1 | //////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id: modstd.lib 14375 2011-08-23 09:29:47Z steidel $"; |
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3 | category = "Commutative Algebra"; |
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4 | info=" |
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5 | LIBRARY: modstd.lib Groebner basis of ideals |
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6 | |
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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 | @* A. Steenpass steenpass@mathematik.uni-kl.de |
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11 | @* S. Steidel steidel@mathematik.uni-kl.de |
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12 | |
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13 | OVERVIEW: |
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14 | |
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15 | A library for computing the Groebner basis of an ideal in the polynomial |
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16 | ring over the rational numbers using modular methods. The procedures are |
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17 | inspired by the following paper: |
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18 | Elizabeth A. Arnold: Modular algorithms for computing Groebner bases. |
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19 | Journal of Symbolic Computation 35, 403-419 (2003). |
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20 | |
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21 | PROCEDURES: |
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22 | modStd(I); standard basis of I using modular methods (chinese remainder) |
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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 | modHenselStd(I); standard basis of I using modular methods (hensel lifting) |
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25 | "; |
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26 | |
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27 | LIB "poly.lib"; |
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28 | LIB "ring.lib"; |
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29 | LIB "parallel.lib"; |
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30 | |
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31 | //////////////////////////////////////////////////////////////////////////////// |
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32 | |
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33 | proc mixedTest() |
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34 | "USAGE: mixedTest(); |
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35 | RETURN: 1 if ordering of basering is mixed, 0 else |
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36 | EXAMPLE: example mixedTest(); shows an example |
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37 | " |
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38 | { |
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39 | int i,p,m; |
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40 | for(i = 1; i <= nvars(basering); i++) |
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41 | { |
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42 | if(var(i) > 1) |
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43 | { |
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44 | p++; |
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45 | } |
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46 | else |
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47 | { |
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48 | m++; |
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49 | } |
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50 | } |
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51 | if((p > 0) && (m > 0)) { return(1); } |
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52 | return(0); |
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53 | } |
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54 | example |
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55 | { "EXAMPLE:"; echo = 2; |
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56 | ring R1 = 0,(x,y,z),dp; |
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57 | mixedTest(); |
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58 | ring R2 = 31,(x(1..4),y(1..3)),(ds(4),lp(3)); |
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59 | mixedTest(); |
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60 | ring R3 = 181,x(1..9),(dp(5),lp(4)); |
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61 | mixedTest(); |
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62 | } |
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63 | |
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64 | //////////////////////////////////////////////////////////////////////////////// |
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65 | |
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66 | static proc redFork(ideal I, ideal J, int n) |
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67 | { |
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68 | attrib(J,"isSB",1); |
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69 | return(reduce(I,J,1)); |
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70 | } |
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71 | |
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72 | //////////////////////////////////////////////////////////////////////////////// |
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73 | |
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74 | proc isIncluded(ideal I, ideal J, list #) |
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75 | "USAGE: isIncluded(I,J); I,J ideals |
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76 | RETURN: 1 if J includes I, |
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77 | @* 0 if there is an element f in I which does not reduce to 0 w.r.t. J. |
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78 | EXAMPLE: example isIncluded; shows an example |
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79 | " |
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80 | { |
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81 | def R = basering; |
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82 | setring R; |
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83 | |
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84 | attrib(J,"isSB",1); |
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85 | int i,j,k; |
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86 | |
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87 | if(size(#) > 0) |
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88 | { |
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89 | int n = #[1]; |
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90 | if(n >= ncols(I)) { n = ncols(I); } |
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91 | if(n > 1) |
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92 | { |
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93 | for(i = 1; i <= n - 1; i++) |
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94 | { |
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95 | //link l(i) = "MPtcp:fork"; |
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96 | link l(i) = "ssi:fork"; |
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97 | open(l(i)); |
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98 | |
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99 | write(l(i), quote(redFork(eval(I[ncols(I)-i]), eval(J), 1))); |
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100 | } |
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101 | |
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102 | int t = timer; |
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103 | if(reduce(I[ncols(I)], J, 1) != 0) |
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104 | { |
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105 | for(i = 1; i <= n - 1; i++) |
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106 | { |
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107 | close(l(i)); |
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108 | } |
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109 | return(0); |
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110 | } |
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111 | t = timer - t; |
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112 | if(t > 60) { t = 60; } |
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113 | int i_sleep = system("sh", "sleep "+string(t)); |
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114 | |
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115 | j = ncols(I) - n; |
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116 | |
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117 | while(j >= 0) |
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118 | { |
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119 | for(i = 1; i <= n - 1; i++) |
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120 | { |
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121 | if(status(l(i), "read", "ready")) |
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122 | { |
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123 | if(read(l(i)) != 0) |
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124 | { |
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125 | for(i = 1; i <= n - 1; i++) |
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126 | { |
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127 | close(l(i)); |
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128 | } |
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129 | return(0); |
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130 | } |
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131 | else |
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132 | { |
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133 | if(j >= 1) |
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134 | { |
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135 | write(l(i), quote(redFork(eval(I[j]), eval(J), 1))); |
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136 | j--; |
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137 | } |
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138 | else |
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139 | { |
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140 | k++; |
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141 | close(l(i)); |
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142 | } |
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143 | } |
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144 | } |
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145 | } |
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146 | if(k == n - 1) |
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147 | { |
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148 | j--; |
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149 | } |
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150 | i_sleep = system("sh", "sleep "+string(t)); |
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151 | } |
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152 | return(1); |
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153 | } |
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154 | } |
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155 | |
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156 | for(i = ncols(I); i >= 1; i--) |
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157 | { |
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158 | if(reduce(I[i],J,1) != 0){ return(0); } |
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159 | } |
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160 | return(1); |
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161 | } |
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162 | example |
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163 | { "EXAMPLE:"; echo = 2; |
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164 | ring r=0,(x,y,z),dp; |
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165 | ideal I = x+1,x+y+1; |
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166 | ideal J = x+1,y; |
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167 | isIncluded(I,J); |
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168 | isIncluded(J,I); |
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169 | isIncluded(I,J,4); |
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170 | |
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171 | ring R = 0, x(1..5), dp; |
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172 | ideal I1 = cyclic(4); |
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173 | ideal I2 = I1,x(5)^2; |
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174 | isIncluded(I1,I2,4); |
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175 | } |
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176 | |
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177 | //////////////////////////////////////////////////////////////////////////////// |
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178 | |
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179 | proc pTestSB(ideal I, ideal J, list L, int variant, list #) |
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180 | "USAGE: pTestSB(I,J,L,variant,#); I,J ideals, L intvec of primes, variant int |
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181 | RETURN: 1 (resp. 0) if for a randomly chosen prime p that is not in L |
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182 | J mod p is (resp. is not) a standard basis of I mod p |
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183 | EXAMPLE: example pTestSB; shows an example |
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184 | " |
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185 | { |
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186 | int i,j,k,p; |
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187 | def R = basering; |
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188 | list r = ringlist(R); |
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189 | |
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190 | while(!j) |
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191 | { |
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192 | j = 1; |
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193 | p = prime(random(1000000000,2134567879)); |
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194 | for(i = 1; i <= size(L); i++) |
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195 | { |
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196 | if(p == L[i]) { j = 0; break; } |
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197 | } |
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198 | if(j) |
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199 | { |
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200 | for(i = 1; i <= ncols(I); i++) |
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201 | { |
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202 | for(k = 2; k <= size(I[i]); k++) |
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203 | { |
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204 | if((denominator(leadcoef(I[i][k])) mod p) == 0) { j = 0; break; } |
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205 | } |
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206 | if(!j){ break; } |
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207 | } |
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208 | } |
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209 | if(j) |
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210 | { |
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211 | if(!primeTest(I,p)) { j = 0; } |
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212 | } |
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213 | } |
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214 | r[1] = p; |
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215 | def @R = ring(r); |
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216 | setring @R; |
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217 | ideal I = imap(R,I); |
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218 | ideal J = imap(R,J); |
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219 | attrib(J,"isSB",1); |
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220 | |
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221 | int t = timer; |
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222 | j = 1; |
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223 | if(isIncluded(I,J) == 0) { j = 0; } |
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224 | |
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225 | if(printlevel >= 11) |
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226 | { |
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227 | "isIncluded(I,J) takes "+string(timer - t)+" seconds"; |
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228 | "j = "+string(j); |
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229 | } |
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230 | |
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231 | t = timer; |
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232 | if(j) |
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233 | { |
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234 | if(size(#) > 0) |
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235 | { |
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236 | ideal K = modpStd(I,p,variant,#[1])[1]; |
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237 | } |
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238 | else |
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239 | { |
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240 | ideal K = groebner(I); |
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241 | } |
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242 | t = timer; |
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243 | if(isIncluded(J,K) == 0) { j = 0; } |
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244 | |
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245 | if(printlevel >= 11) |
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246 | { |
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247 | "isIncluded(J,K) takes "+string(timer - t)+" seconds"; |
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248 | "j = "+string(j); |
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249 | } |
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250 | } |
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251 | setring R; |
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252 | return(j); |
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253 | } |
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254 | example |
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255 | { "EXAMPLE:"; echo = 2; |
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256 | intvec L = 2,3,5; |
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257 | ring r = 0,(x,y,z),dp; |
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258 | ideal I = x+1,x+y+1; |
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259 | ideal J = x+1,y; |
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260 | pTestSB(I,I,L,2); |
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261 | pTestSB(I,J,L,2); |
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262 | } |
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263 | |
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264 | //////////////////////////////////////////////////////////////////////////////// |
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265 | |
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266 | proc deleteUnluckyPrimes(list T, list L, int ho, list #) |
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267 | "USAGE: deleteUnluckyPrimes(T,L,ho,#); T/L list of polys/primes, ho integer |
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268 | RETURN: lists T,L(,M),lT with T/L(/M) list of polys/primes(/type of #), |
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269 | lT ideal |
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270 | NOTE: - if ho = 1, the polynomials in T are homogeneous, else ho = 0, |
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271 | @* - lT is prevalent, i.e. the most appearing leading ideal in T |
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272 | EXAMPLE: example deleteUnluckyPrimes; shows an example |
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273 | " |
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274 | { |
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275 | ho = ((ho)||(ord_test(basering) == -1)); |
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276 | int j,k,c; |
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277 | intvec hl,hc; |
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278 | ideal cT,lT,cK; |
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279 | lT = lead(T[size(T)]); |
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280 | attrib(lT,"isSB",1); |
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281 | if(!ho) |
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282 | { |
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283 | for(j = 1; j < size(T); j++) |
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284 | { |
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285 | cT = lead(T[j]); |
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286 | attrib(cT,"isSB",1); |
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287 | if((size(reduce(cT,lT))!=0)||(size(reduce(lT,cT))!=0)) |
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288 | { |
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289 | cK = cT; |
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290 | c++; |
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291 | } |
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292 | } |
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293 | if(c > size(T) div 2){ lT = cK; } |
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294 | } |
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295 | else |
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296 | { |
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297 | hl = hilb(lT,1); |
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298 | for(j = 1; j < size(T); j++) |
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299 | { |
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300 | cT = lead(T[j]); |
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301 | attrib(cT,"isSB",1); |
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302 | hc = hilb(cT,1); |
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303 | if(hl == hc) |
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304 | { |
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305 | for(k = 1; k <= size(lT); k++) |
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306 | { |
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307 | if(lT[k] < cT[k]) { lT = cT; c++; break; } |
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308 | if(lT[k] > cT[k]) { c++; break; } |
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309 | } |
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310 | } |
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311 | else |
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312 | { |
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313 | if(hc < hl){ lT = cT; hl = hilb(lT,1); c++; } |
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314 | } |
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315 | } |
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316 | } |
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317 | |
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318 | int addList; |
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319 | if(size(#) > 0) { list M = #; addList = 1; } |
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320 | j = 1; |
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321 | attrib(lT,"isSB",1); |
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322 | while((j <= size(T))&&(c > 0)) |
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323 | { |
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324 | cT = lead(T[j]); |
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325 | attrib(cT,"isSB",1); |
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326 | if((size(reduce(cT,lT)) != 0)||(size(reduce(lT,cT)) != 0)) |
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327 | { |
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328 | T = delete(T,j); |
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329 | if(j == 1) |
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330 | { |
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331 | L = L[2..size(L)]; |
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332 | if(addList == 1) { M = M[2..size(M)]; } |
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333 | } |
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334 | else |
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335 | { |
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336 | if(j == size(L)) |
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337 | { |
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338 | L = L[1..size(L)-1]; |
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339 | if(addList == 1) { M = M[1..size(M)-1]; } |
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340 | } |
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341 | else |
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342 | { |
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343 | L = L[1..j-1],L[j+1..size(L)]; |
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344 | if(addList == 1) { M = M[1..j-1],M[j+1..size(M)]; } |
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345 | } |
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346 | } |
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347 | j--; |
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348 | } |
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349 | j++; |
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350 | } |
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351 | |
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352 | for(j = 1; j <= size(L); j++) |
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353 | { |
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354 | L[j] = bigint(L[j]); |
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355 | } |
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356 | |
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357 | if(addList == 0) { return(list(T,L,lT)); } |
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358 | if(addList == 1) { return(list(T,L,M,lT)); } |
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359 | } |
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360 | example |
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361 | { "EXAMPLE:"; echo = 2; |
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362 | list L = 2,3,5,7,11; |
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363 | ring r = 0,(y,x),Dp; |
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364 | ideal I1 = 2y2x,y6; |
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365 | ideal I2 = yx2,y3x,x5,y6; |
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366 | ideal I3 = y2x,x3y,x5,y6; |
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367 | ideal I4 = y2x,11x3y,x5; |
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368 | ideal I5 = y2x,yx3,x5,7y6; |
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369 | list T = I1,I2,I3,I4,I5; |
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370 | deleteUnluckyPrimes(T,L,1); |
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371 | list P = poly(x),poly(x2),poly(x3),poly(x4),poly(x5); |
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372 | deleteUnluckyPrimes(T,L,1,P); |
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373 | } |
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374 | |
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375 | //////////////////////////////////////////////////////////////////////////////// |
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376 | |
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377 | proc primeTest(def II, bigint p) |
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378 | { |
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379 | if(typeof(II) == "string") |
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380 | { |
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381 | execute("ideal I = "+II+";"); |
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382 | } |
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383 | else |
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384 | { |
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385 | ideal I = II; |
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386 | } |
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387 | |
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388 | I = simplify(I, 2); // erase zero generators |
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389 | |
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390 | int i,j; |
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391 | poly f; |
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392 | number cnt; |
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393 | for(i = 1; i <= size(I); i++) |
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394 | { |
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395 | f = cleardenom(I[i]); |
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396 | if(f == 0) { return(0); } |
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397 | cnt = leadcoef(I[i])/leadcoef(f); |
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398 | if((numerator(cnt) mod p) == 0) { return(0); } |
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399 | if((denominator(cnt) mod p) == 0) { return(0); } |
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400 | for(j = size(f); j > 0; j--) |
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401 | { |
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402 | if((leadcoef(f[j]) mod p) == 0) { return(0); } |
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403 | } |
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404 | } |
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405 | return(1); |
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406 | } |
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407 | |
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408 | //////////////////////////////////////////////////////////////////////////////// |
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409 | |
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410 | proc primeList(ideal I, int n, list #) |
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411 | "USAGE: primeList(I,n[,ncores]); ( resp. primeList(I,n[,L,ncores]); ) I ideal, |
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412 | n integer |
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413 | RETURN: the intvec of n greatest primes <= 2147483647 (resp. n greatest primes |
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414 | < L[size(L)] union with L) such that none of these primes divides any |
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415 | coefficient occuring in I |
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416 | NOTE: The number of cores to use can be defined by ncores, default is 1. |
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417 | EXAMPLE: example primList; shows an example |
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418 | " |
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419 | { |
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420 | intvec L; |
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421 | int i,p; |
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422 | int ncores = 1; |
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423 | |
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424 | //----------------- Initialize optional parameter ncores --------------------- |
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425 | if(size(#) > 0) |
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426 | { |
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427 | if(size(#) == 1) |
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428 | { |
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429 | if(typeof(#[1]) == "int") |
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430 | { |
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431 | ncores = #[1]; |
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432 | # = list(); |
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433 | } |
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434 | } |
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435 | else |
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436 | { |
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437 | ncores = #[2]; |
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438 | } |
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439 | } |
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440 | |
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441 | if(size(#) == 0) |
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442 | { |
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443 | p = 2147483647; |
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444 | while(!primeTest(I,p)) |
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445 | { |
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446 | p = prime(p-1); |
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447 | if(p == 2) { ERROR("no more primes"); } |
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448 | } |
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449 | L[1] = p; |
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450 | } |
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451 | else |
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452 | { |
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453 | L = #[1]; |
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454 | p = prime(L[size(L)]-1); |
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455 | while(!primeTest(I,p)) |
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456 | { |
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457 | p = prime(p-1); |
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458 | if(p == 2) { ERROR("no more primes"); } |
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459 | } |
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460 | L[size(L)+1] = p; |
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461 | } |
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462 | if(p == 2) { ERROR("no more primes"); } |
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463 | if(ncores == 1) |
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464 | { |
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465 | for(i = 2; i <= n; i++) |
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466 | { |
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467 | p = prime(p-1); |
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468 | while(!primeTest(I,p)) |
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469 | { |
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470 | p = prime(p-1); |
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471 | if(p == 2) { ERROR("no more primes"); } |
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472 | } |
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473 | L[size(L)+1] = p; |
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474 | } |
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475 | } |
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476 | else |
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477 | { |
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478 | int neededSize = size(L)+n-1;; |
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479 | list parallelResults; |
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480 | list arguments; |
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481 | int neededPrimes = neededSize-size(L); |
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482 | while(neededPrimes > 0) |
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483 | { |
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484 | arguments = list(); |
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485 | for(i = ((neededPrimes div ncores)+1-(neededPrimes%ncores == 0)) |
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486 | *ncores; i > 0; i--) |
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487 | { |
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488 | p = prime(p-1); |
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489 | if(p == 2) { ERROR("no more primes"); } |
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490 | arguments[i] = list("I", p); |
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491 | } |
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492 | parallelResults = parallelWaitAll("primeTest", arguments, |
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493 | list(list(list(ncores)))); |
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494 | for(i = size(arguments); i > 0; i--) |
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495 | { |
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496 | if(parallelResults[i]) |
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497 | { |
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498 | L[size(L)+1] = arguments[i][2]; |
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499 | } |
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500 | } |
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501 | neededPrimes = neededSize-size(L); |
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502 | } |
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503 | if(size(L) > neededSize) |
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504 | { |
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505 | L = L[1..neededSize]; |
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506 | } |
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507 | } |
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508 | return(L); |
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509 | } |
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510 | example |
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511 | { "EXAMPLE:"; echo = 2; |
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512 | ring r = 0,(x,y,z),dp; |
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513 | ideal I = 2147483647x+y, z-181; |
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514 | intvec L = primeList(I,10); |
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515 | size(L); |
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516 | L[1]; |
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517 | L[size(L)]; |
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518 | L = primeList(I,5,L); |
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519 | size(L); |
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520 | L[size(L)]; |
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521 | } |
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522 | |
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523 | //////////////////////////////////////////////////////////////////////////////// |
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524 | |
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525 | static proc liftstd1(ideal I) |
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526 | { |
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527 | def R = basering; |
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528 | list rl = ringlist(R); |
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529 | list ordl = rl[3]; |
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530 | |
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531 | int i; |
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532 | for(i = 1; i <= size(ordl); i++) |
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533 | { |
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534 | if((ordl[i][1] == "C") || (ordl[i][1] == "c")) |
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535 | { |
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536 | ordl = delete(ordl, i); |
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537 | break; |
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538 | } |
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539 | } |
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540 | |
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541 | ordl = insert(ordl, list("c", 0)); |
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542 | rl[3] = ordl; |
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543 | def newR = ring(rl); |
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544 | setring newR; |
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545 | ideal I = imap(R,I); |
---|
546 | |
---|
547 | intvec opt = option(get); |
---|
548 | option(none); |
---|
549 | option(prompt); |
---|
550 | |
---|
551 | module M; |
---|
552 | for(i = 1; i <= size(I); i++) |
---|
553 | { |
---|
554 | M = M + module(I[i]*gen(1) + gen(i+1)); |
---|
555 | M = M + module(gen(i+1)); |
---|
556 | } |
---|
557 | |
---|
558 | module sM = std(M); |
---|
559 | |
---|
560 | ideal sI; |
---|
561 | if(attrib(R,"global")) |
---|
562 | { |
---|
563 | for(i = size(I)+1; i <= size(sM); i++) |
---|
564 | { |
---|
565 | sI[size(sI)+1] = sM[i][1]; |
---|
566 | } |
---|
567 | matrix T = submat(sM,2..nrows(sM),size(I)+1..ncols(sM)); |
---|
568 | } |
---|
569 | else |
---|
570 | { |
---|
571 | //"=========================================================="; |
---|
572 | //"WARNING: Algorithm is not applicable if ordering is mixed."; |
---|
573 | //"=========================================================="; |
---|
574 | for(i = 1; i <= size(sM)-size(I); i++) |
---|
575 | { |
---|
576 | sI[size(sI)+1] = sM[i][1]; |
---|
577 | } |
---|
578 | matrix T = submat(sM,2..nrows(sM),1..ncols(sM)-size(I)); |
---|
579 | } |
---|
580 | |
---|
581 | setring R; |
---|
582 | option(set, opt); |
---|
583 | return(imap(newR,sI),imap(newR,T)); |
---|
584 | } |
---|
585 | example |
---|
586 | { "EXAMPLE:"; echo = 2; |
---|
587 | ring R = 0,(x,y,z),dp; |
---|
588 | poly f = x3+y7+z2+xyz; |
---|
589 | ideal i = jacob(f); |
---|
590 | matrix T; |
---|
591 | ideal sm = liftstd(i,T); |
---|
592 | sm; |
---|
593 | print(T); |
---|
594 | matrix(sm) - matrix(i)*T; |
---|
595 | |
---|
596 | |
---|
597 | ring S = 32003, x(1..5), lp; |
---|
598 | ideal I = cyclic(5); |
---|
599 | ideal sI; |
---|
600 | matrix T; |
---|
601 | sI,T = liftstd1(I); |
---|
602 | matrix(sI) - matrix(I)*T; |
---|
603 | } |
---|
604 | |
---|
605 | //////////////////////////////////////////////////////////////////////////////// |
---|
606 | |
---|
607 | proc modpStd(ideal I, int p, int variant, list #) |
---|
608 | "USAGE: modpStd(I,p,variant,#); I ideal, p integer, variant integer |
---|
609 | ASSUME: If size(#) > 0, then #[1] is an intvec describing the Hilbert series. |
---|
610 | RETURN: ideal - a standard basis of I mod p, integer - p |
---|
611 | NOTE: The procedure computes a standard basis of the ideal I modulo p and |
---|
612 | fetches the result to the basering. If size(#) > 0 the Hilbert driven |
---|
613 | standard basis computation std(.,#[1]) is used instead of groebner. |
---|
614 | The standard basis computation modulo p does also vary depending on the |
---|
615 | integer variant, namely |
---|
616 | @* - variant = 1: std(.,#[1]) resp. groebner, |
---|
617 | @* - variant = 2: groebner, |
---|
618 | @* - variant = 3: homog. - std(.,#[1]) resp. groebner - dehomog., |
---|
619 | @* - variant = 4: fglm. |
---|
620 | EXAMPLE: example modpStd; shows an example |
---|
621 | " |
---|
622 | { |
---|
623 | def R0 = basering; |
---|
624 | list rl = ringlist(R0); |
---|
625 | rl[1] = p; |
---|
626 | def @r = ring(rl); |
---|
627 | setring @r; |
---|
628 | ideal i = fetch(R0,I); |
---|
629 | |
---|
630 | option(redSB); |
---|
631 | |
---|
632 | if(variant == 1) |
---|
633 | { |
---|
634 | if(size(#) > 0) |
---|
635 | { |
---|
636 | i = std(i, #[1]); |
---|
637 | } |
---|
638 | else |
---|
639 | { |
---|
640 | i = groebner(i); |
---|
641 | } |
---|
642 | } |
---|
643 | |
---|
644 | if(variant == 2) |
---|
645 | { |
---|
646 | i = groebner(i); |
---|
647 | } |
---|
648 | |
---|
649 | if(variant == 3) |
---|
650 | { |
---|
651 | list rl = ringlist(@r); |
---|
652 | int nvar@r = nvars(@r); |
---|
653 | |
---|
654 | int k; |
---|
655 | intvec w; |
---|
656 | for(k = 1; k <= nvar@r; k++) |
---|
657 | { |
---|
658 | w[k] = deg(var(k)); |
---|
659 | } |
---|
660 | w[nvar@r + 1] = 1; |
---|
661 | |
---|
662 | rl[2][nvar@r + 1] = "homvar"; |
---|
663 | rl[3][2][2] = w; |
---|
664 | |
---|
665 | def HomR = ring(rl); |
---|
666 | setring HomR; |
---|
667 | ideal i = imap(@r, i); |
---|
668 | i = homog(i, homvar); |
---|
669 | |
---|
670 | if(size(#) > 0) |
---|
671 | { |
---|
672 | if(w == 1) |
---|
673 | { |
---|
674 | i = std(i, #[1]); |
---|
675 | } |
---|
676 | else |
---|
677 | { |
---|
678 | i = std(i, #[1], w); |
---|
679 | } |
---|
680 | } |
---|
681 | else |
---|
682 | { |
---|
683 | i = groebner(i); |
---|
684 | } |
---|
685 | |
---|
686 | i = subst(i, homvar, 1); |
---|
687 | i = simplify(i, 34); |
---|
688 | |
---|
689 | setring @r; |
---|
690 | i = imap(HomR, i); |
---|
691 | i = interred(i); |
---|
692 | kill HomR; |
---|
693 | } |
---|
694 | |
---|
695 | if(variant == 4) |
---|
696 | { |
---|
697 | def R1 = changeord("dp"); |
---|
698 | setring R1; |
---|
699 | ideal i = fetch(@r,i); |
---|
700 | i = std(i); |
---|
701 | setring @r; |
---|
702 | i = fglm(R1,i); |
---|
703 | } |
---|
704 | |
---|
705 | setring R0; |
---|
706 | return(list(fetch(@r,i),p)); |
---|
707 | } |
---|
708 | example |
---|
709 | { "EXAMPLE:"; echo = 2; |
---|
710 | ring r = 0, x(1..4), dp; |
---|
711 | ideal I = cyclic(4); |
---|
712 | int p = 181; |
---|
713 | list P = modpStd(I,p,5); |
---|
714 | P; |
---|
715 | |
---|
716 | int q = 32003; |
---|
717 | list Q = modpStd(I,q,2); |
---|
718 | Q; |
---|
719 | } |
---|
720 | |
---|
721 | ////////////////////////////// main procedures ///////////////////////////////// |
---|
722 | |
---|
723 | proc modStd(ideal I, list #) |
---|
724 | "USAGE: modStd(I); I ideal |
---|
725 | ASSUME: If size(#) > 0, then # contains either 1, 2 or 4 integers such that |
---|
726 | @* - #[1] is the number of available processors for the computation, |
---|
727 | @* - #[2] is an optional parameter for the exactness of the computation, |
---|
728 | if #[2] = 1, the procedure computes a standard basis for sure, |
---|
729 | @* - #[3] is the number of primes until the first lifting, |
---|
730 | @* - #[4] is the constant number of primes between two liftings until |
---|
731 | the computation stops. |
---|
732 | RETURN: a standard basis of I if no warning appears; |
---|
733 | NOTE: The procedure computes a standard basis of I (over the rational |
---|
734 | numbers) by using modular methods. |
---|
735 | By default the procedure computes a standard basis of I for sure, but |
---|
736 | if the optional parameter #[2] = 0, it computes a standard basis of I |
---|
737 | with high probability. |
---|
738 | The procedure distinguishes between different variants for the standard |
---|
739 | basis computation in positive characteristic depending on the ordering |
---|
740 | of the basering, the parameter #[2] and if the ideal I is homogeneous. |
---|
741 | @* - variant = 1, if I is homogeneous, |
---|
742 | @* - variant = 2, if I is not homogeneous, 1-block-ordering, |
---|
743 | @* - variant = 3, if I is not homogeneous, complicated ordering (lp or |
---|
744 | > 1 block), |
---|
745 | @* - variant = 4, if I is not homogeneous, ordering lp, dim(I) = 0. |
---|
746 | EXAMPLE: example modStd; shows an example |
---|
747 | " |
---|
748 | { |
---|
749 | int TT = timer; |
---|
750 | int RT = rtimer; |
---|
751 | |
---|
752 | def R0 = basering; |
---|
753 | list rl = ringlist(R0); |
---|
754 | if((npars(R0) > 0) || (rl[1] > 0)) |
---|
755 | { |
---|
756 | ERROR("Characteristic of basering should be zero, basering should |
---|
757 | have no parameters."); |
---|
758 | } |
---|
759 | |
---|
760 | int index = 1; |
---|
761 | int i,k,c; |
---|
762 | int j = 1; |
---|
763 | int pTest, sizeTest; |
---|
764 | int en = 2134567879; |
---|
765 | int an = 1000000000; |
---|
766 | bigint N; |
---|
767 | |
---|
768 | //-------------------- Initialize optional parameters ------------------------ |
---|
769 | if(size(#) > 0) |
---|
770 | { |
---|
771 | if(size(#) == 1) |
---|
772 | { |
---|
773 | int n1 = #[1]; |
---|
774 | int exactness = 1; |
---|
775 | if(n1 >= 10) |
---|
776 | { |
---|
777 | int n2 = n1 + 1; |
---|
778 | int n3 = n1; |
---|
779 | } |
---|
780 | else |
---|
781 | { |
---|
782 | int n2 = 10; |
---|
783 | int n3 = 10; |
---|
784 | } |
---|
785 | } |
---|
786 | if(size(#) == 2) |
---|
787 | { |
---|
788 | int n1 = #[1]; |
---|
789 | int exactness = #[2]; |
---|
790 | if(n1 >= 10) |
---|
791 | { |
---|
792 | int n2 = n1 + 1; |
---|
793 | int n3 = n1; |
---|
794 | } |
---|
795 | else |
---|
796 | { |
---|
797 | int n2 = 10; |
---|
798 | int n3 = 10; |
---|
799 | } |
---|
800 | } |
---|
801 | if(size(#) == 4) |
---|
802 | { |
---|
803 | int n1 = #[1]; |
---|
804 | int exactness = #[2]; |
---|
805 | if(n1 >= #[3]) |
---|
806 | { |
---|
807 | int n2 = n1 + 1; |
---|
808 | } |
---|
809 | else |
---|
810 | { |
---|
811 | int n2 = #[3]; |
---|
812 | } |
---|
813 | if(n1 >= #[4]) |
---|
814 | { |
---|
815 | int n3 = n1; |
---|
816 | } |
---|
817 | else |
---|
818 | { |
---|
819 | int n3 = #[4]; |
---|
820 | } |
---|
821 | } |
---|
822 | } |
---|
823 | else |
---|
824 | { |
---|
825 | int n1 = 1; |
---|
826 | int exactness = 1; |
---|
827 | int n2 = 10; |
---|
828 | int n3 = 10; |
---|
829 | } |
---|
830 | |
---|
831 | if(printlevel >= 10) |
---|
832 | { |
---|
833 | "n1 = "+string(n1)+", n2 = "+string(n2)+", n3 = "+string(n3) |
---|
834 | +", exactness = "+string(exactness); |
---|
835 | } |
---|
836 | |
---|
837 | //------------------------- Save current options ----------------------------- |
---|
838 | intvec opt = option(get); |
---|
839 | |
---|
840 | option(redSB); |
---|
841 | |
---|
842 | //-------------------- Initialize the list of primes ------------------------- |
---|
843 | int tt = timer; |
---|
844 | int rt = rtimer; |
---|
845 | intvec L = primeList(I,n2,n1); |
---|
846 | if(printlevel >= 10) |
---|
847 | { |
---|
848 | "CPU-time for primeList: "+string(timer-tt)+" seconds."; |
---|
849 | "Real-time for primeList: "+string(rtimer-rt)+" seconds."; |
---|
850 | } |
---|
851 | L[5] = prime(random(an,en)); |
---|
852 | |
---|
853 | //--------------------- Decide which variant to take ------------------------- |
---|
854 | int variant; |
---|
855 | int h = homog(I); |
---|
856 | |
---|
857 | tt = timer; |
---|
858 | rt = rtimer; |
---|
859 | |
---|
860 | if(!mixedTest()) |
---|
861 | { |
---|
862 | if(h) |
---|
863 | { |
---|
864 | variant = 1; |
---|
865 | if(printlevel >= 10) { "variant = 1"; } |
---|
866 | |
---|
867 | rl[1] = L[5]; |
---|
868 | def @r = ring(rl); |
---|
869 | setring @r; |
---|
870 | def @s = changeord("dp"); |
---|
871 | setring @s; |
---|
872 | ideal I = std(fetch(R0,I)); |
---|
873 | intvec hi = hilb(I,1); |
---|
874 | setring R0; |
---|
875 | kill @r,@s; |
---|
876 | } |
---|
877 | else |
---|
878 | { |
---|
879 | string ordstr_R0 = ordstr(R0); |
---|
880 | int neg = 1 - attrib(R0,"global"); |
---|
881 | |
---|
882 | if((find(ordstr_R0, "M") > 0) || (find(ordstr_R0, "a") > 0) || neg) |
---|
883 | { |
---|
884 | variant = 2; |
---|
885 | if(printlevel >= 10) { "variant = 2"; } |
---|
886 | } |
---|
887 | else |
---|
888 | { |
---|
889 | string order; |
---|
890 | if(system("nblocks") <= 2) |
---|
891 | { |
---|
892 | if(find(ordstr_R0, "M") + find(ordstr_R0, "lp") |
---|
893 | + find(ordstr_R0, "rp") <= 0) |
---|
894 | { |
---|
895 | order = "simple"; |
---|
896 | } |
---|
897 | } |
---|
898 | |
---|
899 | if((order == "simple") || (size(rl) > 4)) |
---|
900 | { |
---|
901 | variant = 2; |
---|
902 | if(printlevel >= 10) { "variant = 2"; } |
---|
903 | } |
---|
904 | else |
---|
905 | { |
---|
906 | rl[1] = L[5]; |
---|
907 | def @r = ring(rl); |
---|
908 | setring @r; |
---|
909 | |
---|
910 | def @s = changeord("dp"); |
---|
911 | setring @s; |
---|
912 | ideal I = std(fetch(R0,I)); |
---|
913 | if(dim(I) == 0) |
---|
914 | { |
---|
915 | variant = 4; |
---|
916 | if(printlevel >= 10) { "variant = 4"; } |
---|
917 | } |
---|
918 | else |
---|
919 | { |
---|
920 | variant = 3; |
---|
921 | if(printlevel >= 10) { "variant = 3"; } |
---|
922 | |
---|
923 | int nvar@r = nvars(@r); |
---|
924 | intvec w; |
---|
925 | for(i = 1; i <= nvar@r; i++) |
---|
926 | { |
---|
927 | w[i] = deg(var(i)); |
---|
928 | } |
---|
929 | w[nvar@r + 1] = 1; |
---|
930 | |
---|
931 | list hiRi = hilbRing(fetch(R0,I),w); |
---|
932 | intvec W = hiRi[2]; |
---|
933 | @s = hiRi[1]; |
---|
934 | setring @s; |
---|
935 | |
---|
936 | Id(1) = std(Id(1)); |
---|
937 | intvec hi = hilb(Id(1), 1, W); |
---|
938 | } |
---|
939 | |
---|
940 | setring R0; |
---|
941 | kill @r,@s; |
---|
942 | } |
---|
943 | } |
---|
944 | } |
---|
945 | } |
---|
946 | else |
---|
947 | { |
---|
948 | if(exactness == 1) { return(groebner(I)); } |
---|
949 | if(h) |
---|
950 | { |
---|
951 | variant = 1; |
---|
952 | if(printlevel >= 10) { "variant = 1"; } |
---|
953 | rl[1] = L[5]; |
---|
954 | def @r = ring(rl); |
---|
955 | setring @r; |
---|
956 | def @s = changeord("dp"); |
---|
957 | setring @s; |
---|
958 | ideal I = std(fetch(R0,I)); |
---|
959 | intvec hi = hilb(I,1); |
---|
960 | setring R0; |
---|
961 | kill @r,@s; |
---|
962 | } |
---|
963 | else |
---|
964 | { |
---|
965 | string ordstr_R0 = ordstr(R0); |
---|
966 | int neg = 1 - attrib(R0,"global"); |
---|
967 | |
---|
968 | if((find(ordstr_R0, "M") > 0) || (find(ordstr_R0, "a") > 0) || neg) |
---|
969 | { |
---|
970 | variant = 2; |
---|
971 | if(printlevel >= 10) { "variant = 2"; } |
---|
972 | } |
---|
973 | else |
---|
974 | { |
---|
975 | string order; |
---|
976 | if(system("nblocks") <= 2) |
---|
977 | { |
---|
978 | if(find(ordstr_R0, "M") + find(ordstr_R0, "lp") |
---|
979 | + find(ordstr_R0, "rp") <= 0) |
---|
980 | { |
---|
981 | order = "simple"; |
---|
982 | } |
---|
983 | } |
---|
984 | |
---|
985 | if((order == "simple") || (size(rl) > 4)) |
---|
986 | { |
---|
987 | variant = 2; |
---|
988 | if(printlevel >= 10) { "variant = 2"; } |
---|
989 | } |
---|
990 | else |
---|
991 | { |
---|
992 | variant = 3; |
---|
993 | if(printlevel >= 10) { "variant = 3"; } |
---|
994 | |
---|
995 | rl[1] = L[5]; |
---|
996 | def @r = ring(rl); |
---|
997 | setring @r; |
---|
998 | int nvar@r = nvars(@r); |
---|
999 | intvec w; |
---|
1000 | for(i = 1; i <= nvar@r; i++) |
---|
1001 | { |
---|
1002 | w[i] = deg(var(i)); |
---|
1003 | } |
---|
1004 | w[nvar@r + 1] = 1; |
---|
1005 | |
---|
1006 | list hiRi = hilbRing(fetch(R0,I),w); |
---|
1007 | intvec W = hiRi[2]; |
---|
1008 | def @s = hiRi[1]; |
---|
1009 | setring @s; |
---|
1010 | |
---|
1011 | Id(1) = std(Id(1)); |
---|
1012 | intvec hi = hilb(Id(1), 1, W); |
---|
1013 | |
---|
1014 | setring R0; |
---|
1015 | kill @r,@s; |
---|
1016 | } |
---|
1017 | } |
---|
1018 | } |
---|
1019 | } |
---|
1020 | |
---|
1021 | list P,T1,T2,T3,LL; |
---|
1022 | |
---|
1023 | ideal J,K,H; |
---|
1024 | |
---|
1025 | //----- If there is more than one processor available, we parallelize the ---- |
---|
1026 | //----- main standard basis computations in positive characteristic ---- |
---|
1027 | |
---|
1028 | if(n1 > 1) |
---|
1029 | { |
---|
1030 | ideal I_for_fork = I; |
---|
1031 | export(I_for_fork); // I available for each link |
---|
1032 | |
---|
1033 | //----- Create n1 links l(1),...,l(n1), open all of them and compute --------- |
---|
1034 | //----- standard basis for the primes L[2],...,L[n1 + 1]. --------- |
---|
1035 | |
---|
1036 | for(i = 1; i <= n1; i++) |
---|
1037 | { |
---|
1038 | //link l(i) = "MPtcp:fork"; |
---|
1039 | link l(i) = "ssi:fork"; |
---|
1040 | open(l(i)); |
---|
1041 | if((variant == 1) || (variant == 3)) |
---|
1042 | { |
---|
1043 | write(l(i), quote(modpStd(I_for_fork, eval(L[i + 1]), |
---|
1044 | eval(variant), eval(hi)))); |
---|
1045 | } |
---|
1046 | if((variant == 2) || (variant == 4)) |
---|
1047 | { |
---|
1048 | write(l(i), quote(modpStd(I_for_fork, eval(L[i + 1]), |
---|
1049 | eval(variant)))); |
---|
1050 | } |
---|
1051 | } |
---|
1052 | |
---|
1053 | int t = timer; |
---|
1054 | if((variant == 1) || (variant == 3)) |
---|
1055 | { |
---|
1056 | P = modpStd(I_for_fork, L[1], variant, hi); |
---|
1057 | } |
---|
1058 | if((variant == 2) || (variant == 4)) |
---|
1059 | { |
---|
1060 | P = modpStd(I_for_fork, L[1], variant); |
---|
1061 | } |
---|
1062 | t = timer - t; |
---|
1063 | if(t > 60) { t = 60; } |
---|
1064 | int i_sleep = system("sh", "sleep "+string(t)); |
---|
1065 | T1[1] = P[1]; |
---|
1066 | T2[1] = bigint(P[2]); |
---|
1067 | index++; |
---|
1068 | |
---|
1069 | j = j + n1 + 1; |
---|
1070 | } |
---|
1071 | |
---|
1072 | //-------------- Main standard basis computations in positive ---------------- |
---|
1073 | //---------------------- characteristic start here --------------------------- |
---|
1074 | |
---|
1075 | list arguments_farey, results_farey; |
---|
1076 | |
---|
1077 | while(1) |
---|
1078 | { |
---|
1079 | tt = timer; rt = rtimer; |
---|
1080 | |
---|
1081 | if(printlevel >= 10) { "size(L) = "+string(size(L)); } |
---|
1082 | |
---|
1083 | if(n1 > 1) |
---|
1084 | { |
---|
1085 | while(j <= size(L) + 1) |
---|
1086 | { |
---|
1087 | for(i = 1; i <= n1; i++) |
---|
1088 | { |
---|
1089 | //--- ask if link l(i) is ready otherwise sleep for t seconds --- |
---|
1090 | if(status(l(i), "read", "ready")) |
---|
1091 | { |
---|
1092 | //--- read the result from l(i) --- |
---|
1093 | P = read(l(i)); |
---|
1094 | T1[index] = P[1]; |
---|
1095 | T2[index] = bigint(P[2]); |
---|
1096 | index++; |
---|
1097 | |
---|
1098 | if(j <= size(L)) |
---|
1099 | { |
---|
1100 | if((variant == 1) || (variant == 3)) |
---|
1101 | { |
---|
1102 | write(l(i), quote(modpStd(I_for_fork, eval(L[j]), |
---|
1103 | eval(variant), eval(hi)))); |
---|
1104 | j++; |
---|
1105 | } |
---|
1106 | if((variant == 2) || (variant == 4)) |
---|
1107 | { |
---|
1108 | write(l(i), quote(modpStd(I_for_fork, |
---|
1109 | eval(L[j]), eval(variant)))); |
---|
1110 | j++; |
---|
1111 | } |
---|
1112 | } |
---|
1113 | else |
---|
1114 | { |
---|
1115 | k++; |
---|
1116 | close(l(i)); |
---|
1117 | } |
---|
1118 | } |
---|
1119 | } |
---|
1120 | //--- k describes the number of closed links --- |
---|
1121 | if(k == n1) |
---|
1122 | { |
---|
1123 | j++; |
---|
1124 | } |
---|
1125 | i_sleep = system("sh", "sleep "+string(t)); |
---|
1126 | } |
---|
1127 | } |
---|
1128 | else |
---|
1129 | { |
---|
1130 | while(j <= size(L)) |
---|
1131 | { |
---|
1132 | if((variant == 1) || (variant == 3)) |
---|
1133 | { |
---|
1134 | P = modpStd(I, L[j], variant, hi); |
---|
1135 | } |
---|
1136 | if((variant == 2) || (variant == 4)) |
---|
1137 | { |
---|
1138 | P = modpStd(I, L[j], variant); |
---|
1139 | } |
---|
1140 | |
---|
1141 | T1[index] = P[1]; |
---|
1142 | T2[index] = bigint(P[2]); |
---|
1143 | index++; |
---|
1144 | j++; |
---|
1145 | } |
---|
1146 | } |
---|
1147 | |
---|
1148 | if(printlevel >= 10) |
---|
1149 | { |
---|
1150 | "CPU-time for computing list is "+string(timer - tt)+" seconds."; |
---|
1151 | "Real-time for computing list is "+string(rtimer - rt)+" seconds."; |
---|
1152 | } |
---|
1153 | |
---|
1154 | //------------------------ Delete unlucky primes ----------------------------- |
---|
1155 | //------------- unlucky if and only if the leading ideal is wrong ------------ |
---|
1156 | |
---|
1157 | LL = deleteUnluckyPrimes(T1,T2,h); |
---|
1158 | T1 = LL[1]; |
---|
1159 | T2 = LL[2]; |
---|
1160 | |
---|
1161 | //------------------- Now all leading ideals are the same -------------------- |
---|
1162 | //------------------- Lift results to basering via farey --------------------- |
---|
1163 | |
---|
1164 | tt = timer; rt = rtimer; |
---|
1165 | N = T2[1]; |
---|
1166 | for(i = 2; i <= size(T2); i++) { N = N*T2[i]; } |
---|
1167 | H = chinrem(T1,T2); |
---|
1168 | if(n1 == 1) |
---|
1169 | { |
---|
1170 | J = farey(H,N); |
---|
1171 | } |
---|
1172 | else |
---|
1173 | { |
---|
1174 | for(i = ncols(H); i > 0; i--) |
---|
1175 | { |
---|
1176 | arguments_farey[i] = list(ideal(H[i]), N); |
---|
1177 | } |
---|
1178 | results_farey = parallelWaitAll("farey", arguments_farey, |
---|
1179 | list(list(list(n1)))); |
---|
1180 | for(i = ncols(H); i > 0; i--) |
---|
1181 | { |
---|
1182 | J[i] = results_farey[i][1]; |
---|
1183 | } |
---|
1184 | } |
---|
1185 | if(printlevel >= 10) |
---|
1186 | { |
---|
1187 | "CPU-time for lifting-process is "+string(timer - tt)+" seconds."; |
---|
1188 | "Real-time for lifting-process is "+string(rtimer - rt)+" seconds."; |
---|
1189 | } |
---|
1190 | |
---|
1191 | //---------------- Test if we already have a standard basis of I -------------- |
---|
1192 | |
---|
1193 | tt = timer; rt = rtimer; |
---|
1194 | if((variant == 1) || (variant == 3)) |
---|
1195 | { |
---|
1196 | pTest = pTestSB(I,J,L,variant,hi); |
---|
1197 | } |
---|
1198 | if((variant == 2) || (variant == 4)) |
---|
1199 | { |
---|
1200 | pTest = pTestSB(I,J,L,variant); |
---|
1201 | } |
---|
1202 | |
---|
1203 | if(printlevel >= 10) |
---|
1204 | { |
---|
1205 | "CPU-time for pTest is "+string(timer - tt)+" seconds."; |
---|
1206 | "Real-time for pTest is "+string(rtimer - rt)+" seconds."; |
---|
1207 | } |
---|
1208 | |
---|
1209 | if(pTest) |
---|
1210 | { |
---|
1211 | if(printlevel >= 10) |
---|
1212 | { |
---|
1213 | "CPU-time for computation without final tests is " |
---|
1214 | +string(timer - TT)+" seconds."; |
---|
1215 | "Real-time for computation without final tests is " |
---|
1216 | +string(rtimer - RT)+" seconds."; |
---|
1217 | } |
---|
1218 | |
---|
1219 | attrib(J,"isSB",1); |
---|
1220 | |
---|
1221 | if(exactness == 0) |
---|
1222 | { |
---|
1223 | option(set, opt); |
---|
1224 | if(n1 > 1) { kill I_for_fork; } |
---|
1225 | return(J); |
---|
1226 | } |
---|
1227 | |
---|
1228 | if(exactness == 1) |
---|
1229 | { |
---|
1230 | tt = timer; rt = rtimer; |
---|
1231 | sizeTest = 1 - isIncluded(I,J,n1); |
---|
1232 | |
---|
1233 | if(printlevel >= 10) |
---|
1234 | { |
---|
1235 | "CPU-time for checking if I subset <G> is " |
---|
1236 | +string(timer - tt)+" seconds."; |
---|
1237 | "Real-time for checking if I subset <G> is " |
---|
1238 | +string(rtimer - rt)+" seconds."; |
---|
1239 | } |
---|
1240 | |
---|
1241 | if(sizeTest == 0) |
---|
1242 | { |
---|
1243 | tt = timer; rt = rtimer; |
---|
1244 | K = std(J); |
---|
1245 | |
---|
1246 | if(printlevel >= 10) |
---|
1247 | { |
---|
1248 | "CPU-time for last std-computation is " |
---|
1249 | +string(timer - tt)+" seconds."; |
---|
1250 | "Real-time for last std-computation is " |
---|
1251 | +string(rtimer - rt)+" seconds."; |
---|
1252 | } |
---|
1253 | |
---|
1254 | if(size(reduce(K,J)) == 0) |
---|
1255 | { |
---|
1256 | option(set, opt); |
---|
1257 | if(n1 > 1) { kill I_for_fork; } |
---|
1258 | return(J); |
---|
1259 | } |
---|
1260 | } |
---|
1261 | } |
---|
1262 | } |
---|
1263 | |
---|
1264 | //-------------- We do not already have a standard basis of I ---------------- |
---|
1265 | //----------- Therefore do the main computation for more primes -------------- |
---|
1266 | |
---|
1267 | T1 = H; |
---|
1268 | T2 = N; |
---|
1269 | index = 2; |
---|
1270 | |
---|
1271 | j = size(L) + 1; |
---|
1272 | tt = timer; rt = rtimer; |
---|
1273 | L = primeList(I,n3,L,n1); |
---|
1274 | if(printlevel >= 10) |
---|
1275 | { |
---|
1276 | "CPU-time for primeList: "+string(timer-tt)+" seconds."; |
---|
1277 | "Real-time for primeList: "+string(rtimer-rt)+" seconds."; |
---|
1278 | } |
---|
1279 | |
---|
1280 | if(n1 > 1) |
---|
1281 | { |
---|
1282 | for(i = 1; i <= n1; i++) |
---|
1283 | { |
---|
1284 | open(l(i)); |
---|
1285 | if((variant == 1) || (variant == 3)) |
---|
1286 | { |
---|
1287 | write(l(i), quote(modpStd(I_for_fork, eval(L[j+i-1]), |
---|
1288 | eval(variant), eval(hi)))); |
---|
1289 | } |
---|
1290 | if((variant == 2) || (variant == 4)) |
---|
1291 | { |
---|
1292 | write(l(i), quote(modpStd(I_for_fork, eval(L[j+i-1]), |
---|
1293 | eval(variant)))); |
---|
1294 | } |
---|
1295 | } |
---|
1296 | j = j + n1; |
---|
1297 | k = 0; |
---|
1298 | } |
---|
1299 | } |
---|
1300 | } |
---|
1301 | example |
---|
1302 | { "EXAMPLE:"; echo = 2; |
---|
1303 | ring R1 = 0,(x,y,z,t),dp; |
---|
1304 | ideal I = 3x3+x2+1, 11y5+y3+2, 5z4+z2+4; |
---|
1305 | ideal J = modStd(I); |
---|
1306 | J; |
---|
1307 | I = homog(I,t); |
---|
1308 | J = modStd(I); |
---|
1309 | J; |
---|
1310 | |
---|
1311 | ring R2 = 0,(x,y,z),ds; |
---|
1312 | ideal I = jacob(x5+y6+z7+xyz); |
---|
1313 | ideal J1 = modStd(I,1,0); |
---|
1314 | J1; |
---|
1315 | |
---|
1316 | ring R3 = 0,x(1..4),lp; |
---|
1317 | ideal I = cyclic(4); |
---|
1318 | ideal J1 = modStd(I,1); |
---|
1319 | ideal J2 = modStd(I,1,0); |
---|
1320 | size(reduce(J1,J2)); |
---|
1321 | size(reduce(J2,J1)); |
---|
1322 | } |
---|
1323 | |
---|
1324 | //////////////////////////////////////////////////////////////////////////////// |
---|
1325 | |
---|
1326 | proc modS(ideal I, list L, list #) |
---|
1327 | "USAGE: modS(I,L); I ideal, L intvec of primes |
---|
1328 | if size(#)>0 std is used instead of groebner |
---|
1329 | RETURN: an ideal which is with high probability a standard basis |
---|
1330 | NOTE: This procedure is designed for fast experiments. |
---|
1331 | It is not tested whether the result is a standard basis. |
---|
1332 | It is not tested whether the result generates I. |
---|
1333 | EXAMPLE: example modS; shows an example |
---|
1334 | " |
---|
1335 | { |
---|
1336 | int j; |
---|
1337 | bigint N = 1; |
---|
1338 | def R0 = basering; |
---|
1339 | ideal J; |
---|
1340 | list T; |
---|
1341 | list rl = ringlist(R0); |
---|
1342 | if((npars(R0)>0) || (rl[1]>0)) |
---|
1343 | { |
---|
1344 | ERROR("Characteristic of basering should be zero."); |
---|
1345 | } |
---|
1346 | for(j = 1; j <= size(L); j++) |
---|
1347 | { |
---|
1348 | N = N*L[j]; |
---|
1349 | rl[1] = L[j]; |
---|
1350 | def @r = ring(rl); |
---|
1351 | setring @r; |
---|
1352 | ideal I = fetch(R0,I); |
---|
1353 | if(size(#) > 0) |
---|
1354 | { |
---|
1355 | I = std(I); |
---|
1356 | } |
---|
1357 | else |
---|
1358 | { |
---|
1359 | I = groebner(I); |
---|
1360 | } |
---|
1361 | setring R0; |
---|
1362 | T[j] = fetch(@r,I); |
---|
1363 | kill @r; |
---|
1364 | } |
---|
1365 | L = deleteUnluckyPrimes(T,L,homog(I)); |
---|
1366 | // unlucky if and only if the leading ideal is wrong |
---|
1367 | J = farey(chinrem(L[1],L[2]),N); |
---|
1368 | attrib(J,"isSB",1); |
---|
1369 | return(J); |
---|
1370 | } |
---|
1371 | example |
---|
1372 | { "EXAMPLE:"; echo = 2; |
---|
1373 | list L = 3,5,11,13,181,32003; |
---|
1374 | ring r = 0,(x,y,z,t),dp; |
---|
1375 | ideal I = 3x3+x2+1,11y5+y3+2,5z4+z2+4; |
---|
1376 | I = homog(I,t); |
---|
1377 | ideal J = modS(I,L); |
---|
1378 | J; |
---|
1379 | } |
---|
1380 | |
---|
1381 | //////////////////////////////////////////////////////////////////////////////// |
---|
1382 | |
---|
1383 | proc modHenselStd(ideal I, list #) |
---|
1384 | "USAGE: modHenselStd(I); |
---|
1385 | RETURN: a standard basis of I; |
---|
1386 | NOTE: The procedure computes a standard basis of I (over the rational |
---|
1387 | numbers) by using modular computations and Hensellifting. |
---|
1388 | For further experiments see procedure modS. |
---|
1389 | EXAMPLE: example modHenselStd; shows an example |
---|
1390 | " |
---|
1391 | { |
---|
1392 | int i,j; |
---|
1393 | |
---|
1394 | bigint p = 2134567879; |
---|
1395 | if(size(#)!=0) { p=#[1]; } |
---|
1396 | while(!primeTest(I,p)) |
---|
1397 | { |
---|
1398 | p = prime(random(2000000000,2134567879)); |
---|
1399 | } |
---|
1400 | |
---|
1401 | def R = basering; |
---|
1402 | module F,PrevG,PrevZ,Z2; |
---|
1403 | ideal testG,testG1,G1,G2,G3,Gp; |
---|
1404 | list L = p; |
---|
1405 | list rl = ringlist(R); |
---|
1406 | rl[1] = int(p); |
---|
1407 | |
---|
1408 | def S = ring(rl); |
---|
1409 | setring S; |
---|
1410 | intvec opt = option(get); |
---|
1411 | option(redSB); |
---|
1412 | module Z,M,Z2; |
---|
1413 | ideal I = imap(R,I); |
---|
1414 | ideal Gp,G1,G2,G3; |
---|
1415 | Gp,Z = liftstd1(I); |
---|
1416 | attrib(Gp,"isSB",1); |
---|
1417 | module ZZ = syz(I); |
---|
1418 | attrib(ZZ,"isSB",1); |
---|
1419 | Z = reduce(Z,ZZ); |
---|
1420 | |
---|
1421 | setring R; |
---|
1422 | Gp = imap(S,Gp); |
---|
1423 | PrevZ = imap(S,Z); |
---|
1424 | PrevG = module(Gp); |
---|
1425 | F = module(I); |
---|
1426 | testG = farey(Gp,p); |
---|
1427 | attrib(testG,"isSB",1); |
---|
1428 | while(1) |
---|
1429 | { |
---|
1430 | i++; |
---|
1431 | G1 = ideal(1/(p^i) * sum(F*PrevZ,(-1)*PrevG)); |
---|
1432 | setring S; |
---|
1433 | G1 = imap(R,G1); |
---|
1434 | G2 = reduce(G1,Gp); |
---|
1435 | G3 = sum(G1,(-1)*G2); |
---|
1436 | M = lift(Gp,G3); |
---|
1437 | Z2 = (-1)*Z*M; |
---|
1438 | |
---|
1439 | setring R; |
---|
1440 | G2 = imap(S,G2); |
---|
1441 | Z2 = imap(S,Z2); |
---|
1442 | PrevG = sum(PrevG, module(p^i*G2)); |
---|
1443 | PrevZ = sum(PrevZ, multiply(poly(p^i),Z2)); |
---|
1444 | testG1 = farey(ideal(PrevG),p^(i+1)); |
---|
1445 | attrib(testG1,"isSB",1); |
---|
1446 | if(size(reduce(testG1,testG)) == 0) |
---|
1447 | { |
---|
1448 | if(size(reduce(I,testG1)) == 0) // I is in testG1 |
---|
1449 | { |
---|
1450 | if(pTestSB(I,testG1,L,2)) |
---|
1451 | { |
---|
1452 | G3 = std(testG1); // testG1 is SB |
---|
1453 | if(size(reduce(G3,testG1)) == 0) |
---|
1454 | { |
---|
1455 | option(set, opt); |
---|
1456 | return(G3); |
---|
1457 | } |
---|
1458 | } |
---|
1459 | } |
---|
1460 | } |
---|
1461 | testG = testG1; |
---|
1462 | attrib(testG,"isSB",1); |
---|
1463 | } |
---|
1464 | } |
---|
1465 | example |
---|
1466 | { "EXAMPLE:"; echo = 2; |
---|
1467 | ring r = 0,(x,y,z),dp; |
---|
1468 | ideal I = 3x3+x2+1,11y5+y3+2,5z4+z2+4; |
---|
1469 | ideal J = modHenselStd(I); |
---|
1470 | J; |
---|
1471 | } |
---|
1472 | |
---|
1473 | //////////////////////////////////////////////////////////////////////////////// |
---|
1474 | |
---|
1475 | static proc sum(list #) |
---|
1476 | { |
---|
1477 | if(typeof(#[1])=="ideal") |
---|
1478 | { |
---|
1479 | ideal M; |
---|
1480 | } |
---|
1481 | else |
---|
1482 | { |
---|
1483 | module M; |
---|
1484 | } |
---|
1485 | |
---|
1486 | int i; |
---|
1487 | for(i = 1; i <= ncols(#[1]); i++) { M[i] = #[1][i] + #[2][i]; } |
---|
1488 | return(M); |
---|
1489 | } |
---|
1490 | |
---|
1491 | //////////////////////////////////////////////////////////////////////////////// |
---|
1492 | |
---|
1493 | static proc multiply(poly p, list #) |
---|
1494 | { |
---|
1495 | if(typeof(#[1])=="ideal") |
---|
1496 | { |
---|
1497 | ideal M; |
---|
1498 | } |
---|
1499 | else |
---|
1500 | { |
---|
1501 | module M; |
---|
1502 | } |
---|
1503 | |
---|
1504 | int i; |
---|
1505 | for(i = 1; i <= ncols(#[1]); i++) { M[i] = p * #[1][i]; } |
---|
1506 | return(M); |
---|
1507 | } |
---|
1508 | |
---|
1509 | |
---|
1510 | ////////////////////////////// further examples //////////////////////////////// |
---|
1511 | |
---|
1512 | /* |
---|
1513 | ring r = 0, (x,y,z), lp; |
---|
1514 | poly s1 = 5x3y2z+3y3x2z+7xy2z2; |
---|
1515 | poly s2 = 3xy2z2+x5+11y2z2; |
---|
1516 | poly s3 = 4xyz+7x3+12y3+1; |
---|
1517 | poly s4 = 3x3-4y3+yz2; |
---|
1518 | ideal i = s1, s2, s3, s4; |
---|
1519 | |
---|
1520 | ring r = 0, (x,y,z), lp; |
---|
1521 | poly s1 = 2xy4z2+x3y2z-x2y3z+2xyz2+7y3+7; |
---|
1522 | poly s2 = 2x2y4z+x2yz2-xy2z2+2x2yz-12x+12y; |
---|
1523 | poly s3 = 2y5z+x2y2z-xy3z-xy3+y4+2y2z; |
---|
1524 | poly s4 = 3xy4z3+x2y2z-xy3z+4y3z2+3xyz3+4z2-x+y; |
---|
1525 | ideal i = s1, s2, s3, s4; |
---|
1526 | |
---|
1527 | ring r = 0, (x,y,z), lp; |
---|
1528 | poly s1 = 8x2y2 + 5xy3 + 3x3z + x2yz; |
---|
1529 | poly s2 = x5 + 2y3z2 + 13y2z3 + 5yz4; |
---|
1530 | poly s3 = 8x3 + 12y3 + xz2 + 3; |
---|
1531 | poly s4 = 7x2y4 + 18xy3z2 + y3z3; |
---|
1532 | ideal i = s1, s2, s3, s4; |
---|
1533 | |
---|
1534 | int n = 6; |
---|
1535 | ring r = 0,(x(1..n)),lp; |
---|
1536 | ideal i = cyclic(n); |
---|
1537 | ring s = 0, (x(1..n),t), lp; |
---|
1538 | ideal i = imap(r,i); |
---|
1539 | i = homog(i,t); |
---|
1540 | |
---|
1541 | ring r = 0, (x(1..4),s), (dp(4),dp); |
---|
1542 | poly s1 = 1 + s^2*x(1)*x(3) + s^8*x(2)*x(3) + s^19*x(1)*x(2)*x(4); |
---|
1543 | poly s2 = x(1) + s^8 *x(1)* x(2)* x(3) + s^19* x(2)* x(4); |
---|
1544 | poly s3 = x(2) + s^10*x(3)*x(4) + s^11*x(1)*x(4); |
---|
1545 | 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); |
---|
1546 | poly s5 = x(4) + s^31* x(1)* x(2)* x(3)* x(4); |
---|
1547 | ideal i = s1, s2, s3, s4, s5; |
---|
1548 | |
---|
1549 | ring r = 0, (x,y,z), ds; |
---|
1550 | int a = 16; |
---|
1551 | int b = 15; |
---|
1552 | int c = 4; |
---|
1553 | int t = 1; |
---|
1554 | poly f = x^a+y^b+z^(3*c)+x^(c+2)*y^(c-1)+x^(c-1)*y^(c-1)*z3 |
---|
1555 | +x^(c-2)*y^c*(y2+t*x)^2; |
---|
1556 | ideal i = jacob(f); |
---|
1557 | |
---|
1558 | ring r = 0, (x,y,z), ds; |
---|
1559 | int a = 25; |
---|
1560 | int b = 25; |
---|
1561 | int c = 5; |
---|
1562 | int t = 1; |
---|
1563 | poly f = x^a+y^b+z^(3*c)+x^(c+2)*y^(c-1)+x^(c-1)*y^(c-1)*z3 |
---|
1564 | +x^(c-2)*y^c*(y2+t*x)^2; |
---|
1565 | ideal i = jacob(f),f; |
---|
1566 | |
---|
1567 | ring r = 0, (x,y,z), ds; |
---|
1568 | int a = 10; |
---|
1569 | poly f = xyz*(x+y+z)^2 +(x+y+z)^3 +x^a+y^a+z^a; |
---|
1570 | ideal i = jacob(f); |
---|
1571 | |
---|
1572 | ring r = 0, (x,y,z), ds; |
---|
1573 | int a = 6; |
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1574 | int b = 8; |
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1575 | int c = 10; |
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1576 | int alpha = 5; |
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1577 | int beta = 5; |
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1578 | int t = 1; |
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1579 | poly f = x^a+y^b+z^c+x^alpha*y^(beta-5)+x^(alpha-2)*y^(beta-3) |
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1580 | +x^(alpha-3)*y^(beta-4)*z^2+x^(alpha-4)*y^(beta-4)*(y^2+t*x)^2; |
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1581 | ideal i = jacob(f); |
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1582 | */ |
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1583 | |
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