1 | /////////////////////////////////////////////////////////////////// |
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2 | version="$Id$"; |
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3 | category="Commutative Algebra"; |
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4 | info=" |
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5 | LIBRARY: sing4ti2.lib Communication Interface to 4ti2 |
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6 | |
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7 | AUTHORS: Thomas Kahle , kahle@mis.mpg.de |
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8 | @* Anne Fruehbis-Krueger, anne@math.uni-hannover.de |
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9 | |
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10 | NOTE: This library uses the external program 4ti2 for calculations |
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11 | @* and the standard unix tools sed and awk for conversion of |
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12 | @* the returned result |
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13 | |
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14 | PROCEDURES: |
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15 | markov4ti2(A[,i]) compute Markov basis of given lattice |
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16 | hilbert4ti2(A[,i]) compute Hilbert basis of given lattice |
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17 | graver4ti2(A[,i]) compute Graver basis of given lattice |
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18 | "; |
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19 | |
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20 | |
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21 | proc markov4ti2(matrix A, list #) |
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22 | "USAGE: markov4ti2(A[,i]); |
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23 | @* A=intmat |
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24 | @* i=int |
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25 | ASSUME: - A is a matrix with integer entries which describes the lattice |
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26 | @* as ker(A), if second argument is not present, |
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27 | @* as left image Im(A) = {zA, z \in ZZ^k}(!), if second argument is a positive integer |
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28 | @* - number of variables of basering equals number of columns of A |
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29 | @* (for ker(A)) resp. of rows of A (for Im(A)) |
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30 | CREATE: files sing4ti2.mat, sing4ti2.lat, sing4ti2.mar in the current |
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31 | @* directory (I/O files for communication with 4ti2) |
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32 | NOTE: input rules for 4ti2 also apply to input to this procedure |
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33 | @* hence ker(A)={x|Ax=0} and Im(A)={xA} |
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34 | RETURN: toric ideal specified by Markov basis thereof |
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35 | EXAMPLE: example markov4ti2; shows an example |
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36 | " |
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37 | { |
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38 | //-------------------------------------------------------------------------- |
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39 | // Initialization and Sanity Checks |
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40 | //-------------------------------------------------------------------------- |
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41 | int i,j; |
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42 | int nr=nrows(A); |
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43 | int nc=ncols(A); |
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44 | string fileending="mat"; |
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45 | if (size(#)!=0) |
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46 | { |
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47 | //--- default behaviour: use ker(A) as lattice |
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48 | //--- if #[1]!=0 use Im(A) as lattice |
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49 | if(typeof(#[1])!="int") |
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50 | { |
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51 | ERROR("optional parameter needs to be integer value");\ |
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52 | } |
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53 | if(#[1]!=0) |
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54 | { |
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55 | fileending="lat"; |
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56 | } |
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57 | } |
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58 | //--- we should also be checking whether all entries are indeed integers |
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59 | //--- or whether there are fractions, but in this case the error message |
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60 | //--- of 4ti2 is printed directly |
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61 | if(nvars(basering)!=ncols(A)) |
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62 | { |
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63 | ERROR("number of columns needs to match number of variables"); |
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64 | } |
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65 | //-------------------------------------------------------------------------- |
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66 | // preparing input file for 4ti2 |
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67 | //-------------------------------------------------------------------------- |
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68 | link eing=":w sing4ti2."+fileending; |
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69 | string eingstring=string(nr)+" "+string(nc); |
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70 | write(eing,eingstring); |
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71 | for(i=1;i<=nr;i++) |
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72 | { |
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73 | kill eingstring; |
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74 | string eingstring; |
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75 | for(j=1;j<=nc;j++) |
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76 | { |
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77 | if((deg(A[i,j])>0)||(char(basering)!=0)||(npars(basering)>0)) |
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78 | { |
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79 | ERROR("Input to markov4ti2 needs to be a matrix with integer entries"); |
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80 | } |
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81 | eingstring=eingstring+string(A[i,j])+" "; |
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82 | } |
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83 | write(eing, eingstring); |
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84 | } |
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85 | close(eing); |
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86 | |
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87 | //---------------------------------------------------------------------- |
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88 | // calling 4ti2 and converting output |
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89 | // Singular's string is too clumsy for this, hence we first prepare |
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90 | // using standard unix commands |
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91 | //---------------------------------------------------------------------- |
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92 | j=system("sh","markov sing4ti2"); |
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93 | j=system("sh","awk \'BEGIN{ORS=\",\";}{print $0;}\' sing4ti2.mar | sed s/[\\\ \\\t\\\v\\\f]/,/g | sed s/,+/,/g|sed s/,,/,/g|sed s/,,/,/g > sing4ti2.converted"); |
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94 | if(!defined(keepfiles)) |
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95 | { |
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96 | j=system("sh",("rm -f sing4ti2.mar sing4ti2."+fileending)); |
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97 | } |
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98 | //---------------------------------------------------------------------- |
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99 | // reading output of 4ti2 |
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100 | //---------------------------------------------------------------------- |
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101 | link ausg=":r sing4ti2.converted"; |
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102 | //--- last entry ideal(0) is used to tie the list to the basering |
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103 | //--- it will not be processed any further |
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104 | string ergstr="list erglist="+read(ausg)+ string(ideal(0))+";"; |
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105 | execute(ergstr); |
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106 | ideal toric; |
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107 | poly temppol1,temppol2; |
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108 | for(i=1;i<=erglist[1];i++) |
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109 | { |
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110 | temppol1=1; |
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111 | temppol2=1; |
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112 | for(j=1;j<=erglist[2];j++) |
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113 | { |
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114 | if(erglist[2+(i-1)*erglist[2]+j]>=0) |
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115 | { |
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116 | //--- positive exponents |
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117 | temppol1=temppol1*(var(j)^erglist[2+(i-1)*erglist[2]+j]); |
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118 | } |
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119 | else |
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120 | { |
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121 | //--- negative exponents |
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122 | temppol2=temppol2*(var(j)^(-erglist[2+(i-1)*erglist[2]+j])); |
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123 | } |
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124 | } |
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125 | toric=toric,temppol1-temppol2; |
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126 | } |
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127 | //--- get rid of leading entry 0; |
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128 | toric=toric[2..ncols(toric)]; |
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129 | return(toric); |
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130 | } |
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131 | example |
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132 | {"EXAMPLE:"; |
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133 | echo=2; |
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134 | ring r=0,(x,y,z),dp; |
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135 | matrix M[2][3]=0,1,2,2,1,0; |
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136 | markov4ti2(M); |
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137 | matrix N[1][3]=1,2,1; |
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138 | markov4ti2(N,1); |
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139 | } |
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140 | |
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141 | /////////////////////////////////////////////////////////////////////////////// |
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142 | |
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143 | proc graver4ti2(matrix A, list #) |
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144 | "USAGE: graver4ti2(A[,i]); |
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145 | @* A=intmat |
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146 | @* i=int |
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147 | ASSUME: - A is a matrix with integer entries which describes the lattice |
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148 | @* as ker(A), if second argument is not present, |
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149 | @* as the left image Im(A) = {zA : z \in ZZ^k}, if second argument is a positive integer |
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150 | @* - number of variables of basering equals number of columns of A |
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151 | @* (for ker(A)) resp. of rows of A (for Im(A)) |
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152 | CREATE: temporary files sing4ti2.mat, sing4ti2.lat, sing4ti2.gra |
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153 | @* in the current directory (I/O files for communication with 4ti2) |
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154 | NOTE: input rules for 4ti2 also apply to input to this procedure |
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155 | @* hence ker(A)={x|Ax=0} and Im(A)={xA} |
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156 | RETURN: toric ideal specified by Graver basis thereof |
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157 | EXAMPLE: example graver4ti2; shows an example |
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158 | " |
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159 | { |
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160 | //-------------------------------------------------------------------------- |
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161 | // Initialization and Sanity Checks |
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162 | //-------------------------------------------------------------------------- |
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163 | int i,j; |
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164 | int nr=nrows(A); |
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165 | int nc=ncols(A); |
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166 | string fileending="mat"; |
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167 | if (size(#)!=0) |
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168 | { |
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169 | //--- default behaviour: use ker(A) as lattice |
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170 | //--- if #[1]!=0 use Im(A) as lattice |
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171 | if(typeof(#[1])!="int") |
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172 | { |
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173 | ERROR("optional parameter needs to be integer value");\ |
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174 | } |
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175 | if(#[1]!=0) |
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176 | { |
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177 | fileending="lat"; |
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178 | } |
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179 | } |
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180 | //--- we should also be checking whether all entries are indeed integers |
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181 | //--- or whether there are fractions, but in this case the error message |
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182 | //--- of 4ti2 is printed directly |
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183 | if(nvars(basering)!=ncols(A)) |
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184 | { |
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185 | ERROR("number of columns needs to match number of variables"); |
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186 | } |
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187 | //-------------------------------------------------------------------------- |
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188 | // preparing input file for 4ti2 |
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189 | //-------------------------------------------------------------------------- |
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190 | link eing=":w sing4ti2."+fileending; |
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191 | string eingstring=string(nr)+" "+string(nc); |
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192 | write(eing,eingstring); |
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193 | for(i=1;i<=nr;i++) |
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194 | { |
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195 | kill eingstring; |
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196 | string eingstring; |
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197 | for(j=1;j<=nc;j++) |
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198 | { |
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199 | if((deg(A[i,j])>0)||(char(basering)!=0)||(npars(basering)>0)) |
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200 | { |
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201 | ERROR("Input to graver4ti2 needs to be a matrix with integer entries"); |
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202 | } |
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203 | eingstring=eingstring+string(A[i,j])+" "; |
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204 | } |
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205 | write(eing, eingstring); |
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206 | } |
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207 | close(eing); |
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208 | |
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209 | //---------------------------------------------------------------------- |
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210 | // calling 4ti2 and converting output |
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211 | // Singular's string is too clumsy for this, hence we first prepare |
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212 | // using standard unix commands |
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213 | //---------------------------------------------------------------------- |
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214 | j=system("sh","graver sing4ti2"); |
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215 | j=system("sh","awk \'BEGIN{ORS=\",\";}{print $0;}\' sing4ti2.gra | sed s/[\\\ \\\t\\\v\\\f]/,/g | sed s/,+/,/g |sed s/,,/,/g|sed s/,,/,/g > sing4ti2.converted"); |
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216 | if(!defined(keepfiles)) |
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217 | { |
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218 | j=system("sh",("rm -f sing4ti2.gra sing4ti2."+fileending)); |
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219 | } |
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220 | //---------------------------------------------------------------------- |
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221 | // reading output of 4ti2 |
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222 | //---------------------------------------------------------------------- |
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223 | link ausg=":r sing4ti2.converted"; |
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224 | //--- last entry ideal(0) is used to tie the list to the basering |
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225 | //--- it will not be processed any further |
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226 | string ergstr="list erglist="+read(ausg)+ string(ideal(0))+";"; |
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227 | execute(ergstr); |
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228 | ideal toric; |
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229 | poly temppol1,temppol2; |
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230 | for(i=1;i<=erglist[1];i++) |
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231 | { |
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232 | temppol1=1; |
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233 | temppol2=1; |
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234 | for(j=1;j<=erglist[2];j++) |
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235 | { |
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236 | if(erglist[2+(i-1)*erglist[2]+j]>=0) |
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237 | { |
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238 | //--- positive exponents |
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239 | temppol1=temppol1*(var(j)^erglist[2+(i-1)*erglist[2]+j]); |
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240 | } |
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241 | else |
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242 | { |
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243 | //--- negative exponents |
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244 | temppol2=temppol2*(var(j)^(-erglist[2+(i-1)*erglist[2]+j])); |
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245 | } |
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246 | } |
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247 | toric=toric,temppol1-temppol2; |
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248 | } |
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249 | //--- get rid of leading entry 0; |
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250 | toric=toric[2..ncols(toric)]; |
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251 | return(toric); |
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252 | } |
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253 | example |
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254 | {"EXAMPLE:"; |
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255 | echo=2; |
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256 | ring r=0,(x,y,z,w),dp; |
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257 | matrix M[2][4]=0,1,2,3,3,2,1,0; |
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258 | graver4ti2(M); |
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259 | } |
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260 | |
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261 | /////////////////////////////////////////////////////////////////////////////// |
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262 | |
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263 | proc hilbert4ti2(matrix A, list #) |
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264 | "USAGE: hilbert4ti2(A[,i]); |
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265 | @* A=intmat |
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266 | @* i=int |
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267 | ASSUME: - A is a matrix with integer entries which describes the lattice |
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268 | @* as ker(A), if second argument is not present, |
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269 | @* as the left image Im(A) = {zA : z \in ZZ^k}, if second argument is a positive integer |
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270 | @* - number of variables of basering equals number of columns of A |
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271 | @* (for ker(A)) resp. of rows of A (for Im(A)) |
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272 | CREATE: temporary files sing4ti2.mat, sing4ti2.lat, sing4ti2.mar |
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273 | @* in the current directory (I/O files for communication with 4ti2) |
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274 | NOTE: input rules for 4ti2 also apply to input to this procedure |
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275 | @* hence ker(A)={x|Ax=0} and Im(A)={xA} |
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276 | RETURN: toric ideal specified by Hilbert basis thereof |
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277 | EXAMPLE: example graver4ti2; shows an example |
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278 | " |
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279 | { |
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280 | //-------------------------------------------------------------------------- |
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281 | // Initialization and Sanity Checks |
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282 | //-------------------------------------------------------------------------- |
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283 | int i,j; |
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284 | int nr=nrows(A); |
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285 | int nc=ncols(A); |
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286 | string fileending="mat"; |
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287 | if (size(#)!=0) |
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288 | { |
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289 | //--- default behaviour: use ker(A) as lattice |
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290 | //--- if #[1]!=0 use Im(A) as lattice |
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291 | if(typeof(#[1])!="int") |
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292 | { |
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293 | ERROR("optional parameter needs to be integer value");\ |
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294 | } |
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295 | if(#[1]!=0) |
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296 | { |
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297 | fileending="lat"; |
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298 | } |
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299 | } |
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300 | //--- we should also be checking whether all entries are indeed integers |
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301 | //--- or whether there are fractions, but in this case the error message |
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302 | //--- of 4ti2 is printed directly |
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303 | if(nvars(basering)!=ncols(A)) |
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304 | { |
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305 | ERROR("number of columns needs to match number of variables"); |
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306 | } |
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307 | //-------------------------------------------------------------------------- |
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308 | // preparing input file for 4ti2 |
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309 | //-------------------------------------------------------------------------- |
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310 | link eing=":w sing4ti2."+fileending; |
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311 | string eingstring=string(nr)+" "+string(nc); |
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312 | write(eing,eingstring); |
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313 | for(i=1;i<=nr;i++) |
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314 | { |
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315 | kill eingstring; |
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316 | string eingstring; |
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317 | for(j=1;j<=nc;j++) |
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318 | { |
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319 | if((deg(A[i,j])>0)||(char(basering)!=0)||(npars(basering)>0)) |
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320 | { |
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321 | ERROR("Input to hilbert4ti2 needs to be a matrix with integer entries"); |
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322 | } |
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323 | eingstring=eingstring+string(A[i,j])+" "; |
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324 | } |
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325 | write(eing, eingstring); |
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326 | } |
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327 | close(eing); |
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328 | |
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329 | //---------------------------------------------------------------------- |
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330 | // calling 4ti2 and converting output |
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331 | // Singular's string is too clumsy for this, hence we first prepare |
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332 | // using standard unix commands |
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333 | //---------------------------------------------------------------------- |
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334 | j=system("sh","hilbert sing4ti2"); |
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335 | j=system("sh","awk \'BEGIN{ORS=\",\";}{print $0;}\' sing4ti2.hil | sed s/[\\\ \\\t\\\v\\\f]/,/g | sed s/,+/,/g |sed s/,,/,/g|sed s/,,/,/g > sing4ti2.converted"); |
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336 | if(!defined(keepfiles)) |
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337 | { |
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338 | j=system("sh",("rm -f sing4ti2.hil sing4ti2."+fileending)); |
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339 | } |
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340 | //---------------------------------------------------------------------- |
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341 | // reading output of 4ti2 |
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342 | //---------------------------------------------------------------------- |
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343 | link ausg=":r sing4ti2.converted"; |
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344 | //--- last entry ideal(0) is used to tie the list to the basering |
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345 | //--- it will not be processed any further |
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346 | string ergstr="list erglist="+read(ausg)+ string(ideal(0))+";"; |
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347 | execute(ergstr); |
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348 | ideal toric; |
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349 | poly temppol1,temppol2; |
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350 | for(i=1;i<=erglist[1];i++) |
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351 | { |
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352 | temppol1=1; |
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353 | temppol2=1; |
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354 | for(j=1;j<=erglist[2];j++) |
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355 | { |
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356 | if(erglist[2+(i-1)*erglist[2]+j]>=0) |
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357 | { |
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358 | //--- positive exponents |
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359 | temppol1=temppol1*(var(j)^erglist[2+(i-1)*erglist[2]+j]); |
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360 | } |
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361 | else |
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362 | { |
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363 | //--- negative exponents |
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364 | temppol2=temppol2*(var(j)^(-erglist[2+(i-1)*erglist[2]+j])); |
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365 | } |
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366 | } |
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367 | toric=toric,temppol1-temppol2; |
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368 | } |
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369 | //--- get rid of leading entry 0; |
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370 | toric=toric[2..ncols(toric)]; |
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371 | return(toric); |
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372 | } |
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373 | // A nice example here is the 3x3 Magic Squares |
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374 | example |
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375 | {"EXAMPLE:"; |
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376 | echo=2; |
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377 | ring r=0,(x1,x2,x3,x4,x5,x6,x7,x8,x9),dp; |
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378 | matrix M[7][9]=1,1,1,-1,-1,-1,0,0,0,1,1,1,0,0,0,-1,-1,-1,0,1,1,-1,0,0,-1,0,0,1,0,1,0,-1,0,0,-1,0,1,1,0,0,0,-1,0,0,-1,0,1,1,0,-1,0,0,0,-1,1,1,0,0,-1,0,-1,0,0; |
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379 | hilbert4ti2(M); |
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380 | } |
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381 | |
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382 | ///////////////////////////////////////////////////////////////////////////// |
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383 | |
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