1 | ////////////////////////////////////////////////////////////////////////////// |
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2 | version="version recover.lib 4.0.2.0 30.03.2015 "; // $Id$ |
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3 | category="Algebraic Geometry"; |
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4 | info=" |
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5 | LIBRARY: recover.lib Hybrid numerical/symbolical algorithms for algebraic geometry |
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6 | AUTHOR: Adrian Koch (kocha at rhrk.uni-kl.de) |
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7 | |
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8 | OVERVIEW: In this library you'll find implementations of some of the algorithms presented |
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9 | in the paper listed below: Bertini is used to compute a witness set of a given |
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10 | ideal I. Then a lattice basis reduction algorithm is used to recover exact |
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11 | results from the inexact numerical data. More precisely, we obtain elements |
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12 | of prime components of I, the radical of I, or an elimination ideal of I. |
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13 | |
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14 | NOTE that Bertini may create quite a lot of files in the current directory |
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15 | (or overwrite files which have the same names as the files it wants to create). |
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16 | It also prints information to the screen. |
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17 | |
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18 | The usefulness of the results of the exactness recovery algorithms heavily |
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19 | depends on the quality of the witness set and the quality of the lattice basis |
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20 | reduction algorithm. |
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21 | The procedures requiring a witness set as part of their input use a simple, |
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22 | unsofisticated version of the LLL algorithm. |
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23 | |
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24 | REFERENCES: |
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25 | Daniel Bates, Jonathan Hauenstein, Timothy McCoy, Chris Peterson, and Andrew Sommese; |
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26 | Recovering exact results from inexact numerical data in algebraic geometry; |
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27 | Published in Experimental Mathematics 22(1) on pages 38-50 in 2013 |
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28 | |
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29 | KEYWORDS: numerical algebraic geometry; hybrid algorithms; exactness recovery |
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30 | |
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31 | PROCEDURES: |
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32 | substAll(v,p); poly: ring variables in v substituted by elements of p |
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33 | veronese(d,p); ideal: image of p under the degree d Veronese embedding |
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34 | getRelations(p,..); list of ideals: homogeneous polynomial relations between |
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35 | components of p |
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36 | getRelationsRadical(p,..); modified version of getRelations |
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37 | gaussRowWithoutPerm(M); matrix: a row-reduced form of M |
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38 | gaussColWithoutPerm(M); matrix: a column-reduced form of M |
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39 | getWitnessSet(); extracts the witness set from the file \"main_data\" produced |
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40 | by Bertini |
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41 | writeBertiniInput(J); writes the input-file for bertini with the polynomials in J |
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42 | as functions |
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43 | num_prime_decom(I,..); is supposed to compute a prime decomposition of the radical of I |
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44 | num_prime_decom1(P,..); is supposed to compute a prime decomposition for the ideal |
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45 | represented by the witness point set P |
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46 | num_radical_via_decom(I,..); |
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47 | compute elements of the radical of I by using num_prime_decom |
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48 | num_radical_via_randlincom(I,..); |
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49 | computes elements of the radical of I by using a different method |
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50 | num_radical1(P,..); computes elements of the radical via num_prime_decom1 |
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51 | num_radical2(P,..); computes elements of the radical using a different method |
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52 | num_elim(I,f,..); computes elements of the elimination ideal of I w.r.t. the |
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53 | variables specified by f |
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54 | num_elim1(P,..,v); computes elements of the elimination ideal of the ideal |
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55 | represented by the witness point set P (w.r.t. the variables |
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56 | specified in v) |
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57 | realLLL(M); simple version of the LLL-algorithm;works only over real numbers |
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58 | "; |
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59 | |
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60 | |
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61 | LIB "matrix.lib"; |
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62 | LIB "linalg.lib"; |
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63 | LIB "inout.lib"; |
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64 | LIB "atkins.lib"; |
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65 | |
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66 | ///////////////////////////////////////////////////////////////////////////////////////// |
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67 | /////////////////////// static procs for rounding /////////////////////////////// |
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68 | ///////////////////////////////////////////////////////////////////////////////////////// |
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69 | |
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70 | static proc getposi(string s) |
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71 | {//returns the position of the . in a complex number, or 0 if there is no . in s |
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72 | int i; |
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73 | for(i=1; i<=size(s); i++) |
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74 | { |
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75 | if(s[i] == "."){return(i);} |
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76 | } |
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77 | return(0); |
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78 | } |
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79 | |
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80 | static proc string2digit(string ti) |
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81 | { |
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82 | intvec v=0,1,2,3,4,5,6,7,8,9; |
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83 | int i; |
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84 | for(i=1; i<=size(v); i++) |
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85 | { |
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86 | if( ti == string(v[i]) ) |
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87 | { |
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88 | return(poly(v[i])); |
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89 | } |
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90 | } |
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91 | } |
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92 | |
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93 | static proc string2poly(string t) |
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94 | { |
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95 | poly r=string2digit(t[1]); |
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96 | int i; |
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97 | for(i=2; i<=size(t); i++) |
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98 | { |
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99 | r=r*10+string2digit(t[i]); |
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100 | } |
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101 | return(r); |
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102 | } |
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103 | |
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104 | static proc roundstringpoly(string s, int posi) |
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105 | {//returns the |
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106 | string t; |
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107 | //first check, whether s is negative or not |
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108 | int e=0; |
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109 | if(s[1]=="-") |
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110 | { |
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111 | e=1; |
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112 | t=s[2..(posi-1)];//start at the second symbol (to drop the minus) |
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113 | } |
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114 | else |
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115 | { |
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116 | t=s[1..(posi-1)]; |
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117 | } |
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118 | |
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119 | poly r=string2poly(t);//this is always the rounded-down version of the absolute value |
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120 | //of r |
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121 | //we have to check now, whether we should have rounded up |
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122 | //for that, we check the digit after the . |
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123 | if(string2digit(s[posi+1]) >= 5) |
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124 | { |
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125 | r=r+1; |
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126 | } |
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127 | |
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128 | if(e == 1) |
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129 | {//readjust the sign, if needed |
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130 | r=-r; |
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131 | } |
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132 | return(r); |
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133 | } |
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134 | |
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135 | static proc roundpoly(poly r) |
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136 | { |
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137 | string s=string(r); |
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138 | int posi=getposi(s); |
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139 | if(posi == 0) |
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140 | {//there is no . in r, so r is an integer |
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141 | return(r); |
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142 | } |
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143 | return(roundstringpoly(s, posi)); |
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144 | } |
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145 | |
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146 | ///////////////////////////////////////////////////////////////////////////////////////// |
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147 | ///////////////////////// Veronese embedding //////////////////////////////// |
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148 | ///////////////////////////////////////////////////////////////////////////////////////// |
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149 | |
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150 | proc substAll(poly v, list p) |
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151 | "USAGE: substAll(v,p); poly v, list p |
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152 | RETURN: poly: the polynomial obtained from v by substituting the elements of p for the |
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153 | ring variables |
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154 | NOTE: The list p should have as many elements as there are ring variables. |
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155 | EXAMPLE: example substAll; shows an example |
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156 | " |
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157 | {//substitutes the elements of p for the ring variables |
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158 | //used to obtain the value of the veronese map |
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159 | int i; |
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160 | poly f=v; |
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161 | for(i=1; i<=nvars(basering); i++) |
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162 | { |
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163 | f=subst(f,var(i),p[i]); |
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164 | } |
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165 | return(f); |
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166 | } |
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167 | example |
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168 | { "EXAMPLE:"; echo=2; |
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169 | ring r=0,(x,y,z),dp; |
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170 | poly v=x+y+z; |
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171 | list p=7/11,5/11,-1/11; |
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172 | poly f=substAll(v,p); |
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173 | f; |
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174 | } |
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175 | |
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176 | proc veronese(int d, list p) |
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177 | "USAGE: veronese(d,p); int d, list p |
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178 | RETURN: ideal: the image of the point p under the degree d Veronese embedding |
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179 | NOTE: The list p should have as many elements as there are ring variables. |
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180 | The order of the points in the returned ideal corresponds to the order of the |
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181 | monomials in maxideal(d). |
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182 | SEE ALSO: maxideal |
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183 | EXAMPLE: example veronese; shows an example |
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184 | " |
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185 | {//image of p under the degree d Veronese embedding |
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186 | ideal V=maxideal(d); |
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187 | int i; |
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188 | poly v; |
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189 | int len=size(V); |
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190 | for(i=1; i <= len; i++) |
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191 | { |
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192 | v=V[i]; |
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193 | v=substAll(v,p); |
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194 | V[i]=v; |
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195 | } |
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196 | return(V); |
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197 | } |
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198 | example |
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199 | { "EXAMPLE:"; echo=2; |
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200 | ring R=0,(x,y,z),dp; |
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201 | list p=2,3,5; |
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202 | ideal V=veronese(1,p); |
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203 | V; |
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204 | V=veronese(2,p); |
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205 | V; |
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206 | } |
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207 | |
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208 | static proc veronese_radical(int d, list P) |
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209 | {//returns a random linear combination of the images of the points in P under the |
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210 | //degree d Veronese embedding |
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211 | list p;//one of the points in P |
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212 | ideal Vp;//the Veronese embedding of p |
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213 | int i; |
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214 | for(i=1; i<=size(P); i++) |
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215 | { |
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216 | p=P[i]; |
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217 | Vp=veronese(d,p); |
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218 | P[i]=Vp; |
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219 | } |
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220 | //so we've replaced the points p with their images under the Veronese embedding |
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221 | //now we do a random linear combination of all these images |
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222 | |
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223 | //first, we rand some factors |
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224 | int di=10**7; |
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225 | int de=1; |
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226 | ideal F=poly(random(de,di))/di; |
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227 | poly f; |
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228 | for(i=2; i<=size(P); i++) |
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229 | { |
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230 | f=poly(random(de,di))/di; |
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231 | F=F,f; |
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232 | } |
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233 | |
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234 | //then we compute the linear combination |
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235 | poly v; |
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236 | int j; |
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237 | for(j=1; j<=size(P); j++) |
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238 | { |
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239 | Vp=P[j]; |
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240 | v=v+F[j]*Vp[1]; |
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241 | } |
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242 | ideal V=v; |
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243 | int len=size(maxideal(d)); |
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244 | for(i=2; i<=len; i++) |
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245 | { |
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246 | v=0; |
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247 | for(j=1; j<=size(P); j++) |
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248 | { |
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249 | Vp=P[j]; |
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250 | v=v+F[j]*Vp[i]; |
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251 | } |
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252 | V=V,v; |
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253 | } |
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254 | |
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255 | return(V); |
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256 | } |
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257 | |
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258 | ///////////////////////////////////////////////////////////////////////////////////////// |
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259 | ////////////////////////// some static procs ////////////////////////// |
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260 | ///////////////////////////////////////////////////////////////////////////////////////// |
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261 | |
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262 | static proc randlincom(ideal V, int len) |
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263 | {//produces a random linear combination of the real vectors defined by the real and the |
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264 | //imaginary part of V, respectively |
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265 | //(V is the image of a complex point p under a veronese embedding) |
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266 | poly randre,randim; |
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267 | int di=10**9; |
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268 | int de=1; |
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269 | //we get one of 2(di-de) numbers between (-)de/di and (-)1 |
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270 | randre=(-1)**random(1,2)*poly(random(de,di))/di; |
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271 | randim=(-1)**random(1,2)*poly(random(de,di))/di; |
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272 | |
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273 | ideal lincom=randre*repart(leadcoef(V[1]))+randim*impart(leadcoef(V[1])); |
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274 | |
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275 | int i; |
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276 | for(i=2; i<=len; i++) |
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277 | { |
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278 | lincom=lincom,randre*repart(leadcoef(V[i]))+randim*impart(leadcoef(V[i])); |
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279 | } |
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280 | return(lincom); |
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281 | } |
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282 | |
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283 | static proc getmatrix(ideal V, bigint C, int len) |
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284 | {//constructs the stacked matrix, but with randlincom(V,len) instead of V |
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285 | ideal rl=randlincom(V,len); |
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286 | matrix v=transpose(matrix(rl)); |
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287 | matrix E=diag(1,len); |
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288 | v=C*v; |
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289 | E=concat(E,v); |
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290 | E=transpose(E); |
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291 | return(E); |
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292 | } |
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293 | |
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294 | static proc getpolys(matrix B, int d) |
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295 | {//takes the integer parts* of the columns of B and uses them as coefficients in a |
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296 | //homogeneous poly of degree d |
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297 | //i.e. the first nrows-1 entries |
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298 | ideal V=maxideal(d); |
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299 | poly r=0;//will be one of the relation-polys |
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300 | ideal R;//will contain all the relations |
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301 | intvec rM=1..(nrows(B)-1); |
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302 | intvec cM=1..ncols(B); |
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303 | matrix M=submat(B,rM,cM);//B without the last row |
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304 | //poly nu=poly(10)**(2*d); |
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305 | int i,j; |
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306 | for(i=1; i<=ncols(M); i++) |
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307 | { |
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308 | if(absValue(B[nrows(B),i]) < 10)//if(is_almost_zero(B,i,d)) |
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309 | {//we should check first, if the value of the generated poly in p (i.e. the last |
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310 | //entry of the respective column in B) is "almost" 0 |
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311 | if(1) |
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312 | { |
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313 | for(j=1; j<=size(V); j++) |
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314 | { |
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315 | r=r+M[j,i]*V[j]; |
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316 | } |
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317 | R=R,r; |
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318 | r=0; |
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319 | } |
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320 | } |
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321 | } |
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322 | R=simplify(R,2);//gets rid of the zeroes |
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323 | return(R); |
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324 | } |
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325 | |
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326 | static proc getD(ideal J) |
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327 | { |
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328 | //computes the maximal degree among elements of J |
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329 | int maxdeg,c,i; |
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330 | poly g; |
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331 | for(i=1; i<=size(J); i++) |
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332 | { |
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333 | g=J[i]; |
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334 | c=deg(g); |
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335 | if(c > maxdeg) |
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336 | { |
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337 | maxdeg=c; |
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338 | } |
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339 | } |
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340 | return(maxdeg); |
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341 | } |
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342 | |
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343 | ////////////////////////////////////////////////////////////////////////////////////////// |
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344 | ////////////////////////////////////////////////////////////////////////////////////////// |
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345 | /////////////////////////// use_LLL procedures ////////////////////////////////// |
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346 | ////////////////////////////////////////////////////////////////////////////////////////// |
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347 | ////////////////////////////////////////////////////////////////////////////////////////// |
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348 | |
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349 | |
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350 | static proc mat2list(bigintmat B) |
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351 | { |
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352 | list c;//column of B |
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353 | list M;//the matrix: list of column-lists |
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354 | |
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355 | int i,j; |
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356 | for(i=1; i<=ncols(B); i++) |
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357 | { |
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358 | for(j=1; j<=nrows(B); j++) |
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359 | { |
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360 | c=c+list(B[j,i]); |
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361 | } |
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362 | M=M+list(c); |
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363 | c=list(); |
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364 | } |
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365 | |
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366 | return(M); |
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367 | } |
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368 | |
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369 | static proc list2bigintmat(list L); |
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370 | { |
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371 | int c=size(L); |
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372 | int r=size(L[1]); |
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373 | bigintmat B[r][c]; |
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374 | list Li; |
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375 | int i,j; |
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376 | for(i=1; i<=c; i++) |
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377 | { |
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378 | Li=L[i]; |
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379 | for(j=1; j<=r; j++) |
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380 | { |
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381 | B[j,i]=Li[j]; |
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382 | } |
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383 | } |
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384 | |
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385 | return(B); |
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386 | } |
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387 | |
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388 | static proc bigint2poly(bigint b) |
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389 | { |
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390 | poly p; |
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391 | string bs=string(b); |
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392 | |
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393 | int st=1; |
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394 | int c; |
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395 | if(bs[1] == "-") |
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396 | { |
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397 | st=2; |
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398 | c=1; |
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399 | } |
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400 | |
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401 | |
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402 | int i; |
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403 | for(i=st; i<=size(bs); i++) |
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404 | { |
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405 | p=p*10+string2intdigit(bs[i]); |
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406 | } |
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407 | |
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408 | |
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409 | if(c == 1) |
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410 | { |
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411 | return(-p); |
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412 | } |
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413 | |
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414 | return(p); |
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415 | } |
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416 | |
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417 | static proc bigintmat2matrix(bigintmat B) |
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418 | {//type conversion via matrix(B) does not work |
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419 | int r=nrows(B); |
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420 | int c=ncols(B); |
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421 | matrix M[r][c]; |
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422 | |
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423 | int i,j; |
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424 | for(i=1; i<=r; i++) |
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425 | { |
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426 | for(j=1; j<=c; j++) |
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427 | { |
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428 | M[i,j]=bigint2poly(B[i,j]); |
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429 | } |
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430 | } |
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431 | |
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432 | return(M); |
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433 | } |
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434 | |
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435 | static proc use_LLL(matrix A) |
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436 | { |
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437 | //first, we round the entries in the last row of A |
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438 | int r=nrows(A); |
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439 | int c=ncols(A); |
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440 | int i; |
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441 | for(i=1; i<=c; i++) |
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442 | { |
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443 | A[r,i]=roundpoly(A[r,i]); |
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444 | } |
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445 | |
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446 | //now, all entries of A are integers, but still have type poly |
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447 | //so we convert A to a bigintmat B |
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448 | bigintmat B=mat2bigintmat(A); |
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449 | |
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450 | //apply LLL |
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451 | list M=mat2list(B); |
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452 | list L=LLL(M); |
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453 | B=list2bigintmat(L); |
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454 | |
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455 | return(bigintmat2matrix(B)); |
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456 | } |
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457 | |
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458 | |
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459 | static proc use_LLL_bigintmat(matrix A) |
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460 | {//returns a bigintmat instead of a matrix |
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461 | //first, we round the entries in the last row of A |
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462 | int r=nrows(A); |
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463 | int c=ncols(A); |
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464 | int i; |
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465 | for(i=1; i<=c; i++) |
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466 | { |
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467 | A[r,i]=roundpoly(A[r,i]); |
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468 | } |
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469 | |
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470 | //now, all entries of A are integers, but still have type poly |
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471 | //so we convert A to a bigintmat B |
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472 | bigintmat B=mat2bigintmat(A); |
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473 | |
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474 | //apply LLL |
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475 | list M=mat2list(B); |
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476 | list L=LLL(M); |
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477 | B=list2bigintmat(L); |
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478 | |
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479 | return(B); |
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480 | } |
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481 | |
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482 | static proc use_FLINT_LLL(matrix A) |
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483 | { |
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484 | //first, we round the entries in the last row of A |
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485 | int r=nrows(A); |
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486 | int c=ncols(A); |
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487 | int i; |
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488 | for(i=1; i<=c; i++) |
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489 | { |
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490 | A[r,i]=roundpoly(A[r,i]); |
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491 | } |
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492 | |
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493 | //now, all entries of A are integers, but still have type poly |
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494 | //so we convert A to a bigintmat B |
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495 | bigintmat B=mat2bigintmat(A); |
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496 | |
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497 | //apply LLL |
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498 | bigintmat BB=system("LLL_Flint",B); |
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499 | |
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500 | return(BB); |
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501 | } |
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502 | |
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503 | static proc use_NTL_LLL(matrix A) |
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504 | { |
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505 | //first, we round the entries in the last row of A |
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506 | int r=nrows(A); |
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507 | int c=ncols(A); |
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508 | int i; |
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509 | for(i=1; i<=c; i++) |
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510 | { |
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511 | A[r,i]=roundpoly(A[r,i]); |
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512 | } |
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513 | |
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514 | |
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515 | //now, all entries of A are integers, but still have type poly |
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516 | //so we convert A to a bigintmat B |
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517 | bigintmat B=mat2bigintmat(A); |
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518 | |
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519 | def br=basering; |
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520 | ring newr=0,x,dp; |
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521 | matrix A=bigintmat2matrix(B); |
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522 | |
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523 | //NTL wants the lattice-vectors as row-vectors and returns a matrix of row-vectors |
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524 | A=transpose(A); |
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525 | matrix AA=system("LLL",A); |
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526 | AA=transpose(AA); |
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527 | bigintmat BB=mat2bigintmat(AA); |
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528 | setring br; |
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529 | |
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530 | return(BB); |
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531 | } |
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532 | |
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533 | ////////////////////////////////////////////////////////////////////////////////////////// |
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534 | ////////////////////////////////////////////////////////////////////////////////////////// |
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535 | /////////////////////////// the main procedure(s) ////////////////////////////////// |
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536 | ////////////////////////////////////////////////////////////////////////////////////////// |
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537 | ////////////////////////////////////////////////////////////////////////////////////////// |
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538 | |
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539 | proc getRelations(list p, int D, bigint C) |
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540 | "USAGE: getRelations(p,D,C); list p, int D, bigint C |
---|
541 | RETURN: list K: a list of ideals; the ideals contain homogeneous polynomial relations of |
---|
542 | degree <=D between the components of the point p |
---|
543 | NOTE: This procedure uses only the images of the one point p under the Veronese |
---|
544 | embeddings to find homogeneous polynomial relations. |
---|
545 | SEE ALSO: getRelationsRadical |
---|
546 | EXAMPLE: example getRelations; shows an example |
---|
547 | " |
---|
548 | {//uses degree d Veronese embeddings (for all d<=D) and LLL-algorithm to find |
---|
549 | //(homogeneous) polynomial relations between the entries of p |
---|
550 | //C is the Value with which the Veronese embedding is being multiplied (cf getmatrix) |
---|
551 | |
---|
552 | if(nvars(basering) != size(p) ) |
---|
553 | { |
---|
554 | ERROR("Number of variables not equal to the number of components of p."); |
---|
555 | } |
---|
556 | |
---|
557 | //get the precision |
---|
558 | list RL=ringlist(basering); |
---|
559 | RL=RL[1]; |
---|
560 | RL=RL[2]; |
---|
561 | int Prec=RL[2]; |
---|
562 | |
---|
563 | list P=list(p); |
---|
564 | |
---|
565 | int d,i,len; |
---|
566 | intvec rm; |
---|
567 | ideal vd,Kd; |
---|
568 | list K; |
---|
569 | matrix A,B; |
---|
570 | for(d=1; d<=D; d++) |
---|
571 | { |
---|
572 | vd=veronese(d,p); |
---|
573 | len=size(maxideal(d)); |
---|
574 | A=getmatrix(vd,C,len); |
---|
575 | B=realLLL(A); |
---|
576 | |
---|
577 | Kd=getpolys(B,d); |
---|
578 | |
---|
579 | if(size(Kd) == 0)//i.e. Kd has only zero-entries |
---|
580 | {//then dont add Kd to the list of relations |
---|
581 | d++; |
---|
582 | continue; |
---|
583 | } |
---|
584 | |
---|
585 | |
---|
586 | rm=check_is_zero_lincomradical(Prec,Kd,P); |
---|
587 | for(i=1; i<=size(rm); i++) |
---|
588 | { |
---|
589 | if( rm[i] == 1 ) |
---|
590 | { |
---|
591 | Kd[i] = 0; |
---|
592 | } |
---|
593 | } |
---|
594 | |
---|
595 | Kd=simplify(Kd,2); |
---|
596 | |
---|
597 | |
---|
598 | if(size(Kd) == 0)//i.e. Kd has only zero-entries |
---|
599 | {//then dont add Kd to the list of relations |
---|
600 | d++; |
---|
601 | continue; |
---|
602 | } |
---|
603 | |
---|
604 | K=K+list(Kd); |
---|
605 | } |
---|
606 | return(K); |
---|
607 | } |
---|
608 | example |
---|
609 | { "EXAMPLE:"; echo=2; |
---|
610 | ring r=(complex,50),(x,y,z),dp; |
---|
611 | list p=1,-1,0.5; |
---|
612 | getRelations(p,2,10000); |
---|
613 | } |
---|
614 | |
---|
615 | proc getRelationsRadical(list P, int D, bigint C) |
---|
616 | "USAGE: getRelationsRadical(P,D,C); list P, int D, bigint C |
---|
617 | RETURN: list K: a list of ideals; the ideals contain homogeneous polynomial relations of |
---|
618 | degree <=D between the components of the points in P |
---|
619 | NOTE: This procedure uses random linear combination of the Veronese embeddings of all |
---|
620 | points in P to find homogeneous polynomial relations. |
---|
621 | SEE ALSO: getRelations |
---|
622 | EXAMPLE: example getRelationsRadical; shows an example |
---|
623 | " |
---|
624 | {//here we compute random linear combinations of the degree d Veronese embeddings of the |
---|
625 | //points in P and then proceed as in getRelations to get homogeneous polynomials |
---|
626 | //which vanish on all points in P (with high probability) |
---|
627 | |
---|
628 | if(nvars(basering) != size(P[1]) ) |
---|
629 | { |
---|
630 | ERROR("Number of variables not equal to the number of components of P[1]."); |
---|
631 | } |
---|
632 | |
---|
633 | //get the precision |
---|
634 | list RL=ringlist(basering); |
---|
635 | RL=RL[1]; |
---|
636 | RL=RL[2]; |
---|
637 | int Prec=RL[2]; |
---|
638 | |
---|
639 | int d,i,len; |
---|
640 | intvec rm; |
---|
641 | ideal vd,Kd; |
---|
642 | list K; |
---|
643 | matrix A,B; |
---|
644 | for(d=1; d<=D; d++) |
---|
645 | { |
---|
646 | vd=veronese_radical(d,P); |
---|
647 | len=size(maxideal(d)); |
---|
648 | A=getmatrix(vd,C,len); |
---|
649 | B=realLLL(A); |
---|
650 | Kd=getpolys(B,d); |
---|
651 | |
---|
652 | if(size(Kd) == 0)//i.e. Kd has only zero-entries |
---|
653 | {//then dont add Kd to the list of relations |
---|
654 | d++; |
---|
655 | continue; |
---|
656 | } |
---|
657 | |
---|
658 | |
---|
659 | rm=check_is_zero_lincomradical(Prec,Kd,P); |
---|
660 | for(i=1; i<=size(rm); i++) |
---|
661 | { |
---|
662 | if( rm[i] == 1 ) |
---|
663 | { |
---|
664 | Kd[i] = 0; |
---|
665 | } |
---|
666 | } |
---|
667 | |
---|
668 | Kd=simplify(Kd,2); |
---|
669 | |
---|
670 | |
---|
671 | if(size(Kd) == 0)//i.e. Kd has only zero-entries |
---|
672 | {//then dont add Kd to the list of relations |
---|
673 | d++; |
---|
674 | continue; |
---|
675 | } |
---|
676 | |
---|
677 | K=K+list(Kd); |
---|
678 | } |
---|
679 | return(K); |
---|
680 | } |
---|
681 | example |
---|
682 | { "EXAMPLE:"; echo=2; |
---|
683 | ring r=(complex,50),(x,y,z),dp; |
---|
684 | list p1=1,-1,0.5; |
---|
685 | list p2=1,0,-1; |
---|
686 | list P=list(p1)+list(p2); |
---|
687 | getRelationsRadical(P,2,10**5); |
---|
688 | } |
---|
689 | |
---|
690 | ////////////////////////////////////////////////////////////////////////////////////////// |
---|
691 | ////////////////////////////////////////////////////////////////////////////////////////// |
---|
692 | /////////////////////////// Gauss reduction ////////////////////////////////// |
---|
693 | ////////////////////////////////////////////////////////////////////////////////////////// |
---|
694 | ////////////////////////////////////////////////////////////////////////////////////////// |
---|
695 | |
---|
696 | static proc find_unused_nonzero(matrix M, int j, intvec used) |
---|
697 | {//look in column j of M for a non-zero entry in an unused row |
---|
698 | //if there is one, return its row index |
---|
699 | //if there isn't, return 0 |
---|
700 | int i; |
---|
701 | int r=nrows(M); |
---|
702 | for(i=1; i<=r; i++) |
---|
703 | { |
---|
704 | if(used[i] == 0) |
---|
705 | { |
---|
706 | if(M[i,j] != 0) |
---|
707 | { |
---|
708 | return(i); |
---|
709 | } |
---|
710 | } |
---|
711 | } |
---|
712 | return(0); |
---|
713 | } |
---|
714 | |
---|
715 | proc gaussRowWithoutPerm(matrix M) |
---|
716 | "USAGE: gaussRowWithoutPerm(M); M a matrix of constant polynomials |
---|
717 | RETURN: matrix: basic Gaussian row reduction of M, just without permuting the rows |
---|
718 | EXAMPLE: example gaussRowWithoutPerm; shows an example |
---|
719 | " |
---|
720 | {//M a matrix of constant polys |
---|
721 | int n=ncols(M); |
---|
722 | int r=nrows(M); |
---|
723 | int i,j,k; |
---|
724 | intvec used;//the rows we already used to make entries in other rows 0 |
---|
725 | used[r]=0;//makes it a zero-intvec of length r |
---|
726 | //we dont want to change these used rows anymore and we dont want to use them again |
---|
727 | //entry i will be set to 1 if we used row i already |
---|
728 | for(j=1; j<=n; j++)//go through all columns of M |
---|
729 | { |
---|
730 | //find the first non-zero entry |
---|
731 | i=find_unused_nonzero(M,j,used); |
---|
732 | if(i != 0) |
---|
733 | {//and use it to make all non-pivot entries in the column equal to 0 |
---|
734 | used[i]=1; |
---|
735 | for(k=1; k<=r; k++) |
---|
736 | { |
---|
737 | if(used[k] == 0) |
---|
738 | { |
---|
739 | if(M[k,j] != 0) |
---|
740 | { |
---|
741 | M=addrow(M,i,-M[k,j]/M[i,j],k); |
---|
742 | } |
---|
743 | } |
---|
744 | } |
---|
745 | } |
---|
746 | } |
---|
747 | return(M); |
---|
748 | } |
---|
749 | example |
---|
750 | { "EXAMPLE:"; echo=2; |
---|
751 | ring r=0,x,dp; |
---|
752 | matrix M[5][4]=0,0,2,1,4,5,1,3,0,9,2,0,8,1,0,6,0,9,4,1; |
---|
753 | print(M); |
---|
754 | print(gaussRowWithoutPerm(M)); |
---|
755 | } |
---|
756 | |
---|
757 | proc gaussColWithoutPerm(matrix M) |
---|
758 | "USAGE: gaussColWithoutPerm(M); M a matrix of constant polynomials |
---|
759 | RETURN: matrix: basic Gaussian column reduction of M, just without permuting the columns |
---|
760 | EXAMPLE: example gaussColWithoutPerm; shows an example |
---|
761 | " |
---|
762 | { |
---|
763 | matrix T=transpose(M); |
---|
764 | matrix G=gaussRowWithoutPerm(T); |
---|
765 | return(transpose(G)); |
---|
766 | } |
---|
767 | example |
---|
768 | { "EXAMPLE:"; echo=2; |
---|
769 | ring r=0,x,dp; |
---|
770 | matrix M[3][4]=0,1,0,2,1,2,3,4,1,0,5,0; |
---|
771 | print(M); |
---|
772 | print(gaussColWithoutPerm(M)); |
---|
773 | } |
---|
774 | |
---|
775 | ////////////////////////////////////////////////////////////////////////////////////////// |
---|
776 | ////////////////////////////////////////////////////////////////////////////////////////// |
---|
777 | /////////////////////// static procs needed for minrelations ////////////////////// |
---|
778 | ////////////////////////////////////////////////////////////////////////////////////////// |
---|
779 | ////////////////////////////////////////////////////////////////////////////////////////// |
---|
780 | |
---|
781 | static proc multwithmaxideal(ideal I, int a) |
---|
782 | {//returns the ideal IM containing all products of elements of I and maxideal(a) |
---|
783 | ideal M=maxideal(a); |
---|
784 | int sM=size(M); |
---|
785 | ideal IM=I*M[1]; |
---|
786 | |
---|
787 | int i; |
---|
788 | for(i=2; i<=sM; i++) |
---|
789 | { |
---|
790 | IM=IM,I*M[i]; |
---|
791 | } |
---|
792 | return(IM); |
---|
793 | } |
---|
794 | |
---|
795 | static proc prodofallringvars(int dummy) |
---|
796 | {//returns the product of all ring variables |
---|
797 | poly f=1; |
---|
798 | int i; |
---|
799 | for(i=1; i<=nvars(basering); i++) |
---|
800 | { |
---|
801 | f=f*var(i); |
---|
802 | } |
---|
803 | return(f); |
---|
804 | } |
---|
805 | |
---|
806 | static proc getcoefmat(ideal IM, int m) |
---|
807 | {//computes the matrix of coefficients of the elements of IM |
---|
808 | //the order of the coefficients in each column corresponds to the order of the |
---|
809 | //monomials in maxideal(m); |
---|
810 | matrix Co; |
---|
811 | ideal M=maxideal(m); |
---|
812 | int sM=size(M); |
---|
813 | matrix C[sM][1];//the coeff vector of an element of IM with the coeffs placed at |
---|
814 | //the appropriate positions |
---|
815 | IM=simplify(IM,2);//be sure that size(IM) is the right thing -> get rid of zeroes |
---|
816 | int sIM=size(IM); |
---|
817 | matrix B; |
---|
818 | poly pr=prodofallringvars(1); |
---|
819 | poly g, Coj; |
---|
820 | int i,j,k; |
---|
821 | for(i=1; i<=sIM; i++) |
---|
822 | { |
---|
823 | g=IM[i]; |
---|
824 | Co=coef(g,pr); |
---|
825 | |
---|
826 | //we now have to put the coeffs in the appropriate places (corresponding to the |
---|
827 | //position of the respective monomial in maxideal) |
---|
828 | for(j=1; j<=ncols(Co); j++) |
---|
829 | { |
---|
830 | Coj=Co[1,j];//arranged as row vectors |
---|
831 | //compare the monomials of g with the elements of maxideal(m) |
---|
832 | //and when we find a match, place the coef at the appropriate place in C |
---|
833 | for(k=1; k<=sM; k++) |
---|
834 | { |
---|
835 | if(M[k] == Coj) |
---|
836 | { |
---|
837 | C[k,1]=Co[2,j]; |
---|
838 | break;//we dont need to check any other elements of M |
---|
839 | //since theyre all different |
---|
840 | } |
---|
841 | } |
---|
842 | } |
---|
843 | |
---|
844 | if(i==1) |
---|
845 | { |
---|
846 | B=C; |
---|
847 | C=0; |
---|
848 | i++; |
---|
849 | continue; |
---|
850 | } |
---|
851 | B=concat(B,C); |
---|
852 | C=0;//reset C to the zero vector |
---|
853 | } |
---|
854 | return(B); |
---|
855 | } |
---|
856 | |
---|
857 | static proc getconcatcoefmats(list L) |
---|
858 | {//L the first size(L) entries of K |
---|
859 | //returns the concatenated coef matrices |
---|
860 | //more precisely: let m be the degree of the elements of L[size(L)], then we want |
---|
861 | //to know, which homogenous polynomials of degree m can be written as a combination |
---|
862 | //of polynomials in the ideals contained in L. In particular, we want to know which |
---|
863 | //of the elements of L[size(L)] can be written as a combination of other polys |
---|
864 | //in L and are thereby superfluous (cf superfluousL) |
---|
865 | //what we do here is, we multiply each polynomial (of degree, say, d) in L with a |
---|
866 | //monomial of degree m-d and then store the coefficients of the resulting poly |
---|
867 | //in a matrix |
---|
868 | //(this is rather cumbersome and can probably be improved upon significantly) |
---|
869 | |
---|
870 | matrix B,C; |
---|
871 | ideal IM,I; |
---|
872 | int i,d,m; |
---|
873 | poly l; |
---|
874 | int sL=size(L); |
---|
875 | |
---|
876 | l=L[sL][1];//the polys are homogeneous; deg rising along L; deg same in L[j] |
---|
877 | //for all j |
---|
878 | m=deg(l);//the max degree |
---|
879 | |
---|
880 | if(sL == 1) |
---|
881 | {//then we only consider polys of one certain degree, so we don't have to |
---|
882 | //multiply any of the ideals with any maxideal |
---|
883 | C=getcoefmat(L[1],m); |
---|
884 | return(C);//we dont concatenate anything here, so the initialization of |
---|
885 | //C as the 1x1-zero-matrix is not an issue |
---|
886 | } |
---|
887 | |
---|
888 | for(i=1; i<sL; i++) |
---|
889 | { |
---|
890 | I=L[i]; |
---|
891 | d=deg(I[1]); |
---|
892 | IM=multwithmaxideal(I,m-d); |
---|
893 | B=getcoefmat(IM,m); |
---|
894 | C=concat(C,B);//will again have as first column the zero vector |
---|
895 | } |
---|
896 | |
---|
897 | //if i=sL, the polys in L[i] already have the degree m, so we dont need to multiply |
---|
898 | B=getcoefmat(L[sL],m); |
---|
899 | C=concat(C,B); |
---|
900 | |
---|
901 | //C will contain a zero-column at the beginning, because of the |
---|
902 | //initialization of B as the 1x1-mat with single entry 0 + the way |
---|
903 | //concat handles that situation |
---|
904 | return( submat(C,1..nrows(C),2..ncols(C)) ); |
---|
905 | } |
---|
906 | |
---|
907 | static proc string2intdigit(string ti) |
---|
908 | {//ti a string of size 1, containing an integer digit |
---|
909 | //return the digit |
---|
910 | intvec v=0,1,2,3,4,5,6,7,8,9; |
---|
911 | int i; |
---|
912 | for(i=1; i<=size(v); i++) |
---|
913 | { |
---|
914 | if( ti == string(v[i]) ) |
---|
915 | { |
---|
916 | return(v[i]); |
---|
917 | } |
---|
918 | } |
---|
919 | } |
---|
920 | |
---|
921 | static proc string2bigint(string s) |
---|
922 | { |
---|
923 | string t=s; |
---|
924 | int e=0; |
---|
925 | if(s[1]=="-") |
---|
926 | { |
---|
927 | e=1; |
---|
928 | t=s[2..size(s)];//start at the second symbol (to drop the minus) |
---|
929 | } |
---|
930 | bigint r=string2intdigit(t[1]); |
---|
931 | int i; |
---|
932 | for(i=2; i<=size(t); i++) |
---|
933 | { |
---|
934 | r=r*10+string2intdigit(t[i]); |
---|
935 | } |
---|
936 | |
---|
937 | if(e == 1) |
---|
938 | {//readjust the sign, if needed |
---|
939 | r=-r; |
---|
940 | } |
---|
941 | |
---|
942 | return(r); |
---|
943 | } |
---|
944 | |
---|
945 | static proc mat2bigintmat(matrix M) |
---|
946 | {//M a matrix filled with constant polys of integer value |
---|
947 | //return the corresponding bigintmat |
---|
948 | int c=ncols(M); |
---|
949 | int r=nrows(M); |
---|
950 | bigintmat intM[r][c]; |
---|
951 | int i,j; |
---|
952 | for(i=1; i<=r; i++) |
---|
953 | { |
---|
954 | for(j=1; j<=c; j++) |
---|
955 | { |
---|
956 | intM[i,j]=string2bigint(string(M[i,j])); |
---|
957 | } |
---|
958 | } |
---|
959 | return(intM); |
---|
960 | } |
---|
961 | |
---|
962 | static proc findnonzero(matrix M, int j) |
---|
963 | {//look in column j of M for a non-zero entry |
---|
964 | //if there is one, return its row index |
---|
965 | //if there isn't, return 0 |
---|
966 | int i; |
---|
967 | int r=nrows(M); |
---|
968 | for(i=1; i<=r; i++) |
---|
969 | { |
---|
970 | if(M[i,j] != 0) |
---|
971 | { |
---|
972 | return(i); |
---|
973 | } |
---|
974 | } |
---|
975 | return(0); |
---|
976 | } |
---|
977 | |
---|
978 | ////////////////////////////////////////////////////////////////////////////////////////// |
---|
979 | /////////////////////////////// minrelations /////////////////////////////////// |
---|
980 | ////////////////////////////////////////////////////////////////////////////////////////// |
---|
981 | |
---|
982 | static proc superfluousL(list L) |
---|
983 | {//returns an intvec containing the indices of the elements (of the ideal with highest |
---|
984 | //degree in L) which can be dropped |
---|
985 | intvec sprfls; |
---|
986 | matrix C=getconcatcoefmats(L); |
---|
987 | bigintmat intC=mat2bigintmat(C); |
---|
988 | int l=size(L[size(L)]); |
---|
989 | |
---|
990 | ring ratr=0,x,dp; |
---|
991 | matrix M=bigintmat2matrix(intC); |
---|
992 | M=gaussColWithoutPerm(M); |
---|
993 | int i; |
---|
994 | int c=1;//counts the number of elements (+1) of sprfls |
---|
995 | int k=ncols(M)-l;//the number of cols in M which correspond to polys of lower degree |
---|
996 | //is = ncols(M) - number of elements in L[size(L)] |
---|
997 | for(i=k+1; i<=ncols(M); i++) |
---|
998 | { |
---|
999 | if( findnonzero(M,i) == 0 ) |
---|
1000 | { |
---|
1001 | sprfls[c]=i-k; |
---|
1002 | c++; |
---|
1003 | } |
---|
1004 | } |
---|
1005 | return(sprfls); |
---|
1006 | } |
---|
1007 | |
---|
1008 | static proc minrelations(list K) |
---|
1009 | {//K a list of homogeneous ideals - all individually of "pure degree d" - |
---|
1010 | //ordered from d=1 up to D |
---|
1011 | |
---|
1012 | list L; |
---|
1013 | intvec sprfls; |
---|
1014 | int sj; |
---|
1015 | |
---|
1016 | int i,j; |
---|
1017 | ideal Ki; |
---|
1018 | for(i=1; i<=size(K); i++) |
---|
1019 | { |
---|
1020 | L=K[1..i];//will give the list with one ideal as the only entry, when i=1 |
---|
1021 | //i=1 would make trouble, if K was a list of lists: then L would be the first |
---|
1022 | //list in K |
---|
1023 | |
---|
1024 | sprfls=superfluousL(L); |
---|
1025 | |
---|
1026 | if(sprfls[1] == 0) |
---|
1027 | {//then sprfls returned the intvec v=0; so there are no superfluous elements |
---|
1028 | i++; |
---|
1029 | continue; |
---|
1030 | } |
---|
1031 | |
---|
1032 | Ki=K[i]; |
---|
1033 | for(j=1; j<=size(sprfls); j++) |
---|
1034 | { |
---|
1035 | sj=sprfls[j]; |
---|
1036 | Ki[sj]=0; |
---|
1037 | } |
---|
1038 | Ki=simplify(Ki,2); |
---|
1039 | |
---|
1040 | if( size(Ki) == 0 ) |
---|
1041 | {//then all polys in K[i] can be generated by polys in the K[<i], so we can delete |
---|
1042 | //K[i] from the list |
---|
1043 | |
---|
1044 | K=delete(K,i); |
---|
1045 | continue; |
---|
1046 | |
---|
1047 | //but we dont want to change i |
---|
1048 | //size(K) adjusts itself, so we're fine there |
---|
1049 | } |
---|
1050 | |
---|
1051 | K[i]=Ki; |
---|
1052 | } |
---|
1053 | |
---|
1054 | return(K); |
---|
1055 | } |
---|
1056 | |
---|
1057 | ////////////////////////////////////////////////////////////////////////////////// |
---|
1058 | ////////////////////////////////////////////////////////////////////////////////// |
---|
1059 | //////////////// Procs for Bertini-Singular-Conversation ///////////////////// |
---|
1060 | ////////////////////////////////////////////////////////////////////////////////// |
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1061 | ////////////////////////////////////////////////////////////////////////////////// |
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1062 | |
---|
1063 | proc getWitnessSet() |
---|
1064 | "USAGE: getWitnessSet(); |
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1065 | ASSUME: There is a text-document \"main_data\" in the current directory which was |
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1066 | produced by Bertini. |
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1067 | The basefield is the field of real numbers or the field of complex numbers. |
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1068 | RETURN: list; a list P of lists p_i of numbers: P a set of witness points |
---|
1069 | NOTE: Reads the file \"main_data\", searches the strings containing the witness points, |
---|
1070 | and converts them into floating point numbers. |
---|
1071 | EXAMPLE: example getWitnessSet; shows an example |
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1072 | " |
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1073 | {//goes through the file main_data generated by bertini and returns the witness points |
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1074 | //as a list of complex numbers |
---|
1075 | //(the precision specified in the definition of the basering should* be at least as |
---|
1076 | //high as the precision used by/to be expected from bertini) |
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1077 | string r; |
---|
1078 | list P,p; |
---|
1079 | int i, j; |
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1080 | r=read("main_data"); |
---|
1081 | intvec posi=find_string("Estimated",r); |
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1082 | intvec endpos=find_string("Multiplicity",r); |
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1083 | for(i=1; i<=size(posi); i++) |
---|
1084 | { |
---|
1085 | p=read_point(r,posi[i],endpos[i]); |
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1086 | |
---|
1087 | if( size(p) == 0 ) |
---|
1088 | { |
---|
1089 | ERROR("Bertini nicht erfolgreich"); |
---|
1090 | } |
---|
1091 | |
---|
1092 | P=P+list( convert_p(p) ); |
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1093 | } |
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1094 | |
---|
1095 | return(P); |
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1096 | } |
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1097 | example |
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1098 | { "EXAMPLE:"; echo=2; |
---|
1099 | //First, we write the input file for bertini, then run bertini |
---|
1100 | ring r=0,(x,y,z),dp; |
---|
1101 | ideal I=(x-y)*(y-z)*(x-z); |
---|
1102 | writeBertiniInput(I,40); |
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1103 | system("sh","bertini input"); |
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1104 | |
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1105 | //Then we change the ring and extract the witness set from main_data |
---|
1106 | ring R=(complex,40,i),(x,y,z),dp; |
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1107 | list P=getWitnessSet(); |
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1108 | P; |
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1109 | } |
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1110 | |
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1111 | |
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1112 | static proc get_hom_var_group_str(int dummy) |
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1113 | { |
---|
1114 | string vg=varstr(basering); |
---|
1115 | int i; |
---|
1116 | for(i=1; i<size(vg); i++) |
---|
1117 | { |
---|
1118 | if(vg[i]==",") |
---|
1119 | { |
---|
1120 | vg=vg[1,i]+" "+vg[(i+1),size(vg)-i]; |
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1121 | } |
---|
1122 | |
---|
1123 | if( vg[i] == "(" ) |
---|
1124 | { |
---|
1125 | vg=vg[1,i-1]+vg[(i+1),size(vg)-i]; |
---|
1126 | continue; |
---|
1127 | } |
---|
1128 | |
---|
1129 | if( vg[i] == ")" ) |
---|
1130 | { |
---|
1131 | vg=vg[1,(i-1)]+vg[(i+1),size(vg)-i]; |
---|
1132 | continue; |
---|
1133 | } |
---|
1134 | } |
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1135 | |
---|
1136 | i=size(vg); |
---|
1137 | if(vg[i]==",") |
---|
1138 | { |
---|
1139 | vg=vg[1,i]; |
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1140 | } |
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1141 | |
---|
1142 | if( vg[i] == "(" ) |
---|
1143 | { |
---|
1144 | vg=vg[1,i-1]; |
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1145 | } |
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1146 | |
---|
1147 | if( vg[i] == ")" ) |
---|
1148 | { |
---|
1149 | vg=vg[1,(i-1)]; |
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1150 | } |
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1151 | |
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1152 | vg="hom_variable_group "+vg+";"+newline; |
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1153 | |
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1154 | return(vg); |
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1155 | } |
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1156 | |
---|
1157 | static proc get_declare_function_str(ideal J) |
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1158 | { |
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1159 | string dfs; |
---|
1160 | int i; |
---|
1161 | for(i=1; i<=size(J); i++) |
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1162 | { |
---|
1163 | if(size(dfs) > 0) |
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1164 | { |
---|
1165 | dfs=dfs+", f"+string(i); |
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1166 | } |
---|
1167 | else |
---|
1168 | { |
---|
1169 | dfs=dfs+"f"+string(i); |
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1170 | } |
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1171 | } |
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1172 | |
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1173 | dfs="function "+dfs+";"+newline; |
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1174 | |
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1175 | return(dfs); |
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1176 | } |
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1177 | |
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1178 | static proc remove_brackets(string vg) |
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1179 | {//removes any round brackets from a string |
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1180 | int i; |
---|
1181 | for(i=1; i<size(vg); i++) |
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1182 | { |
---|
1183 | if( vg[i] == "(" ) |
---|
1184 | { |
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1185 | vg=vg[1,i-1]+vg[(i+1),size(vg)-i]; |
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1186 | continue; |
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1187 | } |
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1188 | |
---|
1189 | if( vg[i] == ")" ) |
---|
1190 | { |
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1191 | vg=vg[1,(i-1)]+vg[(i+1),size(vg)-i]; |
---|
1192 | continue; |
---|
1193 | } |
---|
1194 | } |
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1195 | |
---|
1196 | i=size(vg); |
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1197 | |
---|
1198 | if( vg[i] == "(" ) |
---|
1199 | { |
---|
1200 | vg=vg[1,i-1]; |
---|
1201 | } |
---|
1202 | if( vg[i] == ")" ) |
---|
1203 | { |
---|
1204 | vg=vg[1,(i-1)]; |
---|
1205 | } |
---|
1206 | return(vg); |
---|
1207 | } |
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1208 | |
---|
1209 | static proc get_function_str(ideal J) |
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1210 | { |
---|
1211 | string fs, m; |
---|
1212 | matrix C; |
---|
1213 | poly fi; |
---|
1214 | int i,j,k; |
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1215 | string s; |
---|
1216 | for(i=1; i<=size(J); i++) |
---|
1217 | { |
---|
1218 | fs=fs+"f"+string(i)+" = "; |
---|
1219 | fi=J[i]; |
---|
1220 | |
---|
1221 | s=string(fi); |
---|
1222 | s=remove_brackets(s); |
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1223 | |
---|
1224 | fs=fs+s; |
---|
1225 | |
---|
1226 | fs=fs+";"+newline; |
---|
1227 | } |
---|
1228 | return(fs); |
---|
1229 | } |
---|
1230 | |
---|
1231 | static proc get_coef_bound_poly(poly f) |
---|
1232 | { |
---|
1233 | poly pr=prodofallringvars(1); |
---|
1234 | matrix C=coef(f,pr); |
---|
1235 | int c=ncols(C); |
---|
1236 | poly b; |
---|
1237 | int i; |
---|
1238 | for(i=1; i<=c; i++) |
---|
1239 | { |
---|
1240 | b=b+absValue(C[2,i]); |
---|
1241 | } |
---|
1242 | return(b); |
---|
1243 | } |
---|
1244 | |
---|
1245 | static proc get_coef_bound_ideal(ideal J) |
---|
1246 | {//is supposed to compute the maximum among the sums of coefficients in each individual |
---|
1247 | //polynomial in J |
---|
1248 | if(size(J) == 0){ return(1); } |
---|
1249 | |
---|
1250 | J=simplify(J,2); |
---|
1251 | poly b=get_coef_bound_poly(J[1]); |
---|
1252 | poly a; |
---|
1253 | int i; |
---|
1254 | for(i=2; i<=size(J); i++) |
---|
1255 | { |
---|
1256 | a=get_coef_bound_poly(J[i]); |
---|
1257 | if(a > b) |
---|
1258 | { |
---|
1259 | b=a; |
---|
1260 | } |
---|
1261 | } |
---|
1262 | return(string(b)); |
---|
1263 | } |
---|
1264 | |
---|
1265 | static proc get_prec_in_bits(int Prec) |
---|
1266 | {//log_10(2) is approximately 3,3219281 |
---|
1267 | |
---|
1268 | //conversion from decimal digits to bits, rounded up |
---|
1269 | int pb = (3322*Prec div 1000) + 1; |
---|
1270 | |
---|
1271 | int upb=3328;//upper bound on the precision |
---|
1272 | if( pb > upb ) |
---|
1273 | {//bertini allows a maximum of 3328 bits of precision |
---|
1274 | return(upb); |
---|
1275 | } |
---|
1276 | |
---|
1277 | int lowb=64;//lower bound |
---|
1278 | if( pb < lowb) |
---|
1279 | {//bertini requires a minimum of 64 bits of precision |
---|
1280 | //however, using such a low precision is not recommended, since it will |
---|
1281 | //probably not yield any useful results |
---|
1282 | return(lowb); |
---|
1283 | } |
---|
1284 | |
---|
1285 | //bertini wants the precision to be a multiple of 32 |
---|
1286 | pb = pb + 32 - (pb mod 32); |
---|
1287 | return(pb); |
---|
1288 | } |
---|
1289 | |
---|
1290 | |
---|
1291 | proc writeBertiniInput(ideal J, int Prec) |
---|
1292 | "USAGE: writeBertiniInput(J); ideal J |
---|
1293 | RETURN: none; writes the input-file for bertini using the polynomials given by J as |
---|
1294 | functions |
---|
1295 | NOTE: Either creates a file named input in the current directory or overwrites the |
---|
1296 | existing one. |
---|
1297 | If you want to pass different parameters to bertini, you can edit the produced |
---|
1298 | input file or redefine this procedure. |
---|
1299 | EXAMPLE: example writeBertiniInput; shows an example |
---|
1300 | " |
---|
1301 | {//writes the input-file for bertini |
---|
1302 | |
---|
1303 | //we change the ring so that the names of the ring variables are convenient for us |
---|
1304 | def br=basering; |
---|
1305 | int nv=nvars(br); |
---|
1306 | ring r=0,x(1..nv),dp; |
---|
1307 | ideal J=fetch(br,J); |
---|
1308 | |
---|
1309 | |
---|
1310 | link l=":w ./input"; |
---|
1311 | write(l,"CONFIG"); |
---|
1312 | write(l,""); |
---|
1313 | write(l,"TRACKTYPE: 1;"); |
---|
1314 | write(l,"TRACKTOLBEFOREEG: 1e-8;"); |
---|
1315 | write(l,"TRACKTOLDURINGEG: 1e-11;"); |
---|
1316 | write(l,"FINALTOL: 1e-14;"); |
---|
1317 | write(l,""); |
---|
1318 | write(l,""); |
---|
1319 | write(l,"PrintPathProgress: 1;"); |
---|
1320 | write(l,"MPTYPE: 2;"); |
---|
1321 | |
---|
1322 | int pb=get_prec_in_bits(Prec); |
---|
1323 | write(l,"AMPMaxPrec: "+string(pb)+";"); |
---|
1324 | |
---|
1325 | string cb=get_coef_bound_ideal(J); |
---|
1326 | write(l,"COEFFBOUND: "+cb+";"); |
---|
1327 | |
---|
1328 | string db=string(getD(J)); |
---|
1329 | write(l,"DEGREEBOUND: "+db+";"); |
---|
1330 | write(l,""); |
---|
1331 | write(l,"SHARPENDIGITS: "+string(Prec)+";"); |
---|
1332 | |
---|
1333 | write(l,"END;"); |
---|
1334 | write(l,""); |
---|
1335 | write(l,""); |
---|
1336 | |
---|
1337 | |
---|
1338 | write(l,"INPUT"+newline); |
---|
1339 | |
---|
1340 | string vg=get_hom_var_group_str(1); |
---|
1341 | write(l,vg); |
---|
1342 | |
---|
1343 | string dfs=get_declare_function_str(J); |
---|
1344 | write(l,dfs); |
---|
1345 | |
---|
1346 | string fs=get_function_str(J); |
---|
1347 | write(l,fs); |
---|
1348 | |
---|
1349 | write(l,"END;"); |
---|
1350 | } |
---|
1351 | example |
---|
1352 | { "EXAMPLE:"; echo=2; |
---|
1353 | ring r=0,(x,y,z),dp; |
---|
1354 | poly f1=x+y+z; |
---|
1355 | poly f2=x2+xy+y2; |
---|
1356 | ideal I=f1,f2; |
---|
1357 | writeBertiniInput(I,300); |
---|
1358 | } |
---|
1359 | |
---|
1360 | static proc find_string(string F, string S) |
---|
1361 | {//search in string S for the string F |
---|
1362 | //output all the positions in an intvec v |
---|
1363 | string s; |
---|
1364 | intvec v; |
---|
1365 | int c=1;//counts the number of elements of v |
---|
1366 | |
---|
1367 | int i; |
---|
1368 | int a=size(S); |
---|
1369 | int len=size(F); |
---|
1370 | for(i=1; i<=a; i++) |
---|
1371 | { |
---|
1372 | s=S[i,len]; |
---|
1373 | if(F==s) |
---|
1374 | { |
---|
1375 | v[c]=i; |
---|
1376 | c++; |
---|
1377 | } |
---|
1378 | } |
---|
1379 | |
---|
1380 | return(v); |
---|
1381 | } |
---|
1382 | |
---|
1383 | static proc read_point(string r, int po, int endpo) |
---|
1384 | {//reads out a single point from main_data |
---|
1385 | //return as string representing a floating point number split into real and imaginary |
---|
1386 | //part |
---|
1387 | int i, b; |
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1388 | for(i=po; i<=size(r); i++) |
---|
1389 | { |
---|
1390 | if(r[i] == newline) |
---|
1391 | { |
---|
1392 | b=i+1;//b is the first character in the line containing components of the point |
---|
1393 | break; |
---|
1394 | } |
---|
1395 | } |
---|
1396 | |
---|
1397 | list p; |
---|
1398 | string pj; |
---|
1399 | int len, strt; |
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1400 | strt=b; |
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1401 | for(i=b; i<=endpo; i++) |
---|
1402 | { |
---|
1403 | if(r[i] == newline) |
---|
1404 | { |
---|
1405 | len=i-strt; |
---|
1406 | pj=r[strt,len]; |
---|
1407 | p=p+list(pj); |
---|
1408 | strt=i+1; |
---|
1409 | } |
---|
1410 | } |
---|
1411 | |
---|
1412 | return(p); |
---|
1413 | } |
---|
1414 | |
---|
1415 | static proc string2num(string numstr) |
---|
1416 | { |
---|
1417 | number n=0; |
---|
1418 | |
---|
1419 | int c=0; |
---|
1420 | if(numstr[1] == "-") |
---|
1421 | { |
---|
1422 | numstr=numstr[2,size(numstr)-1]; |
---|
1423 | c=1; |
---|
1424 | } |
---|
1425 | |
---|
1426 | int i; |
---|
1427 | for(i=size(numstr); i>=3; i--) |
---|
1428 | { |
---|
1429 | n=n/10+string2intdigit(numstr[i]); |
---|
1430 | } |
---|
1431 | n=n/10+string2intdigit(numstr[1]); |
---|
1432 | |
---|
1433 | if(c==1) |
---|
1434 | { |
---|
1435 | n=-n; |
---|
1436 | } |
---|
1437 | |
---|
1438 | return(n); |
---|
1439 | } |
---|
1440 | |
---|
1441 | static proc string2e(string estr) |
---|
1442 | {//compute the exponent from the scientific notation |
---|
1443 | int e=0; |
---|
1444 | int c=0; |
---|
1445 | if(estr[1] == "-") |
---|
1446 | { |
---|
1447 | c=1; |
---|
1448 | } |
---|
1449 | else |
---|
1450 | { |
---|
1451 | if(estr[1] != "+") |
---|
1452 | { |
---|
1453 | estr="+"+estr; |
---|
1454 | return(string2e(estr)); |
---|
1455 | } |
---|
1456 | } |
---|
1457 | |
---|
1458 | estr=estr[2,size(estr)-1]; |
---|
1459 | |
---|
1460 | int i; |
---|
1461 | for(i=1; i<=size(estr); i++) |
---|
1462 | { |
---|
1463 | e=e*10+string2intdigit(estr[i]); |
---|
1464 | } |
---|
1465 | |
---|
1466 | if(c==1) |
---|
1467 | { |
---|
1468 | e=-e; |
---|
1469 | } |
---|
1470 | |
---|
1471 | return(e); |
---|
1472 | } |
---|
1473 | |
---|
1474 | static proc dismantle_string(string si) |
---|
1475 | {//cuts the string into the real/imaginary parts and their exponents |
---|
1476 | //example of a string si: |
---|
1477 | //1.124564280901713e+00 -2.550064206873323e-01 |
---|
1478 | int e1,e2; |
---|
1479 | number im,re; |
---|
1480 | string prt;//the currently considered part of the string |
---|
1481 | int i, len; |
---|
1482 | int strt=1; |
---|
1483 | for(i=1; i<=size(si); i++) |
---|
1484 | { |
---|
1485 | if( si[i] == "e" ) |
---|
1486 | { |
---|
1487 | len=i-strt; |
---|
1488 | prt=si[strt,len]; |
---|
1489 | re=string2num(prt); |
---|
1490 | break; |
---|
1491 | } |
---|
1492 | } |
---|
1493 | |
---|
1494 | strt=i+1;//start at the character coming after "e" |
---|
1495 | for(i=strt; i<=size(si); i++) |
---|
1496 | { |
---|
1497 | if( si[i] == " " ) |
---|
1498 | { |
---|
1499 | len=i-strt; |
---|
1500 | prt=si[strt,len]; |
---|
1501 | e1=string2e(prt); |
---|
1502 | break; |
---|
1503 | } |
---|
1504 | } |
---|
1505 | |
---|
1506 | strt=i+1;//start at the character coming after " " |
---|
1507 | for(i=strt; i<=size(si); i++) |
---|
1508 | { |
---|
1509 | if( si[i] == "e" ) |
---|
1510 | { |
---|
1511 | len=i-strt; |
---|
1512 | prt=si[strt,len]; |
---|
1513 | im=string2num(prt); |
---|
1514 | break; |
---|
1515 | } |
---|
1516 | } |
---|
1517 | |
---|
1518 | strt=i+1;//start at the character coming after "e" |
---|
1519 | len=size(si)-strt+1; |
---|
1520 | prt=si[strt,len]; |
---|
1521 | e2=string2e(prt); |
---|
1522 | |
---|
1523 | number ten=10; |
---|
1524 | if(0)//e1 < -1000 |
---|
1525 | { |
---|
1526 | re=0; |
---|
1527 | } |
---|
1528 | else |
---|
1529 | { |
---|
1530 | re=re*(ten^e1); |
---|
1531 | } |
---|
1532 | |
---|
1533 | if(0)//e2 < -1000 |
---|
1534 | { |
---|
1535 | im=0; |
---|
1536 | } |
---|
1537 | else |
---|
1538 | { |
---|
1539 | im=im*(ten^e2); |
---|
1540 | } |
---|
1541 | number n=re + IUnit*im; |
---|
1542 | |
---|
1543 | return(n); |
---|
1544 | } |
---|
1545 | |
---|
1546 | |
---|
1547 | static proc convert_p(list p) |
---|
1548 | {//p a list of strings representing the components of the point p |
---|
1549 | //converts the list of strings to a list of numbers |
---|
1550 | |
---|
1551 | //interesting: apparently, since p is a list of strings to begin with, it is not |
---|
1552 | //bound to the basering, so it will exist in the ring r, as well. But, as we change |
---|
1553 | //the entries of p from type string to type number/poly, it gets bound to the ring r, |
---|
1554 | //so it doesnt exist in br anymore. Hence, we have do define list p=fetch. |
---|
1555 | |
---|
1556 | |
---|
1557 | //we change the ring, so that we know, what the imaginary unit is called, define the |
---|
1558 | //points over that ring and then fetch them to the original ring |
---|
1559 | def br=basering; |
---|
1560 | list l=ringlist(br); |
---|
1561 | l[1][3]="IUnit"; |
---|
1562 | def r=ring(l); |
---|
1563 | setring r; |
---|
1564 | |
---|
1565 | string si; |
---|
1566 | number pi; |
---|
1567 | int i; |
---|
1568 | for(i=1; i<=size(p); i++) |
---|
1569 | { |
---|
1570 | pi=dismantle_string(p[i]); |
---|
1571 | p[i]=pi; |
---|
1572 | } |
---|
1573 | |
---|
1574 | setring br; |
---|
1575 | list p=fetch(r,p); |
---|
1576 | |
---|
1577 | return(p); |
---|
1578 | } |
---|
1579 | |
---|
1580 | |
---|
1581 | static proc getP_plus_posis(int dummy) |
---|
1582 | {//goes through the file main_data generated by bertini and returns the witness points |
---|
1583 | //as a list of complex numbers |
---|
1584 | //(the precision specified in the definition of the basering should* be at least as |
---|
1585 | //high as the precision used by/to be expected from bertini) |
---|
1586 | string r; |
---|
1587 | list P,p; |
---|
1588 | int i, j; |
---|
1589 | r=read("main_data"); |
---|
1590 | intvec posi=find_string("Estimated",r); |
---|
1591 | intvec endpos=find_string("Multiplicity",r); |
---|
1592 | for(i=1; i<=size(posi); i++) |
---|
1593 | { |
---|
1594 | p=read_point(r,posi[i],endpos[i]); |
---|
1595 | |
---|
1596 | if( size(p) == 0 ) |
---|
1597 | { |
---|
1598 | ERROR("Bertini nicht erfolgreich"); |
---|
1599 | } |
---|
1600 | |
---|
1601 | P=P+list( convert_p(p) ); |
---|
1602 | } |
---|
1603 | return(posi, endpos, P); |
---|
1604 | } |
---|
1605 | |
---|
1606 | |
---|
1607 | static proc getPi_from_main_data(int i, intvec posi, intvec endpos) |
---|
1608 | {//gets only the i-th point in main_data; is used by check_is_zero |
---|
1609 | string r; |
---|
1610 | list P,p; |
---|
1611 | int j; |
---|
1612 | r=read("main_data"); |
---|
1613 | p=read_point(r,posi[i],endpos[i]); |
---|
1614 | |
---|
1615 | if( size(p) == 0 ) |
---|
1616 | { |
---|
1617 | ERROR("Bertini nicht erfolgreich"); |
---|
1618 | } |
---|
1619 | |
---|
1620 | P=P+list( convert_p(p) ); |
---|
1621 | |
---|
1622 | return(P); |
---|
1623 | } |
---|
1624 | |
---|
1625 | ////////////////////////////////////////////////////////////////////////////////////////// |
---|
1626 | ////////////////////////////////////////////////////////////////////////////////////////// |
---|
1627 | ////////////////////////////// Applications ////////////////////////////////// |
---|
1628 | ////////////////////////////////////////////////////////////////////////////////////////// |
---|
1629 | ////////////////////////////////////////////////////////////////////////////////////////// |
---|
1630 | |
---|
1631 | |
---|
1632 | ////////////////////////////////////////////////////////////////////////////////////////// |
---|
1633 | ////////////////// static procs to get the relations from the ////////////////////////// |
---|
1634 | ////////////////// complex to the rational numbers ////////////////// |
---|
1635 | ////////////////////////////////////////////////////////////////////////////////////////// |
---|
1636 | |
---|
1637 | static proc get_relations_as_bigintmats(list p, int D, bigint C) |
---|
1638 | {//uses degree d Veronese embeddings (for all d<=D) and LLL-algorithm to find |
---|
1639 | //(homogeneous) polynomial relations between the entries of p |
---|
1640 | //C is the Value with which the Veronese embedding is being multiplied (cf getmatrix) |
---|
1641 | |
---|
1642 | //returns the list of the bigintmats computed by the LLL-algorithm |
---|
1643 | //these are then processed further by get_relations_over_rationals after a switch |
---|
1644 | //of rings in the level above |
---|
1645 | |
---|
1646 | if(nvars(basering) != size(p) ) |
---|
1647 | { |
---|
1648 | ERROR("Number of variables not equal to the number of components of p."); |
---|
1649 | } |
---|
1650 | |
---|
1651 | int d,len; |
---|
1652 | list mats; |
---|
1653 | ideal vd; |
---|
1654 | matrix A; |
---|
1655 | bigintmat B; |
---|
1656 | for(d=1; d<=D; d++) |
---|
1657 | { |
---|
1658 | vd=veronese(d,p); |
---|
1659 | len=size(maxideal(d)); |
---|
1660 | A=getmatrix(vd,C,len); |
---|
1661 | //B=use_FLINT_LLL(A); |
---|
1662 | B=use_NTL_LLL(A); |
---|
1663 | //B=use_LLL_bigintmat(A); |
---|
1664 | |
---|
1665 | mats=mats+list(B); |
---|
1666 | } |
---|
1667 | return(mats); |
---|
1668 | } |
---|
1669 | |
---|
1670 | static proc get_relations_radical_as_bigintmats(list P, int D, bigint C) |
---|
1671 | {//is to get_relations_as_bigintmats what get_relationsRadical is to get_relations |
---|
1672 | //ie uses a random linear combination of the Veronese embeddings of all points in P |
---|
1673 | //in order to get polynomials which vanish over all points simultaneously |
---|
1674 | |
---|
1675 | int d,len; |
---|
1676 | list mats; |
---|
1677 | ideal vd; |
---|
1678 | matrix A; |
---|
1679 | bigintmat B; |
---|
1680 | for(d=1; d<=D; d++) |
---|
1681 | { |
---|
1682 | vd=veronese_radical(d,P); |
---|
1683 | len=size(maxideal(d)); |
---|
1684 | A=getmatrix(vd,C,len); |
---|
1685 | //B=use_FLINT_LLL(A); |
---|
1686 | B=use_NTL_LLL(A); |
---|
1687 | //B=use_LLL_bigintmat(A); |
---|
1688 | |
---|
1689 | mats=mats+list(B); |
---|
1690 | } |
---|
1691 | return(mats); |
---|
1692 | } |
---|
1693 | |
---|
1694 | static proc check_is_zero(int Prec, ideal Kd, intvec posi, intvec endpos, int k) |
---|
1695 | { |
---|
1696 | def br=basering; |
---|
1697 | int n=nvars(basering); |
---|
1698 | ring R=(complex,Prec,IUnit),x(1..n),dp; |
---|
1699 | ideal I=fetch(br,Kd); |
---|
1700 | list P=getPi_from_main_data(k, posi, endpos); |
---|
1701 | list p; |
---|
1702 | poly v; |
---|
1703 | number eps=number(10)**(5-Prec); |
---|
1704 | number a; |
---|
1705 | int i,j,c; |
---|
1706 | int len = size(I); |
---|
1707 | intvec rm; |
---|
1708 | rm[len]=0; |
---|
1709 | for(i=1; i<=len; i++) |
---|
1710 | { |
---|
1711 | for(j=1;j<=size(P); j++) |
---|
1712 | { |
---|
1713 | p=P[j]; |
---|
1714 | v=substAll(I[i],p); |
---|
1715 | a=number(v); |
---|
1716 | a=absValue(repart(a))+absValue(impart(a)); |
---|
1717 | //v=v*( poly(10)**(Prec-10) ); |
---|
1718 | if( a > eps) |
---|
1719 | { |
---|
1720 | rm[i] = 1; |
---|
1721 | break; |
---|
1722 | } |
---|
1723 | } |
---|
1724 | } |
---|
1725 | return(rm); |
---|
1726 | } |
---|
1727 | |
---|
1728 | |
---|
1729 | static proc get_relations_over_rationals(int D, int Prec, list mats, intvec posi, |
---|
1730 | intvec endpos, int k) |
---|
1731 | {//finds the relations by passing the bigintmats to getpolys |
---|
1732 | //returns a list of ideals containing the corresponding polynomials |
---|
1733 | bigintmat B; |
---|
1734 | int d; |
---|
1735 | list K; |
---|
1736 | ideal Kd; |
---|
1737 | intvec rm; |
---|
1738 | int i; |
---|
1739 | |
---|
1740 | for(d=1; d<=D; d++) |
---|
1741 | { |
---|
1742 | B=mats[d]; |
---|
1743 | Kd=getpolys(bigintmat2matrix(B),d); |
---|
1744 | if(size(Kd) != 0) |
---|
1745 | { |
---|
1746 | rm=check_is_zero(Prec,Kd,posi,endpos,k); |
---|
1747 | |
---|
1748 | for(i=1; i<=size(rm); i++) |
---|
1749 | { |
---|
1750 | if( rm[i] == 1 ) |
---|
1751 | { |
---|
1752 | Kd[i] = 0; |
---|
1753 | } |
---|
1754 | } |
---|
1755 | Kd=simplify(Kd,2); |
---|
1756 | } |
---|
1757 | |
---|
1758 | if(size(Kd) == 0)//i.e. Kd has only zero-entries |
---|
1759 | {//then dont add Kd to the list of relations |
---|
1760 | d++; |
---|
1761 | continue; |
---|
1762 | } |
---|
1763 | K=K+list(Kd); |
---|
1764 | } |
---|
1765 | return(K); |
---|
1766 | } |
---|
1767 | |
---|
1768 | |
---|
1769 | static proc getP_from_known_posis(intvec posi, intvec endpos) |
---|
1770 | {//goes through the file main_data generated by bertini and returns the witness points |
---|
1771 | //as a list of complex numbers |
---|
1772 | //(the precision specified in the definition of the basering should* be at least as |
---|
1773 | //high as the precision used by/to be expected from bertini) |
---|
1774 | string r; |
---|
1775 | list P,p; |
---|
1776 | int i, j; |
---|
1777 | r=read("main_data"); |
---|
1778 | for(i=1; i<=size(posi); i++) |
---|
1779 | { |
---|
1780 | p=read_point(r,posi[i],endpos[i]); |
---|
1781 | |
---|
1782 | if( size(p) == 0 ) |
---|
1783 | { |
---|
1784 | ERROR("Bertini nicht erfolgreich"); |
---|
1785 | } |
---|
1786 | |
---|
1787 | P=P+list( convert_p(p) ); |
---|
1788 | } |
---|
1789 | return(P); |
---|
1790 | } |
---|
1791 | |
---|
1792 | static proc check_is_zero_lincomradical(int Prec, ideal I, list P) |
---|
1793 | { |
---|
1794 | //altered ckeck_is_zero for the linear-combination-of-Veronese-embeddings version |
---|
1795 | //of the procedures |
---|
1796 | list p; |
---|
1797 | poly v; |
---|
1798 | number eps=number(10)**(5-Prec); |
---|
1799 | number a; |
---|
1800 | int i,j,c; |
---|
1801 | int len = size(I); |
---|
1802 | intvec rm; |
---|
1803 | rm[len]=0; |
---|
1804 | for(i=1; i<=len; i++) |
---|
1805 | { |
---|
1806 | for(j=1;j<=size(P); j++) |
---|
1807 | { |
---|
1808 | p=P[j]; |
---|
1809 | v=substAll(I[i],p); |
---|
1810 | a=number(v); |
---|
1811 | a=absValue(repart(a))+absValue(impart(a)); |
---|
1812 | //v=v*( poly(10)**(Prec-10) ); |
---|
1813 | if( a > eps) |
---|
1814 | { |
---|
1815 | rm[i] = 1; |
---|
1816 | break; |
---|
1817 | } |
---|
1818 | } |
---|
1819 | } |
---|
1820 | return(rm); |
---|
1821 | } |
---|
1822 | |
---|
1823 | |
---|
1824 | static proc get_relations_lincomradical_over_rationals(int D, int Prec, list mats, |
---|
1825 | intvec posi, intvec endpos) |
---|
1826 | {//finds the relations by passing the bigintmats to getpolys |
---|
1827 | //returns a list of ideals containing the corresponding polynomials |
---|
1828 | bigintmat B; |
---|
1829 | int d; |
---|
1830 | list K; |
---|
1831 | ideal Kd; |
---|
1832 | intvec rm; |
---|
1833 | int i; |
---|
1834 | |
---|
1835 | |
---|
1836 | //set up the ring to check whether the supposed relations have value zero at |
---|
1837 | //all the witness points |
---|
1838 | def br=basering; |
---|
1839 | int n=nvars(br); |
---|
1840 | ring cr=(complex,Prec,IUnit),x(1..n),dp; |
---|
1841 | list P=getP_from_known_posis(posi, endpos); |
---|
1842 | ideal I; |
---|
1843 | int le; |
---|
1844 | |
---|
1845 | setring br; |
---|
1846 | for(d=1; d<=D; d++) |
---|
1847 | { |
---|
1848 | B=mats[d]; |
---|
1849 | Kd=getpolys(bigintmat2matrix(B),d); |
---|
1850 | |
---|
1851 | //go to the complex ring to see which candidate relations should be removed |
---|
1852 | setring cr; |
---|
1853 | I=fetch(br,Kd); |
---|
1854 | le=size(I); |
---|
1855 | if(le != 0) |
---|
1856 | { |
---|
1857 | rm=check_is_zero_lincomradical(Prec,I,P); |
---|
1858 | } |
---|
1859 | |
---|
1860 | //remove from the ideal over the rational numbers |
---|
1861 | setring br; |
---|
1862 | |
---|
1863 | if(le != 0) |
---|
1864 | { |
---|
1865 | for(i=1; i<=size(rm); i++) |
---|
1866 | { |
---|
1867 | if( rm[i] == 1 ) |
---|
1868 | { |
---|
1869 | Kd[i] = 0; |
---|
1870 | } |
---|
1871 | } |
---|
1872 | Kd=simplify(Kd,2); |
---|
1873 | } |
---|
1874 | |
---|
1875 | if(size(Kd) == 0)//i.e. Kd has only zero-entries |
---|
1876 | {//then dont add Kd to the list of relations |
---|
1877 | d++; |
---|
1878 | continue; |
---|
1879 | } |
---|
1880 | K=K+list(Kd); |
---|
1881 | } |
---|
1882 | return(K); |
---|
1883 | } |
---|
1884 | |
---|
1885 | ////////////////////////////////////////////////////////////////////////////////////////// |
---|
1886 | ////////////////////////////// num_prime_decom ///////////////////////////////// |
---|
1887 | ////////////////////////////////////////////////////////////////////////////////////////// |
---|
1888 | |
---|
1889 | proc num_prime_decom(ideal I, int D, int Prec) |
---|
1890 | "USAGE: num_prime_decom(I,D); ideal I, int D |
---|
1891 | D a bound to the degree of the elements of the components of a prime |
---|
1892 | decomposition of I. |
---|
1893 | RETURN: list of ideals: each of the ideals a prime component of the radical of I |
---|
1894 | REMARKS: Uses Bertini. |
---|
1895 | NOTE: Should only be called from a ring over the rational numbers. |
---|
1896 | EXAMPLE: example num_prime_decom; shows an example |
---|
1897 | " |
---|
1898 | {//App. 3.1: computes a prime decomposition of the radical of I |
---|
1899 | //returns a list of ideals, each of them a prime component |
---|
1900 | def br=basering; |
---|
1901 | int n=nvars(br); |
---|
1902 | list K;//will contain the relations over the basering |
---|
1903 | list Q;//will contain the components |
---|
1904 | ideal M; |
---|
1905 | |
---|
1906 | writeBertiniInput(I,Prec); |
---|
1907 | |
---|
1908 | //move to a ring over the complex numbers to get the points computed by bertini |
---|
1909 | ring Ri=(complex,Prec,IUnit),x(1..n),dp; |
---|
1910 | system("sh","bertini input"); |
---|
1911 | list P; |
---|
1912 | intvec posi, endpos; |
---|
1913 | (posi, endpos, P)=getP_plus_posis(1); |
---|
1914 | int sP=size(P); |
---|
1915 | |
---|
1916 | |
---|
1917 | bigint C=bigint(10)**Prec;//digits of precision |
---|
1918 | list p, mats; |
---|
1919 | int i,j; |
---|
1920 | for(i=1; i<=sP; i++) |
---|
1921 | { |
---|
1922 | setring Ri; |
---|
1923 | |
---|
1924 | //compute the relations (with LLL, NTL_LLL or FLINT_LLL) in the form of bigintmats |
---|
1925 | p=P[i]; |
---|
1926 | mats=get_relations_as_bigintmats(p,D,C); |
---|
1927 | |
---|
1928 | |
---|
1929 | //move to br again to obtain the relation-polynomials over the rational numbers |
---|
1930 | setring br; |
---|
1931 | K=get_relations_over_rationals(D, Prec, mats, posi, endpos, i); |
---|
1932 | |
---|
1933 | if(size(K) == 0)//ie K the empty list |
---|
1934 | { |
---|
1935 | i++; |
---|
1936 | continue; |
---|
1937 | } |
---|
1938 | |
---|
1939 | K=minrelations(K); |
---|
1940 | //K is now the list of ideals containing min gens in the respective degrees |
---|
1941 | //now, we put these min gens in one ideal |
---|
1942 | M=K[1]; |
---|
1943 | for(j=2; j<=size(K); j++) |
---|
1944 | { |
---|
1945 | M=M+K[j]; |
---|
1946 | } |
---|
1947 | |
---|
1948 | Q=Q+list(M); |
---|
1949 | } |
---|
1950 | |
---|
1951 | return(Q); |
---|
1952 | } |
---|
1953 | example |
---|
1954 | { "EXAMPLE:"; echo=2; |
---|
1955 | ring R=0,(x,y,z),dp; |
---|
1956 | ideal I=(x+y)*(y+2z), (x+y)*(x-3z); |
---|
1957 | int D=2; |
---|
1958 | int Prec=300; |
---|
1959 | num_prime_decom(I,D,Prec); |
---|
1960 | |
---|
1961 | //Let us compare that to the result of primdecSY: |
---|
1962 | primdecSY(I); |
---|
1963 | } |
---|
1964 | |
---|
1965 | proc num_prime_decom1(list P, int D, bigint C) |
---|
1966 | "USAGE: num_prime_decom1(P,D,C); list P, int D, bigint C |
---|
1967 | P a list of lists representing a witness point set representing an ideal I |
---|
1968 | D should be a bound to the degree of the elements of the components of the |
---|
1969 | prime decomposition of I |
---|
1970 | C the number with which the images of the Veronese embeddings are multiplied |
---|
1971 | RETURN: list of ideals: each of the ideals a prime component of the radical of I |
---|
1972 | NOTE: Should only be called from a ring over the complex numbers. |
---|
1973 | EXAMPLE: example num_prime_decom1; shows an example |
---|
1974 | " |
---|
1975 | {//P a list of lists containing the witness points |
---|
1976 | //returns (or is supposed to return) a list containing the prime components |
---|
1977 | //of the radical of the ideal which is represented by the witness points in P |
---|
1978 | list p,K,Q; |
---|
1979 | int i,j; |
---|
1980 | ideal M; |
---|
1981 | for(i=1; i<=size(P); i++) |
---|
1982 | { |
---|
1983 | p=P[i]; |
---|
1984 | K=getRelations(p,D,C); |
---|
1985 | |
---|
1986 | if(size(K) == 0)//ie K the empty list |
---|
1987 | { |
---|
1988 | i++; |
---|
1989 | continue; |
---|
1990 | } |
---|
1991 | |
---|
1992 | K=minrelations(K); |
---|
1993 | //K is now the list of ideals containing min gens in the respective degrees |
---|
1994 | //now, we put these min gens in one ideal |
---|
1995 | M=K[1]; |
---|
1996 | for(j=2; j<=size(K); j++) |
---|
1997 | { |
---|
1998 | M=M+K[j]; |
---|
1999 | } |
---|
2000 | |
---|
2001 | Q=Q+list(M); |
---|
2002 | } |
---|
2003 | return(Q); |
---|
2004 | } |
---|
2005 | example |
---|
2006 | { "EXAMPLE:"; echo=2; |
---|
2007 | //First, we compute a prime decomposition of the ideal I=x+y; |
---|
2008 | ring R1=(complex,300,IUnit),(x,y),dp; |
---|
2009 | list p1=1,-1; |
---|
2010 | list P=list(p1); |
---|
2011 | int D=2; |
---|
2012 | bigint C=bigint(10)**300; |
---|
2013 | num_prime_decom1(P,D,C); |
---|
2014 | |
---|
2015 | |
---|
2016 | //Now, we try to obtain a prime decomposition of the ideal I=(x+y)*(y+2z), (x+y)*(x-3z); |
---|
2017 | ring R2=(complex,20,IUnit),(x,y,z),dp; |
---|
2018 | p1=1.7381623928,-1.7381623928,0.2819238763; |
---|
2019 | list p2=-3.578512854,2.385675236,-1.192837618; |
---|
2020 | P=p1,p2; |
---|
2021 | num_prime_decom1(P,D,10000); |
---|
2022 | |
---|
2023 | //Now, we look at the result of a purely symbolic algorithm |
---|
2024 | ring r2=0,(x,y,z),dp; |
---|
2025 | ideal I=(x+y)*(y+2z), (x+y)*(x-3z); |
---|
2026 | primdecSY(I); |
---|
2027 | |
---|
2028 | //If you compare the results, you may find that they don't match. |
---|
2029 | //Most likely, the hybrid algorithm got the second component wrong. This is due to the |
---|
2030 | //way the algorithm looks for homogeneous polynomial relations, and the specific version |
---|
2031 | //of the LLL algorithm used here (an implementation into Singular of a rather simple |
---|
2032 | //version which allows real input). It looks in degree 1, finds one relation and is |
---|
2033 | //thereafter unable to see a second one. Then it moves on to degree 2 and finds |
---|
2034 | //relations containing degree-1 relations as a factor. |
---|
2035 | } |
---|
2036 | |
---|
2037 | |
---|
2038 | ////////////////////////////////////////////////////////////////////////////////////////// |
---|
2039 | //////////////////////////////// num_radical /////////////////////////////////// |
---|
2040 | ////////////////////////////////////////////////////////////////////////////////////////// |
---|
2041 | |
---|
2042 | |
---|
2043 | proc num_radical_via_decom(ideal I, int D, int Prec) |
---|
2044 | "USAGE: num_radical_via_decom(I,D); ideal I, int D |
---|
2045 | D a bound to the degree of the elements of the components. |
---|
2046 | RETURN: ideal: the radical of I |
---|
2047 | REMARKS: Uses Bertini. |
---|
2048 | This procedure merely calls num_prime_decom with the same input and then |
---|
2049 | intersects the returned components. |
---|
2050 | NOTE: Should only be called from a ring over the rational numbers. |
---|
2051 | SEE ALSO: num_prime_decom, num_radical_via_randlincom |
---|
2052 | EXAMPLE: example num_radical_via_decom; shows an example |
---|
2053 | " |
---|
2054 | {//check p.14/15, App. 3.2 |
---|
2055 | list Q=num_prime_decom(I,D,Prec); |
---|
2056 | ideal interQ=1; |
---|
2057 | int i; |
---|
2058 | for(i=1; i<=size(Q); i++) |
---|
2059 | { |
---|
2060 | interQ=intersect(interQ,Q[i]); |
---|
2061 | } |
---|
2062 | return(interQ); |
---|
2063 | } |
---|
2064 | example |
---|
2065 | { "EXAMPLE:"; echo=2; |
---|
2066 | //First, we attempt to compute the radical via the hybrid algorithm. |
---|
2067 | ring R=0,(x,y,z),dp; |
---|
2068 | ideal I=(x+y)^2*(y+2z)^3, (x+y)^3*(x-3z)^2; |
---|
2069 | int D=2; |
---|
2070 | int Prec=300; |
---|
2071 | ideal numRad=num_radical_via_decom(I,D,Prec); |
---|
2072 | numRad; |
---|
2073 | |
---|
2074 | //Then we compute the radical symbolically and compare the results. |
---|
2075 | ideal Rad=radical(I); |
---|
2076 | Rad; |
---|
2077 | |
---|
2078 | reduce(Rad,std(numRad)); |
---|
2079 | |
---|
2080 | reduce(numRad,std(Rad)); |
---|
2081 | } |
---|
2082 | |
---|
2083 | proc num_radical_via_randlincom(ideal I, int D, int Prec) |
---|
2084 | "USAGE: num_radical_via_randlincom(I,D); ideal I, int D |
---|
2085 | D a bound to the degree of the elements of the components. |
---|
2086 | RETURN: ideal: the radical of I |
---|
2087 | REMARKS: Uses Bertini. |
---|
2088 | Instead of using the images of the Veronese embeddings of each individual witness |
---|
2089 | point, this procedure first computes a random linear combination of those images |
---|
2090 | and searches for homogeneous polynomial relations for this linear combination. |
---|
2091 | NOTE: Should only be called from a ring over the rational numbers. |
---|
2092 | SEE ALSO: num_radical_via_decom |
---|
2093 | EXAMPLE: example num_radical_via_randlincom; shows an example |
---|
2094 | " |
---|
2095 | {//check p.14/15, App. 3.2 |
---|
2096 | |
---|
2097 | bigint C=bigint(10)**Prec;//digits of precision |
---|
2098 | def br=basering; |
---|
2099 | int n=nvars(br); |
---|
2100 | |
---|
2101 | writeBertiniInput(I,Prec); |
---|
2102 | |
---|
2103 | //move to a ring over the complex numbers to get the points computed by bertini |
---|
2104 | ring Ri=(complex,Prec,IUnit),x(1..n),dp; |
---|
2105 | system("sh","bertini input"); |
---|
2106 | list P; |
---|
2107 | intvec posi, endpos; |
---|
2108 | (posi, endpos, P)=getP_plus_posis(1); |
---|
2109 | list mats=get_relations_radical_as_bigintmats(P,D,C); |
---|
2110 | |
---|
2111 | setring br; |
---|
2112 | list K=get_relations_lincomradical_over_rationals(D,Prec,mats,posi,endpos); |
---|
2113 | |
---|
2114 | ideal Q; |
---|
2115 | |
---|
2116 | if(size(K) > 0) |
---|
2117 | { |
---|
2118 | K=minrelations(K); |
---|
2119 | Q=K[1]; |
---|
2120 | int i; |
---|
2121 | for(i=2; i<=size(K); i++) |
---|
2122 | { |
---|
2123 | Q=Q,K[i]; |
---|
2124 | } |
---|
2125 | } |
---|
2126 | |
---|
2127 | return(Q); |
---|
2128 | } |
---|
2129 | example |
---|
2130 | { "EXAMPLE:"; echo=2; |
---|
2131 | //First, we attempt to compute the radical via the hybrid algorithm. |
---|
2132 | ring R=0,(x,y,z),dp; |
---|
2133 | ideal I=(x+y)^2*(y+2z)^3, (x+y)^3*(x-3z)^2; |
---|
2134 | int D=2; |
---|
2135 | int Prec=300; |
---|
2136 | ideal numRad=num_radical_via_randlincom(I,D,Prec); |
---|
2137 | numRad; |
---|
2138 | |
---|
2139 | //Then we compute the radical symbolically and compare the results. |
---|
2140 | ideal Rad=radical(I); |
---|
2141 | Rad; |
---|
2142 | |
---|
2143 | reduce(Rad,std(numRad)); |
---|
2144 | |
---|
2145 | reduce(numRad,std(Rad)); |
---|
2146 | } |
---|
2147 | |
---|
2148 | proc num_radical1(list P, int D, bigint C) |
---|
2149 | "USAGE: num_radical1(P,D,C); list P, int D, bigint C |
---|
2150 | P a list of lists representing a witness point set representing an ideal I |
---|
2151 | D should be a bound to the degree of the elements of the components |
---|
2152 | C the number with which the images of the Veronese embeddings are multiplied |
---|
2153 | RETURN: list of ideals: each of the ideals a prime component of the radical of I |
---|
2154 | REMARKS: This procedure merely calls num_prime_decom1 with the same input and then |
---|
2155 | intersects the returned components. |
---|
2156 | NOTE: Should only be called from a ring over the complex numbers. |
---|
2157 | SEE ALSO: num_prime_decom1, num_radical2 |
---|
2158 | EXAMPLE: example num_radical1; shows an example |
---|
2159 | " |
---|
2160 | {//computes the radical via num_prime_decom (intersecting the obtained prime decom) |
---|
2161 | list Q=num_prime_decom1(P,D,C); |
---|
2162 | ideal interQ=1; |
---|
2163 | int i; |
---|
2164 | for(i=1; i<=size(Q); i++) |
---|
2165 | { |
---|
2166 | interQ=intersect(interQ,Q[i]); |
---|
2167 | } |
---|
2168 | return(interQ); |
---|
2169 | } |
---|
2170 | example |
---|
2171 | { "EXAMPLE:"; echo=2; |
---|
2172 | //First, we write the input file for bertini and compute the radical symbolically. |
---|
2173 | ring r=0,(x,y,z),dp; |
---|
2174 | ideal I=4xy2-4z3,-2x2y+5xz2; |
---|
2175 | ideal Rad=radical(I); |
---|
2176 | writeBertiniInput(I,100); |
---|
2177 | |
---|
2178 | //Then we attempt to compute the radical via the hybrid algorithm. |
---|
2179 | ring R=(complex,100,i),(x,y,z),dp; |
---|
2180 | system("sh","bertini input"); |
---|
2181 | list P=getWitnessSet(); |
---|
2182 | int D=2; |
---|
2183 | bigint C=bigint(10)**30; |
---|
2184 | ideal Rad1=num_radical1(P,D,C); |
---|
2185 | |
---|
2186 | //Lastly, we compare the results. |
---|
2187 | Rad1; |
---|
2188 | |
---|
2189 | ideal Rad=fetch(r,Rad); |
---|
2190 | Rad; |
---|
2191 | |
---|
2192 | reduce(Rad,std(Rad1)); |
---|
2193 | |
---|
2194 | reduce(Rad1,std(Rad)); |
---|
2195 | } |
---|
2196 | |
---|
2197 | proc num_radical2(list P, int D, bigint C) |
---|
2198 | "USAGE: num_radical2(P,D,C); list P, int D, bigint C |
---|
2199 | P a list of lists representing a witness point set representing an ideal I |
---|
2200 | D should be a bound to the degree of the elements of the components |
---|
2201 | C the number with which the images of the Veronese embeddings are multiplied |
---|
2202 | RETURN: list of ideals: each of the ideals a prime component of the radical of I |
---|
2203 | REMARKS: Instead of using the images of the Veronese embeddings of each individual witness |
---|
2204 | point, this procedure first computes a random linear combination of those images |
---|
2205 | and searches for homogeneous polynomial relations for this linear combination. |
---|
2206 | NOTE: Should only be called from a ring over the complex numbers. |
---|
2207 | SEE ALSO: num_radical1 |
---|
2208 | EXAMPLE: example num_radical2; shows an example |
---|
2209 | " |
---|
2210 | {//computes the radical via getRelationsRadical |
---|
2211 | list K=getRelationsRadical(P,D,C); |
---|
2212 | K=minrelations(K); |
---|
2213 | K; |
---|
2214 | //unite the elements of K into one ideal |
---|
2215 | ideal Q=K[1]; |
---|
2216 | int i; |
---|
2217 | for(i=2; i<=size(K); i++) |
---|
2218 | { |
---|
2219 | Q=Q,K[i]; |
---|
2220 | } |
---|
2221 | |
---|
2222 | return(Q); |
---|
2223 | } |
---|
2224 | example |
---|
2225 | { "EXAMPLE:"; echo=2; |
---|
2226 | //First, we write the input file for bertini and compute the radical symbolically. |
---|
2227 | ring r=0,(x,y,z),dp; |
---|
2228 | ideal I=4xy2-4z3,-2x2y+5xz2; |
---|
2229 | ideal Rad=radical(I); |
---|
2230 | writeBertiniInput(I,100); |
---|
2231 | |
---|
2232 | //Then we attempt to compute the radical via the hybrid algorithm. |
---|
2233 | ring R=(complex,100,i),(x,y,z),dp; |
---|
2234 | system("sh","bertini input"); |
---|
2235 | list P=getWitnessSet(); |
---|
2236 | int D=2; |
---|
2237 | bigint C=bigint(10)**30; |
---|
2238 | ideal Rad2=num_radical2(P,D,C); |
---|
2239 | |
---|
2240 | //Lastly, we compare the results. |
---|
2241 | Rad2; |
---|
2242 | |
---|
2243 | ideal Rad=fetch(r,Rad); |
---|
2244 | Rad; |
---|
2245 | |
---|
2246 | reduce(Rad,std(Rad2)); |
---|
2247 | |
---|
2248 | reduce(Rad2,std(Rad)); |
---|
2249 | } |
---|
2250 | |
---|
2251 | ////////////////////////////////////////////////////////////////////////////////////////// |
---|
2252 | ///////////////////////////////// num_elim ///////////////////////////////////// |
---|
2253 | ////////////////////////////////////////////////////////////////////////////////////////// |
---|
2254 | |
---|
2255 | static proc project_p(list p, intvec projvec) |
---|
2256 | {//projects a single point p onto the components specified in projvec |
---|
2257 | list pr;//the projection |
---|
2258 | int i,k; |
---|
2259 | for(i=1; i<=size(projvec); i++) |
---|
2260 | { |
---|
2261 | k=projvec[i]; |
---|
2262 | pr=pr+list(p[k]); |
---|
2263 | } |
---|
2264 | return(pr); |
---|
2265 | } |
---|
2266 | |
---|
2267 | static proc project_P(list P, intvec projvec) |
---|
2268 | {//projects the points in P onto the components specified in projvec |
---|
2269 | list p;//elements of P |
---|
2270 | list Pr;//the list of projections |
---|
2271 | list pr;//projection of a point p |
---|
2272 | |
---|
2273 | int i; |
---|
2274 | for(i=1; i<=size(P); i++) |
---|
2275 | { |
---|
2276 | p=P[i]; |
---|
2277 | pr=project_p(p,projvec); |
---|
2278 | Pr=Pr+list(pr); |
---|
2279 | } |
---|
2280 | return(Pr); |
---|
2281 | } |
---|
2282 | |
---|
2283 | static proc get_projection_intvec(intvec elvec) |
---|
2284 | {//computes the intvec containing the indices of the variables which are not to be |
---|
2285 | //eliminated |
---|
2286 | int nv=nvars(basering); |
---|
2287 | intvec projvec; |
---|
2288 | int i,j,c,count;//count counts the elements of projvec |
---|
2289 | for(i=1; i<=nv; i++) |
---|
2290 | { |
---|
2291 | c=1; |
---|
2292 | for(j=1; j<=size(elvec); j++) |
---|
2293 | { |
---|
2294 | if(i == elvec[j]) |
---|
2295 | { |
---|
2296 | c=0; |
---|
2297 | break; |
---|
2298 | } |
---|
2299 | } |
---|
2300 | |
---|
2301 | //if i is not among the elements of elvec, store it in projvec |
---|
2302 | if(c == 1) |
---|
2303 | { |
---|
2304 | count++; |
---|
2305 | projvec[count]=i; |
---|
2306 | } |
---|
2307 | } |
---|
2308 | return(projvec); |
---|
2309 | } |
---|
2310 | |
---|
2311 | static proc get_elvec(poly f) |
---|
2312 | {//computes the elimination intvec from a product of ring variables |
---|
2313 | if(size(f) != 1) |
---|
2314 | { |
---|
2315 | ERROR("f must be a product of ringvariables, i.e. a monomial."); |
---|
2316 | } |
---|
2317 | |
---|
2318 | int n=nvars(basering); |
---|
2319 | intvec elvec; |
---|
2320 | int i, c; |
---|
2321 | for(i=1; i<=n; i++) |
---|
2322 | { |
---|
2323 | if( f/var(i) != 0) |
---|
2324 | { |
---|
2325 | c++; |
---|
2326 | elvec[c]=i; |
---|
2327 | } |
---|
2328 | } |
---|
2329 | return(elvec); |
---|
2330 | } |
---|
2331 | |
---|
2332 | proc num_elim(ideal I, poly f, int D, int Prec) |
---|
2333 | "USAGE: num_elim(I,f,D); ideal I, poly f, int D |
---|
2334 | f the product of the ring variables you want to eliminate |
---|
2335 | D a bound to the degree of the elements of the components |
---|
2336 | RETURN: ideal: the ideal obtained from I by eliminating the variables specified in f |
---|
2337 | REMARKS: This procedure uses Bertini to compute a set of witness points for I, projects |
---|
2338 | them onto the components corresponding to the variables specified in f and then |
---|
2339 | proceeds as num_radical_via_randlincom. |
---|
2340 | NOTE: Should only be called from a ring over the rational numbers. |
---|
2341 | EXAMPLE: example num_elim; shows an example |
---|
2342 | " |
---|
2343 | {//App. 3.3 |
---|
2344 | bigint C=bigint(10)**Prec;//digits of precision |
---|
2345 | |
---|
2346 | //first, get elvec and projvec |
---|
2347 | intvec elvec=get_elvec(f); |
---|
2348 | intvec projvec=get_projection_intvec(elvec); |
---|
2349 | writeBertiniInput(I,Prec); |
---|
2350 | |
---|
2351 | //define the ring with eliminated variables |
---|
2352 | //we have to compute the relations over this ring, since the number of variables |
---|
2353 | //must be the same as the number of components of the projected point |
---|
2354 | def br=basering; |
---|
2355 | list l=ringlist(br); |
---|
2356 | |
---|
2357 | int i; |
---|
2358 | for(i=size(elvec); i>=1; i--) |
---|
2359 | { |
---|
2360 | l[2]=delete(l[2],elvec[i]); |
---|
2361 | } |
---|
2362 | |
---|
2363 | def brel=ring(l); |
---|
2364 | int n=nvars(brel); |
---|
2365 | |
---|
2366 | //move to a ring over the complex numbers to get the points computed by bertini |
---|
2367 | ring Ri=(complex,Prec,IUnit),x(1..n),dp; |
---|
2368 | system("sh","bertini input"); |
---|
2369 | list P; |
---|
2370 | intvec posi, endpos; |
---|
2371 | (posi, endpos, P)=getP_plus_posis(1); |
---|
2372 | list Pr=project_P(P,projvec); |
---|
2373 | list mats=get_relations_radical_as_bigintmats(Pr,D,C); |
---|
2374 | |
---|
2375 | setring brel; |
---|
2376 | list K=get_relations_lincomradical_over_rationals(D,Prec,mats,posi,endpos); |
---|
2377 | |
---|
2378 | ideal R; |
---|
2379 | if(size(K) > 0) |
---|
2380 | { |
---|
2381 | K=minrelations(K); |
---|
2382 | R=K[1]; |
---|
2383 | for(i=2; i<=size(K); i++) |
---|
2384 | { |
---|
2385 | R=R,K[i]; |
---|
2386 | } |
---|
2387 | } |
---|
2388 | |
---|
2389 | setring br; |
---|
2390 | ideal R=imap(brel,R); |
---|
2391 | |
---|
2392 | return(R); |
---|
2393 | } |
---|
2394 | example |
---|
2395 | { "EXAMPLE:"; echo=2; |
---|
2396 | ring r=0,(x,y,z),dp; |
---|
2397 | poly f1=x-y; |
---|
2398 | poly f2=z*(x+3y); |
---|
2399 | poly f3=z*(x2+y2); |
---|
2400 | ideal I=f1,f2,f3; |
---|
2401 | |
---|
2402 | //First, we attempt to compute the elimination ideal with the hybrid algorithm. |
---|
2403 | ideal E1=num_elim(I,z,3,200); |
---|
2404 | |
---|
2405 | //Now, we compute the elimination ideal symbolically. |
---|
2406 | ideal E2=elim(I,z); |
---|
2407 | |
---|
2408 | //Lastly, we compare the results. |
---|
2409 | E1; |
---|
2410 | E2; |
---|
2411 | } |
---|
2412 | |
---|
2413 | proc num_elim1(list P, int D, bigint C, intvec elvec) |
---|
2414 | "USAGE: num_elim1(P,D,C,v); list P, int D, bigint C, intvec v |
---|
2415 | P a list of lists representing a witness point set representing an ideal J |
---|
2416 | D should be a bound to the degree of the elements of the components |
---|
2417 | C the number with which the images of the Veronese embeddings are multiplied |
---|
2418 | v an intvec specifying the numbers/positions of the variables to be eliminated |
---|
2419 | RETURN: ideal: the ideal obtained from J by eliminating the variables specified in v |
---|
2420 | REMARKS: This procedure just canonically projects the witness points onto the components |
---|
2421 | specified in the intvec v and then applies num_radical1 to the resulting points. |
---|
2422 | NOTE: Should only be called from a ring over the complex numbers. |
---|
2423 | EXAMPLE: example num_elim1; shows an example |
---|
2424 | " |
---|
2425 | {//let J be the ideal represented by the witness points in P |
---|
2426 | //returns (or is supposed to return) the prime decomposition of the radical of the |
---|
2427 | //elimination ideal of J |
---|
2428 | //(where we eliminate the variables with the indices specified in elvec) |
---|
2429 | |
---|
2430 | //Note that, since we are in a homogeneous setting eliminating all variables |
---|
2431 | //is quite simple, since we only have to decide, whether its the 0-ideal or the |
---|
2432 | //whole ring. This procedure won't work in that case. |
---|
2433 | |
---|
2434 | intvec projvec=get_projection_intvec(elvec); |
---|
2435 | list Pr=project_P(P,projvec); |
---|
2436 | |
---|
2437 | //We now have to change the ring we work over: we delete the variables which are |
---|
2438 | //to be eliminated. -> The number of variables and the number of components in |
---|
2439 | //the projected point are the same. Then we can apply our procedure and imap the |
---|
2440 | //results to our original ring, since we didnt change the names of the variables. |
---|
2441 | def br=basering; |
---|
2442 | list l=ringlist(br); |
---|
2443 | |
---|
2444 | int i; |
---|
2445 | for(i=size(elvec); i>=1; i--) |
---|
2446 | { |
---|
2447 | l[2]=delete(l[2],elvec[i]); |
---|
2448 | } |
---|
2449 | |
---|
2450 | def r=ring(l); |
---|
2451 | setring r; |
---|
2452 | list Pr=fetch(br,Pr); |
---|
2453 | |
---|
2454 | ideal R=num_radical1(Pr,D,C); |
---|
2455 | |
---|
2456 | setring br; |
---|
2457 | ideal R=imap(r,R); |
---|
2458 | |
---|
2459 | return(R); |
---|
2460 | } |
---|
2461 | example |
---|
2462 | { "EXAMPLE:"; echo=2; |
---|
2463 | //First, we write the input file for bertini and compute the elimination ideal |
---|
2464 | //symbolically. |
---|
2465 | ring r=0,(x,y,z),dp; |
---|
2466 | poly f1=x-y; |
---|
2467 | poly f2=z*(x+3y); |
---|
2468 | poly f3=z*(x2+y2); |
---|
2469 | ideal J=f1,f2,f3; |
---|
2470 | ideal E2=elim(J,z); |
---|
2471 | writeBertiniInput(J,100); |
---|
2472 | |
---|
2473 | //Then we attempt to compute the elimination ideal via the hybrid algorithm. |
---|
2474 | ring R=(complex,100,i),(x,y,z),dp; |
---|
2475 | system("sh","bertini input"); |
---|
2476 | list P=getWitnessSet(); |
---|
2477 | intvec v=3; |
---|
2478 | bigint C=bigint(10)**25; |
---|
2479 | ideal E1=num_elim1(P,2,C,v); |
---|
2480 | |
---|
2481 | //Lastly, we compare the results. |
---|
2482 | E1; |
---|
2483 | setring r; |
---|
2484 | E2; |
---|
2485 | } |
---|
2486 | |
---|
2487 | /////////////////////////////////////////////////////////////////////////////////////// |
---|
2488 | /////////////////////////////////////////////////////////////////////////////////////// |
---|
2489 | //////////////////////////// lattice basis reduction ////////////////////////////// |
---|
2490 | /////////////////////////////////////////////////////////////////////////////////////// |
---|
2491 | /////////////////////////////////////////////////////////////////////////////////////// |
---|
2492 | //An implementation of a simple LLL algorithm |
---|
2493 | //Works with real numbers |
---|
2494 | //Is only used by those procedures which require the user to provide a witness set, |
---|
2495 | // instead of calling Bertini to compute one. |
---|
2496 | |
---|
2497 | static proc eucl(int m, vector u) |
---|
2498 | {//the square of the Euclidean norm of u |
---|
2499 | poly e=inner_product(u,u); |
---|
2500 | return(e); |
---|
2501 | } |
---|
2502 | |
---|
2503 | static proc red(int i, module B, module U) |
---|
2504 | { |
---|
2505 | int j; |
---|
2506 | poly r; |
---|
2507 | for(j=i-1; j>=1; j--) |
---|
2508 | { |
---|
2509 | r=roundpoly(U[i][j]); |
---|
2510 | B[i]=B[i]-r*B[j]; |
---|
2511 | U[i]=U[i]-r*U[j]; |
---|
2512 | } |
---|
2513 | return(B,matrix(U)); |
---|
2514 | } |
---|
2515 | |
---|
2516 | static proc initBBsU(matrix M) |
---|
2517 | {//the columns of M a basis of a lattice over R |
---|
2518 | int m=nrows(M); |
---|
2519 | int c=ncols(M); |
---|
2520 | module B=M; |
---|
2521 | module Bs=M; |
---|
2522 | poly f,k,u; |
---|
2523 | matrix U=diag(1,c); |
---|
2524 | int i,j; |
---|
2525 | for(i=1; i<=c; i++) |
---|
2526 | { |
---|
2527 | for(j=1; j<=i-1; j++) |
---|
2528 | { |
---|
2529 | f=inner_product(B[i],Bs[j]); |
---|
2530 | k=inner_product(Bs[j],Bs[j]); |
---|
2531 | u=f/k; |
---|
2532 | U[j,i]=u; |
---|
2533 | Bs[i]=Bs[i]-u*Bs[j]; |
---|
2534 | } |
---|
2535 | (B,U)=red(i,B,U); |
---|
2536 | } |
---|
2537 | return(B,Bs,U); |
---|
2538 | } |
---|
2539 | |
---|
2540 | static proc mymax(int i, int k) |
---|
2541 | { |
---|
2542 | if(i >= k) |
---|
2543 | { |
---|
2544 | return(i); |
---|
2545 | } |
---|
2546 | return(k); |
---|
2547 | } |
---|
2548 | |
---|
2549 | proc realLLL(matrix M) |
---|
2550 | "USAGE: realLLL(M); matrix M |
---|
2551 | ASSUME: The columns of M represent a basis of a lattice. |
---|
2552 | The groundfield is the field of real number or the field of complex numbers, the |
---|
2553 | elements of M are real numbers. |
---|
2554 | RETURN: matrix: the columns representing an LLL-reduced basis of the lattice given by M |
---|
2555 | EXAMPLE: example realLLL; shows an example |
---|
2556 | " |
---|
2557 | { |
---|
2558 | int n=ncols(M); |
---|
2559 | int m=nrows(M); |
---|
2560 | matrix U; |
---|
2561 | module B,Bs; |
---|
2562 | poly f,k,u; |
---|
2563 | (B,Bs,U)=initBBsU(M); |
---|
2564 | int i=1; |
---|
2565 | int j; |
---|
2566 | while(i<n) |
---|
2567 | { |
---|
2568 | //check whether there is an i sth eucl(Bs[i,i]) <= 4/3*euclid(Bs[i+1,i]) |
---|
2569 | //if so, thats fine |
---|
2570 | //if not, b_i and b_i+1 are swapped + we do the necessary changes in Bs and U |
---|
2571 | if(inner_product(B[i],B[i]) <= (301/300)*inner_product(B[i+1],B[i+1])) |
---|
2572 | { |
---|
2573 | i++; |
---|
2574 | } |
---|
2575 | else |
---|
2576 | { |
---|
2577 | Bs[i+1]=Bs[i+1]+U[i,i+1]*Bs[i]; |
---|
2578 | f=inner_product(B[i],Bs[i+1]); |
---|
2579 | k=inner_product(Bs[i+1],Bs[i+1]); |
---|
2580 | u=f/k; |
---|
2581 | U[i,i]=u; |
---|
2582 | U[i+1,i]=1; |
---|
2583 | U[i,i+1]=1; |
---|
2584 | U[i+1,i+1]=0; |
---|
2585 | Bs[i]=Bs[i]-U[i,i]*Bs[i+1]; |
---|
2586 | U=permcol(U,i,i+1); |
---|
2587 | Bs=permcol(Bs,i,i+1); |
---|
2588 | B=permcol(B,i,i+1); |
---|
2589 | |
---|
2590 | for(j=i+2; j<=n; j++) |
---|
2591 | { |
---|
2592 | f=inner_product(B[j],Bs[i]); |
---|
2593 | k=inner_product(Bs[i],Bs[i]); |
---|
2594 | u=f/k; |
---|
2595 | U[i,j]=u; |
---|
2596 | |
---|
2597 | f=inner_product(B[j],Bs[i+1]); |
---|
2598 | k=inner_product(Bs[i+1],Bs[i+1]); |
---|
2599 | u=f/k; |
---|
2600 | U[i+1,j]=u; |
---|
2601 | } |
---|
2602 | |
---|
2603 | if(absValue(U[i,i+1]) > 1/2) |
---|
2604 | { |
---|
2605 | (B,U)=red(i+1,B,U); |
---|
2606 | } |
---|
2607 | i=mymax(i-1,1); |
---|
2608 | } |
---|
2609 | } |
---|
2610 | return(B); |
---|
2611 | } |
---|
2612 | example |
---|
2613 | { "EXAMPLE:"; echo=2; |
---|
2614 | ring r=(real,50),x,dp; |
---|
2615 | matrix M[5][4]= |
---|
2616 | 1,0,0,0, |
---|
2617 | 0,1,0,0, |
---|
2618 | 0,0,1,0, |
---|
2619 | 0,0,0,1, |
---|
2620 | 5*81726716.91827716, 817267.1691827716, poly(10)**30, 13*81726716.91827716; |
---|
2621 | matrix L=realLLL(M); |
---|
2622 | print(L); |
---|
2623 | } |
---|