1 | /////////////////////////////////////////////////////////////////////////////// |
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2 | version="version standard.lib 4.1.2.0 Feb_2019 "; |
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3 | category="Commutative Algebra"; |
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4 | info=" |
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5 | LIBRARY: rstandard.lib Computes Janet bases and border bases for ideals |
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6 | |
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7 | AUTHORS: Shamsa Kanwal lotus_zone16@yahoo.com |
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8 | Gerhard Pfister pfister@mathematik.uni-kl.de |
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9 | |
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10 | OVERVIEW: Computing Janet bases and border bases for any ordering |
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11 | using the idea of r-standard bases (defined by V. Gerdt) |
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12 | |
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13 | REFERENCES: |
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14 | [1] A. Kehrein, M. Kreuzer, L. Robbiano: An algebrists view on border bases, in: |
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15 | A. Dickenstein and I. Emiris (eds.), Solving Polynomial Equations: |
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16 | Foundations, Algorithms and Applications, Springer, Heidelberg 2005, 169-202. |
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17 | |
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18 | [2] V.P. Gerdt: Involute Algorithms for Computing Groebner Bases, In |
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19 | Computational Commutative and Non-Computational Algebra Geometry, |
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20 | S.Conjocaru, G. Pfister and V. Ufnarovski (Eds.), NATO Science Series,105 |
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21 | Press 2005, 199-255. |
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22 | |
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23 | |
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24 | PROCEDURES: |
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25 | borderBasis(I); computes a border basis of the ideal I |
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26 | modBorder(I); computes a border basis of the ideal I using modular methods |
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27 | rJanet(I); computes a Janet basis of the ideal I |
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28 | modJanet(I); computes a Janet basis of the ideal I using modular methods |
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29 | |
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30 | KEYWORDS: border basis; janet basis |
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31 | "; |
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32 | LIB "general.lib"; |
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33 | LIB "modstd.lib"; |
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34 | |
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35 | //================================= procedures ============================== |
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36 | //================================= border bases ============================== |
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37 | |
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38 | |
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39 | static proc orderIdeal(ideal I) |
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40 | { |
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41 | //returns the order ideal, the corners and the border of the |
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42 | //order ideal |
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43 | I=std(I); |
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44 | int d=vdim(I); |
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45 | if(d==-1){ERROR(" ideal is not zero dimensional ");} |
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46 | ideal O=kbase(I); //this is the order ideal |
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47 | ideal C=lead(I); //this are the corners |
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48 | ideal B=border(C); //this is the border of the order ideal |
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49 | list L=O,C,B,I; |
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50 | return(L); |
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51 | } |
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52 | |
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53 | static proc border(ideal L) |
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54 | { |
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55 | //L a monomial ideal |
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56 | //returns the border of the order ideal corresponding to L |
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57 | ideal O=kbase(L); |
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58 | ideal Omax=maxO(O); |
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59 | ideal B=L; |
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60 | poly p; |
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61 | list C; |
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62 | int i,j,k; |
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63 | for(i=1;i<=size(L);i++) |
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64 | { |
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65 | for(j=1;j<=size(Omax);j++) |
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66 | { |
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67 | for(k=1;k<=nvars(basering);k++) |
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68 | { |
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69 | p=(var(k)*Omax[j])/L[i]; |
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70 | if(p!=0){break;} |
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71 | } |
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72 | if(k<=nvars(basering)) |
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73 | { |
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74 | C=coneM(p); |
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75 | for(k=1;k<=size(C);k++) |
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76 | { |
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77 | B[size(B)+1]=C[k]*L[i]; |
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78 | } |
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79 | } |
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80 | } |
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81 | } |
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82 | B=simplify(B,4); |
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83 | B=simplify(B,2); |
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84 | B=sort(B)[1]; |
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85 | return(B); |
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86 | } |
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87 | |
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88 | static proc coneM(poly p) |
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89 | { |
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90 | //computes all monomials different from 1 dividing p |
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91 | list L=p; |
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92 | int i,j,k; |
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93 | poly q; |
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94 | k=1; |
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95 | ideal I; |
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96 | while(k) |
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97 | { |
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98 | k=0; |
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99 | i++; |
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100 | I=maxideal(i); |
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101 | for(j=1;j<=size(I);j++) |
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102 | { |
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103 | q=p/I[j]; |
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104 | if((q!=0)&&(q!=1)) |
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105 | { |
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106 | k=1; |
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107 | L[size(L)+1]=q; |
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108 | } |
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109 | } |
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110 | } |
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111 | return(L); |
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112 | } |
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113 | |
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114 | static proc maxO(ideal O) |
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115 | { |
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116 | //O is a monomial ideal |
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117 | //computes the maximal elements in O with respect to division |
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118 | def R=basering; |
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119 | def S=changeord(list(list("ds",nvars(R)))); |
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120 | setring S; |
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121 | ideal O=imap(R,O); |
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122 | O=sort(O)[1]; |
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123 | setring R; |
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124 | O=imap(S,O); |
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125 | int i=1; |
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126 | int j; |
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127 | int n=size(O); |
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128 | while(i<ncols(O)) |
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129 | { |
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130 | j=i+1; |
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131 | while(j<=ncols(O)) |
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132 | { |
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133 | if(O[j]!=0) |
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134 | { |
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135 | if(O[i]/O[j]!=0) |
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136 | { |
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137 | O[j]=0; |
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138 | } |
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139 | } |
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140 | j++; |
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141 | } |
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142 | O=simplify(O,2); |
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143 | i++; |
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144 | } |
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145 | return(O); |
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146 | } |
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147 | |
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148 | ///////////////////////////////////////////////////////////////////////////// |
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149 | proc borderBasis(ideal I) |
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150 | "USAGE: borderBasis(I); I is an ideal. |
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151 | RETURN: ideal, a border basis for I. |
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152 | PURPOSE: Computes a border basis for the ideal given by the generators in I. |
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153 | SEE ALSO: modBorder |
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154 | KEYWORDS: order ideal, border basis |
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155 | EXAMPLE: example borderBasis; shows an example |
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156 | " |
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157 | { |
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158 | list L=orderIdeal(I); |
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159 | I=L[4]; |
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160 | ideal B=L[3]; |
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161 | poly P; |
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162 | int i; |
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163 | if(ringlist(basering)[3][1][1]=="ds"||ringlist(basering)[3][1][1]=="Ds") |
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164 | { |
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165 | for(i=1;i<=size(B);i++) |
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166 | { |
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167 | P=system("reduce_bound",B[i],I,deg(highcorner(I))+1); |
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168 | B[i]=B[i]-P; |
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169 | } |
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170 | } |
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171 | else |
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172 | { |
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173 | if(ord_test(basering)) |
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174 | { |
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175 | for(i=1;i<=size(B);i++) |
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176 | { |
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177 | P=reduce(B[i],I); |
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178 | B[i]=B[i]-P; |
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179 | } |
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180 | } |
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181 | else |
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182 | { |
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183 | L=division(B[i],I); |
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184 | B[i]=L[3][1,1]*B[i]-L[2][1]; |
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185 | } |
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186 | } |
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187 | attrib(B,"isSB",1); |
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188 | return(B); |
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189 | } |
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190 | example |
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191 | { "EXAMPLE:"; |
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192 | ring R=32003,(x,y,z),ds; |
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193 | poly f=x3y+x5+x3y2+2x2y3+x2yz2+xy5+x12+y16+z20; |
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194 | ideal i= jacob(f); |
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195 | i=i,f; |
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196 | ideal j=borderBasis(i); j; |
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197 | } |
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198 | |
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199 | ///////////////////////////////////////////////////////////////////////////// |
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200 | |
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201 | proc modBorder(ideal I, list #) |
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202 | "USAGE: modBorder(I,i); I is an ideal, i an integer. |
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203 | RETURN: ideal, a border basis for I using modular methods. |
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204 | PURPOSE: Computes a border basis for the ideal given by the generators in I |
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205 | using modular techniques. |
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206 | If second argument is 0 then the result is not verified. |
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207 | SEE ALSO: borderBasis |
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208 | KEYWORDS: order ideal, border basis |
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209 | EXAMPLE: example modBorder; shows an example |
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210 | " |
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211 | { |
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212 | int exactness = 1; |
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213 | if (size(#) > 0) { |
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214 | exactness = #[1]; |
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215 | } |
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216 | /* save options */ |
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217 | intvec opt = option(get); |
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218 | option(redSB); |
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219 | /* choose the right command */ |
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220 | string command = "borderBasis"; |
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221 | /* call modular() */ |
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222 | if (exactness) { |
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223 | I = modular(command, list(I), Modstd::primeTest_std, |
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224 | Modstd::deleteUnluckyPrimes_std, Modstd::pTest_std,finalTest_std_new); |
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225 | } |
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226 | else { |
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227 | I = modular(command, list(I), Modstd::primeTest_std, |
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228 | Modstd::deleteUnluckyPrimes_std,Modstd::pTest_std); |
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229 | } |
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230 | /* return the result */ |
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231 | attrib(I, "isSB", 1); |
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232 | option(set, opt); |
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233 | return(I); |
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234 | } |
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235 | example |
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236 | { "EXAMPLE:"; |
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237 | ring R=0,(x,y,z),ds; |
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238 | poly f=x3y+x5+x3y2+2x2y3+x2yz2+xy5+x12+y16+z20; |
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239 | ideal i= jacob(f); |
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240 | i=i,f; |
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241 | ideal j=modBorder(i,1); j; |
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242 | |
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243 | ring S=0,(x,y,z),ds; |
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244 | ideal i= |
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245 | 3x2+6xy+3y2+6xz+6yz+3z2+3x2yz+4xy2z+y3z+4xyz2+2y2z2+yz3+10x9, |
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246 | 3x2+6xy+3y2+6xz+6yz+3z2+x3z+4x2yz+3xy2z+2x2z2+4xyz2+xz3+10y9, |
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247 | 3x2+6xy+3y2+6xz+6yz+3z2+x3y+2x2y2+xy3+4x2yz+4xy2z+3xyz2+10z9; |
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248 | ideal j=modBorder(i,0); j; |
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249 | } |
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250 | |
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251 | //=================================== Janet bases ============================== |
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252 | |
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253 | |
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254 | ///////////////////////////////////////////////////////////////////////////// |
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255 | proc rJanet(ideal I) |
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256 | "USAGE: rJanet(I); I is an ideal. |
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257 | RETURN: ideal, a Janet basis for I. |
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258 | PURPOSE: Computes a Janet basis for the ideal given by the generators in I for any ordering. |
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259 | SEE ALSO: modJanet |
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260 | KEYWORDS: Janet division, Janet basis |
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261 | EXAMPLE: example rJanet; shows an example |
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262 | " |
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263 | { |
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264 | def R=basering; |
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265 | int i,d; |
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266 | poly p; |
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267 | |
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268 | I=std(I); |
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269 | d=dim(I); |
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270 | if(I[1]==1){return(ideal(1));} |
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271 | ideal J=lead(I); |
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272 | intvec v; |
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273 | v[nvars(R)]=0; |
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274 | v=v+1; |
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275 | list rl=ringlist(R); |
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276 | rl[3]=list(list("dp",v)); |
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277 | ring S=ring(rl); |
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278 | ideal J=imap(R,J); |
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279 | J=janet(J); |
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280 | J=simplify(J,1+2+4+8+16); |
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281 | setring R; |
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282 | J=imap(S,J); |
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283 | list L; |
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284 | poly P; |
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285 | if((ringlist(basering)[3][1][1]=="ds"||ringlist(basering)[3][1][1]=="Ds")&&(d==0)) |
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286 | { |
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287 | for(i=1;i<=size(J);i++) |
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288 | { |
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289 | P=system("reduce_bound",J[i],I,deg(highcorner(I))+1); |
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290 | J[i]=J[i]-P; |
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291 | } |
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292 | } |
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293 | else |
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294 | { |
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295 | if(ord_test(basering)) |
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296 | { |
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297 | for(i=1;i<=size(J);i++) |
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298 | { |
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299 | P=reduce(J[i],I); |
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300 | J[i]=J[i]-P; |
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301 | } |
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302 | } |
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303 | else |
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304 | { |
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305 | L=division(J[i],I); |
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306 | J[i]=L[3][1,1]*J[i]-L[2][1]; |
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307 | } |
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308 | } |
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309 | attrib(J,"isSB",1); |
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310 | return(J); |
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311 | } |
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312 | example |
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313 | { "EXAMPLE:"; |
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314 | ring R= 32003,(a,b,c,d,e,f,g),dp; |
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315 | ideal i= |
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316 | a+b+c+d+e+f+g, |
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317 | ab+bc+cd+de+ef+fg+ga, |
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318 | abc+bcd+cde+fde+efg+fga+gab, |
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319 | abcd+bcde+cdef+defg+efga+fgab+gabc, |
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320 | abcde+bcdef+cdefg+defga+efgab+fgabc+gabcd, |
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321 | abcdef+bcdefg+cdefga+defgab+efgabc+fgabcd+gabcde, |
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322 | abcdefg-1; |
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323 | ideal j = rJanet(i); j; |
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324 | } |
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325 | |
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326 | ///////////////////////////////////////////////////////////////////////////// |
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327 | proc modJanet(ideal I, list #) |
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328 | "USAGE: modJanet(I,i); I is an ideal, i an integer (optional). |
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329 | RETURN: ideal, a Janet basis for I using modular methods. |
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330 | PURPOSE: Computes a Janet basis for the ideal given by the generators in I |
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331 | using modular techniques. |
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332 | If second argument is 0 then the result is not verified. |
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333 | SEE ALSO: rJanet |
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334 | KEYWORDS: Janet basis |
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335 | EXAMPLE: example Janet; shows an example |
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336 | " |
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337 | { |
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338 | int exactness = 1; |
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339 | if (size(#) > 0) { |
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340 | exactness = #[1]; |
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341 | } |
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342 | /* save options */ |
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343 | intvec opt = option(get); |
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344 | option(redSB); |
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345 | /* choose the right command */ |
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346 | string command = "rJanet"; |
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347 | /* call modular() */ |
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348 | if (exactness) { |
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349 | I = modular(command, list(I), Modstd::primeTest_std, |
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350 | Modstd::deleteUnluckyPrimes_std, Modstd::pTest_std,finalTest_std_new); |
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351 | } |
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352 | else { |
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353 | I = modular(command, list(I), Modstd::primeTest_std, |
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354 | Modstd::deleteUnluckyPrimes_std,Modstd::pTest_std); |
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355 | } |
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356 | /* return the result */ |
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357 | attrib(I, "isSB", 1); |
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358 | option(set, opt); |
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359 | return(I); |
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360 | } |
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361 | example |
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362 | { "EXAMPLE:"; |
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363 | ring R=0,(t,x,y,z),ds; |
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364 | ideal i= |
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365 | 5t3x2z+2t2y3x5, |
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366 | 7y+4x2y+y2x+2zt, |
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367 | 3tz+3yz2+2yz4; |
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368 | ideal j=modJanet(i); j; |
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369 | |
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370 | ring S=0,(x,y,z),dp; |
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371 | poly p1 =x2y*(47x5y7z3+28xy5z8+63+91x5y3z7); |
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372 | poly p2 =xyz*(57y6+21x2yz9+51y2z2+15x2z4); |
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373 | poly p3 =xy4z*(74y+32x6z7+53x5y2z+17x2y3z); |
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374 | poly p4 =y3z*(21x2z6+32x10y6z5+23x5y5z7+27y2); |
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375 | poly p5 =xz*(36y2z2+81x9y10+19x2y5z4+79x4z6); |
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376 | ideal i =p1,p2,p3,p4,p5; |
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377 | ideal j=modJanet(i,0); j; |
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378 | } |
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379 | ///////////////////////////////////////////////////////////////////////////// |
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380 | static proc minimizeS(ideal I) |
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381 | { |
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382 | int i,j; |
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383 | ideal K; |
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384 | ideal J=std(lead(I)); |
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385 | I=sort(I)[1]; |
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386 | J=sort(J)[1]; |
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387 | while(i<size(J)) |
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388 | { |
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389 | i++; |
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390 | while(j<size(I)) |
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391 | { |
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392 | j++; |
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393 | if(leadmonom(I[j])==J[i]) |
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394 | { |
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395 | K[size(K)+1]=I[j]; |
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396 | break; |
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397 | } |
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398 | } |
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399 | } |
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400 | return(K); |
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401 | } |
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402 | |
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403 | static proc finalTest_std_new(string command, alias list args, ideal result) |
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404 | { |
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405 | result=minimizeS(result); |
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406 | attrib(result,"isSB",1); |
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407 | return(Modstd::finalTest_std(command, args,result)); |
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408 | } |
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409 | |
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410 | ///////////////////////////////////////////////////////////////////////////// |
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411 | |
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412 | /* |
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413 | //===== Examples (http://symbolicdata.org/XMLResources/IntPS/): ============ |
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414 | |
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415 | //Cyclic_7.xml |
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416 | ring r18= 32003,(a,b,c,d,e,f,g),dp; |
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417 | ideal i= |
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418 | a+b+c+d+e+f+g, |
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419 | ab+bc+cd+de+ef+fg+ga, |
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420 | abc+bcd+cde+fde+efg+fga+gab, |
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421 | abcd+bcde+cdef+defg+efga+fgab+gabc, |
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422 | abcde+bcdef+cdefg+defga+efgab+fgabc+gabcd, |
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423 | abcdef+bcdefg+cdefga+defgab+efgabc+fgabcd+gabcde, |
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424 | abcdefg-1; |
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425 | |
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426 | // Milnor1.xml |
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427 | ring s8=32003,(x,y,z),dp; |
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428 | int a =60; |
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429 | int b =40; |
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430 | int c =20; |
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431 | int t =1; |
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432 | poly f= x^a+y^b+z^(3*c)+x^(c+2)*y^(c-1)+x^(c-1)*y^(c-1)*z3+x^(c-2)*y^c*(y2+t*x)^2; |
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433 | ideal i= jacob(f); |
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434 | |
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435 | // Tjurina1.xml |
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436 | ring s10=32003,(x,y,z),dp; |
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437 | int a=30; |
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438 | poly f =xyz*(x+y+z)^2 +(x+y+z)^3 +x^a+y^a+z^a; |
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439 | ideal i= jacob(f); |
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440 | i=i,f; |
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441 | |
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442 | // random1.xml |
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443 | ring s15=32003,(x,y,z),dp; |
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444 | poly p1 =x2y*(47x5y7z3+28xy5z8+63+91x5y3z7); |
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445 | poly p2 =xyz*(57y6+21x2yz9+51y2z2+15x2z4); |
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446 | poly p3 =xy4z*(74y+32x6z7+53x5y2z+17x2y3z); |
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447 | poly p4 =y3z*(21x2z6+32x10y6z5+23x5y5z7+27y2); |
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448 | poly p5 =xz*(36y2z2+81x9y10+19x2y5z4+79x4z6); |
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449 | ideal i =p1,p2,p3,p4,p5; |
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450 | |
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451 | // random2.xml |
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452 | ring s16=32003,(x,y,z),dp; |
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453 | poly p1 =xy2z*(47x6y6z2+28y5+63y3z3+91xyz6+57z2); |
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454 | poly p2 =x3y2z*(21y4z7+51z9+15x4y2z7+74x7y7z5+32x7y8); |
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455 | poly p3 =xy*(53x+17y2z3+21x5y5z2+32x4y7z8+23x2y4z5); |
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456 | poly p4 =z*(27x7y4z6+36x3yz4+81x3y7z+19y5+79x5z6); |
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457 | ideal i =p1,p2,p3,p4; |
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458 | |
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459 | // Singular.schwarz_6.xml (dehomogenized: h=1) |
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460 | ring r16= 32003,(x(1..6)),lp; |
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461 | poly s1= -1*x(1)^2-1*x(1)-2*x(2)*x(6)-2*x(3)*x(5)-1*x(4)^2; |
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462 | poly s2= -2*x(1)*x(2)-1*x(2)-2*x(3)*x(6)-2*x(4)*x(5); |
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463 | poly s3= -2*x(1)*x(3)-1*x(2)^2-1*x(3)-2*x(4)*x(6)-1*x(5)^2; |
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464 | poly s4= -2*x(1)*x(4)-2*x(2)*x(3)-1*x(4)-2*x(5)*x(6); |
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465 | poly s5= -2*x(1)*x(5)-2*x(2)*x(4)-1*x(3)^2-1*x(5)-1*x(6)^2; |
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466 | poly s6= -2*x(1)*x(6)-2*x(2)*x(5)-2*x(3)*x(4)-1*x(6); |
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467 | ideal i =s1,s2,s3,s4,s5,s6; |
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468 | |
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469 | //Twomat3.xml |
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470 | ring r1=32003,(a,b,c,d,e,f,g,h,i,A,B,C,D,E,F,G,H,I),dp; |
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471 | ideal i= |
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472 | -dB-gC+bD+cG, |
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473 | -bA+aB-eB-hC+bE+cH, |
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474 | -cA-fB+aC-iC+bF+cI, |
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475 | dA-aD+eD-dE-gF+fG, |
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476 | dB-bD-hF+fH, |
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477 | dC-cD-fE+eF-iF+fI, |
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478 | gA+hD-aG+iG-dH-gI, |
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479 | gB+hE-bG-eH+iH-hIgC+hF-cG-fH; |
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480 | |
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481 | // Klein.xml |
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482 | ring r12 = 32003,(x,y,z,t,u),dp; |
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483 | ideal i= x6+y6+522*(x^5*y-x*y^5)-10005*(x^4*y^2+x^2*y^4)-z, |
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484 | -(x^4+y^4)+228*(x^3*y-x*y^3)-494*x^2*y^2-t, |
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485 | x*y*(x*2+11*x*y-y^2)*5-u; |
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486 | |
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487 | // Meintjes.xml |
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488 | ring r15 = 32003,(x,y,z,t,u,a,b,c,d,e,f,g),dp; |
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489 | ideal i= |
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490 | xy+x-3u, |
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491 | yz2+2y2a+ytb+yzd+2xy+yc-ug+x, |
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492 | 2yz2+yzd+2z2f+ze-8u, |
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493 | ytb+2t2+4ug, |
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494 | yz2+y2a+ytb+yzd+z2f+xy+t2+yc+ze+x-1; |
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495 | |
---|
496 | // Wilfred.xml |
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497 | ring w7=32003,(x(1..16)),dp; |
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498 | ideal i= |
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499 | x(1)+x(2)+x(3)+x(4), |
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500 | x(1)^2*x(2)^2+x(2)*x(3)*x(5)*x(6), |
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501 | x(3)*x(4)*x(9)*x(10)+x(4)*x(5)*x(13)*x(14), |
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502 | x(1)*x(2)^2*x(3)+x(2)*x(3)*x(6)*x(7), |
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503 | x(3)*x(4)*x(11)*x(12)+x(4)*x(5)*x(15)*x(16), |
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504 | x(7)*x(8)*x(9)*x(10)+x(8)*x(9)*x(13)*x(14), |
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505 | x(7)*x(8)*x(11)*x(12)+x(8)*x(9)*x(15)*x(16), |
---|
506 | x(1)*x(2)*x(9)*x(10)+x(5)*x(6)*x(10)*x(11), |
---|
507 | x(2)*x(3)*x(9)*x(10)+x(6)*x(7)*x(10)*x(11), |
---|
508 | x(11)*x(12)*x(15)*x(16)+x(1)*x(15)*x(16)^2, |
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509 | x(4)*x(5)*x(13)*x(14)+x(8)*x(9)*x(14)*x(15)+x(12)*x(13)*x(15)*x(16)+ |
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510 | x(1)^2*x(16)^2; |
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511 | |
---|
512 | // Behnke.xml |
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513 | ring bn=32003,(a,b,c,d,e),dp; |
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514 | int n=7; |
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515 | ideal i= |
---|
516 | a^n-b^n, |
---|
517 | b^n-c^n, |
---|
518 | c^n-d^n, |
---|
519 | d^n-e^n, |
---|
520 | a^(n-1)*b+b^(n-1)*c+c^(n-1)*d+d^(n-1)*e+e^(n-1)*a; |
---|
521 | |
---|
522 | //Paris.ilias13.xml |
---|
523 | ring r=32003,(S1,s1,d1,S2,D2,s2,d2),dp; |
---|
524 | ideal i=-8*d1*D2+2*S2*s2-2*D2*d2-8*S2-4*s2+16, |
---|
525 | 8*d1*D2-2*S2*s2+2*D2*d2+8*S2+4*s2-16, |
---|
526 | -8*d1*S2-16*s1*D2+2*D2*s2-2*S2*d2+16*d1+8*D2+4*d2, |
---|
527 | 8*d1*S2+16*s1*D2-2*D2*s2+2*S2*d2-16*d1-8*D2-4*d2, |
---|
528 | -8*S1*S2+4*S2^2-20*D2^2+16*S1-16, |
---|
529 | 8*S1*S2-4*S2^2+20*D2^2-16*S1+16, |
---|
530 | -16*d1^2-8*s1*s2+s2^2-8*d1*d2-d2^2+32*s1-16, |
---|
531 | 16*d1^2+8*s1*s2-s2^2+8*d1*d2+d2^2-32*s1+16, |
---|
532 | -16*S1*d1+8*d1*S2-10*D2*s2-4*S1*d2+2*S2*d2+16*d1+40*D2+4*d2, |
---|
533 | 16*S1*d1-8*d1*S2+10*D2*s2+4*S1*d2-2*S2*d2-16*d1-40*D2-4*d2, |
---|
534 | -32*S1*s1+16*s1*S2+40*d1*D2+4*S1*s2-2*S2*s2+10*D2*d2+16*S1+32*s1-8*S2-4*s2-16, |
---|
535 | 32*S1*s1-16*s1*S2-40*d1*D2-4*S1*s2+2*S2*s2-10*D2*d2-16*S1-32*s1+8*S2+4*s2+16, |
---|
536 | S2^2*s2^3-2*S2*D2*s2^3+D2^2*s2^3+3*S2^2*s2^2*d2-6*S2*D2*s2^2*d2+3*D2^2*s2^2*d2 |
---|
537 | +3*S2^2*s2*d2^2-6*S2*D2*s2*d2^2+3*D2^2*s2*d2^2+S2^2*d2^3-2*S2*D2*d2^3 |
---|
538 | +D2^2*d2^3-32, |
---|
539 | S2^2*s2^3+2*S2*D2*s2^3+D2^2*s2^3-3*S2^2*s2^2*d2-6*S2*D2*s2^2*d2-3*D2^2*s2^2*d2 |
---|
540 | +3*S2^2*s2*d2^2+6*S2*D2*s2*d2^2+3*D2^2*s2*d2^2-S2^2*d2^3-2*S2*D2*d2^3 |
---|
541 | -D2^2*d2^3-32, |
---|
542 | S1^2*s1^3-3*S1^2*s1^2*d1+3*S1^2*s1*d1^2-S1^2*d1^3+4*S1*s1^3*D2-12*S1*s1^2*d1*D2 |
---|
543 | +12*S1*s1*d1^2*D2-4*S1*d1^3*D2+4*s1^3*D2^2-12*s1^2*d1*D2^2+12*s1*d1^2*D2^2 |
---|
544 | -4*d1^3*D2^2-32, |
---|
545 | S1^2*s1^3+3*S1^2*s1^2*d1+3*S1^2*s1*d1^2+S1^2*d1^3-4*S1*s1^3*D2-12*S1*s1^2*d1*D2 |
---|
546 | -12*S1*s1*d1^2*D2-4*S1*d1^3*D2+4*s1^3*D2^2+12*s1^2*d1*D2^2+12*s1*d1^2*D2^2 |
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547 | +4*d1^3*D2^2-32; |
---|
548 | |
---|
549 | |
---|
550 | // Pfister_2.xlm |
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551 | ring r=32003,(x2,x3,x4,x5,y1,y2,y3,y4,z1,z2,z3,z4,z5),dp; |
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552 | ideal i=x2^3+x3^2+x4^2+x5^2+y1^2+y2^2+y3^2+y4^2+z1^2+z2^2+z3^2+z4^2+z5^2, |
---|
553 | y1^2+z1^2-1,x2^2+y2^2+z2^2-1,x3^2+y3^2+z3^2-1,x4^2+y4^2+z4^2-1, |
---|
554 | x5^2+z5^2-1,y1*y2+z1*z2,x2*x3+y2*y3+z2*z3,x3*x4+y3*y4+z3*z4,x4*x5+z4*z5, |
---|
555 | x2+x3+x4+x5+1,y1+y2+y3+y4-1,z1+z2+z3+z4+z5; |
---|
556 | |
---|
557 | // Singular.gerhard_3.xml |
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558 | ring r=32003,(w,x,y,z),dp; |
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559 | ideal i=3*w^5*x^5*y^5*z^2+18*z^17, |
---|
560 | 5*w^10*x^8*y^4+6*w^11*x^6*y^5+16*w^10*x^5*y^7+10*w^9*x^4*y^9+23*y^22+5*w^10*x^5*y^4*z^3, |
---|
561 | 24*x^23+8*w^11*x^7*y^5+6*w^12*x^5*y^6+10*w^11*x^4*y^8+4*w^10*x^3*y^10+5*w^11*x^4*y^5*z^3; |
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562 | |
---|
563 | //Singular.gerhard_1.xml (dehomogenized w=1) |
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564 | ring r1=32003,(t,x,y,z),ds; |
---|
565 | ideal i= |
---|
566 | 5t3x2z+2t2y3x5, |
---|
567 | 7y+4x2y+y2x+2zt, |
---|
568 | 3tz+3yz2+2yz4; |
---|
569 | |
---|
570 | // Milnor3.xml |
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571 | ring r3=32003,(x,y,z),ds; |
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572 | int a =33; |
---|
573 | int b =30; |
---|
574 | int c =10; |
---|
575 | int t =1; |
---|
576 | poly f =x^a+y^b+z^(3*c)+x^(c+2)*y^(c-1)+x^(c-1)*y^(c-1)*z3+x^(c-2)*y^c*(y2+t*x)^2; |
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577 | ideal i= jacob(f); |
---|
578 | |
---|
579 | // Tjurina1.xml |
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580 | ring r4=0,(x,y,z),ds; |
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581 | int a=30; |
---|
582 | poly f =xyz*(x+y+z)^2 +(x+y+z)^3 +x^a+y^a+z^a; |
---|
583 | ideal i= jacob(f); |
---|
584 | |
---|
585 | // Milnor4.xml |
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586 | ring r6=32003,(x,y,z),ds; |
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587 | int a =12; |
---|
588 | int b =16; |
---|
589 | int c =20; |
---|
590 | int alpha =5; |
---|
591 | int beta= 5; |
---|
592 | int t= 1; |
---|
593 | poly f =x^a+y^b+z^c+x^alpha*y^(beta-5)+x^(alpha-2)*y^(beta-3) |
---|
594 | +x^(alpha-3)*y^(beta-4)*z^2+x^(alpha-4)*y^(beta-4)*(y^2+t*x)^2; |
---|
595 | ideal i= jacob(f); |
---|
596 | i=i,f; |
---|
597 | |
---|
598 | // Steidel_1.xml |
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599 | ring r=0,(x,y,z),ds; |
---|
600 | ideal i=3*x^2+10*x^9+6*x*y+3*y^2+6*x*z+6*y*z+3*x^2*y*z+4*x*y^2*z+y^3*z |
---|
601 | +3*z^2+4*x*y*z^2+2*y^2*z^2+y*z^3, |
---|
602 | 3*x^2+6*x*y+3*y^2+10*y^9+6*x*z+x^3*z+6*y*z+4*x^2*y*z+3*x*y^2*z+3*z^2 |
---|
603 | +2*x^2*z^2+4*x*y*z^2+x*z^3, |
---|
604 | 3*x^2+6*x*y+x^3*y+3*y^2+2*x^2*y^2+x*y^3+6*x*z+6*y*z+4*x^2*y*z |
---|
605 | +4*x*y^2*z+3*z^2+3*x*y*z^2+10*z^9; |
---|
606 | |
---|
607 | */ |
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