second argument is a format control string. Additional |
---|
1878 | arguments may be required, depending on the content of the |
---|
1879 | control string. A series of output characters is generated as |
---|
1880 | directed by the control string; these characters are |
---|
1881 | written to the link l. |
---|
1882 | The control string @code{fmt} is simply text to be copied, except |
---|
1883 | that the string may contain conversion specifications.@* |
---|
1884 | Type @code{help print;} for a listing of valid conversion |
---|
1885 | specifications. As an addition to the conversions of @code{print}, |
---|
1886 | the @code{%n} and @code{%2} conversion specification does not |
---|
1887 | consume an additional argument, but simply generates a newline |
---|
1888 | character. |
---|
1889 | NOTE: If one of the additional arguments is a list, then it should be |
---|
1890 | enclosed once more into a @code{list()} command, since passing |
---|
1891 | a list as an argument flattens the list by one level. |
---|
1892 | SEE ALSO: sprintf, printf, print, string |
---|
1893 | EXAMPLE : example fprintf; shows an example |
---|
1894 | " |
---|
1895 | { |
---|
1896 | write(l, sprintf(fmt, #)); |
---|
1897 | } |
---|
1898 | example |
---|
1899 | { "EXAMPLE:"; echo=2; |
---|
1900 | ring r=0,(x,y,z),dp; |
---|
1901 | module m=[1,y],[0,x+z]; |
---|
1902 | intmat M=betti(mres(m,0)); |
---|
1903 | list l=r,m,M; |
---|
1904 | link li=""; // link to stdout |
---|
1905 | fprintf(li,"s:%s,l:%l",1,2); |
---|
1906 | fprintf(li,"s:%s",l); |
---|
1907 | fprintf(li,"s:%s",list(l)); |
---|
1908 | fprintf(li,"2l:%2l",list(l)); |
---|
1909 | fprintf(li,"%p",list(l)); |
---|
1910 | fprintf(li,"%;",list(l)); |
---|
1911 | fprintf(li,"%b",M); |
---|
1912 | } |
---|
1913 | |
---|
1914 | ////////////////////////////////////////////////////////////////////////// |
---|
1915 | |
---|
1916 | /* |
---|
1917 | proc minres(list #) |
---|
1918 | { |
---|
1919 | if (size(#) == 2) |
---|
1920 | { |
---|
1921 | if (typeof(#[1]) == "ideal" || typeof(#[1]) == "module") |
---|
1922 | { |
---|
1923 | if (typeof(#[2] == "int")) |
---|
1924 | { |
---|
1925 | return (res(#[1],#[2],1)); |
---|
1926 | } |
---|
1927 | } |
---|
1928 | } |
---|
1929 | |
---|
1930 | if (typeof(#[1]) == "resolution") |
---|
1931 | { |
---|
1932 | return minimizeres(#[1]); |
---|
1933 | } |
---|
1934 | else |
---|
1935 | { |
---|
1936 | return minimizeres(#); |
---|
1937 | } |
---|
1938 | |
---|
1939 | } |
---|
1940 | */ |
---|
1941 | /////////////////////////////////////////////////////////////////////////////// |
---|
1942 | |
---|
1943 | proc weightKB(def stc, int dd, list wim) |
---|
1944 | "SYNTAX: @code{weightKB (} module_expression@code{,} int_expression @code{,} |
---|
1945 | list_expression @code{)}@* |
---|
1946 | @code{weightKB (} ideal_expression@code{,} int_expression@code{,} |
---|
1947 | list_expression @code{)} |
---|
1948 | RETURN: the same as the input type of the first argument |
---|
1949 | PURPOSE: If @code{I,d,wim} denotes the three arguments then weightKB |
---|
1950 | computes the weighted degree- @code{d} part of a vector space basis |
---|
1951 | (consisting of monomials) of the quotient ring, resp. of the |
---|
1952 | quotient module, modulo @code{I} w.r.t. weights given by @code{wim} |
---|
1953 | The information about the weights is given as a list of two intvec: |
---|
1954 | @code{wim[1]} weights for all variables (positive), |
---|
1955 | @code{wim[2]} weights for the module generators. |
---|
1956 | NOTE: This is a generalization of the command @code{kbase} with the same |
---|
1957 | first two arguments. |
---|
1958 | SEE ALSO: kbase |
---|
1959 | EXAMPLE: example weightKB; shows an example |
---|
1960 | " |
---|
1961 | { |
---|
1962 | if(checkww(wim)){ERROR("wrong weights";);} |
---|
1963 | kbclass(); |
---|
1964 | wwtop=wim[1]; |
---|
1965 | stc=interred(lead(stc)); |
---|
1966 | if(typeof(stc)=="ideal") |
---|
1967 | { |
---|
1968 | stdtop=stc; |
---|
1969 | ideal out=widkbase(dd); |
---|
1970 | delkbclass(); |
---|
1971 | out=simplify(out,2); // delete 0 |
---|
1972 | return(out); |
---|
1973 | } |
---|
1974 | list mbase=kbprepare(stc); |
---|
1975 | module mout; |
---|
1976 | int im,ii; |
---|
1977 | if(size(wim)>1){mmtop=wim[2];} |
---|
1978 | else{mmtop=0;} |
---|
1979 | for(im=size(mbase);im>0;im--) |
---|
1980 | { |
---|
1981 | stdtop=mbase[im]; |
---|
1982 | if(im>size(mmtop)){ii=dd;} |
---|
1983 | else{ii=dd-mmtop[im];} |
---|
1984 | mout=mout+widkbase(ii)*gen(im); |
---|
1985 | } |
---|
1986 | delkbclass(); |
---|
1987 | mout=simplify(mout,2); // delete 0 |
---|
1988 | return(mout); |
---|
1989 | } |
---|
1990 | example |
---|
1991 | { "EXAMPLE:"; echo=2; |
---|
1992 | ring R=0, (x,y), wp(1,2); |
---|
1993 | weightKB(ideal(0),3,intvec(1,2)); |
---|
1994 | } |
---|
1995 | |
---|
1996 | /////////////////////////////////////////////////////////////////////////////// |
---|
1997 | |
---|
1998 | proc datetime() |
---|
1999 | "SYNTAX: @code{datetime ()} |
---|
2000 | RETURN: string |
---|
2001 | PURPOSE: return the curent date and time as a string |
---|
2002 | EXAMPLE: example datetime; shows an example |
---|
2003 | " |
---|
2004 | { |
---|
2005 | return(read("|: date")); |
---|
2006 | } |
---|
2007 | example |
---|
2008 | { "EXAMPLE:"; echo=2; |
---|
2009 | datetime(); |
---|
2010 | } |
---|
2011 | |
---|
2012 | /////////////////////////////////////////////////////////////////////////////// |
---|
2013 | // construct global values |
---|
2014 | static proc kbclass() |
---|
2015 | { |
---|
2016 | intvec wwtop,mmtop; |
---|
2017 | export (wwtop,mmtop); |
---|
2018 | ideal stdtop,kbtop; |
---|
2019 | export (stdtop,kbtop); |
---|
2020 | } |
---|
2021 | // delete global values |
---|
2022 | static proc delkbclass() |
---|
2023 | { |
---|
2024 | kill wwtop,mmtop; |
---|
2025 | kill stdtop,kbtop; |
---|
2026 | } |
---|
2027 | // select parts of the modul |
---|
2028 | static proc kbprepare(module mstc) |
---|
2029 | { |
---|
2030 | list rr; |
---|
2031 | ideal kk; |
---|
2032 | int i1,i2; |
---|
2033 | mstc=transpose(mstc); |
---|
2034 | for(i1=ncols(mstc);i1>0;i1--) |
---|
2035 | { |
---|
2036 | kk=0; |
---|
2037 | for(i2=nrows(mstc[i1]);i2>0;i2--) |
---|
2038 | { |
---|
2039 | kk=kk+mstc[i1][i2]; |
---|
2040 | } |
---|
2041 | rr[i1]=kk; |
---|
2042 | } |
---|
2043 | return(rr); |
---|
2044 | } |
---|
2045 | // check for weights |
---|
2046 | static proc checkww(list vv) |
---|
2047 | { |
---|
2048 | if(typeof(vv[1])!="intvec"){return(1);} |
---|
2049 | intvec ww=vv[1]; |
---|
2050 | int mv=nvars(basering); |
---|
2051 | if(size(ww)<mv){return(1);} |
---|
2052 | while(mv>0) |
---|
2053 | { |
---|
2054 | if(ww[mv]<=0){return(1);} |
---|
2055 | mv--; |
---|
2056 | } |
---|
2057 | if(size(vv)>1) |
---|
2058 | { |
---|
2059 | if(typeof(vv[2])!="intvec"){return(1);} |
---|
2060 | } |
---|
2061 | return(0); |
---|
2062 | } |
---|
2063 | /////////////////////////////////////////////////////// |
---|
2064 | // The "Caller" for ideals |
---|
2065 | // dd - the degree of the result |
---|
2066 | static proc widkbase(int dd) |
---|
2067 | { |
---|
2068 | if((size(stdtop)==1)&&(deg(stdtop[1])==0)){return(0);} |
---|
2069 | if(dd<=0) |
---|
2070 | { |
---|
2071 | if(dd<0){return(0);} |
---|
2072 | else{return(1);} |
---|
2073 | } |
---|
2074 | int m1,m2; |
---|
2075 | m1=nvars(basering); |
---|
2076 | while(wwtop[m1]>dd) |
---|
2077 | { |
---|
2078 | m1--; |
---|
2079 | if(m1==0){return(0);} |
---|
2080 | } |
---|
2081 | attrib(stdtop,"isSB",1); |
---|
2082 | poly mo=1; |
---|
2083 | if(m1==1) |
---|
2084 | { |
---|
2085 | m2=dd div wwtop[1]; |
---|
2086 | if((m2*wwtop[1])==dd) |
---|
2087 | { |
---|
2088 | mo=var(1)^m2; |
---|
2089 | if(reduce(mo,stdtop)==mo){return(mo);} |
---|
2090 | else{return(0);} |
---|
2091 | } |
---|
2092 | } |
---|
2093 | kbtop=0; |
---|
2094 | m2=dd; |
---|
2095 | weightmon(m1-1,m2,mo); |
---|
2096 | while(m2>=wwtop[m1]) |
---|
2097 | { |
---|
2098 | m2=m2-wwtop[m1]; |
---|
2099 | mo=var(m1)*mo; |
---|
2100 | if(m2==0) |
---|
2101 | { |
---|
2102 | if((mo!=0) and (reduce(mo,stdtop)==mo)) |
---|
2103 | { |
---|
2104 | kbtop[ncols(kbtop)+1]=mo; |
---|
2105 | return(kbtop); |
---|
2106 | } |
---|
2107 | } |
---|
2108 | weightmon(m1-1,m2,mo); |
---|
2109 | } |
---|
2110 | return(kbtop); |
---|
2111 | } |
---|
2112 | ///////////////////////////////////////////////////////// |
---|
2113 | // the recursive procedure |
---|
2114 | // va - number of the variable |
---|
2115 | // drest - rest of the degree |
---|
2116 | // mm - the candidate |
---|
2117 | static proc weightmon(int va, int drest, poly mm) |
---|
2118 | { |
---|
2119 | while(wwtop[va]>drest) |
---|
2120 | { |
---|
2121 | va--; |
---|
2122 | if(va==0){return();} |
---|
2123 | } |
---|
2124 | int m2; |
---|
2125 | if(va==1) |
---|
2126 | { |
---|
2127 | m2=drest div wwtop[1]; |
---|
2128 | if((m2*wwtop[1])==drest) |
---|
2129 | { |
---|
2130 | mm=var(1)^m2*mm; |
---|
2131 | if ((mm!=0) and (reduce(mm,stdtop)==mm)) |
---|
2132 | { |
---|
2133 | kbtop[ncols(kbtop)+1]=mm; |
---|
2134 | } |
---|
2135 | } |
---|
2136 | return(); |
---|
2137 | } |
---|
2138 | m2=drest; |
---|
2139 | if ((mm!=0) and (reduce(mm,stdtop)==mm)) |
---|
2140 | { |
---|
2141 | weightmon(va-1,m2,mm); |
---|
2142 | } |
---|
2143 | while(m2>=wwtop[va]) |
---|
2144 | { |
---|
2145 | m2=m2-wwtop[va]; |
---|
2146 | mm=var(va)*mm; |
---|
2147 | if(m2==0) |
---|
2148 | { |
---|
2149 | if ((mm!=0) and (reduce(mm,stdtop)==mm)) |
---|
2150 | { |
---|
2151 | kbtop[ncols(kbtop)+1]=mm; |
---|
2152 | return(); |
---|
2153 | } |
---|
2154 | } |
---|
2155 | if ((mm!=0) and (reduce(mm,stdtop)==mm)) |
---|
2156 | { |
---|
2157 | weightmon(va-1,m2,mm); |
---|
2158 | } |
---|
2159 | } |
---|
2160 | return(); |
---|
2161 | } |
---|
2162 | example |
---|
2163 | { "EXAMPLE:"; echo=2; |
---|
2164 | ring r=0,(x,y,z),dp; |
---|
2165 | ideal i = x6,y4,xyz; |
---|
2166 | intvec w = 2,3,6; |
---|
2167 | weightKB(i, 12, list(w)); |
---|
2168 | } |
---|
2169 | |
---|
2170 | /////////////////////////////////////////////////////////////////////////////// |
---|
2171 | proc max(def i,def j) |
---|
2172 | "SYNTAX: max (i, j) |
---|
2173 | TYPE: same as type of i resp. j |
---|
2174 | PURPOSE: returns the maximum for any 2 arguments of a type |
---|
2175 | for which '>' is defined |
---|
2176 | SEE ALSO: min |
---|
2177 | EXAMPLE: example max; shows an example" |
---|
2178 | { |
---|
2179 | if(i>j){return(i);} |
---|
2180 | return(j); |
---|
2181 | } |
---|
2182 | example |
---|
2183 | { "EXAMPLE:"; echo=2; |
---|
2184 | max(2,3); |
---|
2185 | max(4,3); |
---|
2186 | } |
---|
2187 | /////////////////////////////////////////////////////////////////////////////// |
---|
2188 | proc min(def i,def j) |
---|
2189 | "SYNTAX: min (i, j) |
---|
2190 | TYPE: same as type of i resp. j |
---|
2191 | PURPOSE: returns the minimum for any 2 arguments of a type |
---|
2192 | for which '>' is defined |
---|
2193 | SEE ALSO: max |
---|
2194 | EXAMPLE: example min; shows an example" |
---|
2195 | { |
---|
2196 | if(i>j){return(j);} |
---|
2197 | return(i); |
---|
2198 | } |
---|
2199 | example |
---|
2200 | { "EXAMPLE:"; echo=2; |
---|
2201 | min(2,3); |
---|
2202 | min(4,3); |
---|
2203 | } |
---|
2204 | |
---|
2205 | |
---|
2206 | /////////////////////////////////////////////////////////////////////////////// |
---|
2207 | /* |
---|
2208 | Versuche: |
---|
2209 | /////////////////////////////////////////////////////////////////////////////// |
---|
2210 | proc downsizeSB (def I, list #) |
---|
2211 | "USAGE: downsizeSB(I [,l]); I ideal, l list of integers [default: l=0] |
---|
2212 | RETURN: intvec, say v, with v[j] either 1 or 0. We have v[j]=1 if |
---|
2213 | leadmonom(I[j]) is divisible by some leadmonom(I[k]) or if |
---|
2214 | leadmonom(i[j]) == leadmonom(i[k]) and l[j] >= l[k], with k!=j. |
---|
2215 | PURPOSE: The procedure is applied in a situation where the standard basis |
---|
2216 | computation in the basering R is done via a conversion through an |
---|
2217 | overring Phelp with additional variables and where a direct |
---|
2218 | imap from Phelp to R is too expensive. |
---|
2219 | Assume Phelp is created by the procedure @code{par2varRing} or |
---|
2220 | @code{hilbRing} and IPhelp is a SB in Phelp [ with l[j]= |
---|
2221 | length(IPhelp(j)) or any other integer reflecting the complexity |
---|
2222 | of a IPhelp[j] ]. Let I = lead(IPhelp) mapped to R and compute |
---|
2223 | v = downsizeSB(imap(Phelp,I),l) in R. Then, if Ihelp[j] is deleted |
---|
2224 | for all j with v[j]=1, we can apply imap to the remaining generators |
---|
2225 | of Ihelp and still get SB in R (in general not minimal). |
---|
2226 | EXAMPLE: example downsizeSB; shows an example" |
---|
2227 | { |
---|
2228 | int k,j; |
---|
2229 | intvec v,l; |
---|
2230 | poly M,N,W; |
---|
2231 | int c=size(I); |
---|
2232 | if( size(#) != 0 ) |
---|
2233 | { |
---|
2234 | if ( typeof(#[1]) == "intvec" ) |
---|
2235 | { |
---|
2236 | l = #[1]; |
---|
2237 | } |
---|
2238 | else |
---|
2239 | { |
---|
2240 | ERROR("// 2nd argument must be an intvec"); |
---|
2241 | } |
---|
2242 | } |
---|
2243 | |
---|
2244 | l[c+1]=0; |
---|
2245 | v[c]=0; |
---|
2246 | |
---|
2247 | j=0; |
---|
2248 | while(j<c-1) |
---|
2249 | { |
---|
2250 | j++; |
---|
2251 | M = leadmonom(I[j]); |
---|
2252 | if( M != 0 ) |
---|
2253 | { |
---|
2254 | for( k=j+1; k<=c; k++ ) |
---|
2255 | { |
---|
2256 | N = leadmonom(I[k]); |
---|
2257 | if( N != 0 ) |
---|
2258 | { |
---|
2259 | if( (M==N) && (l[j]>l[k]) ) |
---|
2260 | { |
---|
2261 | I[j]=0; |
---|
2262 | v[j]=1; |
---|
2263 | break; |
---|
2264 | } |
---|
2265 | if( (M==N) && (l[j]<=l[k]) || N/M != 0 ) |
---|
2266 | { |
---|
2267 | I[k]=0; |
---|
2268 | v[k]=1; |
---|
2269 | } |
---|
2270 | } |
---|
2271 | } |
---|
2272 | } |
---|
2273 | } |
---|
2274 | return(v); |
---|
2275 | } |
---|
2276 | example |
---|
2277 | { "EXAMPLE:"; echo = 2; |
---|
2278 | ring r = 0,(x,y,z,t),(dp(3),dp); |
---|
2279 | ideal i = x+y+z+t,xy+yz+xt+zt,xyz+xyt+xzt+yzt,xyzt-t4; |
---|
2280 | ideal Id = std(i); |
---|
2281 | ideal I = lead(Id); I; |
---|
2282 | ring S = (0,t),(x,y,z),dp; |
---|
2283 | downsizeSB(imap(r,I)); |
---|
2284 | //Id[5] can be deleted, we still have a SB of i in the ring S |
---|
2285 | |
---|
2286 | ring R = (0,x),(y,z,u),lp; |
---|
2287 | ideal i = x+y+z+u,xy+xu+yz+zu,xyz+xyu+xzu+yzu,xyzu-1; |
---|
2288 | def Phelp = par2varRing()[1]; |
---|
2289 | setring Phelp; |
---|
2290 | ideal IPhelp = std(imap(R,i)); |
---|
2291 | ideal I = lead(IPhelp); |
---|
2292 | setring R; |
---|
2293 | ideal I = imap(Phelp,I); I; |
---|
2294 | intvec v = downsizeSB(I); v; |
---|
2295 | } |
---|
2296 | /////////////////////////////////////////////////////////////////////////// |
---|
2297 | // PROBLEM: Die Prozedur funktioniert nur fuer Ringe die global bekannt |
---|
2298 | // sind, also interaktiv, aber nicht aus einer Prozedur. |
---|
2299 | // Z.B. funktioniert example imapDownsize; nicht |
---|
2300 | |
---|
2301 | proc imapDownsize (string R, string I) |
---|
2302 | "SYNTAX: @code{imapDownsize (} string @code{,} string @code{)} *@ |
---|
2303 | First string must be the string of the name of a ring, second |
---|
2304 | string must be the string of the name of an object in the ring. |
---|
2305 | TYPE: same type as the object with name the second string |
---|
2306 | PURPOSE: maps the object given by the second string to the basering. |
---|
2307 | If R resp. I are the first resp. second string, then |
---|
2308 | imapDownsize(R,I) is equivalent to simplify(imap(`R`,`I`),34). |
---|
2309 | NOTE: imapDownsize is usually faster than imap if `I` is large and if |
---|
2310 | simplify has a great effect, since the procedure maps only those |
---|
2311 | generators from `I` which are not killed by simplify( - ,34). |
---|
2312 | This is useful if `I` is a standard bases for a block ordering of |
---|
2313 | `R` and if some variables from the last block in `R` are mapped |
---|
2314 | to parameters. Then the returned result is a standard basis in |
---|
2315 | the basering. |
---|
2316 | SEE ALSO: imap, fetch, map |
---|
2317 | EXAMPLE: example imapDownsize; shows an example" |
---|
2318 | { |
---|
2319 | def BR = basering; |
---|
2320 | int k; |
---|
2321 | |
---|
2322 | setring `R`; |
---|
2323 | def @leadI@ = lead(`I`); |
---|
2324 | int s = ncols(@leadI@); |
---|
2325 | setring BR; |
---|
2326 | ideal @leadI@ = simplify(imap(`R`,@leadI@),32); |
---|
2327 | intvec vi; |
---|
2328 | for (k=1; k<=s; k++) |
---|
2329 | { |
---|
2330 | vi[k] = @leadI@[k]==0; |
---|
2331 | } |
---|
2332 | kill @leadI@; |
---|
2333 | |
---|
2334 | setring `R`; |
---|
2335 | kill @leadI@; |
---|
2336 | for (k=1; k<=s; k++) |
---|
2337 | { |
---|
2338 | if( vi[k]==1 ) |
---|
2339 | { |
---|
2340 | `I`[k]=0; |
---|
2341 | } |
---|
2342 | } |
---|
2343 | `I` = simplify(`I`,2); |
---|
2344 | |
---|
2345 | setring BR; |
---|
2346 | return(imap(`R`,`I`)); |
---|
2347 | } |
---|
2348 | example |
---|
2349 | { "EXAMPLE:"; echo = 2; |
---|
2350 | ring r = 0,(x,y,z,t),(dp(3),dp); |
---|
2351 | ideal i = x+y+z+t,xy+yz+xt+zt,xyz+xyt+xzt+yzt,xyzt-1; |
---|
2352 | i = std(i); i; |
---|
2353 | |
---|
2354 | ring s = (0,t),(x,y,z),dp; |
---|
2355 | imapDownsize("r","i"); //i[5] is omitted since lead(i[2]) | lead(i[5]) |
---|
2356 | } |
---|
2357 | /////////////////////////////////////////////////////////////////////////////// |
---|
2358 | //die folgende proc war fuer groebner mit fglm vorgesehen, ist aber zu teuer. |
---|
2359 | //Um die projektive Dimension korrekt zu berechnen, muss man aber teuer |
---|
2360 | //voerher ein SB bzgl. einer Gradordnung berechnen und dann homogenisieren. |
---|
2361 | //Sonst koennen hoeherdimensionale Komponenten in Unendlich entstehen |
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2362 | |
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2363 | proc projInvariants(ideal i,list #) |
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2364 | "SYNTAX: @code{projInvariants (} ideal_expression @code{)} @* |
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2365 | @code{projInvariants (} ideal_expression@code{,} list of string_expres sions@code{)} |
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2366 | TYPE: list, say L, with L[1] and L[2] of type int and L[3] of type intvec |
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2367 | PURPOSE: Computes the (projective) dimension (L[1]), degree (L[2]) and the |
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2368 | first Hilbert series (L[3], as intvec) of the homogenized ideal |
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2369 | in the ring given by the procedure @code{hilbRing} with global |
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2370 | ordering dp (resp. wp if the variables have weights >1) |
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2371 | If an argument of type string @code{\"std\"} resp. @code{\"slimgb\"} |
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2372 | is given, the standard basis computatuion uses @code{std} or |
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2373 | @code{slimgb}, otherwise a heuristically chosen method (default) |
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2374 | NOTE: Homogenized means weighted homogenized with respect to the weights |
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2375 | w[i] of the variables var(i) of the basering. The returned dimension, |
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2376 | degree and Hilbertseries are the respective invariants of the |
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2377 | projective variety defined by the homogenized ideal. The dimension |
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2378 | is equal to the (affine) dimension of the ideal in the basering |
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2379 | (degree and Hilbert series make only sense for homogeneous ideals). |
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2380 | SEE ALSO: dim, dmult, hilb |
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2381 | KEYWORDS: dimension, degree, Hilbert function |
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2382 | EXAMPLE: example projInvariants; shows an example" |
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2383 | { |
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2384 | def P = basering; |
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2385 | int p_opt; |
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2386 | string s_opt = option(); |
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2387 | if (find(option(), "prot")) { p_opt = 1; } |
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2388 | |
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2389 | //---------------- check method and clear denomintors -------------------- |
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2390 | int k; |
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2391 | string method; |
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2392 | for (k=1; k<=size(#); k++) |
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2393 | { |
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2394 | if (typeof(#[k]) == "string") |
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2395 | { |
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2396 | method = method + "," + #[k]; |
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2397 | } |
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2398 | } |
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2399 | |
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2400 | if (npars(P) > 0) //clear denominators of parameters |
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2401 | { |
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2402 | for( k=ncols(i); k>0; k-- ) |
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2403 | { |
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2404 | i[k]=cleardenom(i[k]); |
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2405 | } |
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2406 | } |
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2407 | |
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2408 | //------------------------ change to hilbRing ---------------------------- |
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2409 | list hiRi = hilbRing(i); |
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2410 | intvec W = hiRi[2]; |
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2411 | def Philb = hiRi[1]; //note: Philb is no qring and the predefined |
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2412 | setring Philb; //ideal Id(1) in Philb is homogeneous |
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2413 | int di, de; //for dimension, degree |
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2414 | intvec hi; //for hilbert series |
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2415 | |
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2416 | //-------- compute Hilbert function of homogenized ideal in Philb --------- |
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2417 | //Philb has only 1 block. There are three cases |
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2418 | |
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2419 | string algorithm; //possibilities: std, slimgb, stdorslimgb |
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2420 | //define algorithm: |
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2421 | if( find(method,"std") && !find(method,"slimgb") ) |
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2422 | { |
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2423 | algorithm = "std"; |
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2424 | } |
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2425 | if( find(method,"slimgb") && !find(method,"std") ) |
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2426 | { |
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2427 | algorithm = "slimgb"; |
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2428 | } |
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2429 | if( find(method,"std") && find(method,"slimgb") || |
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2430 | (!find(method,"std") && !find(method,"slimgb")) ) |
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2431 | { |
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2432 | algorithm = "stdorslimgb"; |
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2433 | } |
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2434 | |
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2435 | if ( algorithm=="std" || ( algorithm=="stdorslimgb" && char(P)>0 ) ) |
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2436 | { |
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2437 | if (p_opt) {"std in ring " + string(Philb);} |
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2438 | Id(1) = std(Id(1)); |
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2439 | di = dim(Id(1))-1; |
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2440 | de = mult(Id(1)); |
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2441 | hi = hilb( Id(1),1,W ); |
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2442 | } |
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2443 | if ( algorithm=="slimgb" || ( algorithm=="stdorslimgb" && char(P)==0 ) ) |
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2444 | { |
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2445 | if (p_opt) {"slimgb in ring " + string(Philb);} |
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2446 | Id(1) = slimgb(Id(1)); |
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2447 | di = dim( Id(1) ); |
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2448 | if (di > -1) |
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2449 | { |
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2450 | di = di-1; |
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2451 | } |
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2452 | de = mult( Id(1) ); |
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2453 | hi = hilb( Id(1),1,W ); |
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2454 | } |
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2455 | kill Philb; |
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2456 | list L = di,de,hi; |
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2457 | return(L); |
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2458 | } |
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2459 | example |
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2460 | { "EXAMPLE:"; echo = 2; |
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2461 | ring r = 32003,(x,y,z),lp; |
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2462 | ideal i = y2-xz,x2-z; |
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2463 | projInvariants(i); |
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2464 | |
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2465 | ring R = (0),(x,y,z,u,v),lp; |
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2466 | //minpoly = x2+1; |
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2467 | ideal i = x2+1,x2+y+z+u+v,xyzuv-1; |
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2468 | projInvariants(i); |
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2469 | qring S =std(x2+1); |
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2470 | ideal i = imap(R,i); |
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2471 | projInvariants(i); |
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2472 | } |
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2473 | |
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2474 | */ |
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2475 | /////////////////////////////////////////////////////////////////////////////// |
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2476 | // EXAMPLES |
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2477 | /////////////////////////////////////////////////////////////////////////////// |
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2478 | /* |
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2479 | example stdfglm; |
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2480 | example stdhilb; |
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2481 | example groebner; |
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2482 | example res; |
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2483 | example sprintf; |
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2484 | example fprintf; |
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2485 | example printf; |
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2486 | example weightKB; |
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2487 | example qslimgb; |
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2488 | example par2varRing; |
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2489 | */ |
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2490 | static proc mod_init() |
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2491 | { |
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2492 | //int pagelength=24; |
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2493 | //exportto(Top,pagelength); |
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2494 | } |
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