1 | //GP, last modified 23.10.06 |
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2 | /////////////////////////////////////////////////////////////////////////////// |
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3 | version="$Id: modstd.lib,v 1.2 2007-01-09 12:44:33 pfister Exp $"; |
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4 | category="Commutative Algebra"; |
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5 | info=" |
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6 | LIBRARY: modstd.lib Grobner basis of ideals |
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7 | AUTHORS: A. Hashemi, Amir.Hashemi@lip6.fr |
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8 | @* G. Pfister pfister@mathematik.uni-kl.de |
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9 | @* H. Schoenemann hannes@mathematik.uni-kl.de |
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10 | |
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11 | NOTE: |
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12 | A library for computing the Grobner basis of an ideal in the polynomial |
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13 | ring over the rational numbers using modular methods.The procedures are |
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14 | inspired by the following paper: |
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15 | Elizabeth A. Arnold: |
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16 | Modular Algorithms for Computing Groebner Bases , Journal of Symbolic |
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17 | Computation , April 2003, Volume 35, (4), p. 403-419. |
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18 | |
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19 | |
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20 | |
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21 | PROCEDURES: |
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22 | modStd(I); compute a standard basis of I using modular methods |
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23 | modS(I,L); liftings to Q of standard bases of I mod p for p in L |
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24 | primeList(n); list of n primes <= 2134567879 in decreasing order |
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25 | "; |
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26 | |
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27 | LIB "poly.lib"; |
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28 | LIB "krypto.lib"; |
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29 | /////////////////////////////////////////////////////////////////////////////// |
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30 | proc modStd(ideal I) |
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31 | "USAGE: modStd(I); |
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32 | RETURN: a standard basis of I if no warning appears; |
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33 | NOTE: the procedure computes a standard basis of I (over the |
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34 | rational numbers) by using modular methods. If a |
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35 | warning appears then the result is a standard basis with no defined |
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36 | relation to I; this is a sign that not enough prime numbers have |
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37 | been used. For further experiments see procedure modS. |
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38 | EXAMPLE: example modStd; shows an example |
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39 | " |
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40 | { |
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41 | int aa=timer; |
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42 | def R0=basering; |
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43 | list rl=ringlist(R0); |
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44 | if((npars(R0)>0)||(rl[1]>0)) |
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45 | { |
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46 | ERROR("characteristic of basering should be zero"); |
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47 | } |
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48 | int l,j,k,q; |
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49 | int en=2134567879; |
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50 | int an=1000000000; |
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51 | intvec hi,hl,hc,hpl,hpc; |
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52 | list T,TT; |
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53 | list L=primeList(5); |
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54 | L[6]=prime(random(an,en)); |
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55 | ideal J,cT,lT,K; |
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56 | ideal I0=I; |
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57 | int h=homog(I); |
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58 | if((!h)&&(ord_test(R0)==0)) |
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59 | { |
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60 | ERROR("input is not homogeneous and ordering is not local"); |
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61 | } |
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62 | if(h) |
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63 | { |
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64 | execute("ring gn="+string(L[6])+",x(1.."+string(nvars(R0))+"),dp;"); |
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65 | ideal I=fetch(R0,I); |
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66 | ideal J=std(I); |
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67 | hi=hilb(J,1); |
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68 | setring R0; |
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69 | } |
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70 | for (j=1;j<=size(L);j++) |
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71 | { |
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72 | rl[1]=L[j]; |
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73 | def oro=ring(rl); |
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74 | setring oro; |
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75 | ideal I=fetch(R0,I); |
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76 | option(redSB); |
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77 | if(h) |
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78 | { |
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79 | ideal I1=std(I,hi); |
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80 | } |
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81 | else |
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82 | { |
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83 | if(ord_test(R0)==-1) |
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84 | { |
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85 | ideal I1=std(I); |
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86 | } |
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87 | else |
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88 | { |
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89 | matrix M; |
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90 | ideal I1=liftstd(I,M); |
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91 | } |
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92 | } |
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93 | setring R0; |
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94 | T[j]=fetch(oro,I1); |
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95 | kill oro; |
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96 | } |
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97 | //================= delete unlucky primes ==================== |
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98 | // unlucky iff the leading ideal is wrong |
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99 | list LL=deleteUnluckyPrimes(T,L); |
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100 | T=LL[1]; |
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101 | L=LL[2]; |
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102 | lT=LL[3]; |
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103 | //============ now all leading ideals are the same ============ |
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104 | for(j=1;j<=ncols(T[1]);j++) |
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105 | { |
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106 | for(k=1;k<=size(L);k++) |
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107 | { |
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108 | TT[k]=T[k][j]; |
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109 | } |
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110 | J[j]=liftPoly(TT,L); |
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111 | } |
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112 | //=========== chooses more primes up to the moment the result becomes stable |
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113 | while(1) |
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114 | { |
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115 | k=0; |
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116 | q=prime(random(an,en)); |
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117 | while(k<size(L)) |
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118 | { |
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119 | k++; |
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120 | if(L[k]==q) |
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121 | { |
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122 | k=0; |
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123 | q=prime(random(an,en)); |
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124 | } |
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125 | } |
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126 | L[size(L)+1]=q; |
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127 | rl[1]=L[size(L)]; |
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128 | def @r=ring(rl); |
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129 | setring @r; |
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130 | ideal i=fetch(R0,I); |
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131 | option(redSB); |
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132 | if(h) |
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133 | { |
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134 | i=std(i,hi); |
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135 | } |
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136 | else |
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137 | { |
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138 | if(ord_test(R0)==-1) |
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139 | { |
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140 | i=std(i); |
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141 | } |
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142 | else |
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143 | { |
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144 | matrix M; |
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145 | i=liftstd(i,M); |
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146 | } |
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147 | } |
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148 | setring R0; |
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149 | T[size(T)+1]=fetch(@r,i); |
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150 | kill @r; |
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151 | cT=lead(T[size(T)]); |
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152 | attrib(cT,"isSB",1); |
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153 | if((size(reduce(cT,lT))!=0)||(size(reduce(lT,cT))!=0)) |
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154 | { |
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155 | T=delete(T,size(T)); |
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156 | L=delete(L,size(L)); |
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157 | k=0; |
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158 | } |
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159 | else |
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160 | { |
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161 | for(j=1;j<=ncols(T[1]);j++) |
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162 | { |
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163 | for(k=1;k<=size(L);k++) |
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164 | { |
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165 | TT[k]=T[k][j]; |
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166 | } |
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167 | K[j]=liftPoly(TT,L); |
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168 | } |
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169 | k=1; |
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170 | for(j=1;j<=size(K);j++) |
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171 | { |
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172 | if(K[j]-J[j]!=0) |
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173 | { |
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174 | k=0; |
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175 | J=K; |
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176 | break; |
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177 | } |
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178 | } |
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179 | } |
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180 | if(k){break;} |
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181 | } |
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182 | //============ test for standard basis and I=J ======= |
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183 | "computed";timer-aa;aa=timer; |
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184 | J=std(J); |
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185 | I0=reduce(I0,J); |
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186 | if(size(I0)>0){"WARNING: The input ideal is not contained |
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187 | in the ideal generated by the standardbasis";} |
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188 | attrib(J,"isSB",1); |
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189 | "verification";timer-aa; |
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190 | return(J); |
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191 | } |
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192 | example |
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193 | { "EXAMPLE:"; echo = 2; |
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194 | ring r=0,(x,y,z),dp; |
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195 | ideal I=3x3+x2+1,11y5+y3+2,5z4+z2+4; |
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196 | ideal J=modStd(I); |
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197 | J; |
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198 | } |
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199 | /////////////////////////////////////////////////////////////////////////////// |
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200 | proc modS(ideal I, list L, list #) |
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201 | "USAGE: modS(I,L); I ideal, L list of primes |
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202 | if size(#)>0 std is used instead of groebner |
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203 | RETURN: an ideal which is with high probability a standard basis |
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204 | NOTE: This procedure is designed for fast experiments. |
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205 | It is not tested whether the result is a standard basis. |
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206 | It is not tested whether the result generates I. |
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207 | EXAMPLE: example modS; shows an example |
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208 | " |
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209 | { |
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210 | int j,k; |
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211 | list T,TT; |
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212 | def R0=basering; |
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213 | ideal J,cT,lT,K; |
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214 | ideal I0=I; |
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215 | list rl=ringlist(R0); |
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216 | if((npars(R0)>0)||(rl[1]>0)) |
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217 | { |
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218 | ERROR("characteristic of basering should be zero"); |
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219 | } |
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220 | for (j=1;j<=size(L);j++) |
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221 | { |
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222 | rl[1]=L[j]; |
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223 | def @r=ring(rl); |
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224 | setring @r; |
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225 | ideal i=fetch(R0,I); |
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226 | option(redSB); |
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227 | if(size(#)>0) |
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228 | { |
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229 | i=std(i); |
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230 | } |
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231 | else |
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232 | { |
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233 | i=groebner(i) |
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234 | } |
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235 | setring R0; |
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236 | T[j]=fetch(@r,i); |
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237 | kill @r; |
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238 | } |
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239 | //================= delete unlucky primes ==================== |
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240 | // unlucky iff the leading ideal is wrong |
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241 | list LL=deleteUnluckyPrimes(T,L); |
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242 | T=LL[1]; |
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243 | L=LL[2]; |
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244 | //============ now all leading ideals are the same ============ |
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245 | for(j=1;j<=ncols(T[1]);j++) |
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246 | { |
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247 | for(k=1;k<=size(L);k++) |
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248 | { |
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249 | TT[k]=T[k][j]; |
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250 | } |
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251 | J[j]=liftPoly(TT,L); |
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252 | } |
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253 | attrib(J,"isSB",1); |
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254 | return(J); |
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255 | } |
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256 | example |
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257 | { "EXAMPLE:"; echo = 2; |
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258 | list L=3,5,11,13,181; |
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259 | ring r=0,(x,y,z),dp; |
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260 | ideal I=3x3+x2+1,11y5+y3+2,5z4+z2+4; |
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261 | ideal J=modS(I,L); |
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262 | J; |
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263 | } |
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264 | /////////////////////////////////////////////////////////////////////////////// |
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265 | proc deleteUnluckyPrimes(list T,list L) |
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266 | "USAGE: deleteUnluckyPrimes(T,L);T list of polys, L list of primes |
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267 | RETURN: list L,T with T list of polys, L list of primes |
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268 | EXAMPLE: example deleteUnluckyPrimes; shows an example |
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269 | NOTE: works only for homogeneous ideals with global orderings or |
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270 | for ideals with local orderings |
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271 | " |
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272 | { |
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273 | int j,k; |
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274 | intvec hl,hc; |
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275 | ideal cT,lT; |
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276 | |
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277 | lT=lead(T[size(T)]); |
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278 | attrib(lT,"isSB",1); |
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279 | hl=hilb(lT,1); |
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280 | for (j=1;j<size(T);j++) |
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281 | { |
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282 | cT=lead(T[j]); |
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283 | attrib(cT,"isSB",1); |
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284 | hc=hilb(cT,1); |
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285 | if(hl==hc) |
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286 | { |
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287 | for(k=1;k<=size(lT);k++) |
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288 | { |
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289 | if(lT[k]<cT[k]){lT=cT;break;} |
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290 | if(lT[k]>cT[k]){break;} |
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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(hc<hl){lT=cT;hl=hilb(lT,1);} |
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296 | } |
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297 | } |
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298 | j=1; |
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299 | attrib(lT,"isSB",1); |
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300 | while(j<=size(T)) |
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301 | { |
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302 | cT=lead(T[j]); |
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303 | attrib(cT,"isSB",1); |
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304 | if((size(reduce(cT,lT))!=0)||(size(reduce(lT,cT))!=0)) |
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305 | { |
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306 | T=delete(T,j); |
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307 | L=delete(L,j); |
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308 | j--; |
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309 | } |
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310 | j++; |
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311 | } |
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312 | return(list(T,L,lT)); |
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313 | } |
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314 | example |
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315 | { "EXAMPLE:"; echo = 2; |
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316 | list L=2,3,5,7,11; |
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317 | ring r=0,(y,x),Dp; |
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318 | ideal I1=y2x,y6; |
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319 | ideal I2=yx2,y3x,x5,y6; |
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320 | ideal I3=y2x,x3y,x5,y6; |
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321 | ideal I4=y2x,x3y,x5; |
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322 | ideal I5=y2x,yx3,x5,y6; |
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323 | list T=I1,I2,I3,I4,I5; |
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324 | list TT=deleteUnluckyPrimes(T,L); |
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325 | TT; |
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326 | } |
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327 | /////////////////////////////////////////////////////////////////////////////// |
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328 | proc liftPoly(list T, list L) |
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329 | "USAGE: liftPoly(T,L); T list of polys, L list of primes |
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330 | RETURN: poly p in Q[x] such that p mod L[i]=T[i] |
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331 | EXAMPLE: example liftPoly; shows an example |
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332 | " |
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333 | { |
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334 | poly result; |
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335 | int i; |
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336 | poly p; |
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337 | list TT; |
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338 | number n; |
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339 | |
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340 | number N=L[1]; |
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341 | for(i=2;i<=size(L);i++) |
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342 | { |
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343 | N=N*L[i]; |
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344 | } |
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345 | while(1) |
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346 | { |
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347 | p=leadmonom(T[1]); |
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348 | for(i=2;i<=size(T);i++) |
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349 | { |
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350 | if(leadmonom(T[i])>p) |
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351 | { |
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352 | p=leadmonom(T[i]); |
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353 | } |
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354 | } |
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355 | if (p==0) {return(result);} |
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356 | for(i=1;i<=size(T);i++) |
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357 | { |
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358 | if(p==leadmonom(T[i])) |
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359 | { |
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360 | TT[i]=leadcoef(T[i]); |
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361 | T[i]=T[i]-lead(T[i]); |
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362 | } |
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363 | else |
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364 | { |
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365 | TT[i]=0; |
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366 | } |
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367 | } |
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368 | n=chineseR(TT,L,N); |
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369 | n=Farey(N,n); |
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370 | result=result+n*p; |
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371 | } |
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372 | } |
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373 | example |
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374 | { "EXAMPLE:"; echo = 2; |
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375 | ring R = 0,(x,y),dp; |
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376 | list L=32003,181,241,499; |
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377 | list T=x2+7000x+13000,x2+100x+147y+40,x2+120x+191y+10,x2+x+67y+100; |
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378 | liftPoly(T,L); |
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379 | } |
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380 | /////////////////////////////////////////////////////////////////////////////// |
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381 | proc fareyIdeal(ideal I,list L) |
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382 | { |
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383 | poly result,p; |
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384 | int i,j; |
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385 | number n; |
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386 | number N=L[1]; |
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387 | for(i=2;i<=size(L);i++) |
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388 | { |
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389 | N=N*L[i]; |
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390 | } |
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391 | |
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392 | for(i=1;i<=size(I);i++) |
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393 | { |
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394 | p=I[i]; |
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395 | result=lead(p); |
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396 | while(1) |
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397 | { |
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398 | if (p==0) {break;} |
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399 | p=p-lead(p); |
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400 | n=Farey(N,leadcoef(p)); |
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401 | result=result+n*leadmonom(p); |
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402 | } |
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403 | I[i]=result; |
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404 | } |
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405 | return(I); |
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406 | } |
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407 | /////////////////////////////////////////////////////////////////////////////// |
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408 | proc Farey (number P, number N) |
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409 | "USAGE: Farey (P,N); P, N number; |
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410 | RETURN: a rational number a/b such that a/b=N mod P |
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411 | and |a|,|b|<(P/2)^{1/2} |
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412 | " |
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413 | { |
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414 | if (P<0){P=-P;} |
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415 | if (N<0){N=N+P;} |
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416 | number A,B,C,D,E; |
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417 | E=P; |
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418 | B=1; |
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419 | while (N!=0) |
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420 | { |
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421 | if (2*N^2<P) |
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422 | { |
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423 | return(N/B); |
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424 | } |
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425 | D=E mod N; |
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426 | C=A-(E-E mod N)/N*B; |
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427 | E=N; |
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428 | N=D; |
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429 | A=B; |
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430 | B=C; |
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431 | } |
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432 | return(0); |
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433 | } |
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434 | example |
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435 | { "EXAMPLE:"; echo = 2; |
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436 | ring R = 0,x,dp; |
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437 | Farey(32003,12345); |
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438 | } |
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439 | /////////////////////////////////////////////////////////////////////////////// |
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440 | proc chineseR(list T,list L,number N) |
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441 | "USAGE: chineseR(T,L); |
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442 | RETURN: x such that x = T[i] mod L[i] |
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443 | NOTE: chinese remainder theorem |
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444 | EXAMPLE:example chineseR; shows an example |
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445 | " |
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446 | { |
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447 | number x; |
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448 | if(size(L)==1) |
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449 | { |
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450 | x=T[1] mod L[1]; |
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451 | return(x); |
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452 | } |
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453 | int i; |
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454 | int n=size(L); |
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455 | list M; |
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456 | for(i=1;i<=n;i++) |
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457 | { |
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458 | M[i]=N/L[i]; |
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459 | } |
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460 | list S=eexgcdN(M); |
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461 | for(i=1;i<=n;i++) |
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462 | { |
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463 | x=x+S[i]*M[i]*T[i]; |
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464 | } |
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465 | x=x mod N; |
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466 | return(x); |
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467 | } |
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468 | example |
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469 | { "EXAMPLE:"; echo = 2; |
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470 | ring R = 0,x,dp; |
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471 | chineseR(list(24,15,7),list(2,3,5)); |
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472 | } |
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473 | |
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474 | /////////////////////////////////////////////////////////////////////////////// |
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475 | proc primeList(int n) |
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476 | "USAGE: primeList(n); |
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477 | RETURN: the list of n greatest primes <= 2134567879 |
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478 | EXAMPLE:example primList; shows an example |
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479 | " |
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480 | { |
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481 | list L; |
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482 | int i; |
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483 | int p=2134567879; |
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484 | for(i=1;i<=n;i++) |
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485 | { |
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486 | L[i]=p; |
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487 | p=prime(p-1); |
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488 | } |
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489 | return(L); |
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490 | } |
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491 | example |
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492 | { "EXAMPLE:"; echo = 2; |
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493 | list L=primeList(10); |
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494 | size(L); |
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495 | L[size(L)]; |
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496 | } |
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497 | /////////////////////////////////////////////////////////////////////////////// |
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498 | /* |
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499 | ring r=0,(x,y,z),lp; |
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500 | poly s1 = 5x3y2z+3y3x2z+7xy2z2; |
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501 | poly s2 = 3xy2z2+x5+11y2z2; |
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502 | poly s3 = 4xyz+7x3+12y3+1; |
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503 | poly s4 = 3x3-4y3+yz2; |
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504 | ideal i = s1, s2, s3, s4; |
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505 | |
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506 | ring r=0,(x,y,z),lp; |
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507 | poly s1 = 2xy4z2+x3y2z-x2y3z+2xyz2+7y3+7; |
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508 | poly s2 = 2x2y4z+x2yz2-xy2z2+2x2yz-12x+12y; |
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509 | poly s3 = 2y5z+x2y2z-xy3z-xy3+y4+2y2z; |
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510 | poly s4 = 3xy4z3+x2y2z-xy3z+4y3z2+3xyz3+4z2-x+y; |
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511 | ideal i = s1, s2, s3, s4; |
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512 | |
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513 | ring r=0,(x,y,z),lp; |
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514 | poly s1 = 8x2y2 + 5xy3 + 3x3z + x2yz; |
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515 | poly s2 = x5 + 2y3z2 + 13y2z3 + 5yz4; |
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516 | poly s3 = 8x3 + 12y3 + xz2 + 3; |
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517 | poly s4 = 7x2y4 + 18xy3z2 + y3z3; |
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518 | ideal i = s1, s2, s3, s4; |
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519 | |
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520 | int n = 6; |
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521 | ring r = 0,(x(1..n)),lp; |
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522 | ideal i = cyclic(n); |
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523 | ring s=0,(x(1..n),t),lp; |
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524 | ideal i=imap(r,i); |
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525 | i=homog(i,t); |
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526 | |
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527 | ring r=0,(x(1..4),s),(dp(4),dp); |
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528 | poly s1 =1 + s^2*x(1)*x(3) + s^8*x(2)*x(3) + s^19*x(1)*x(2)*x(4); |
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529 | poly s2 = x(1) + s^8 *x(1)* x(2)* x(3) + s^19* x(2)* x(4); |
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530 | poly s3 = x(2) + s^10*x(3)*x(4) + s^11*x(1)*x(4); |
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531 | poly s4 = x(3) + s^4*x(1)*x(2) + s^19*x(1)*x(3)*x(4) +s^24*x(2)*x(3)*x(4); |
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532 | poly s5 = x(4) + s^31* x(1)* x(2)* x(3)* x(4); |
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533 | ideal i = s1, s2, s3, s4, s5; |
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534 | |
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535 | ring r=0,(x,y,z),ds; |
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536 | int a =16; |
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537 | int b =15; |
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538 | int c =4; |
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539 | int t =1; |
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540 | 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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541 | ideal i= jacob(f); |
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542 | |
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543 | ring r=0,(x,y,z),ds; |
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544 | int a =25; |
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545 | int b =25; |
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546 | int c =5; |
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547 | int t =1; |
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548 | 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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549 | ideal i= jacob(f),f; |
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550 | |
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551 | ring r=0,(x,y,z),ds; |
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552 | int a=10; |
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553 | poly f =xyz*(x+y+z)^2 +(x+y+z)^3 +x^a+y^a+z^a; |
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554 | ideal i= jacob(f); |
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555 | |
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556 | ring r=0,(x,y,z),ds; |
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557 | int a =6; |
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558 | int b =8; |
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559 | int c =10; |
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560 | int alpha =5; |
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561 | int beta= 5; |
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562 | int t= 1; |
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563 | poly f =x^a+y^b+z^c+x^alpha*y^(beta-5)+x^(alpha-2)*y^(beta-3)+x^(alpha-3)*y^(beta-4)*z^2+x^(alpha-4)*y^(beta-4)*(y^2+t*x)^2; |
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564 | ideal i= jacob(f); |
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565 | |
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566 | */ |
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