1 | version="$Id: polymake.lib,v 1.9 2008-08-29 15:16:49 keilen Exp $"; |
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2 | category="Tropical Geometry"; |
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3 | info=" |
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4 | LIBRARY: polymake.lib Computations with polytopes and fans, interface to polymake and TOPCOM |
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5 | AUTHOR: Thomas Markwig, email: keilen@mathematik.uni-kl.de |
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6 | |
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7 | WARNING: |
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8 | Most procedures will not work unless polymake or topcom is installed and |
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9 | if so, they will only work with the operating system LINUX! |
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10 | For more detailed information see IMPORTANT NOTE respectively consult the |
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11 | help string of the procedures. |
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12 | |
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13 | IMPORTANT NOTE: |
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14 | Even though this is a Singular library for computing polytopes and fans such |
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15 | as the Newton polytope or the Groebner fan of a polynomial, most of the hard |
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16 | computations are NOT done by Singular but by the program |
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17 | @* - polymake by Ewgenij Gawrilow, TU Berlin and Michael Joswig, TU Darmstadt |
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18 | @* (see http://www.math.tu-berlin.de/polymake/), |
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19 | @* respectively (only in the procedure triangularions) by the program |
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20 | @* - topcom by Joerg Rambau, Universitaet Bayreuth |
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21 | @* (see http://www.uni-bayreuth.de/departments/wirtschaftsmathematik/rambau/TOPCOM); |
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22 | @* this library should rather be seen as an interface which allows to use a (very limited) |
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23 | number of options which polymake respectively topcom offers to compute with polytopes |
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24 | and fans and to make the results available in Singular for further computations; |
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25 | moreover, the user familiar with Singular does not have to learn the syntax of polymake |
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26 | or topcom, if the options offered here are sufficient for his purposes. |
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27 | @* Note, though, that the procedures concerned with planar polygons are independent of |
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28 | both, polymake and topcom. |
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29 | |
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30 | PROCEDURES USING POLYMAKE: |
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31 | polymakePolytope(list) computes the vertices of a polytope using polymake |
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32 | newtonPolytope(poly) computes the Newton polytope of the polynomial |
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33 | newtonPolytopeLP(poly) computes the lattice points of the Newton polytope of the polynomial |
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34 | normalFan(intmat,intmat,list) computes the normal fan of a polytope |
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35 | groebnerFan(poly) computes the Groebner fan of a polynomial |
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36 | intmatToPolymake(intmat,string) transforms an integer matrix into polymake format |
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37 | polymakeToIntmat(string,string) transforms polymake output into an integer matrix |
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38 | |
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39 | PROCEDURES USING TOPCOM: |
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40 | triangulations(list) computes all triangulations of a marked polytope |
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41 | secondaryPolytope(list) computes the secondary polytope of a marked polytope |
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42 | |
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43 | PROCEDURES USING POLYMAKE AND TOPCOM: |
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44 | secondaryFan(list) computes the secondary fan of a marked polytope |
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45 | |
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46 | PROCEDURES CONERNED WITH PLANAR POLYGONS: |
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47 | cycleLength(list,intvec) computes the cycleLength of cycle dual to list with interior point intvec |
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48 | splitPolygon(list) splits a marked polygon into vertices, facets and interior points |
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49 | eta(list,list) computes the eta-vector of a triangulation |
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50 | findOrientedBoundary(list) computes the boundary of the convex hull of a list of lattice points |
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51 | cyclePoints(list,list,int) computes lattice points connected to a lattice point in a triangulation |
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52 | latticeArea(list) computes the lattice area of a polygon |
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53 | picksFormula(list) computes the ingrediants of Pick's formula for a polygon |
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54 | ellipticNF(list) computes the normal form of an elliptic polygon |
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55 | ellipticNFDB(int) displays the 16 normal forms of elliptic polygons |
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56 | |
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57 | AUXILARY PROCEDURES: |
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58 | polymakeKeepTmpFiles(int) determines whether the files created in /tmp should be kept |
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59 | |
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60 | KEYWORDS: polytope; fan; secondary fan; secondary polytope; polymake; |
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61 | Newton polytope; Groebner fan |
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62 | |
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63 | "; |
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64 | |
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65 | //////////////////////////////////////////////////////////////////////////////////// |
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66 | /// Auxilary Static Procedures in this Library |
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67 | //////////////////////////////////////////////////////////////////////////////////// |
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68 | /// - scalarproduct |
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69 | /// - intmatcoldelete |
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70 | /// - intmatconcat |
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71 | /// - sortlist |
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72 | /// - minInList |
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73 | /// - stringdelete |
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74 | /// - abs |
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75 | /// - commondenominator |
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76 | /// - maxPosInIntvec |
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77 | /// - maxPosInIntmat |
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78 | /// - sortintvec |
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79 | /// - matrixtointmat |
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80 | ///////////////////////////////////////////////////////////////////////////////////// |
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81 | |
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82 | //////////////////////////////////////////////////////////////////////////////////// |
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83 | LIB "poly.lib"; |
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84 | LIB "linalg.lib"; |
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85 | LIB "random.lib"; |
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86 | //////////////////////////////////////////////////////////////////////////////////// |
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87 | |
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88 | |
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89 | //////////////////////////////////////////////////////////////////////////////////////// |
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90 | /// PROCEDURES USING POLYMAKE |
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91 | //////////////////////////////////////////////////////////////////////////////////////// |
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92 | |
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93 | proc polymakePolytope (intmat polytope,list #) |
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94 | "USAGE: polymakePolytope(polytope[,#]); polytope list, # string |
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95 | ASSUME: each row of polytope gives the coordinates of a lattice point of a polytope |
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96 | with their affine coordinates as given by the output of secondaryPolytope |
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97 | PURPOSE: the procedure calls polymake to compute the vertices of the polytope as well |
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98 | as its dimension and information on its facets |
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99 | RETURN: list, L with four entries |
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100 | @* L[1] : an integer matrix whose rows are the coordinates of vertices |
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101 | of the polytope |
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102 | @* L[2] : the dimension of the polytope |
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103 | @* L[3] : a list whose ith entry explains to which vertices the ith vertex |
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104 | of the Newton polytope is connected -- i.e. L[3][i] is an integer |
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105 | vector and an entry k in there means that the vertex L[1][i] is |
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106 | connected to the vertex L[1][k] |
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107 | @* L[4] : an integer matrix whose rows mulitplied by (1,var(1),...,var(nvar)) give |
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108 | a linear system of equations describing the affine hull of the polytope, |
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109 | i.e. the smallest affine space containing the polytope |
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110 | NOTE: - for its computations the procedure calls the program polymake by Ewgenij Gawrilow, |
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111 | TU Berlin and Michael Joswig, TU Darmstadt; it therefore is necessary that |
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112 | this program is installed in order to use this procedure; |
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113 | see http://www.math.tu-berlin.de/polymake/ |
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114 | @* - note that in the vertex edge graph we have changed the polymake convention which |
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115 | starts indexing its vertices by zero while we start with one ! |
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116 | @* - the procedure creates the file /tmp/polytope.polymake which contains the polytope |
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117 | in polymake format; if you wish to use this for further computations with polymake, |
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118 | you have to use the procedure polymakeKeepTmpFiles in before |
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119 | @* - moreover, the procedure creates the file /tmp/polytope.output which it deletes |
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120 | again before ending |
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121 | @* - it is possible to give as an optional second argument as string which then will be |
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122 | used instead of 'polytope' in the name of the polymake output file |
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123 | EXAMPLE: example polymakePolytope; shows an example" |
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124 | { |
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125 | // the header for the file secendarypolytope.polymake |
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126 | string sp="_application polytope |
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127 | _version 2.2 |
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128 | _type RationalPolytope |
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129 | |
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130 | POINTS |
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131 | "; |
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132 | int i,j; |
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133 | // set the name for the polymake output file |
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134 | if (size(#)>0) |
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135 | { |
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136 | if (typeof(#[1])=="string") |
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137 | { |
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138 | string dateiname=#[1]; |
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139 | } |
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140 | else |
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141 | { |
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142 | string dateiname="polytope"; |
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143 | } |
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144 | } |
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145 | else |
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146 | { |
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147 | string dateiname="polytope"; |
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148 | } |
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149 | // create the lattice point list for polymake |
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150 | sp=sp+intmatToPolymake(polytope,"points"); |
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151 | // initialise dateiname.polymake and compute the vertices |
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152 | write(":w /tmp/"+dateiname+".polymake",sp); |
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153 | system("sh","cd /tmp; polymake "+dateiname+".polymake VERTICES > "+dateiname+".output"); |
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154 | string vertices=read("/tmp/"+dateiname+".output"); |
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155 | system("sh","/bin/rm /tmp/"+dateiname+".output"); |
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156 | intmat np=polymakeToIntmat(vertices,"affine"); |
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157 | // compute the dimension |
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158 | system("sh","cd /tmp; polymake "+dateiname+".polymake DIM > "+dateiname+".output"); |
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159 | string pdim=read("/tmp/"+dateiname+".output"); |
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160 | system("sh","/bin/rm /tmp/"+dateiname+".output"); |
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161 | pdim=pdim[5,size(pdim)-6]; |
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162 | execute("int nd="+pdim+";"); |
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163 | // compute the vertex-edge graph |
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164 | system("sh","cd /tmp; polymake "+dateiname+".polymake GRAPH > "+dateiname+".output"); |
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165 | string vertexedgegraph=read("/tmp/"+dateiname+".output"); |
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166 | system("sh","/bin/rm /tmp/"+dateiname+".output"); |
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167 | vertexedgegraph=vertexedgegraph[7,size(vertexedgegraph)-8]; |
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168 | string newveg; |
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169 | for (i=1;i<=size(vertexedgegraph);i++) |
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170 | { |
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171 | if (vertexedgegraph[i]=="{") |
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172 | { |
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173 | newveg=newveg+"intvec("; |
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174 | } |
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175 | else |
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176 | { |
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177 | if (vertexedgegraph[i]=="}") |
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178 | { |
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179 | newveg=newveg+"),"; |
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180 | } |
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181 | else |
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182 | { |
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183 | if (vertexedgegraph[i]==" ") |
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184 | { |
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185 | newveg=newveg+","; |
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186 | } |
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187 | else |
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188 | { |
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189 | newveg=newveg+vertexedgegraph[i]; |
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190 | } |
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191 | } |
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192 | } |
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193 | } |
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194 | newveg=newveg[1,size(newveg)-1]; |
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195 | execute("list nveg="+newveg+";"); |
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196 | // raise each entry in nveg by one |
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197 | for (i=1;i<=size(nveg);i++) |
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198 | { |
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199 | for (j=1;j<=size(nveg[i]);j++) |
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200 | { |
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201 | nveg[i][j]=nveg[i][j]+1; |
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202 | } |
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203 | } |
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204 | // compute the affine hull |
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205 | system("sh","cd /tmp; polymake "+dateiname+".polymake AFFINE_HULL > "+dateiname+".output"); |
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206 | string equations=read("/tmp/"+dateiname+".output"); |
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207 | system("sh","/bin/rm /tmp/"+dateiname+".output"); |
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208 | if (size(equations)>14) |
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209 | { |
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210 | intmat neq=polymakeToIntmat(equations,"cleardenom"); |
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211 | } |
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212 | else |
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213 | { |
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214 | intmat neq[1][ncols(polytope)+1]; |
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215 | } |
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216 | // delete the tmp-files, if polymakekeeptmpfiles is not set |
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217 | if (defined(polymakekeeptmpfiles)==0) |
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218 | { |
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219 | system("sh","/bin/rm /tmp/"+dateiname+".polymake"); |
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220 | } |
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221 | // return the files |
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222 | return(list(np,nd,nveg,neq)); |
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223 | } |
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224 | example |
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225 | { |
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226 | "EXAMPLE:"; |
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227 | echo=2; |
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228 | // the lattice points of the unit square in the plane |
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229 | list points=intvec(0,0),intvec(0,1),intvec(1,0),intvec(1,1); |
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230 | // the secondary polytope of this lattice point configuration is computed |
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231 | intmat secpoly=secondaryPolytope(points)[1]; |
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232 | list np=polymakePolytope(secpoly); |
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233 | // the vertices of the secondary polytope are: |
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234 | np[1]; |
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235 | // its dimension is |
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236 | np[2]; |
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237 | // np[3] contains information how the vertices are connected to each other, |
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238 | // e.g. the first vertex (number 0) is connected to the second one |
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239 | np[3][1]; |
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240 | // the affine hull has the equation |
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241 | ring r=0,x(1..4),dp; |
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242 | matrix M[5][1]=1,x(1),x(2),x(3),x(4); |
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243 | np[4]*M; |
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244 | } |
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245 | |
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246 | proc newtonPolytope (poly f,list #) |
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247 | "USAGE: newtonPolytope(f[,#]); f poly, # string |
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248 | RETURN: list, L with four entries |
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249 | @* L[1] : an integer matrix whose rows are the coordinates of vertices |
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250 | of the Newton polytope of f |
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251 | @* L[2] : the dimension of the Newton polytope of f |
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252 | @* L[3] : a list whose ith entry explains to which vertices the ith vertex |
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253 | of the Newton polytope is connected -- i.e. L[3][i] is an integer |
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254 | vector and an entry k in there means that the vertex L[1][i] is |
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255 | connected to the vertex L[1][k] |
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256 | @* L[4] : an integer matrix whose rows mulitplied by (1,var(1),...,var(nvar)) give |
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257 | a linear system of equations describing the affine hull of the Newton |
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258 | polytope, i.e. the smallest affine space containing the Newton polytope |
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259 | NOTE: - if we replace the first column of L[4] by zeros, i.e. if we move the affine hull to |
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260 | the origin, then we get the equations for the orthogonal comploment of the linearity |
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261 | space of the normal fan dual to the Newton polytope, i.e. we get the EQUATIONS that |
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262 | we need as input for polymake when computing the normal fan |
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263 | @* - the procedure calls for its computation polymake by Ewgenij Gawrilow, |
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264 | TU Berlin and Michael Joswig, so it only works if polymake is installed; |
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265 | see http://www.math.tu-berlin.de/polymake/ |
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266 | @* - the procedure creates the file /tmp/newtonPolytope.polymake which contains the polytope |
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267 | in polymake format and which can be used for further computations with polymake |
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268 | @* - moreover, the procedure creates the file /tmp/newtonPolytope.output which it deletes |
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269 | again before ending |
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270 | @* - it is possible to give as an optional second argument as string which then will be |
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271 | used instead of 'newtonPolytope' in the name of the polymake output file |
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272 | EXAMPLE: example newtonPolytope; shows an example" |
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273 | { |
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274 | int i,j; |
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275 | // compute the list of exponent vectors of the polynomial, which are the lattice points |
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276 | // whose convex hull is the Newton polytope of f |
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277 | intmat exponents[size(f)][nvars(basering)]; |
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278 | while (f!=0) |
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279 | { |
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280 | i++; |
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281 | exponents[i,1..nvars(basering)]=leadexp(f); |
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282 | f=f-lead(f); |
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283 | } |
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284 | if (size(#)==0) |
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285 | { |
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286 | #[1]="newtonPolytope"; |
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287 | } |
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288 | // call polymakePolytope with exponents |
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289 | return(polymakePolytope(exponents,#)); |
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290 | } |
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291 | example |
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292 | { |
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293 | "EXAMPLE:"; |
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294 | echo=2; |
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295 | ring r=0,(x,y,z),dp; |
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296 | matrix M[4][1]=1,x,y,z; |
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297 | poly f=y3+x2+xy+2xz+yz+z2+1; |
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298 | // the Newton polytope of f is |
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299 | list np=newtonPolytope(f); |
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300 | // the vertices of the Newton polytope are: |
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301 | np[1]; |
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302 | // its dimension is |
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303 | np[2]; |
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304 | // np[3] contains information how the vertices are connected to each other, |
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305 | // e.g. the first vertex (number 0) is connected to the second, third and |
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306 | // fourth vertex |
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307 | np[3][1]; |
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308 | ////////////////////////// |
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309 | f=x2-y3; |
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310 | // the Newton polytope of f is |
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311 | np=newtonPolytope(f); |
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312 | // the vertices of the Newton polytope are: |
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313 | np[1]; |
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314 | // its dimension is |
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315 | np[2]; |
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316 | // the Newton polytope is contained in the affine space given |
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317 | // by the equations |
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318 | np[4]*M; |
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319 | } |
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320 | |
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321 | proc newtonPolytopeLP (poly f) |
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322 | "USAGE: newtonPolytopeLP(f); f poly |
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323 | RETURN: list, the exponent vectors of the monomials occuring in f, i.e. the lattice |
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324 | points of the Newton polytope of f |
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325 | EXAMPLE: example normalFan; shows an example" |
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326 | { |
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327 | list np; |
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328 | int i=1; |
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329 | while (f!=0) |
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330 | { |
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331 | np[i]=leadexp(f); |
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332 | f=f-lead(f); |
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333 | i++; |
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334 | } |
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335 | return(np); |
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336 | } |
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337 | example |
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338 | { |
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339 | "EXAMPLE:"; |
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340 | echo=2; |
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341 | ring r=0,(x,y,z),dp; |
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342 | poly f=y3+x2+xy+2xz+yz+z2+1; |
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343 | // the lattice points of the Newton polytope of f are |
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344 | newtonPolytopeLP(f); |
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345 | } |
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346 | |
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347 | proc normalFan (intmat vertices,intmat affinehull,list graph,int er,list #) |
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348 | "USAGE: normalFan (vert,aff,graph,rays,[,#]); vert,aff intmat, graph list, rays int, # string |
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349 | ASSUME: - vert is an integer matrix whose rows are the coordinate of the vertices of |
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350 | a convex lattice polygon; |
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351 | @* - aff describes the affine hull of this polytope, i.e. |
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352 | the smallest affine space containing it, in the following sense: |
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353 | denote by n the number of columns of vert, then multiply aff by (1,x(1),...,x(n)) |
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354 | and set the resulting terms to zero in order to get the equations for the affine hull; |
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355 | @* - the ith entry of graph is an integer vector describing to which vertices |
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356 | the ith vertex is connected, i.e. a k as entry means that the vertex vert[i] is |
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357 | connected to vert[k]; |
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358 | @* - the integer rays is either one (if the extreme rays should be computed) or zero (otherwise) |
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359 | RETURN: list, the ith entry of L[1] contains information about the cone in the normal fan |
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360 | dual to the ith vertex of the polytope |
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361 | @* L[1][i][1] = integer matrix representing the inequalities which describe the |
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362 | cone dual to the ith vertex |
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363 | @* L[1][i][2] = a list which contains the inequalities represented by L[i][1] |
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364 | as a list of strings, where we use the variables x(1),...,x(n) |
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365 | @* L[1][i][3] = only present if 'er' is set to 1; in that case it is an interger matrix |
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366 | whose rows are the extreme rays of the cone |
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367 | @* L[2] = is an integer matrix whose rows span the linearity space of the fan, |
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368 | i.e. the linear space which is contained in each cone |
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369 | NOTE: - the procedure calls for its computation polymake by Ewgenij Gawrilow, |
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370 | TU Berlin and Michael Joswig, so it only works if polymake is installed; |
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371 | see http://www.math.tu-berlin.de/polymake/ |
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372 | @* - in the optional argument # it is possible to hand over other names for the |
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373 | variables to be used -- be carful, the format must be correct and that is |
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374 | not tested, e.g. if you want the variable names to be u00,u10,u01,u11 |
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375 | then you must hand over the string u11,u10,u01,u11 |
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376 | EXAMPLE: example normalFan; shows an example" |
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377 | { |
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378 | list ineq; // stores the inequalities of the cones |
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379 | int i,j,k; |
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380 | // we work over the following ring |
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381 | execute("ring ineqring=0,x(1.."+string(ncols(vertices))+"),lp;"); |
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382 | string greatersign=">"; |
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383 | // create the variable names |
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384 | if (size(#)>0) |
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385 | { |
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386 | if (typeof(#[1])=="string") |
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387 | { |
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388 | kill ineqring; |
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389 | execute("ring ineqring=0,("+#[1]+"),lp;"); |
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390 | } |
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391 | if (size(#)>1) |
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392 | { |
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393 | greatersign="<"; |
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394 | } |
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395 | } |
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396 | ////////////////////////////////////////////////////////////////// |
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397 | // Compute first the inequalities of the cones |
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398 | ////////////////////////////////////////////////////////////////// |
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399 | matrix VAR[1][ncols(vertices)]=maxideal(1); |
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400 | matrix EXP[ncols(vertices)][1]; |
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401 | poly p,pl,pr; |
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402 | // consider all vertices of the polytope |
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403 | for (i=1;i<=nrows(vertices);i++) |
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404 | { |
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405 | // first we produce for each vertex in the polytope |
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406 | // the inequalities describing the dual cone in the normal fan |
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407 | list pp; // contain strings representing the inequalities describing the normal cone |
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408 | intmat ie[size(graph[i])][ncols(vertices)]; // contains the inequalities as rows |
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409 | // consider all the vertices to which the ith vertex in the polytope is connected by an edge |
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410 | for (j=1;j<=size(graph[i]);j++) |
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411 | { |
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412 | // produce the vector ie_j pointing from the jth vertex to the ith vertex; |
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413 | // this will be the jth inequality for the cone in the normal fan dual to the ith vertex |
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414 | ie[j,1..ncols(vertices)]=vertices[i,1..ncols(vertices)]-vertices[graph[i][j],1..ncols(vertices)]; |
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415 | EXP=ie[j,1..ncols(vertices)]; |
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416 | // build a linear polynomial with the entries of ie_j as coefficients |
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417 | p=(VAR*EXP)[1,1]; |
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418 | pl,pr=0,0; |
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419 | // separate the terms with positive coefficients in p from those with negative coefficients |
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420 | for (k=1;k<=size(p);k++) |
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421 | { |
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422 | if (leadcoef(p[k])<0) |
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423 | { |
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424 | pr=pr-p[k]; |
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425 | } |
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426 | else |
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427 | { |
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428 | pl=pl+p[k]; |
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429 | } |
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430 | } |
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431 | // build the string which represents the jth inequality for the cone dual to the ith vertex |
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432 | // as polynomial inequality of type string, and store this in the list pp as jth entry |
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433 | pp[j]=string(pl)+" "+greatersign+" "+string(pr); |
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434 | } |
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435 | // all inequalities for the ith vertex are stored in the list ineq |
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436 | ineq[i]=list(ie,pp); |
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437 | kill ie,pp; // kill certain lists |
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438 | } |
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439 | // remove the first column of affine hull to compute the linearity space |
---|
440 | intmat linearity=intmatcoldelete(affinehull,1); |
---|
441 | ////////////////////////////////////////////////////////////////// |
---|
442 | // Compute next the extreme rays of the cones |
---|
443 | ////////////////////////////////////////////////////////////////// |
---|
444 | if (er==1) |
---|
445 | { |
---|
446 | list extremerays; // keeps the result |
---|
447 | string polymake; // keeps polymake output |
---|
448 | // the header for ineq.polymake |
---|
449 | string head="_application polytope |
---|
450 | _version 2.2 |
---|
451 | _type RationalPolytope |
---|
452 | |
---|
453 | INEQUALITIES |
---|
454 | "; |
---|
455 | // the tail for both polymake files |
---|
456 | string tail=" |
---|
457 | EQUATIONS |
---|
458 | "; |
---|
459 | tail=tail+intmatToPolymake(linearity,"rays"); |
---|
460 | string ungleichungen; // keeps the inequalities for the polymake code |
---|
461 | intmat M; // the matrix keeping the inequalities |
---|
462 | // create the file ineq.output |
---|
463 | write(":w /tmp/ineq.output",""); |
---|
464 | int dimension; // keeps the dimension of the intersection the bad cones with the u11tobeseencone |
---|
465 | for (i=1;i<=size(ineq);i++) |
---|
466 | { |
---|
467 | i,". Cone of ",nrows(vertices); // indicate how many vertices have been dealt with |
---|
468 | ungleichungen=intmatToPolymake(ineq[i][1],"rays"); |
---|
469 | // write the inequalities to ineq.polymake and call polymake |
---|
470 | write(":w /tmp/ineq.polymake",head+ungleichungen+tail); |
---|
471 | ungleichungen=""; // clear ungleichungen |
---|
472 | system("sh","cd /tmp; /bin/rm ineq.output; polymake ineq.polymake VERTICES > ineq.output"); |
---|
473 | // read the result of polymake |
---|
474 | polymake=read("/tmp/ineq.output"); |
---|
475 | intmat VERT=polymakeToIntmat(polymake,"affine"); |
---|
476 | extremerays[i]=VERT; |
---|
477 | kill VERT; |
---|
478 | } |
---|
479 | for (i=1;i<=size(ineq);i++) |
---|
480 | { |
---|
481 | ineq[i]=ineq[i]+list(extremerays[i]); |
---|
482 | } |
---|
483 | } |
---|
484 | // delete the tmp-files, if polymakekeeptmpfiles is not set |
---|
485 | if (defined(polymakekeeptmpfiles)==0) |
---|
486 | { |
---|
487 | system("sh","/bin/rm /tmp/ineq.polymake"); |
---|
488 | system("sh","/bin/rm /tmp/ineq.output"); |
---|
489 | } |
---|
490 | // get the linearity space |
---|
491 | return(list(ineq,linearity)); |
---|
492 | } |
---|
493 | example |
---|
494 | { |
---|
495 | "EXAMPLE:"; |
---|
496 | echo=2; |
---|
497 | ring r=0,(x,y,z),dp; |
---|
498 | matrix M[4][1]=1,x,y,z; |
---|
499 | poly f=y3+x2+xy+2xz+yz+z2+1; |
---|
500 | // the Newton polytope of f is |
---|
501 | list np=newtonPolytope(f); |
---|
502 | // the Groebner fan of f, i.e. the normal fan of the Newton polytope |
---|
503 | list gf=normalFan(np[1],np[4],np[3],1,"x,y,z"); |
---|
504 | // the number of cones in the Groebner fan of f is: |
---|
505 | size(gf[1]); |
---|
506 | // the inequalities of the first cone as matrix are: |
---|
507 | print(gf[1][1][1]); |
---|
508 | // the inequalities of the first cone as string are: |
---|
509 | print(gf[1][1][2]); |
---|
510 | // the rows of the following matrix are the extreme rays of the first cone: |
---|
511 | print(gf[1][1][3]); |
---|
512 | // each cone contains the linearity space spanned by: |
---|
513 | print(gf[2]); |
---|
514 | } |
---|
515 | |
---|
516 | proc groebnerFan (poly f,list #) |
---|
517 | "USAGE: groebnerFan(f[,#]); f poly, # string |
---|
518 | RETURN: list, the ith entry of L[1] contains information about the ith cone in the Groebner fan |
---|
519 | dual to the ith vertex in the Newton polytope of the f |
---|
520 | @* L[1][i][1] = integer matrix representing the inequalities which describe the cone |
---|
521 | @* L[1][i][2] = a list which contains the inequalities represented by L[1][i][1] |
---|
522 | as a list of strings |
---|
523 | @* L[1][i][3] = an interger matrix whose rows are the extreme rays of the cone |
---|
524 | @* L[2] = is an integer matrix whose rows span the linearity space of the fan, |
---|
525 | i.e. the linear space which is contained in each cone |
---|
526 | @* L[3] = the Newton polytope of f in the format of the procedure newtonPolytope |
---|
527 | @* L[4] = integer matrix where each row represents the exponet vector of one monomial |
---|
528 | occuring in the input polynomial |
---|
529 | NOTE: - if you have alread computed the Newton polytope of f then you might want |
---|
530 | to use the procedure normalFan instead in order to avoid doing costly computation |
---|
531 | twice |
---|
532 | @* - the procedure calls for its computation polymake by Ewgenij Gawrilow, |
---|
533 | TU Berlin and Michael Joswig, so it only works if polymake is installed; |
---|
534 | see http://www.math.tu-berlin.de/polymake/ |
---|
535 | @* - the procedure creates the file /tmp/newtonPolytope.polymake which contains the |
---|
536 | Newton polytope of f in polymake format and which can be used for further |
---|
537 | computations with polymake |
---|
538 | @* - it is possible to give as an optional second argument as string which then will be |
---|
539 | used instead of 'newtonPolytope' in the name of the polymake output file |
---|
540 | EXAMPLE: example groebnerFan; shows an example" |
---|
541 | { |
---|
542 | int i,j; |
---|
543 | // compute the list of exponent vectors of the polynomial, which are the lattice points |
---|
544 | // whose convex hull is the Newton polytope of f |
---|
545 | intmat exponents[size(f)][nvars(basering)]; |
---|
546 | while (f!=0) |
---|
547 | { |
---|
548 | i++; |
---|
549 | exponents[i,1..nvars(basering)]=leadexp(f); |
---|
550 | f=f-lead(f); |
---|
551 | } |
---|
552 | if (size(#)==0) |
---|
553 | { |
---|
554 | #[1]="newtonPolytope"; |
---|
555 | } |
---|
556 | // call polymakePolytope with exponents |
---|
557 | list newtonp=polymakePolytope(exponents,"newtonPolytope"); |
---|
558 | // get the variables as string |
---|
559 | string variablen; |
---|
560 | for (i=1;i<=nvars(basering);i++) |
---|
561 | { |
---|
562 | variablen=variablen+string(var(i))+","; |
---|
563 | } |
---|
564 | variablen=variablen[1,size(variablen)-1]; |
---|
565 | // call normalFan in order to compute the Groebner fan |
---|
566 | list gf=normalFan(newtonp[1],newtonp[4],newtonp[3],1,variablen); |
---|
567 | // append newtonp to gf |
---|
568 | gf[3]=newtonp; |
---|
569 | // append the exponent vectors to gf |
---|
570 | gf[4]=exponents; |
---|
571 | return(gf); |
---|
572 | } |
---|
573 | example |
---|
574 | { |
---|
575 | "EXAMPLE:"; |
---|
576 | echo=2; |
---|
577 | ring r=0,(x,y,z),dp; |
---|
578 | matrix M[4][1]=1,x,y,z; |
---|
579 | poly f=y3+x2+xy+2xz+yz+z2+1; |
---|
580 | // the Newton polytope of f is |
---|
581 | list gf=groebnerFan(f); |
---|
582 | // the exponent vectors of f are ordered as follows |
---|
583 | gf[4]; |
---|
584 | // the first cone of the groebner fan has the inequalities |
---|
585 | gf[1][1][1]; |
---|
586 | // as a string they look like |
---|
587 | gf[1][1][2]; |
---|
588 | // and it has the extreme rays |
---|
589 | print(gf[1][1][3]); |
---|
590 | // the linearity space is spanned by |
---|
591 | print(gf[2]); |
---|
592 | // the vertices of the Newton polytope are: |
---|
593 | gf[3][1]; |
---|
594 | // its dimension is |
---|
595 | gf[3][2]; |
---|
596 | // np[3] contains information how the vertices are connected to each other, |
---|
597 | // e.g. the 1st vertex is connected to the 2nd, 3rd and 4th vertex |
---|
598 | gf[3][3][1]; |
---|
599 | } |
---|
600 | |
---|
601 | |
---|
602 | |
---|
603 | proc intmatToPolymake (intmat M,string art) |
---|
604 | "USAGE: intmatToPolymake(M,art); M intmat, art string |
---|
605 | ASSUME: - M is an integer matrix which should be transformed into polymake format; |
---|
606 | @* - art is one of the following strings: |
---|
607 | @* + 'rays' : indicating that a first column of 0's should be added |
---|
608 | @* + 'points' : indicating that a first column of 1's should be added |
---|
609 | RETURN: string, the matrix is transformed in a string and a first column has been added |
---|
610 | EXAMPLE: example intmatToPolymake; shows an example" |
---|
611 | { |
---|
612 | if (art=="rays") |
---|
613 | { |
---|
614 | string anf="0 "; |
---|
615 | } |
---|
616 | else |
---|
617 | { |
---|
618 | string anf="1 "; |
---|
619 | } |
---|
620 | string sp; |
---|
621 | int i,j; |
---|
622 | // create the lattice point list for polymake |
---|
623 | for (i=1;i<=nrows(M);i++) |
---|
624 | { |
---|
625 | sp=sp+anf; |
---|
626 | for (j=1;j<=ncols(M);j++) |
---|
627 | { |
---|
628 | sp=sp+string(M[i,j])+" "; |
---|
629 | if (j==ncols(M)) |
---|
630 | { |
---|
631 | sp=sp+" |
---|
632 | "; |
---|
633 | } |
---|
634 | } |
---|
635 | } |
---|
636 | return(sp); |
---|
637 | } |
---|
638 | example |
---|
639 | { |
---|
640 | "EXAMPLE:"; |
---|
641 | echo=2; |
---|
642 | intmat M[3][4]=1,2,3,4,5,6,7,8,9,10,11,12; |
---|
643 | intmatToPolymake(M,"rays"); |
---|
644 | intmatToPolymake(M,"points"); |
---|
645 | } |
---|
646 | |
---|
647 | proc polymakeToIntmat (string pm,string art) |
---|
648 | "USAGE: polymakeToIntmat(pm,art); pm, art string |
---|
649 | ASSUME: pm is the result of calling polymake with one 'argument' like VERTICES, AFFINE_HULL, etc., |
---|
650 | so that the first row of the string is the name of the corresponding 'argument' and |
---|
651 | the further rows contain the result which consists of vectors either over the integers |
---|
652 | or over the rationals |
---|
653 | RETURN: intmat, the rows of the matrix are basically the vectors in pm from the second row on |
---|
654 | where each row has been multiplied with the lowest common multiple of the |
---|
655 | denominators of its entries so as to be an integer matrix; moreover, |
---|
656 | if art=='affine', then the first column is omitted since we only want affine |
---|
657 | coordinates |
---|
658 | EXAMPLE: example polymakeToIntmat; shows an example" |
---|
659 | { |
---|
660 | // we need a line break |
---|
661 | string zeilenumbruch=" |
---|
662 | "; |
---|
663 | // remove the 'argment' name, i.e. the first row of pm |
---|
664 | while (pm[1]!=zeilenumbruch) |
---|
665 | { |
---|
666 | pm=stringdelete(pm,1); |
---|
667 | } |
---|
668 | pm=stringdelete(pm,1); |
---|
669 | // find out how many entries each vector has, namely one more than 'spaces' in a row |
---|
670 | int i=1; |
---|
671 | int s=1; |
---|
672 | int z=1; |
---|
673 | while (pm[i]!=zeilenumbruch) |
---|
674 | { |
---|
675 | if (pm[i]==" ") |
---|
676 | { |
---|
677 | s++; |
---|
678 | } |
---|
679 | i++; |
---|
680 | } |
---|
681 | // if we want to have affine coordinates |
---|
682 | if (art=="affine") |
---|
683 | { |
---|
684 | s--; // then there is one column less |
---|
685 | // and the entry of the first column (in the first row) has to be removed |
---|
686 | while (pm[1]!=" ") |
---|
687 | { |
---|
688 | pm=stringdelete(pm,1); |
---|
689 | } |
---|
690 | pm=stringdelete(pm,1); |
---|
691 | } |
---|
692 | // we add two line breaks at the end in order to have this as a stopping criterion |
---|
693 | pm=pm+zeilenumbruch+zeilenumbruch; |
---|
694 | // we now have to work through each row |
---|
695 | for (i=1;i<=size(pm);i++) |
---|
696 | { |
---|
697 | // if there are two consecutive line breaks we are done |
---|
698 | if ((pm[i]==zeilenumbruch) and (pm[i+1]==zeilenumbruch)) |
---|
699 | { |
---|
700 | i=size(pm)+1; |
---|
701 | } |
---|
702 | else |
---|
703 | { |
---|
704 | // a line break has to be replaced by a comma |
---|
705 | if (pm[i]==zeilenumbruch) |
---|
706 | { |
---|
707 | z++; |
---|
708 | pm[i]=","; |
---|
709 | // if we want to have affine coordinates, then we have to delete the first entry in each row |
---|
710 | if (art=="affine") |
---|
711 | { |
---|
712 | while (pm[i+1]!=" ") |
---|
713 | { |
---|
714 | pm=stringdelete(pm,i+1); |
---|
715 | } |
---|
716 | pm=stringdelete(pm,i+1); |
---|
717 | } |
---|
718 | } |
---|
719 | // a space has to be replaced by a comma |
---|
720 | if (pm[i]==" ") |
---|
721 | { |
---|
722 | pm[i]=","; |
---|
723 | } |
---|
724 | } |
---|
725 | } |
---|
726 | // if we have introduced superflous commata at the end, they should be removed |
---|
727 | while (pm[size(pm)]==",") |
---|
728 | { |
---|
729 | pm=stringdelete(pm,size(pm)); |
---|
730 | } |
---|
731 | // since the matrix could be over the rationals, we need a ring with rational coefficients |
---|
732 | ring zwischering=0,x,lp; |
---|
733 | // create the matrix with the elements of pm as entries |
---|
734 | execute("matrix mm["+string(z)+"]["+string(s)+"]="+pm+";"); |
---|
735 | // transform this into an integer matrix |
---|
736 | matrix M[1][ncols(mm)]; // takes a row of mm |
---|
737 | int cm; // takes a lowest common multiple |
---|
738 | // multiply each row by an integer such that its entries are integers |
---|
739 | for (int j=1;j<=nrows(mm);j++) |
---|
740 | { |
---|
741 | M=mm[j,1..ncols(mm)]; |
---|
742 | cm=commondenominator(M); |
---|
743 | for (i=1;i<=ncols(mm);i++) |
---|
744 | { |
---|
745 | mm[j,i]=cm*mm[j,i]; |
---|
746 | } |
---|
747 | } |
---|
748 | // transform the matrix mm into an integer matrix |
---|
749 | execute("intmat im["+string(z)+"]["+string(s)+"]="+string(mm)+";"); |
---|
750 | return(im); |
---|
751 | } |
---|
752 | example |
---|
753 | { |
---|
754 | "EXAMPLE:"; |
---|
755 | echo=2; |
---|
756 | // this is the usual output of some polymake computation |
---|
757 | string pm="VERTICES |
---|
758 | 0 1 3 5/3 1/3 -1 -23/3 -1/3 5/3 1/3 1 |
---|
759 | 0 1 3 -23/3 5/3 1 5/3 1/3 1/3 -1/3 -1 |
---|
760 | 0 1 1 1/3 -1/3 -1 5/3 1/3 -23/3 5/3 3 |
---|
761 | 0 1 1 5/3 -23/3 3 1/3 5/3 -1/3 1/3 -1 |
---|
762 | 0 1 -1 1/3 5/3 3 -1/3 -23/3 1/3 5/3 1 |
---|
763 | 0 1 -1 -1/3 1/3 1 1/3 5/3 5/3 -23/3 3 |
---|
764 | 0 1 -1 1 3 -5 -1 3 -1 1 -1 |
---|
765 | 0 1 -1 -1 -1 -1 1 1 3 3 -5 |
---|
766 | 0 1 -5 3 1 -1 3 -1 1 -1 -1 |
---|
767 | |
---|
768 | "; |
---|
769 | intmat PM=polymakeToIntmat(pm,"affine"); |
---|
770 | // note that the first column has been removed, since we asked for |
---|
771 | // affine coordinates, and the denominators have been cleared |
---|
772 | print(PM); |
---|
773 | } |
---|
774 | |
---|
775 | //////////////////////////////////////////////////////////////////////////////////////// |
---|
776 | /// PROCEDURES USING TOPCOM |
---|
777 | //////////////////////////////////////////////////////////////////////////////////////// |
---|
778 | |
---|
779 | proc triangulations (list polygon) |
---|
780 | "USAGE: triangulations(polygon); list polygon |
---|
781 | ASSUME: polygon is a list of integer vectors of the same size representing the affine |
---|
782 | coordinates of the lattice points |
---|
783 | PURPOSE: the procedure considers the marked polytope given as the convex hull of |
---|
784 | the lattice points and with these lattice points as markings; it then |
---|
785 | computes all possible triangulations of this marked polytope |
---|
786 | RETURN: list, each entry corresponds to one triangulation and the ith entry is |
---|
787 | itself a list of integer vectors of size three, where each integer |
---|
788 | vector defines one triangle in the triangulation by telling which |
---|
789 | points of the input are the vertices of the triangle |
---|
790 | NOTE: - the procedure calls for its computations the program points2triangs |
---|
791 | from the program topcom by Joerg Rambau, Universitaet Bayreuth; it |
---|
792 | therefore is necessary that this program is installed in order to use this |
---|
793 | procedure; see |
---|
794 | @* http://www.uni-bayreuth.de/departments/wirtschaftsmathematik/rambau/TOPCOM |
---|
795 | @* - the procedure creates the files /tmp/triangulationsinput and /tmp/triangulationsoutput; |
---|
796 | the former is used as input for points2triangs and the latter is its output |
---|
797 | containing the triangulations of corresponding to points in the format |
---|
798 | of points2triangs; if you wish to use this for further computations with topcom, |
---|
799 | you have to use the procedure polymakeKeepTmpFiles in before |
---|
800 | @* - note that an integer i in an integer vector representing a triangle refers to |
---|
801 | the ith lattice point, i.e. polygon[i]; this convention is different from |
---|
802 | TOPCOM's convention, where i would refer to the i-1st lattice point |
---|
803 | EXAMPLE: example triangulations; shows an example" |
---|
804 | { |
---|
805 | int i,j; |
---|
806 | // prepare the input for points2triangs by writing the input polygon in the |
---|
807 | // necessary format |
---|
808 | string spi="["; |
---|
809 | for (i=1;i<=size(polygon);i++) |
---|
810 | { |
---|
811 | polygon[i][size(polygon[i])+1]=1; |
---|
812 | spi=spi+"["+string(polygon[i])+"]"; |
---|
813 | if (i<size(polygon)) |
---|
814 | { |
---|
815 | spi=spi+","; |
---|
816 | } |
---|
817 | } |
---|
818 | spi=spi+"]"; |
---|
819 | write(":w /tmp/triangulationsinput",spi); |
---|
820 | // call points2triangs |
---|
821 | system("sh","cd /tmp; points2triangs < triangulationsinput > triangulationsoutput"); |
---|
822 | string p2t=read("/tmp/triangulationsoutput"); // takes the result of points2triangs |
---|
823 | // delete the tmp-files, if polymakekeeptmpfiles is not set |
---|
824 | if (defined(polymakekeeptmpfiles)==0) |
---|
825 | { |
---|
826 | system("sh","cd /tmp; rm -f triangulationsinput; rm -f triangulationsoutput"); |
---|
827 | } |
---|
828 | // preprocessing of p2t if points2triangs is version >= 0.15 brings p2t to the format of version 0.14 |
---|
829 | string np2t; // takes the triangulations in Singular format |
---|
830 | for (i=1;i<=size(p2t)-2;i++) |
---|
831 | { |
---|
832 | if ((p2t[i]==":") and (p2t[i+1]=="=") and (p2t[i+2]=="[")) |
---|
833 | { |
---|
834 | np2t=np2t+p2t[i]+p2t[i+1]; |
---|
835 | i=i+3; |
---|
836 | while (p2t[i]!=":") |
---|
837 | { |
---|
838 | i=i+1; |
---|
839 | } |
---|
840 | } |
---|
841 | else |
---|
842 | { |
---|
843 | if ((p2t[i]=="]") and (p2t[i+1]==";")) |
---|
844 | { |
---|
845 | np2t=np2t+p2t[i+1]; |
---|
846 | i=i+1; |
---|
847 | } |
---|
848 | else |
---|
849 | { |
---|
850 | np2t=np2t+p2t[i]; |
---|
851 | } |
---|
852 | } |
---|
853 | } |
---|
854 | if (p2t[size(p2t)-1]=="]") |
---|
855 | { |
---|
856 | np2t=np2t+p2t[size(p2t)]; |
---|
857 | } |
---|
858 | else |
---|
859 | { |
---|
860 | if (np2t[size(np2t)]!=";") |
---|
861 | { |
---|
862 | np2t=np2t+p2t[size(p2t)-1]+p2t[size(p2t)]; |
---|
863 | } |
---|
864 | } |
---|
865 | p2t=np2t; |
---|
866 | np2t=""; |
---|
867 | // transform the points2triangs output of version 0.14 into Singular format |
---|
868 | for (i=1;i<=size(p2t);i++) |
---|
869 | { |
---|
870 | if (p2t[i]=="=") |
---|
871 | { |
---|
872 | np2t=np2t+p2t[i]+"list("; |
---|
873 | i++; |
---|
874 | } |
---|
875 | else |
---|
876 | { |
---|
877 | if (p2t[i]!=":") |
---|
878 | { |
---|
879 | if ((p2t[i]=="}") and (p2t[i+1]=="}")) |
---|
880 | { |
---|
881 | np2t=np2t+"))"; |
---|
882 | i++; |
---|
883 | } |
---|
884 | else |
---|
885 | { |
---|
886 | if (p2t[i]=="{") |
---|
887 | { |
---|
888 | np2t=np2t+"intvec("; |
---|
889 | } |
---|
890 | else |
---|
891 | { |
---|
892 | if (p2t[i]=="}") |
---|
893 | { |
---|
894 | np2t=np2t+")"; |
---|
895 | } |
---|
896 | else |
---|
897 | { |
---|
898 | np2t=np2t+p2t[i]; |
---|
899 | } |
---|
900 | } |
---|
901 | } |
---|
902 | } |
---|
903 | } |
---|
904 | } |
---|
905 | list T; |
---|
906 | execute(np2t); |
---|
907 | // raise each index by one |
---|
908 | for (i=1;i<=size(T);i++) |
---|
909 | { |
---|
910 | for (j=1;j<=size(T[i]);j++) |
---|
911 | { |
---|
912 | T[i][j]=T[i][j]+1; |
---|
913 | } |
---|
914 | } |
---|
915 | return(T); |
---|
916 | } |
---|
917 | example |
---|
918 | { |
---|
919 | "EXAMPLE:"; |
---|
920 | echo=2; |
---|
921 | // the lattice points of the unit square in the plane |
---|
922 | list polygon=intvec(0,0),intvec(0,1),intvec(1,0),intvec(1,1); |
---|
923 | // the triangulations of this lattice point configuration are computed |
---|
924 | list triang=triangulations(polygon); |
---|
925 | triang; |
---|
926 | } |
---|
927 | |
---|
928 | proc secondaryPolytope (list polygon,list #) |
---|
929 | "USAGE: secondaryPolytope(polygon[,#]); list polygon, list # |
---|
930 | ASSUME: - polygon is a list of integer vectors of the same size representing the affine |
---|
931 | coordinates of lattice points |
---|
932 | @* - if the triangulations of the corresponding polygon have already been computed |
---|
933 | with the procedure triangulations then these can be given as a second (optional) |
---|
934 | argument in order to avoid doing this computation again |
---|
935 | PURPOSE: the procedure considers the marked polytope given as the convex hull of |
---|
936 | the lattice points and with these lattice points as markings; it then |
---|
937 | computes the lattice points of the secondary polytope given by this |
---|
938 | marked polytope which correspond to the triangulations computed by |
---|
939 | the procedure triangulations |
---|
940 | RETURN: list, say L, such that: |
---|
941 | @* L[1] = intmat, each row gives the affine coordinates of a lattice point |
---|
942 | in the secondary polytope given by the marked |
---|
943 | polytope corresponding to polygon |
---|
944 | @* L[2] = the list of corresponding triangulations |
---|
945 | NOTE: if the triangluations are not handed over as optional argument the procedure calls |
---|
946 | for its computation of these triangulations the program points2triangs |
---|
947 | from the program topcom by Joerg Rambau, Universitaet Bayreuth; it |
---|
948 | therefore is necessary that this program is installed in order to use this |
---|
949 | procedure; see |
---|
950 | @* http://www.uni-bayreuth.de/departments/wirtschaftsmathematik/rambau/TOPCOM |
---|
951 | EXAMPLE: example secondaryPolytope; shows an example" |
---|
952 | { |
---|
953 | // compute the triangulations of the point configuration with points2triangs |
---|
954 | if (size(#)==0) |
---|
955 | { |
---|
956 | list triangs=triangulations(polygon); |
---|
957 | } |
---|
958 | else |
---|
959 | { |
---|
960 | list triangs=#; |
---|
961 | } |
---|
962 | int i,j,k,l; |
---|
963 | intmat N[2][2]; // is used to compute areas of triangles |
---|
964 | intvec vertex; // stores a point in the secondary polytope as intermediate result |
---|
965 | int eintrag; |
---|
966 | int halt; |
---|
967 | intmat secpoly[size(triangs)][size(polygon)]; // stores all lattice points of the secondary polytope |
---|
968 | // consider each triangulation and compute the corresponding point in the secondary polytope |
---|
969 | for (i=1;i<=size(triangs);i++) |
---|
970 | { |
---|
971 | // for each triangulation we have to compute the coordinates corresponding to each marked point |
---|
972 | for (j=1;j<=size(polygon);j++) |
---|
973 | { |
---|
974 | eintrag=0; |
---|
975 | // for each marked point we have to consider all triangles in the triangulation |
---|
976 | // which involve this particular point |
---|
977 | for (k=1;k<=size(triangs[i]);k++) |
---|
978 | { |
---|
979 | halt=0; |
---|
980 | for (l=1;(l<=3) and (halt==0);l++) |
---|
981 | { |
---|
982 | if (triangs[i][k][l]==j) |
---|
983 | { |
---|
984 | halt=1; |
---|
985 | N[1,1]=polygon[triangs[i][k][3]][1]-polygon[triangs[i][k][1]][1]; |
---|
986 | N[1,2]=polygon[triangs[i][k][2]][1]-polygon[triangs[i][k][1]][1]; |
---|
987 | N[2,1]=polygon[triangs[i][k][3]][2]-polygon[triangs[i][k][1]][2]; |
---|
988 | N[2,2]=polygon[triangs[i][k][2]][2]-polygon[triangs[i][k][1]][2]; |
---|
989 | eintrag=eintrag+abs(det(N)); |
---|
990 | } |
---|
991 | } |
---|
992 | } |
---|
993 | vertex[j]=eintrag; |
---|
994 | } |
---|
995 | secpoly[i,1..size(polygon)]=vertex; |
---|
996 | } |
---|
997 | return(list(secpoly,triangs)); |
---|
998 | } |
---|
999 | example |
---|
1000 | { |
---|
1001 | "EXAMPLE:"; |
---|
1002 | echo=2; |
---|
1003 | // the lattice points of the unit square in the plane |
---|
1004 | list polygon=intvec(0,0),intvec(0,1),intvec(1,0),intvec(1,1); |
---|
1005 | // the secondary polytope of this lattice point configuration is computed |
---|
1006 | list secpoly=secondaryPolytope(polygon); |
---|
1007 | // the points in the secondary polytope |
---|
1008 | print(secpoly[1]); |
---|
1009 | // the corresponding triangulations |
---|
1010 | secpoly[2]; |
---|
1011 | } |
---|
1012 | |
---|
1013 | //////////////////////////////////////////////////////////////////////////////////////// |
---|
1014 | /// PROCEDURES USING POLYMAKE AND TOPCOM |
---|
1015 | //////////////////////////////////////////////////////////////////////////////////////// |
---|
1016 | |
---|
1017 | proc secondaryFan (list polygon,list #) |
---|
1018 | "USAGE: secondaryFan(polygon[,#]); list polygon, list # |
---|
1019 | ASSUME: - polygon is a list of integer vectors of the same size representing the affine |
---|
1020 | coordinates of lattice points |
---|
1021 | @* - if the triangulations of the corresponding polygon have already been computed |
---|
1022 | with the procedure triangulations then these can be given as a second (optional) |
---|
1023 | argument in order to avoid doing this computation again |
---|
1024 | PURPOSE: the procedure considers the marked polytope given as the convex hull of |
---|
1025 | the lattice points and with these lattice points as markings; it then |
---|
1026 | computes the lattice points of the secondary polytope given by this |
---|
1027 | marked polytope which correspond to the triangulations computed by |
---|
1028 | the procedure triangulations |
---|
1029 | RETURN: list, the ith entry of L[1] contains information about the ith cone in the |
---|
1030 | secondary fan of the polygon, i.e. the cone dual to the ith vertex of the |
---|
1031 | secondary polytope |
---|
1032 | @* L[1][i][1] = integer matrix representing the inequalities which describe the |
---|
1033 | cone dual to the ith vertex |
---|
1034 | @* L[1][i][2] = a list which contains the inequalities represented by L[i][1] |
---|
1035 | as a list of strings, where we use the variables x(1),...,x(n) |
---|
1036 | @* L[1][i][3] = only present if 'er' is set to 1; in that case it is an interger matrix |
---|
1037 | whose rows are the extreme rays of the cone |
---|
1038 | @* L[2] = is an integer matrix whose rows span the linearity space of the fan, |
---|
1039 | i.e. the linear space which is contained in each cone |
---|
1040 | @* L[3] = the secondary polytope in the format of the procedure polymakePolytope |
---|
1041 | @* L[4] = the list of triangulations corresponding to the vertices |
---|
1042 | of the secondary polytope |
---|
1043 | NOTE: - the procedure calls for its computation polymake by Ewgenij Gawrilow, |
---|
1044 | TU Berlin and Michael Joswig, so it only works if polymake is installed; |
---|
1045 | see http://www.math.tu-berlin.de/polymake/ |
---|
1046 | @* - in the optional argument # it is possible to hand over other names for the |
---|
1047 | variables to be used -- be carful, the format must be correct and that is |
---|
1048 | not tested, e.g. if you want the variable names to be u00,u10,u01,u11 |
---|
1049 | then you must hand over the string u11,u10,u01,u11 |
---|
1050 | @* - if the triangluations are not handed over as optional argument the procedure calls |
---|
1051 | for its computation of these triangulations the program points2triangs |
---|
1052 | from the program topcom by Joerg Rambau, Universitaet Bayreuth; it |
---|
1053 | therefore is necessary that this program is installed in order to use this |
---|
1054 | procedure; see |
---|
1055 | @* http://www.uni-bayreuth.de/departments/wirtschaftsmathematik/rambau/TOPCOM |
---|
1056 | EXAMPLE: example secondaryFan; shows an example" |
---|
1057 | { |
---|
1058 | if (size(#)==0) |
---|
1059 | { |
---|
1060 | list triang=triangulations(polygon); |
---|
1061 | } |
---|
1062 | else |
---|
1063 | { |
---|
1064 | list triang=#[1]; |
---|
1065 | } |
---|
1066 | list sp=secondaryPolytope(polygon,triang); |
---|
1067 | list spp=polymakePolytope(sp[1]); |
---|
1068 | list sf=normalFan(spp[1],spp[4],spp[3],1); |
---|
1069 | return(list(sf[1],sf[2],spp,triang)); |
---|
1070 | } |
---|
1071 | example |
---|
1072 | { |
---|
1073 | "EXAMPLE:"; |
---|
1074 | echo=2; |
---|
1075 | // the lattice points of the unit square in the plane |
---|
1076 | list polygon=intvec(0,0),intvec(0,1),intvec(1,0),intvec(1,1); |
---|
1077 | // the secondary polytope of this lattice point configuration is computed |
---|
1078 | list secfan=secondaryFan(polygon); |
---|
1079 | // the number of cones in the secondary fan of the polygon |
---|
1080 | size(secfan[1]); |
---|
1081 | // the inequalities of the first cone as matrix are: |
---|
1082 | print(secfan[1][1][1]); |
---|
1083 | // the inequalities of the first cone as string are: |
---|
1084 | print(secfan[1][1][2]); |
---|
1085 | // the rows of the following matrix are the extreme rays of the first cone: |
---|
1086 | print(secfan[1][1][3]); |
---|
1087 | // each cone contains the linearity space spanned by: |
---|
1088 | print(secfan[2]); |
---|
1089 | // the points in the secondary polytope |
---|
1090 | print(secfan[3][1]); |
---|
1091 | // the corresponding triangulations |
---|
1092 | secfan[4]; |
---|
1093 | } |
---|
1094 | |
---|
1095 | |
---|
1096 | //////////////////////////////////////////////////////////////////////////////////////// |
---|
1097 | /// PROCEDURES CONCERNED WITH PLANAR POLYGONS |
---|
1098 | //////////////////////////////////////////////////////////////////////////////////////// |
---|
1099 | |
---|
1100 | proc cycleLength (list boundary,intvec interior) |
---|
1101 | "USAGE: cycleLength(boundary,interior); list boundary, intvec interior |
---|
1102 | ASSUME: boundary is a list of integer vectors describing a cycle in some convex lattice |
---|
1103 | polygon around the lattice point interior ordered clock wise |
---|
1104 | RETURN: string, the cycle length of the corresponding cycle in the dual tropical curve |
---|
1105 | EXAMPLE: example cycleLength; shows an example" |
---|
1106 | { |
---|
1107 | int j; |
---|
1108 | // create a ring whose variables are indexed by the points in boundary resp. by interior |
---|
1109 | string rst="ring cyclering=0,(u"+string(interior[1])+string(interior[2]); |
---|
1110 | for (j=1;j<=size(boundary);j++) |
---|
1111 | { |
---|
1112 | rst=rst+",u"+string(boundary[j][1])+string(boundary[j][2]); |
---|
1113 | } |
---|
1114 | rst=rst+"),lp;"; |
---|
1115 | execute(rst); |
---|
1116 | // add the first and second point at the end of boundary |
---|
1117 | boundary[size(boundary)+1]=boundary[1]; |
---|
1118 | boundary[size(boundary)+1]=boundary[2]; |
---|
1119 | poly cl,summand; // takes the cycle length |
---|
1120 | matrix N1[2][2]; // used to compute the area of a triangle |
---|
1121 | matrix N2[2][2]; // used to compute the area of a triangle |
---|
1122 | matrix N3[2][2]; // used to compute the area of a triangle |
---|
1123 | // for each original point in the boundary compute its contribution to the cycle |
---|
1124 | for (j=2;j<=size(boundary)-1;j++) |
---|
1125 | { |
---|
1126 | N1=boundary[j-1]-interior,boundary[j]-interior; |
---|
1127 | N2=boundary[j]-interior,boundary[j+1]-interior; |
---|
1128 | N3=boundary[j+1]-interior,boundary[j-1]-interior; |
---|
1129 | execute("summand=-u"+string(boundary[j][1])+string(boundary[j][2])+"+u"+string(interior[1])+string(interior[2])+";"); |
---|
1130 | summand=summand*(det(N1)+det(N2)+det(N3))/(det(N1)*det(N2)); |
---|
1131 | cl=cl+summand; |
---|
1132 | } |
---|
1133 | return(string(cl)); |
---|
1134 | } |
---|
1135 | example |
---|
1136 | { |
---|
1137 | "EXAMPLE:"; |
---|
1138 | echo=2; |
---|
1139 | // the integer vectors in boundary are lattice points on the boundary |
---|
1140 | // of a convex lattice polygon in the plane |
---|
1141 | list boundary=intvec(0,0),intvec(0,1),intvec(0,2),intvec(2,2), |
---|
1142 | intvec(2,1),intvec(2,0); |
---|
1143 | // interior is a lattice point in the interior of this lattice polygon |
---|
1144 | intvec interior=1,1; |
---|
1145 | // compute the general cycle length of a cycle of the corresponding cycle |
---|
1146 | // in the dual tropical curve, note that (0,1) and (2,1) do not contribute |
---|
1147 | cycleLength(boundary,interior); |
---|
1148 | } |
---|
1149 | |
---|
1150 | proc splitPolygon (list markings) |
---|
1151 | "USAGE: splitPolygon (markings); markings list |
---|
1152 | ASSUME: markings is a list of integer vectors representing lattice points in the plane |
---|
1153 | which we consider as the marked points of the convex lattice polytope spanned by them |
---|
1154 | PURPOSE: split the marked points in the vertices, the points on the facets which are not vertices, |
---|
1155 | and the interior points |
---|
1156 | RETURN: list, L consisting of three lists |
---|
1157 | @* L[1] : represents the vertices the polygon ordered clockwise |
---|
1158 | @* L[1][i][1] = intvec, the coordinates of the ith vertex |
---|
1159 | @* L[1][i][2] = int, the position of L[1][i][1] in markings |
---|
1160 | @* L[2][i] : represents the lattice points on the facet of the polygon with |
---|
1161 | endpoints L[1][i] and L[1][i+1] (i considered modulo size(L[1])) |
---|
1162 | @* L[2][i][j][1] = intvec, the coordinates of the jth lattice point on that facet |
---|
1163 | @* L[2][i][j][2] = int, the position of L[2][i][j][1] in markings |
---|
1164 | @* L[3] : represents the interior lattice points of the polygon |
---|
1165 | @* L[3][i][1] = intvec, the coordinates of the ith interior point |
---|
1166 | @* L[3][i][2] = int, the position of L[3][i][1] in markings |
---|
1167 | EXAMPLE: example splitPolygon; shows an example" |
---|
1168 | { |
---|
1169 | list vert; // stores the result |
---|
1170 | // compute the boundary of the polygon in an oriented way |
---|
1171 | list pb=findOrientedBoundary(markings); |
---|
1172 | // the vertices are just the second entry of pb |
---|
1173 | vert[1]=pb[2]; |
---|
1174 | int i,j,k; // indices |
---|
1175 | list boundary; // stores the points on the facets of the polygon which are not vertices |
---|
1176 | // append to the boundary points as well as to the vertices the first vertex a second time |
---|
1177 | pb[1]=pb[1]+list(pb[1][1]); |
---|
1178 | pb[2]=pb[2]+list(pb[2][1]); |
---|
1179 | // for each vertex find all points on the facet of the polygon with this vertex |
---|
1180 | // and the next vertex as endpoints |
---|
1181 | int z=2; |
---|
1182 | for (i=1;i<=size(vert[1]);i++) |
---|
1183 | { |
---|
1184 | j=1; |
---|
1185 | list facet; // stores the points on this facet which are not vertices |
---|
1186 | // while the next vertex is not reached, store the boundary lattice point |
---|
1187 | while (pb[1][z]!=pb[2][i+1]) |
---|
1188 | { |
---|
1189 | facet[j]=pb[1][z]; |
---|
1190 | j++; |
---|
1191 | z++; |
---|
1192 | } |
---|
1193 | // store the points on the ith facet as boundary[i] |
---|
1194 | boundary[i]=facet; |
---|
1195 | kill facet; |
---|
1196 | z++; |
---|
1197 | } |
---|
1198 | // store the information on the boundary in vert[2] |
---|
1199 | vert[2]=boundary; |
---|
1200 | // find the remaining points in the input which are not on the boundary by checking |
---|
1201 | // for each point in markings if it is contained in pb[1] |
---|
1202 | list interior=markings; |
---|
1203 | for (i=size(interior);i>=1;i--) |
---|
1204 | { |
---|
1205 | for (j=1;j<=size(pb[1])-1;j++) |
---|
1206 | { |
---|
1207 | if (interior[i]==pb[1][j]) |
---|
1208 | { |
---|
1209 | interior=delete(interior,i); |
---|
1210 | j=size(pb[1]); |
---|
1211 | } |
---|
1212 | } |
---|
1213 | } |
---|
1214 | // store the interior points in vert[3] |
---|
1215 | vert[3]=interior; |
---|
1216 | // add to each point in vert the index which it gets from its position in the input markings; |
---|
1217 | // do so for ver[1] |
---|
1218 | for (i=1;i<=size(vert[1]);i++) |
---|
1219 | { |
---|
1220 | j=1; |
---|
1221 | while (markings[j]!=vert[1][i]) |
---|
1222 | { |
---|
1223 | j++; |
---|
1224 | } |
---|
1225 | vert[1][i]=list(vert[1][i],j); |
---|
1226 | } |
---|
1227 | // do so for ver[2] |
---|
1228 | for (i=1;i<=size(vert[2]);i++) |
---|
1229 | { |
---|
1230 | for (k=1;k<=size(vert[2][i]);k++) |
---|
1231 | { |
---|
1232 | j=1; |
---|
1233 | while (markings[j]!=vert[2][i][k]) |
---|
1234 | { |
---|
1235 | j++; |
---|
1236 | } |
---|
1237 | vert[2][i][k]=list(vert[2][i][k],j); |
---|
1238 | } |
---|
1239 | } |
---|
1240 | // do so for ver[3] |
---|
1241 | for (i=1;i<=size(vert[3]);i++) |
---|
1242 | { |
---|
1243 | j=1; |
---|
1244 | while (markings[j]!=vert[3][i]) |
---|
1245 | { |
---|
1246 | j++; |
---|
1247 | } |
---|
1248 | vert[3][i]=list(vert[3][i],j); |
---|
1249 | } |
---|
1250 | return(vert); |
---|
1251 | } |
---|
1252 | example |
---|
1253 | { |
---|
1254 | "EXAMPLE:"; |
---|
1255 | echo=2; |
---|
1256 | // the lattice polygon spanned by the points (0,0), (3,0) and (0,3) |
---|
1257 | // with all integer points as markings |
---|
1258 | list polygon=intvec(1,1),intvec(3,0),intvec(2,0),intvec(1,0), |
---|
1259 | intvec(0,0),intvec(2,1),intvec(0,1),intvec(1,2), |
---|
1260 | intvec(0,2),intvec(0,3); |
---|
1261 | // split the polygon in its vertices, its facets and its interior points |
---|
1262 | list sp=splitPolygon(polygon); |
---|
1263 | // the vertices |
---|
1264 | sp[1]; |
---|
1265 | // the points on facets which are not vertices |
---|
1266 | sp[2]; |
---|
1267 | // the interior points |
---|
1268 | sp[3]; |
---|
1269 | } |
---|
1270 | |
---|
1271 | |
---|
1272 | proc eta (list triang,list polygon) |
---|
1273 | "USAGE: eta(triang,polygon); triang, polygon list |
---|
1274 | ASSUME: polygon has the format of the output of splitPolygon, i.e. it is a list with three |
---|
1275 | entries describing a convex lattice polygon in the following way: |
---|
1276 | @* polygon[1] : is a list of lists; for each i the entry polygon[1][i][1] is a lattice point which is |
---|
1277 | a vertex of the lattice polygon, and polygon[1][i][2] is an integer assigned to |
---|
1278 | this lattice point as identifying index |
---|
1279 | @* polygon[2] : is a list of lists; for each vertex of the polygon, i.e. for each entry in polygon[1], |
---|
1280 | it contains a list polygon[2][i], which contains the lattice points on the facet |
---|
1281 | with endpoints polygon[1][i] and polygon[1][i+1] - i considered mod size(polygon[1]); |
---|
1282 | each such lattice point contributes an entry polygon[2][i][j][1] which is an integer |
---|
1283 | vector giving the coordinate of the lattice point and an entry polygon[2][i][j][2] |
---|
1284 | which is the identifying index |
---|
1285 | @* polygon[3] : is a list of lists, where each entry corresponds to a lattice point in the |
---|
1286 | interior of the polygon, with polygon[3][j][1] being the coordinates of the point |
---|
1287 | and polygon[3][j][2] being the identifying index; |
---|
1288 | @* triang is a list of integer vectors all of size three describing a triangulation of the |
---|
1289 | polygon described by polygon; if an entry of triang is the vector (i,j,k) then the triangle |
---|
1290 | is build by the vertices with indices i, j and k |
---|
1291 | RETURN: intvec, the integer vector eta describing that vertex of the Newton polytope discriminant |
---|
1292 | of the polygone whose dual cone in the Groebner fan contains the cone of the |
---|
1293 | secondary fan of the polygon corresponding to the given triangulation |
---|
1294 | NOTE: for a better description of eta see either Gelfand, Kapranov, Zelevinski: Discriminants, |
---|
1295 | Resultants and multidimensional Determinants. Chapter 10. |
---|
1296 | EXAMPLE: example eta; shows an example" |
---|
1297 | { |
---|
1298 | int i,j,k,l,m,n; // index variables |
---|
1299 | list ordpolygon; // stores the lattice points in the order used in the triangulation |
---|
1300 | list triangarea; // stores the areas of the triangulations |
---|
1301 | intmat N[2][2]; // used to compute triangle areas |
---|
1302 | // 1) store the lattice points in the order used in the triangulation |
---|
1303 | // go first through all vertices of the polytope |
---|
1304 | for (j=1;j<=size(polygon[1]);j++) |
---|
1305 | { |
---|
1306 | ordpolygon[polygon[1][j][2]]=polygon[1][j][1]; |
---|
1307 | } |
---|
1308 | // then consider all inner points |
---|
1309 | for (j=1;j<=size(polygon[3]);j++) |
---|
1310 | { |
---|
1311 | ordpolygon[polygon[3][j][2]]=polygon[3][j][1]; |
---|
1312 | } |
---|
1313 | // finally consider all lattice points on the boundary which are not vertices |
---|
1314 | for (j=1;j<=size(polygon[2]);j++) |
---|
1315 | { |
---|
1316 | for (i=1;i<=size(polygon[2][j]);i++) |
---|
1317 | { |
---|
1318 | ordpolygon[polygon[2][j][i][2]]=polygon[2][j][i][1]; |
---|
1319 | } |
---|
1320 | } |
---|
1321 | // 2) compute for each triangle in the triangulation the area of the triangle |
---|
1322 | for (i=1;i<=size(triang);i++) |
---|
1323 | { |
---|
1324 | // Note that the ith lattice point in orderedpolygon has the number i-1 in the triangulation! |
---|
1325 | N=ordpolygon[triang[i][1]]-ordpolygon[triang[i][2]],ordpolygon[triang[i][1]]-ordpolygon[triang[i][3]]; |
---|
1326 | triangarea[i]=abs(det(N)); |
---|
1327 | } |
---|
1328 | intvec ETA; // stores the eta_ij |
---|
1329 | int etaij; // stores the part of eta_ij during computations which comes from triangle areas |
---|
1330 | int seitenlaenge; // stores the part of eta_ij during computations which comes from boundary facets |
---|
1331 | list seiten; // stores the lattice points on facets of the polygon |
---|
1332 | intvec v; // used to compute a facet length |
---|
1333 | // 3) store first in seiten[i] all lattice points on the facet connecting the ith vertex, |
---|
1334 | // i.e. polygon[1][i], with the i+1st vertex, i.e. polygon[1][i+1], where we replace i+1 |
---|
1335 | // 1 if i=size(polygon[1]); |
---|
1336 | // then append the last entry of seiten once more at the very beginning of seiten, so |
---|
1337 | // that the index is shifted by one |
---|
1338 | for (i=1;i<=size(polygon[1]);i++) |
---|
1339 | { |
---|
1340 | if (i<size(polygon[1])) |
---|
1341 | { |
---|
1342 | seiten[i]=list(polygon[1][i])+polygon[2][i]+list(polygon[1][i+1]); |
---|
1343 | } |
---|
1344 | else |
---|
1345 | { |
---|
1346 | seiten[i]=list(polygon[1][i])+polygon[2][i]+list(polygon[1][1]); |
---|
1347 | } |
---|
1348 | } |
---|
1349 | seiten=insert(seiten,seiten[size(seiten)],0); |
---|
1350 | // 4) compute the eta_ij for all vertices of the polygon |
---|
1351 | for (j=1;j<=size(polygon[1]);j++) |
---|
1352 | { |
---|
1353 | // the vertex itself contributes a 1 |
---|
1354 | etaij=1; |
---|
1355 | // check for each triangle in the triangulation ... |
---|
1356 | for (k=1;k<=size(triang);k++) |
---|
1357 | { |
---|
1358 | // ... if the vertex is actually a vertex of the triangle ... |
---|
1359 | if ((polygon[1][j][2]==triang[k][1]) or (polygon[1][j][2]==triang[k][2]) or (polygon[1][j][2]==triang[k][3])) |
---|
1360 | { |
---|
1361 | // ... if so, add the area of the triangle to etaij |
---|
1362 | etaij=etaij+triangarea[k]; |
---|
1363 | // then check if that triangle has a facet which is contained in one of the |
---|
1364 | // two facets of the polygon which are adjecent to the given vertex ... |
---|
1365 | // these two facets are seiten[j] and seiten[j+1] |
---|
1366 | for (n=j;n<=j+1;n++) |
---|
1367 | { |
---|
1368 | // check for each lattice point in the facet of the polygon ... |
---|
1369 | for (l=1;l<=size(seiten[n]);l++) |
---|
1370 | { |
---|
1371 | // ... and for each lattice point in the triangle ... |
---|
1372 | for (m=1;m<=size(triang[k]);m++) |
---|
1373 | { |
---|
1374 | // ... if they coincide and are not the vertex itself ... |
---|
1375 | if ((seiten[n][l][2]==triang[k][m]) and (seiten[n][l][2]!=polygon[1][j][2])) |
---|
1376 | { |
---|
1377 | // if so, then compute the vector pointing from this lattice point to the vertex |
---|
1378 | v=polygon[1][j][1]-seiten[n][l][1]; |
---|
1379 | // and the lattice length of this vector has to be subtracted from etaij |
---|
1380 | etaij=etaij-abs(gcd(v[1],v[2])); |
---|
1381 | } |
---|
1382 | } |
---|
1383 | } |
---|
1384 | } |
---|
1385 | } |
---|
1386 | } |
---|
1387 | // store etaij in the list |
---|
1388 | ETA[polygon[1][j][2]]=etaij; |
---|
1389 | } |
---|
1390 | // 5) compute the eta_ij for all lattice points on the facets of the polygon which are not vertices, |
---|
1391 | // these are the lattice points in polygon[2][1] to polygon[2][size(polygon[1])] |
---|
1392 | for (i=1;i<=size(polygon[2]);i++) |
---|
1393 | { |
---|
1394 | for (j=1;j<=size(polygon[2][i]);j++) |
---|
1395 | { |
---|
1396 | // initialise etaij |
---|
1397 | etaij=0; |
---|
1398 | // initialise seitenlaenge |
---|
1399 | seitenlaenge=0; |
---|
1400 | // check for each triangle in the triangulation ... |
---|
1401 | for (k=1;k<=size(triang);k++) |
---|
1402 | { |
---|
1403 | // ... if the vertex is actually a vertex of the triangle ... |
---|
1404 | if ((polygon[2][i][j][2]==triang[k][1]) or (polygon[2][i][j][2]==triang[k][2]) or (polygon[2][i][j][2]==triang[k][3])) |
---|
1405 | { |
---|
1406 | // ... if so, add the area of the triangle to etaij |
---|
1407 | etaij=etaij+triangarea[k]; |
---|
1408 | // then check if that triangle has a facet which is contained in the |
---|
1409 | // facet of the polygon which contains the lattice point in question, |
---|
1410 | // this is the facet seiten[i+1]; |
---|
1411 | // check for each lattice point in the facet of the polygon ... |
---|
1412 | for (l=1;l<=size(seiten[i+1]);l++) |
---|
1413 | { |
---|
1414 | // ... and for each lattice point in the triangle ... |
---|
1415 | for (m=1;m<=size(triang[k]);m++) |
---|
1416 | { |
---|
1417 | // ... if they coincide and are not the vertex itself ... |
---|
1418 | if ((seiten[i+1][l][2]==triang[k][m]) and (seiten[i+1][l][2]!=polygon[2][i][j][2])) |
---|
1419 | { |
---|
1420 | // if so, then compute the vector pointing from this lattice point to the vertex |
---|
1421 | v=polygon[2][i][j][1]-seiten[i+1][l][1]; |
---|
1422 | // and the lattice length of this vector contributes to seitenlaenge |
---|
1423 | seitenlaenge=seitenlaenge+abs(gcd(v[1],v[2])); |
---|
1424 | } |
---|
1425 | } |
---|
1426 | } |
---|
1427 | } |
---|
1428 | } |
---|
1429 | // if the lattice point was a vertex of any triangle in the triangulation ... |
---|
1430 | if (etaij!=0) |
---|
1431 | { |
---|
1432 | // then eta_ij is the sum of the triangle areas minus seitenlaenge |
---|
1433 | ETA[polygon[2][i][j][2]]=etaij-seitenlaenge; |
---|
1434 | } |
---|
1435 | else |
---|
1436 | { |
---|
1437 | // otherwise it is just zero |
---|
1438 | ETA[polygon[2][i][j][2]]=0; |
---|
1439 | } |
---|
1440 | } |
---|
1441 | } |
---|
1442 | // 4) compute the eta_ij for all inner lattice points of the polygon |
---|
1443 | for (j=1;j<=size(polygon[3]);j++) |
---|
1444 | { |
---|
1445 | // initialise etaij |
---|
1446 | etaij=0; |
---|
1447 | // check for each triangle in the triangulation ... |
---|
1448 | for (k=1;k<=size(triang);k++) |
---|
1449 | { |
---|
1450 | // ... if the vertex is actually a vertex of the triangle ... |
---|
1451 | if ((polygon[3][j][2]==triang[k][1]) or (polygon[3][j][2]==triang[k][2]) or (polygon[3][j][2]==triang[k][3])) |
---|
1452 | { |
---|
1453 | // ... if so, add the area of the triangle to etaij |
---|
1454 | etaij=etaij+triangarea[k]; |
---|
1455 | } |
---|
1456 | } |
---|
1457 | // store etaij in ETA |
---|
1458 | ETA[polygon[3][j][2]]=etaij; |
---|
1459 | } |
---|
1460 | return(ETA); |
---|
1461 | } |
---|
1462 | example |
---|
1463 | { |
---|
1464 | "EXAMPLE:"; |
---|
1465 | echo=2; |
---|
1466 | // the lattice polygon spanned by the points (0,0), (3,0) and (0,3) |
---|
1467 | // with all integer points as markings |
---|
1468 | list polygon=intvec(1,1),intvec(3,0),intvec(2,0),intvec(1,0), |
---|
1469 | intvec(0,0),intvec(2,1),intvec(0,1),intvec(1,2), |
---|
1470 | intvec(0,2),intvec(0,3); |
---|
1471 | // split the polygon in its vertices, its facets and its interior points |
---|
1472 | list sp=splitPolygon(polygon); |
---|
1473 | // define a triangulation by connecting the only interior point |
---|
1474 | // with the vertices |
---|
1475 | list triang=intvec(1,2,5),intvec(1,5,10),intvec(1,5,10); |
---|
1476 | // compute the eta-vector of this triangulation |
---|
1477 | eta(triang,sp); |
---|
1478 | } |
---|
1479 | |
---|
1480 | proc findOrientedBoundary (list polygon) |
---|
1481 | "USAGE: findOrientedBoundary(polygon); polygon list |
---|
1482 | ASSUME: polygon is a list of integer vectors defining integer lattice points in the plane |
---|
1483 | RETURN: list, l with the followin interpretation |
---|
1484 | @* l[1] = list of integer vectors such that the polygonal path defined by |
---|
1485 | these is the boundary of the convex hull of the lattice points in polygon |
---|
1486 | @* l[2] = list, the redundant points in l[1] have been removed |
---|
1487 | EXAMPLE: example findOrientedBoundary; shows an example" |
---|
1488 | { |
---|
1489 | // Order the vertices such that passing from one to the next we travel along |
---|
1490 | // the boundary of the convex hull of the vertices clock wise |
---|
1491 | int d,k,i,j; |
---|
1492 | intmat D[2][2]; |
---|
1493 | ///////////////////////////////////// |
---|
1494 | // Treat first the pathological cases that the polygon is not two-dimensional: |
---|
1495 | ///////////////////////////////////// |
---|
1496 | // if the polygon is empty or only one point or a line segment of two points |
---|
1497 | if (size(polygon)<=2) |
---|
1498 | { |
---|
1499 | return(list(polygon,polygon)); |
---|
1500 | } |
---|
1501 | // check is the polygon is only a line segment given by more than two points; |
---|
1502 | // for this first compute sum of the absolute values of the determinants of the matrices whose |
---|
1503 | // rows are the vectors pointing from the first to the second point and from the |
---|
1504 | // the first point to the ith point for i=3,...,size(polygon); if this sum is zero |
---|
1505 | // then the polygon is a line segment and we have to find its end points |
---|
1506 | d=0; |
---|
1507 | for (i=3;i<=size(polygon);i++) |
---|
1508 | { |
---|
1509 | D=polygon[2]-polygon[1],polygon[i]-polygon[1]; |
---|
1510 | d=d+abs(det(D)); |
---|
1511 | } |
---|
1512 | if (d==0) // then polygon is a line segment |
---|
1513 | { |
---|
1514 | intmat laenge[size(polygon)][size(polygon)]; |
---|
1515 | intvec mp; |
---|
1516 | // for this collect first all vectors pointing from one lattice point to the next, |
---|
1517 | // compute their pairwise angles and their lengths |
---|
1518 | for (i=1;i<=size(polygon)-1;i++) |
---|
1519 | { |
---|
1520 | for (j=i+1;j<=size(polygon);j++) |
---|
1521 | { |
---|
1522 | mp=polygon[i]-polygon[j]; |
---|
1523 | laenge[i,j]=abs(gcd(mp[1],mp[2])); |
---|
1524 | } |
---|
1525 | } |
---|
1526 | mp=maxPosInIntmat(laenge); |
---|
1527 | list endpoints=polygon[mp[1]],polygon[mp[2]]; |
---|
1528 | intvec abstand; |
---|
1529 | for (i=1;i<=size(polygon);i++) |
---|
1530 | { |
---|
1531 | abstand[i]=0; |
---|
1532 | if (i<mp[1]) |
---|
1533 | { |
---|
1534 | abstand[i]=laenge[i,mp[1]]; |
---|
1535 | } |
---|
1536 | if (i>mp[1]) |
---|
1537 | { |
---|
1538 | abstand[i]=laenge[mp[1],i]; |
---|
1539 | } |
---|
1540 | } |
---|
1541 | polygon=sortlistbyintvec(polygon,abstand); |
---|
1542 | return(list(polygon,endpoints)); |
---|
1543 | } |
---|
1544 | /////////////////////////////////////////////////////////////// |
---|
1545 | list orderedvertices; // stores the vertices in an ordered way |
---|
1546 | list minimisedorderedvertices; // stores the vertices in an ordered way; redundant ones removed |
---|
1547 | list comparevertices; // stores vertices which should be compared to the testvertex |
---|
1548 | orderedvertices[1]=polygon[1]; // set the starting vertex |
---|
1549 | minimisedorderedvertices[1]=polygon[1]; // set the starting vertex |
---|
1550 | intvec testvertex=polygon[1]; // the vertex to which the others have to be compared |
---|
1551 | intvec startvertex=polygon[1]; // keep the starting vertex to test, when the end is reached |
---|
1552 | int endtest; // is set to one, when the end is reached |
---|
1553 | int startvertexfound;// is 1, once for some testvertex a candidate for the next vertex has been found |
---|
1554 | polygon=delete(polygon,1); // delete the testvertex |
---|
1555 | intvec v,w; |
---|
1556 | int l=1; // counts the vertices |
---|
1557 | // the basic idea is that a vertex can be the next one on the boundary if all other vertices |
---|
1558 | // ly to the right of the vector v pointing from the testvertex to this one; this can be tested |
---|
1559 | // by checking if the determinant of the 2x2-matrix with first column v and second column the vector w, |
---|
1560 | // pointing from the testvertex to the new vertex, is non-positive; if this is the case for all |
---|
1561 | // new vertices, then the one in consideration is a possible choice for the next vertex on the boundary |
---|
1562 | // and it is stored in naechste; we can then order the candidates according to their distance from |
---|
1563 | // the testvertex; then they occur on the boundary in that order! |
---|
1564 | while (endtest==0) |
---|
1565 | { |
---|
1566 | list naechste; // stores the possible choices for the next vertex |
---|
1567 | k=1; |
---|
1568 | for (i=1;i<=size(polygon);i++) |
---|
1569 | { |
---|
1570 | d=0; // stores the value of the determinant of (v,w) |
---|
1571 | v=polygon[i]-testvertex; // points from the testvertex to the ith vertex |
---|
1572 | comparevertices=delete(polygon,i); // we needn't compare v to itself |
---|
1573 | // we should compare v to the startvertex-testvertex; in the first calling of the loop |
---|
1574 | // this is irrelevant since the difference will be zero; however, later on it will |
---|
1575 | // be vital, since we delete the vertices which we have already tested from the list |
---|
1576 | // of all vertices, and when all vertices on the boundary have been found we would |
---|
1577 | // therefore find a vertex in the interior as candidate; but always testing against |
---|
1578 | // the starting vertex, this can not happen |
---|
1579 | comparevertices[size(comparevertices)+1]=startvertex; |
---|
1580 | for (j=1;(j<=size(comparevertices)) and (d<=0);j++) |
---|
1581 | { |
---|
1582 | w=comparevertices[j]-testvertex; // points form the testvertex to the jth vertex |
---|
1583 | D=v,w; |
---|
1584 | d=det(D); |
---|
1585 | } |
---|
1586 | if (d<=0) // if all determinants are non-positive, then the ith vertex is a candidate |
---|
1587 | { |
---|
1588 | naechste[k]=list(polygon[i],i,scalarproduct(v,v)); // we store the vertex, its position, and its |
---|
1589 | k++; // distance from the testvertex |
---|
1590 | } |
---|
1591 | } |
---|
1592 | if (size(naechste)>0) // then a candidate for the next vertex has been found |
---|
1593 | { |
---|
1594 | startvertexfound=1; // at least once a candidate has been found |
---|
1595 | naechste=sortlist(naechste,3); //we order the candidates according to their distance from testvertex; |
---|
1596 | for (j=1;j<=size(naechste);j++) // then we store them in this order in orderedvertices |
---|
1597 | { |
---|
1598 | l++; |
---|
1599 | orderedvertices[l]=naechste[j][1]; |
---|
1600 | } |
---|
1601 | testvertex=naechste[size(naechste)][1]; // we store the last one as next testvertex; |
---|
1602 | minimisedorderedvertices[size(minimisedorderedvertices)+1]=testvertex; // store the next corner of NSD |
---|
1603 | naechste=sortlist(naechste,2); // then we reorder the vertices according to their position |
---|
1604 | for (j=size(naechste);j>=1;j--) // and we delete them from the vertices |
---|
1605 | { |
---|
1606 | polygon=delete(polygon,naechste[j][2]); |
---|
1607 | } |
---|
1608 | } |
---|
1609 | else // that means either that the vertex was inside the polygon, |
---|
1610 | { // or that we have reached the last vertex on the boundary of the polytope |
---|
1611 | if (startvertexfound==0) // the vertex was in the interior; we delete it and start all over again |
---|
1612 | { |
---|
1613 | orderedvertices[1]=polygon[1]; |
---|
1614 | minimisedorderedvertices[1]=polygon[1]; |
---|
1615 | testvertex=polygon[1]; |
---|
1616 | startvertex=polygon[1]; |
---|
1617 | polygon=delete(polygon,1); |
---|
1618 | } |
---|
1619 | else // we have reached the last vertex on the boundary of the polytope and can stop |
---|
1620 | { |
---|
1621 | endtest=1; |
---|
1622 | } |
---|
1623 | } |
---|
1624 | kill naechste; |
---|
1625 | } |
---|
1626 | // test if the first vertex in minimisedorderedvertices is on the same line with the second and |
---|
1627 | // the last, i.e. if we started our search in the middle of a face; if so, delete it |
---|
1628 | v=minimisedorderedvertices[2]-minimisedorderedvertices[1]; |
---|
1629 | w=minimisedorderedvertices[size(minimisedorderedvertices)]-minimisedorderedvertices[1]; |
---|
1630 | D=v,w; |
---|
1631 | if (det(D)==0) |
---|
1632 | { |
---|
1633 | minimisedorderedvertices=delete(minimisedorderedvertices,1); |
---|
1634 | } |
---|
1635 | // test if the first vertex in minimisedorderedvertices is on the same line with the two |
---|
1636 | // last ones, i.e. if we started our search at the end of a face; if so, delete it |
---|
1637 | v=minimisedorderedvertices[size(minimisedorderedvertices)-1]-minimisedorderedvertices[1]; |
---|
1638 | w=minimisedorderedvertices[size(minimisedorderedvertices)]-minimisedorderedvertices[1]; |
---|
1639 | D=v,w; |
---|
1640 | if (det(D)==0) |
---|
1641 | { |
---|
1642 | minimisedorderedvertices=delete(minimisedorderedvertices,size(minimisedorderedvertices)); |
---|
1643 | } |
---|
1644 | return(list(orderedvertices,minimisedorderedvertices)); |
---|
1645 | } |
---|
1646 | example |
---|
1647 | { |
---|
1648 | "EXAMPLE:"; |
---|
1649 | echo=2; |
---|
1650 | // the following lattice points in the plane define a polygon |
---|
1651 | list polygon=intvec(0,0),intvec(3,1),intvec(1,0),intvec(2,0), |
---|
1652 | intvec(1,1),intvec(3,2),intvec(1,2),intvec(2,3), |
---|
1653 | intvec(2,4); |
---|
1654 | // we compute its boundary |
---|
1655 | list boundarypolygon=findOrientedBoundary(polygon); |
---|
1656 | // the points on the boundary ordered clockwise are boundarypolygon[1] |
---|
1657 | boundarypolygon[1]; |
---|
1658 | // the vertices of the boundary are boundarypolygon[2] |
---|
1659 | boundarypolygon[2]; |
---|
1660 | } |
---|
1661 | |
---|
1662 | |
---|
1663 | proc cyclePoints (list triang,list points,int pt) |
---|
1664 | "USAGE: cyclePoints(triang,points,pt) triang,points list, pt int |
---|
1665 | ASSUME: - points is a list of integer vectors describing the lattice points of a marked polygon; |
---|
1666 | @* - triang is a list of integer vectors describing a triangulation of the marked polygon |
---|
1667 | in the sense that an integer vector of the form (i,j,k) describes the triangle formed |
---|
1668 | by polygon[i], polygon[j] and polygon[k]; |
---|
1669 | @* - pt is an integer between 1 and size(points), singling out a lattice point among |
---|
1670 | the marked points |
---|
1671 | PURPOSE: consider the convex lattice polygon, say P, spanned by all lattice points in points which |
---|
1672 | in the triangulation triang are connected to the point points[pt]; the procedure |
---|
1673 | computes all marked points in points which ly on the boundary of that polygon, ordered |
---|
1674 | clockwise |
---|
1675 | RETURN: list, of integer vectors which are the coordinates of the lattice points on |
---|
1676 | the boundary of the above mentioned polygon P, if this polygon is not the |
---|
1677 | empty set (that would be the case if points[pt] is not a vertex of any |
---|
1678 | triangle in the triangulation); otherwise return the empty list |
---|
1679 | EXAMPLE: example cyclePoints; shows an example" |
---|
1680 | { |
---|
1681 | int i,j; // indices |
---|
1682 | list v; // saves the indices of lattice points connected to the interior point in the triangulation |
---|
1683 | // save all points in triangulations containing pt in v |
---|
1684 | for (i=1;i<=size(triang);i++) |
---|
1685 | { |
---|
1686 | if ((triang[i][1]==pt) or (triang[i][2]==pt) or (triang[i][3]==pt)) |
---|
1687 | { |
---|
1688 | j++; |
---|
1689 | v[3*j-2]=triang[i][1]; |
---|
1690 | v[3*j-1]=triang[i][2]; |
---|
1691 | v[3*j]=triang[i][3]; |
---|
1692 | } |
---|
1693 | } |
---|
1694 | if (size(v)==0) |
---|
1695 | { |
---|
1696 | return(list()); |
---|
1697 | } |
---|
1698 | // remove pt itself and redundancies in v |
---|
1699 | for (i=size(v);i>=1;i--) |
---|
1700 | { |
---|
1701 | j=1; |
---|
1702 | while ((j<i) and (v[i]!=v[j])) |
---|
1703 | { |
---|
1704 | j++; |
---|
1705 | } |
---|
1706 | if ((j<i) or (v[i]==pt)) |
---|
1707 | { |
---|
1708 | v=delete(v,i); |
---|
1709 | } |
---|
1710 | } |
---|
1711 | // save in pts the coordinates of the points with indices in v |
---|
1712 | list pts; |
---|
1713 | for (i=1;i<=size(v);i++) |
---|
1714 | { |
---|
1715 | pts[i]=points[v[i]]; |
---|
1716 | } |
---|
1717 | // consider the convex polytope spanned by the points in pts, find the points on the |
---|
1718 | // boundary and order them clockwise |
---|
1719 | return(findOrientedBoundary(pts)[1]); |
---|
1720 | } |
---|
1721 | example |
---|
1722 | { |
---|
1723 | "EXAMPLE:"; |
---|
1724 | echo=2; |
---|
1725 | // the lattice polygon spanned by the points (0,0), (3,0) and (0,3) |
---|
1726 | // with all integer points as markings |
---|
1727 | list points=intvec(1,1),intvec(3,0),intvec(2,0),intvec(1,0), |
---|
1728 | intvec(0,0),intvec(2,1),intvec(0,1),intvec(1,2), |
---|
1729 | intvec(0,2),intvec(0,3); |
---|
1730 | // define a triangulation |
---|
1731 | list triang=intvec(1,2,5),intvec(1,5,7),intvec(1,7,9),intvec(8,9,10), |
---|
1732 | intvec(1,8,9),intvec(1,2,8); |
---|
1733 | // compute the points connected to (1,1) in triang |
---|
1734 | cyclePoints(triang,points,1); |
---|
1735 | } |
---|
1736 | |
---|
1737 | proc latticeArea (list polygon) |
---|
1738 | "USAGE: latticeArea(polygon); polygon list |
---|
1739 | ASSUME: polygon is a list of integer vectors in the plane |
---|
1740 | RETURN: int, the lattice area of the convex hull of the lattice points in polygon, |
---|
1741 | i.e. twice the Euclidean area |
---|
1742 | EXAMPLE: example polygonlatticeArea; shows an example" |
---|
1743 | { |
---|
1744 | list pg=findOrientedBoundary(polygon)[2]; |
---|
1745 | int area; |
---|
1746 | intmat M[2][2]; |
---|
1747 | for (int i=2;i<=size(pg)-1;i++) |
---|
1748 | { |
---|
1749 | M[1,1..2]=pg[i]-pg[1]; |
---|
1750 | M[2,1..2]=pg[i+1]-pg[1]; |
---|
1751 | area=area+abs(det(M)); |
---|
1752 | } |
---|
1753 | return(area); |
---|
1754 | } |
---|
1755 | example |
---|
1756 | { |
---|
1757 | "EXAMPLE:"; |
---|
1758 | echo=2; |
---|
1759 | // define a polygon with lattice area 5 |
---|
1760 | list polygon=intvec(1,2),intvec(1,0),intvec(2,0),intvec(1,1), |
---|
1761 | intvec(2,1),intvec(0,0); |
---|
1762 | latticeArea(polygon); |
---|
1763 | } |
---|
1764 | |
---|
1765 | proc picksFormula (list polygon) |
---|
1766 | "USAGE: picksFormula(polygon); polygon list |
---|
1767 | ASSUME: polygon is a list of integer vectors in the plane and consider their convex hull C |
---|
1768 | RETURN: list, L of three integersthe |
---|
1769 | @* L[1] : the lattice area of C, i.e. twice the Euclidean area |
---|
1770 | @* L[2] : the number of lattice points on the boundary of C |
---|
1771 | @* L[3] : the number of interior lattice points of C |
---|
1772 | NOTE: the integers in L are related by Pick's formula, namely: L[1]=L[2]+2*L[3]-2 |
---|
1773 | EXAMPLE: example picksFormula; shows an example" |
---|
1774 | { |
---|
1775 | list pg=findOrientedBoundary(polygon)[2]; |
---|
1776 | int area,bdpts,i; |
---|
1777 | intmat M[2][2]; |
---|
1778 | // compute the lattice area of the polygon, i.e. twice the Euclidean area |
---|
1779 | for (i=2;i<=size(pg)-1;i++) |
---|
1780 | { |
---|
1781 | M[1,1..2]=pg[i]-pg[1]; |
---|
1782 | M[2,1..2]=pg[i+1]-pg[1]; |
---|
1783 | area=area+abs(det(M)); |
---|
1784 | } |
---|
1785 | // compute the number of lattice points on the boundary |
---|
1786 | intvec edge; |
---|
1787 | pg[size(pg)+1]=pg[1]; |
---|
1788 | for (i=1;i<=size(pg)-1;i++) |
---|
1789 | { |
---|
1790 | edge=pg[i]-pg[i+1]; |
---|
1791 | bdpts=bdpts+abs(gcd(edge[1],edge[2])); |
---|
1792 | } |
---|
1793 | // Pick's formula says that the lattice area A, the number g of interior points and |
---|
1794 | // the number b of boundary points are connected by the formula: A=b+2g-2 |
---|
1795 | return(list(area,bdpts,(area-bdpts+2)/2)); |
---|
1796 | } |
---|
1797 | example |
---|
1798 | { |
---|
1799 | "EXAMPLE:"; |
---|
1800 | echo=2; |
---|
1801 | // define a polygon with lattice area 5 |
---|
1802 | list polygon=intvec(1,2),intvec(1,0),intvec(2,0),intvec(1,1), |
---|
1803 | intvec(2,1),intvec(0,0); |
---|
1804 | list pick=picksFormula(polygon); |
---|
1805 | // the lattice area of the polygon is: |
---|
1806 | pick[1]; |
---|
1807 | // the number of lattice points on the boundary is: |
---|
1808 | pick[2]; |
---|
1809 | // the number of interior lattice points is: |
---|
1810 | pick[3]; |
---|
1811 | // the number's are related by Pick's formula: |
---|
1812 | pick[1]-pick[2]-2*pick[3]+2; |
---|
1813 | } |
---|
1814 | |
---|
1815 | proc ellipticNF (list polygon) |
---|
1816 | "USAGE: ellipticNF(polygon); polygon list |
---|
1817 | ASSUME: polygon is a list of integer vectors in the plane such that their convex hull C |
---|
1818 | has precisely one interior lattice point; i.e. C is the Newton polygon of an |
---|
1819 | elliptic curve |
---|
1820 | PURPOSE: compute the normal form of the polygon with respect to the unimodular affine |
---|
1821 | transformations T=A*x+v; there are sixteen different normal forms |
---|
1822 | (see e.g. Bjorn Poonen, Fernando Rodriguez-Villegas: Lattice Polygons and |
---|
1823 | the number 12. Amer. Math. Monthly 107 (2000), no. 3, 238--250.) |
---|
1824 | RETURN: list, L such that |
---|
1825 | @* L[1] : list whose entries are the vertices of the normal form of the polygon |
---|
1826 | @* L[2] : the matrix A of the unimodular transformation |
---|
1827 | @* L[3] : the translation vector v of the unimodular transformation |
---|
1828 | @* L[4] : list such that the ith entry is the image of polygon[i] under the |
---|
1829 | unimodular transformation T |
---|
1830 | EXAMPLE: example ellipticNF; shows an example" |
---|
1831 | { |
---|
1832 | int i; // index |
---|
1833 | intvec edge; // stores the vector of an edge |
---|
1834 | intvec boundary; // stores the lattice lengths of the edges of the Newton cylce |
---|
1835 | // find the vertices of the Newton cycle and order it clockwise |
---|
1836 | list pg=findOrientedBoundary(polygon)[2]; |
---|
1837 | // check if there is precisely one interior point in the Newton polygon |
---|
1838 | if (picksFormula(pg)[3]!=1) |
---|
1839 | { |
---|
1840 | ERROR("The polygon has not precisely one interior point!"); |
---|
1841 | } |
---|
1842 | // insert the first vertex at the end once again |
---|
1843 | pg[size(pg)+1]=pg[1]; |
---|
1844 | // compute the number of lattice points on each edge |
---|
1845 | for (i=1;i<=size(pg)-1;i++) |
---|
1846 | { |
---|
1847 | edge=pg[i]-pg[i+1]; |
---|
1848 | boundary[i]=1+abs(gcd(edge[1],edge[2])); |
---|
1849 | } |
---|
1850 | // store the values of boundary once more adding the first two at the end |
---|
1851 | intvec tboundary=boundary,boundary[1],boundary[2]; |
---|
1852 | // sort boundary in an asecending way |
---|
1853 | intvec sbd=sortintvec(boundary); |
---|
1854 | // find the first edge having the maximal number of lattice points |
---|
1855 | int max=maxPosInIntvec(boundary); |
---|
1856 | // some computations have to be done over the rationals |
---|
1857 | ring transformationring=0,x,lp; |
---|
1858 | intvec trans; // stores the vector by which we have to translate the polygon |
---|
1859 | intmat A[2][2]; // stores the matrix by which we have to transform the polygon |
---|
1860 | matrix M[3][3]; // stores the projective coordinates of the points which are to be transformed |
---|
1861 | matrix N[3][3]; // stores the projective coordinates of the points to which M is to be transformed |
---|
1862 | intmat T[3][3]; // stores the unimodular affine transformation in projective form |
---|
1863 | // add the second point of pg once again at the end |
---|
1864 | pg=insert(pg,pg[2],size(pg)); |
---|
1865 | // if there is only one edge which has the maximal number of lattice points, then M should be: |
---|
1866 | M=pg[max],1,pg[max+1],1,pg[max+2],1; |
---|
1867 | // consider the 16 different cases which can occur: |
---|
1868 | // Case 1: |
---|
1869 | if (sbd==intvec(2,2,2)) |
---|
1870 | { |
---|
1871 | N=0,1,1,1,2,1,2,0,1; |
---|
1872 | } |
---|
1873 | // Case 2: |
---|
1874 | if (sbd==intvec(2,2,3)) |
---|
1875 | { |
---|
1876 | N=2,0,1,0,0,1,1,2,1; |
---|
1877 | } |
---|
1878 | // Case 3: |
---|
1879 | if (sbd==intvec(2,3,4)) |
---|
1880 | { |
---|
1881 | // here the orientation of the Newton polygon is important ! |
---|
1882 | if (tboundary[max+1]==3) |
---|
1883 | { |
---|
1884 | N=3,0,1,0,0,1,0,2,1; |
---|
1885 | } |
---|
1886 | else |
---|
1887 | { |
---|
1888 | N=0,0,1,3,0,1,0,2,1; |
---|
1889 | } |
---|
1890 | } |
---|
1891 | // Case 4: |
---|
1892 | if (sbd==intvec(3,3,5)) |
---|
1893 | { |
---|
1894 | N=4,0,1,0,0,1,0,2,1; |
---|
1895 | } |
---|
1896 | // Case 5: |
---|
1897 | if (sbd==intvec(4,4,4)) |
---|
1898 | { |
---|
1899 | N=3,0,1,0,0,1,0,3,1; |
---|
1900 | } |
---|
1901 | // Case 6+7: |
---|
1902 | if (sbd==intvec(2,2,2,2)) |
---|
1903 | { |
---|
1904 | // there are two different polygons which has four edges all of length 2, |
---|
1905 | // but only one of them has two edges whose direction vectors form a matrix |
---|
1906 | // of determinant 3 |
---|
1907 | A=pg[1]-pg[2],pg[3]-pg[2]; |
---|
1908 | while ((max<4) and (det(A)!=3)) |
---|
1909 | { |
---|
1910 | max++; |
---|
1911 | A=pg[max]-pg[max+1],pg[max+2]-pg[max+1]; |
---|
1912 | } |
---|
1913 | // Case 6: |
---|
1914 | if (det(A)==3) |
---|
1915 | { |
---|
1916 | M=pg[max],1,pg[max+1],1,pg[max+2],1; |
---|
1917 | N=1,0,1,0,2,1,2,1,1; |
---|
1918 | } |
---|
1919 | // Case 7: |
---|
1920 | else |
---|
1921 | { |
---|
1922 | N=2,1,1,1,0,1,0,1,1; |
---|
1923 | } |
---|
1924 | } |
---|
1925 | // Case 8: |
---|
1926 | if (sbd==intvec(2,2,2,3)) |
---|
1927 | { |
---|
1928 | // the orientation of the polygon is important |
---|
1929 | A=pg[max]-pg[max+1],pg[max+2]-pg[max+1]; |
---|
1930 | if (det(A)==2) |
---|
1931 | { |
---|
1932 | N=2,0,1,0,0,1,0,1,1; |
---|
1933 | } |
---|
1934 | else |
---|
1935 | { |
---|
1936 | N=0,0,1,2,0,1,1,2,1; |
---|
1937 | } |
---|
1938 | } |
---|
1939 | // Case 9: |
---|
1940 | if (sbd==intvec(2,2,3,3)) |
---|
1941 | { |
---|
1942 | // if max==1, then the 5th entry in tboundary is the same as the first |
---|
1943 | if (max==1) |
---|
1944 | { |
---|
1945 | max=5; |
---|
1946 | } |
---|
1947 | // if boundary=3,2,2,3 then set max=4 |
---|
1948 | if (tboundary[max+1]!=3) |
---|
1949 | { |
---|
1950 | max=4; |
---|
1951 | } |
---|
1952 | M=pg[max],1,pg[max+1],1,pg[max+2],1; |
---|
1953 | // the orientation of the polygon matters |
---|
1954 | A=pg[max-1]-pg[max],pg[max+1]-pg[max]; |
---|
1955 | if (det(A)==4) |
---|
1956 | { |
---|
1957 | N=2,0,1,0,0,1,0,2,1; |
---|
1958 | } |
---|
1959 | else |
---|
1960 | { |
---|
1961 | N=0,2,1,0,0,1,2,0,1; |
---|
1962 | } |
---|
1963 | } |
---|
1964 | // Case 10: |
---|
1965 | if (sbd==intvec(2,2,3,4)) |
---|
1966 | { |
---|
1967 | // the orientation of the polygon matters |
---|
1968 | if (tboundary[max+1]==3) |
---|
1969 | { |
---|
1970 | N=3,0,1,0,0,1,0,2,1; |
---|
1971 | } |
---|
1972 | else |
---|
1973 | { |
---|
1974 | N=0,0,1,3,0,1,2,1,1; |
---|
1975 | } |
---|
1976 | } |
---|
1977 | // Case 11: |
---|
1978 | if (sbd==intvec(2,3,3,4)) |
---|
1979 | { |
---|
1980 | N=3,0,1,0,0,1,0,2,1; |
---|
1981 | } |
---|
1982 | // Case 12: |
---|
1983 | if (sbd==intvec(3,3,3,3)) |
---|
1984 | { |
---|
1985 | N=2,0,1,0,0,1,0,2,1; |
---|
1986 | } |
---|
1987 | // Case 13: |
---|
1988 | if (sbd==intvec(2,2,2,2,2)) |
---|
1989 | { |
---|
1990 | // compute the angles of the polygon vertices |
---|
1991 | intvec dt; |
---|
1992 | for (i=1;i<=5;i++) |
---|
1993 | { |
---|
1994 | A=pg[i]-pg[i+1],pg[i+2]-pg[i+1]; |
---|
1995 | dt[i]=det(A); |
---|
1996 | } |
---|
1997 | dt[6]=dt[1]; |
---|
1998 | // find the vertex to be mapped to (0,1) |
---|
1999 | max=1; |
---|
2000 | while ((dt[max]!=2) or (dt[max+1]!=2)) |
---|
2001 | { |
---|
2002 | max++; |
---|
2003 | } |
---|
2004 | M=pg[max],1,pg[max+1],1,pg[max+2],1; |
---|
2005 | N=0,1,1,1,2,1,2,1,1; |
---|
2006 | } |
---|
2007 | // Case 14: |
---|
2008 | if (sbd==intvec(2,2,2,2,3)) |
---|
2009 | { |
---|
2010 | N=2,0,1,0,0,1,0,1,1; |
---|
2011 | } |
---|
2012 | // Case 15: |
---|
2013 | if (sbd==intvec(2,2,2,3,3)) |
---|
2014 | { |
---|
2015 | // find the vertix to be mapped to (2,0) |
---|
2016 | if (tboundary[max+1]!=3) |
---|
2017 | { |
---|
2018 | max=5; |
---|
2019 | M=pg[max],1,pg[max+1],1,pg[max+2],1; |
---|
2020 | } |
---|
2021 | N=2,0,1,0,0,1,0,2,1; |
---|
2022 | } |
---|
2023 | // Case 16: |
---|
2024 | if (sbd==intvec(2,2,2,2,2,2)) |
---|
2025 | { |
---|
2026 | N=2,0,1,1,0,1,0,1,1; |
---|
2027 | } |
---|
2028 | // we have to transpose the matrices M and N |
---|
2029 | M=transpose(M); |
---|
2030 | N=transpose(N); |
---|
2031 | // compute the unimodular affine transformation, which is of the form |
---|
2032 | // A11 A12 | T1 |
---|
2033 | // A21 A22 | T2 |
---|
2034 | // 0 0 | 1 |
---|
2035 | T=matrixtointmat(N*inverse(M)); |
---|
2036 | // the upper-left 2x2-block is A |
---|
2037 | A=T[1..2,1..2]; |
---|
2038 | // the upper-right 2x1-block is the translation vector |
---|
2039 | trans=T[1,3],T[2,3]; |
---|
2040 | // transform now the lattice points of the polygon with respect to A and T |
---|
2041 | list nf; |
---|
2042 | for (i=1;i<=size(polygon);i++) |
---|
2043 | { |
---|
2044 | intmat V[2][1]=polygon[i]; |
---|
2045 | V=A*V; |
---|
2046 | nf[i]=intvec(V[1,1]+trans[1],V[2,1]+trans[2]); |
---|
2047 | kill V; |
---|
2048 | } |
---|
2049 | return(list(findOrientedBoundary(nf)[2],A,trans,nf)); |
---|
2050 | } |
---|
2051 | example |
---|
2052 | { |
---|
2053 | "EXAMPLE:"; |
---|
2054 | echo=2; |
---|
2055 | ring r=0,(x,y),dp; |
---|
2056 | // the Newton polygon of the following polynomial |
---|
2057 | // has precisely one interior point |
---|
2058 | poly f=x22y11+x19y10+x17y9+x16y9+x12y7+x9y6+x7y5+x2y3; |
---|
2059 | list polygon=newtonPolytopeLP(f); |
---|
2060 | // its lattice points are |
---|
2061 | polygon; |
---|
2062 | // find its normal form |
---|
2063 | list nf=ellipticNF(polygon); |
---|
2064 | // the vertices of the normal form are |
---|
2065 | nf[1]; |
---|
2066 | // it has been transformed by the unimodular affine transformation A*x+v |
---|
2067 | // with matrix A |
---|
2068 | nf[2]; |
---|
2069 | // and translation vector v |
---|
2070 | nf[3]; |
---|
2071 | // the 3rd lattice point ... |
---|
2072 | polygon[3]; |
---|
2073 | // ... has been transformed to |
---|
2074 | nf[4][3]; |
---|
2075 | } |
---|
2076 | |
---|
2077 | |
---|
2078 | proc ellipticNFDB (int n,list #) |
---|
2079 | "USAGE: ellipticNFDB(n[,#]); n int, # list |
---|
2080 | ASSUME: n is an intger between 1 and 16 |
---|
2081 | PURPOSE: this is a database storing the 16 normal forms of planar polygons with |
---|
2082 | precisely one interior point up to unimodular affine transformations |
---|
2083 | @* (see e.g. Bjorn Poonen, Fernando Rodriguez-Villegas: Lattice Polygons and |
---|
2084 | the number 12. Amer. Math. Monthly 107 (2000), no. 3, 238--250.) |
---|
2085 | RETURN: list, L such that |
---|
2086 | @* L[1] : list whose entries are the vertices of the nth normal form |
---|
2087 | @* L[2] : list whose entries are all the lattice points of the nth normal form |
---|
2088 | @* L[3] : only present if the optional parameter # is present, and then |
---|
2089 | it is a polynomial in the variables (x,y) whose Newton polygon |
---|
2090 | is the nth normal form |
---|
2091 | NOTE: the optional parameter is only allowed if the basering has the variables x and y |
---|
2092 | EXAMPLE: example ellipticNFDB; shows an example" |
---|
2093 | { |
---|
2094 | if ((n<1) or (n>16)) |
---|
2095 | { |
---|
2096 | ERROR("n is not between 1 and 16."); |
---|
2097 | } |
---|
2098 | if (size(#)>0) |
---|
2099 | { |
---|
2100 | if ((defined(x)==0) or (defined(y)==0)) |
---|
2101 | { |
---|
2102 | ERROR("The variables x and y are not defined."); |
---|
2103 | } |
---|
2104 | } |
---|
2105 | if ((defined(x)==0) or (defined(y)==0)) |
---|
2106 | { |
---|
2107 | ring nfring=0,(x,y),dp; |
---|
2108 | } |
---|
2109 | // store the normal forms as polynomials |
---|
2110 | list nf=x2+y+xy2,x2+x+1+xy2,x3+x2+x+1+y2+y,x4+x3+x2+x+1+y2+y+x2y,x3+x2+x+1+x2y+y+xy2+y2+y3, |
---|
2111 | x2+x+x2y+y2,x2y+x+y+xy2,x2+x+1+y+xy2,x2+x+1+y+xy2+y2,x3+x2+x+1+x2y+y+y2,x3+x2+x+1+x2y+y+xy2+y2, |
---|
2112 | x2+x+1+x2y+y+x2y2+xy2+y2,x+1+x2y+y+xy2,x2+x+1+x2y+y+xy2,x2+x+1+x2y+y+xy2+y2,x2+x+x2y+y+xy2+y2; |
---|
2113 | list pg=newtonPolytopeLP(nf[n]); |
---|
2114 | if (size(#)==0) |
---|
2115 | { |
---|
2116 | return(list(findOrientedBoundary(pg)[2],pg)); |
---|
2117 | } |
---|
2118 | else |
---|
2119 | { |
---|
2120 | return(list(findOrientedBoundary(pg)[2],pg,nf[n])); |
---|
2121 | } |
---|
2122 | } |
---|
2123 | example |
---|
2124 | { |
---|
2125 | "EXAMPLE:"; |
---|
2126 | echo=2; |
---|
2127 | list nf=ellipticNFDB(5); |
---|
2128 | // the vertices of the 5th normal form are |
---|
2129 | nf[1]; |
---|
2130 | // its lattice points are |
---|
2131 | nf[2]; |
---|
2132 | } |
---|
2133 | |
---|
2134 | |
---|
2135 | ///////////////////////////////////////////////////////////////////////////////// |
---|
2136 | /// AUXILARY PROCEDURES |
---|
2137 | ///////////////////////////////////////////////////////////////////////////////// |
---|
2138 | |
---|
2139 | proc polymakeKeepTmpFiles (int i) |
---|
2140 | "USAGE: polymakeKeepTmpFiles(int i); i int |
---|
2141 | PURPOSE: some procedures create files in the directory /tmp which are used for |
---|
2142 | computations with polymake respectively topcom; these will be removed |
---|
2143 | when the corresponding procedure is left; however, it might be desireable |
---|
2144 | to keep them for further computations with either polymake or topcom; this |
---|
2145 | can be achieved by this procedure; call the procedure as: |
---|
2146 | @* - polymakeKeepTmpFiles(1); - then the files will be kept |
---|
2147 | @* - polymakeKeepTmpFiles(0); - then the files will be removed in the future |
---|
2148 | RETURN: none" |
---|
2149 | { |
---|
2150 | if (i==1) |
---|
2151 | { |
---|
2152 | int polymakekeeptmpfiles; |
---|
2153 | export polymakekeeptmpfiles; |
---|
2154 | } |
---|
2155 | else |
---|
2156 | { |
---|
2157 | if (defined(polymakekeeptmpfiles)) |
---|
2158 | { |
---|
2159 | kill polymakekeeptmpfiles; |
---|
2160 | } |
---|
2161 | } |
---|
2162 | } |
---|
2163 | |
---|
2164 | |
---|
2165 | ///////////////////////////////////////////////////////////////////////////////// |
---|
2166 | ///////////////////////////////////////////////////////////////////////////////// |
---|
2167 | /// AUXILARY PROCEDURES, WHICH ARE DECLARED STATIC |
---|
2168 | ///////////////////////////////////////////////////////////////////////////////// |
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2169 | ///////////////////////////////////////////////////////////////////////////////// |
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2170 | /// - scalarproduct |
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2171 | /// - intmatcoldelete |
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2172 | /// - intmatconcat |
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2173 | /// - sortlist |
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2174 | /// - minInList |
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2175 | /// - stringdelete |
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2176 | /// - abs |
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2177 | /// - commondenominator |
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2178 | /// - maxPosInIntvec |
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2179 | /// - maxPosInIntmat |
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2180 | /// - sortintvec |
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2181 | /// - matrixtointmat |
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2182 | ///////////////////////////////////////////////////////////////////////////////// |
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2183 | |
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2184 | static proc scalarproduct (intvec w,intvec v) |
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2185 | "USAGE: scalarproduct(w,v); w,v intvec |
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2186 | ASSUME: w and v are integer vectors of the same length |
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2187 | RETURN: int, the scalarproduct of v and w |
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2188 | NOTE: the procedure is called by findOrientedBoundary" |
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2189 | { |
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2190 | int sp; |
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2191 | for (int i=1;i<=size(w);i++) |
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2192 | { |
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2193 | sp=sp+v[i]*w[i]; |
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2194 | } |
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2195 | return(sp); |
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2196 | } |
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2197 | |
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2198 | static proc intmatcoldelete (intmat w,int i) |
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2199 | "USAGE: intmatcoldelete(w,i); w intmat, i int |
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2200 | RETURN: intmat, the integer matrix w with the ith comlumn deleted |
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2201 | NOTE: the procedure is called by intmatsort and normalFan" |
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2202 | { |
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2203 | if ((i<1) or (i>ncols(w)) or (ncols(w)==1)) |
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2204 | { |
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2205 | return(w); |
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2206 | } |
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2207 | if (i==1) |
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2208 | { |
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2209 | intmat M[nrows(w)][ncols(w)-1]=w[1..nrows(w),2..ncols(w)]; |
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2210 | return(M); |
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2211 | } |
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2212 | if (i==ncols(w)) |
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2213 | { |
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2214 | intmat M[nrows(w)][ncols(w)-1]=w[1..nrows(w),1..ncols(w)-1]; |
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2215 | return(M); |
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2216 | } |
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2217 | else |
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2218 | { |
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2219 | intmat M[nrows(w)][i-1]=w[1..nrows(w),1..i-1]; |
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2220 | intmat N[nrows(w)][ncols(w)-i]=w[1..nrows(w),i+1..ncols(w)]; |
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2221 | return(intmatconcat(M,N)); |
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2222 | } |
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2223 | } |
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2224 | |
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2225 | static proc intmatconcat (intmat M,intmat N) |
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2226 | "USAGE: intmatconcat(M,N); M,N intmat |
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2227 | RETURN: intmat, M and N concatenated |
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2228 | NOTE: the procedure is called by intmatcoldelete and sortintmat" |
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2229 | { |
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2230 | if (nrows(M)>=nrows(N)) |
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2231 | { |
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2232 | int m=nrows(M); |
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2233 | |
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2234 | } |
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2235 | else |
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2236 | { |
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2237 | int m=nrows(N); |
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2238 | } |
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2239 | intmat P[m][ncols(M)+ncols(N)]; |
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2240 | P[1..nrows(M),1..ncols(M)]=M[1..nrows(M),1..ncols(M)]; |
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2241 | P[1..nrows(N),ncols(M)+1..ncols(M)+ncols(N)]=N[1..nrows(N),1..ncols(N)]; |
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2242 | return(P); |
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2243 | } |
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2244 | |
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2245 | static proc sortlist (list v,int pos) |
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2246 | "USAGE: sortlist(v,pos); v list, pos int |
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2247 | RETURN: list, the list L ordered in an ascending way according to the pos-th entries |
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2248 | NOTE: called by tropicalCurve" |
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2249 | { |
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2250 | if(size(v)==1) |
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2251 | { |
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2252 | return(v); |
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2253 | } |
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2254 | list w=minInList(v,pos); |
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2255 | v=delete(v,w[2]); |
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2256 | v=sortlist(v,pos); |
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2257 | v=list(w[1])+v; |
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2258 | return(v); |
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2259 | } |
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2260 | |
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2261 | static proc minInList (list v,int pos) |
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2262 | "USAGE: minInList(v,pos); v list, pos int |
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2263 | RETURN: list, (v[i],i) such that v[i][pos] is minimal |
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2264 | NOTE: called by sortlist" |
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2265 | { |
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2266 | int min=v[1][pos]; |
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2267 | int minpos=1; |
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2268 | for (int i=2;i<=size(v);i++) |
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2269 | { |
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2270 | if (v[i][pos]<min) |
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2271 | { |
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2272 | min=v[i][pos]; |
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2273 | minpos=i; |
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2274 | } |
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2275 | } |
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2276 | return(list(v[minpos],minpos)); |
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2277 | } |
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2278 | |
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2279 | static proc stringdelete (string w,int i) |
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2280 | "USAGE: stringdelete(w,i); w string, i int |
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2281 | RETURN: string, the string w with the ith component deleted |
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2282 | NOTE: the procedure is called by texnumber and choosegfanvector" |
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2283 | { |
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2284 | if ((i>size(w)) or (i<=0)) |
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2285 | { |
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2286 | return(w); |
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2287 | } |
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2288 | if ((size(w)==1) and (i==1)) |
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2289 | { |
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2290 | return(""); |
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2291 | |
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2292 | } |
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2293 | if (i==1) |
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2294 | { |
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2295 | return(w[2..size(w)]); |
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2296 | } |
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2297 | if (i==size(w)) |
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2298 | { |
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2299 | return(w[1..size(w)-1]); |
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2300 | } |
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2301 | else |
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2302 | { |
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2303 | string erg=w[1..i-1],w[i+1..size(w)]; |
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2304 | return(erg); |
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2305 | } |
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2306 | } |
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2307 | |
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2308 | static proc abs (def n) |
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2309 | "USAGE: abs(n); n poly or int |
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2310 | RETURN: poly or int, the absolute value of n" |
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2311 | { |
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2312 | if (n>=0) |
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2313 | { |
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2314 | return(n); |
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2315 | } |
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2316 | else |
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2317 | { |
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2318 | return(-n); |
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2319 | } |
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2320 | } |
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2321 | |
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2322 | static proc commondenominator (matrix M) |
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2323 | "USAGE: commondenominator(M); M matrix |
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2324 | ASSUME: the base ring has characteristic zero |
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2325 | RETURN: int, the lowest common multiple of the denominators of the leading coefficients |
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2326 | of the entries in M |
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2327 | NOTE: the procedure is called from polymakeToIntmat" |
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2328 | { |
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2329 | int i,j; |
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2330 | int kgV=1; |
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2331 | // successively build the lowest common multiple of the denominators of the leading coefficients |
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2332 | // of the entries in M |
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2333 | for (i=1;i<=nrows(M);i++) |
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2334 | { |
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2335 | for (j=1;j<=ncols(M);j++) |
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2336 | { |
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2337 | kgV=lcm(kgV,int(denominator(leadcoef(M[i,j])))); |
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2338 | } |
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2339 | } |
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2340 | return(kgV); |
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2341 | } |
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2342 | |
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2343 | static proc maxPosInIntvec (intvec v) |
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2344 | "USAGE: maxPosInIntvec(v); v intvec |
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2345 | RETURN: int, the first position of a maximal entry in v |
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2346 | NOTE: called by sortintmat" |
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2347 | { |
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2348 | int max=v[1]; |
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2349 | int maxpos=1; |
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2350 | for (int i=2;i<=size(v);i++) |
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2351 | { |
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2352 | if (v[i]>max) |
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2353 | { |
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2354 | max=v[i]; |
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2355 | maxpos=i; |
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2356 | } |
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2357 | } |
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2358 | return(maxpos); |
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2359 | } |
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2360 | |
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2361 | static proc maxPosInIntmat (intmat v) |
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2362 | "USAGE: maxPosInIntmat(v); v intmat |
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2363 | ASSUME: v has a unique maximal entry |
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2364 | RETURN: intvec, the position (i,j) of the maximal entry in v |
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2365 | NOTE: called by findOrientedBoundary" |
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2366 | { |
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2367 | int max=v[1,1]; |
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2368 | intvec maxpos=1,1; |
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2369 | int i,j; |
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2370 | for (i=1;i<=nrows(v);i++) |
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2371 | { |
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2372 | for (j=1;j<=ncols(v);j++) |
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2373 | { |
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2374 | if (v[i,j]>max) |
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2375 | { |
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2376 | max=v[i,j]; |
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2377 | maxpos=i,j; |
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2378 | } |
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2379 | } |
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2380 | } |
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2381 | return(maxpos); |
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2382 | } |
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2383 | |
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2384 | static proc sortintvec (intvec w) |
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2385 | "USAGE: sortintvec(v); v intvec |
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2386 | RETURN: intvec, the entries of v are ordered in an ascending way |
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2387 | NOTE: called from ellipticNF" |
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2388 | { |
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2389 | int j,k,stop; |
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2390 | intvec v=w[1]; |
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2391 | for (j=2;j<=size(w);j++) |
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2392 | { |
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2393 | k=1; |
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2394 | stop=0; |
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2395 | while ((k<=size(v)) and (stop==0)) |
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2396 | { |
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2397 | if (v[k]<w[j]) |
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2398 | { |
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2399 | k++; |
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2400 | } |
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2401 | else |
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2402 | { |
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2403 | stop=1; |
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2404 | } |
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2405 | } |
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2406 | if (k==size(v)+1) |
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2407 | { |
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2408 | v=v,w[j]; |
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2409 | } |
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2410 | else |
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2411 | { |
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2412 | if (k==1) |
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2413 | { |
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2414 | v=w[j],v; |
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2415 | } |
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2416 | else |
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2417 | { |
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2418 | v=v[1..k-1],w[j],v[k..size(v)]; |
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2419 | } |
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2420 | } |
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2421 | } |
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2422 | return(v); |
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2423 | } |
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2424 | |
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2425 | static proc sortlistbyintvec (list L,intvec w) |
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2426 | "USAGE: sortlistbyintvec(L,w); L list, w intvec |
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2427 | RETURN: list, the entries of L are ordered such that the corresponding reordering of |
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2428 | w would order w in an ascending way |
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2429 | NOTE: called from ellipticNF" |
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2430 | { |
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2431 | int j,k,stop; |
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2432 | intvec v=w[1]; |
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2433 | list LL=L[1]; |
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2434 | for (j=2;j<=size(w);j++) |
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2435 | { |
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2436 | k=1; |
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2437 | stop=0; |
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2438 | while ((k<=size(v)) and (stop==0)) |
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2439 | { |
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2440 | if (v[k]<w[j]) |
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2441 | { |
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2442 | k++; |
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2443 | } |
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2444 | else |
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2445 | { |
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2446 | stop=1; |
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2447 | } |
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2448 | } |
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2449 | if (k==size(v)+1) |
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2450 | { |
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2451 | v=v,w[j]; |
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2452 | LL=insert(LL,L[j],size(LL)); |
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2453 | } |
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2454 | else |
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2455 | { |
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2456 | if (k==1) |
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2457 | { |
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2458 | v=w[j],v; |
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2459 | LL=insert(LL,L[j]); |
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2460 | } |
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2461 | else |
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2462 | { |
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2463 | v=v[1..k-1],w[j],v[k..size(v)]; |
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2464 | LL=insert(LL,L[j],k-1); |
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2465 | } |
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2466 | } |
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2467 | } |
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2468 | return(LL); |
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2469 | } |
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2470 | |
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2471 | static proc matrixtointmat (matrix MM) |
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2472 | "USAGE: matrixtointmat(v); MM matrix |
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2473 | ASSUME: MM is a matrix with only integers as entries |
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2474 | RETURN: intmat, the matrix MM has been transformed to type intmat |
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2475 | NOTE: called from ellipticNF" |
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2476 | { |
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2477 | intmat M[nrows(MM)][ncols(MM)]=M; |
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2478 | int i,j; |
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2479 | for (i=1;i<=nrows(M);i++) |
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2480 | { |
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2481 | for (j=1;j<=ncols(M);j++) |
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2482 | { |
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2483 | execute("M["+string(i)+","+string(j)+"]="+string(MM[i,j])+";"); |
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2484 | } |
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2485 | } |
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2486 | return(M); |
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2487 | } |
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2488 | |
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2489 | ////////////////////////////////////////////////////////////////////////////// |
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2490 | |
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2491 | static proc polygonToCoordinates (list points) |
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2492 | "USAGE: polygonToCoordinates(points); points list |
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2493 | ASSUME: points is a list of integer vectors each of size two describing the |
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2494 | marked points of a convex lattice polygon like the output of polygonDB |
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2495 | RETURN: list, the first entry is a string representing the coordinates corresponding |
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2496 | to the latticpoints seperated by commata |
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2497 | the second entry is a list where the ith entry is a string representing |
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2498 | the coordinate of corresponding to the ith lattice point |
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2499 | the third entry is the latex format of the first entry |
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2500 | NOTE: the procedure is called by fan" |
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2501 | { |
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2502 | string coord; |
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2503 | list coords; |
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2504 | string latex; |
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2505 | for (int i=1;i<=size(points);i++) |
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2506 | { |
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2507 | coords[i]="u"+string(points[i][1])+string(points[i][2]); |
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2508 | coord=coord+coords[i]+","; |
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2509 | latex=latex+"u_{"+string(points[i][1])+string(points[i][2])+"},"; |
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2510 | } |
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2511 | coord=coord[1,size(coord)-1]; |
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2512 | latex=latex[1,size(latex)-1]; |
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2513 | return(list(coord,coords,latex)); |
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2514 | } |
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2515 | |
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2516 | |
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2517 | /* |
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2518 | proc ADeterminant (list polygon,list #) |
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2519 | { |
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2520 | list triangs=triangulations(polygon); |
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2521 | list sppg=splitPolygon(polygon); |
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2522 | list etavectors; |
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2523 | int i,j; |
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2524 | int stop; |
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2525 | for (i=1;i<=size(triangs);i++) |
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2526 | { |
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2527 | etavectors[i]=eta(triangs[i],sppg); |
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2528 | } |
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2529 | size(etavectors); |
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2530 | |
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2531 | for (i=size(etavectors);i>=2;i--) |
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2532 | { |
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2533 | stop=0; |
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2534 | for (j=1;(j<i) and (stop==0);j++) |
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2535 | { |
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2536 | if (etavectors[i]==etavectors[j]) |
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2537 | { |
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2538 | etavectors=delete(etavectors,i); |
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2539 | stop=1; |
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2540 | } |
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2541 | } |
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2542 | } |
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2543 | size(etavectors); |
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2544 | if (size(#)>0) |
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2545 | { |
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2546 | execute("ring ADring=(0,a(1.."+string(size(etavectors))+")),("+polygonToCoordinates(polygon)[1]+"),lp;"); |
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2547 | list terme; |
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2548 | poly ad,term; |
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2549 | matrix XE[1][1]; |
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2550 | for (i=1;i<=size(etavectors);i++) |
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2551 | { |
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2552 | term=1; |
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2553 | for (j=1;j<=nvars(basering);j++) |
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2554 | { |
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2555 | term=term*var(j)^etavectors[i][j]; |
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2556 | } |
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2557 | terme[i]=term; |
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2558 | ad=ad+a(i)*term; |
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2559 | } |
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2560 | matrix M[size(etavectors)][nvars(basering)]; |
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2561 | for (i=1;i<=size(etavectors);i++) |
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2562 | { |
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2563 | } |
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2564 | return(list(etavectors,string(ad))); |
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2565 | |
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2566 | |
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2567 | } |
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2568 | |
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2569 | return(etavectors); |
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2570 | } |
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2571 | |
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2572 | proc adsub () |
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2573 | { |
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2574 | ring r=0,(x,y,u00,u10,u20,u01,u11,u02,a(1..5)),dp; |
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2575 | poly f1=(3x-y+1)*(2x+y+1); |
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2576 | poly f2=(7x+2y-1)*(x+y); |
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2577 | poly f3=(x-y-2)*(x+y+3); |
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2578 | poly f4=(17x-11y+3)*(x+7y-2); |
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2579 | poly f5=(x+2y-7)*(3x+3y-1); |
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2580 | poly f6=(2x+12y-17)*(33x-3y-1); |
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2581 | matrix M1=coeffs(f1,ideal(1,x,x2,y,xy,y2)); |
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2582 | matrix M2=coeffs(f2,ideal(1,x,x2,y,xy,y2)); |
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2583 | matrix M3=coeffs(f3,ideal(1,x,x2,y,xy,y2)); |
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2584 | matrix M4=coeffs(f4,ideal(1,x,x2,y,xy,y2)); |
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2585 | matrix M5=coeffs(f5,ideal(1,x,x2,y,xy,y2)); |
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2586 | matrix M6=coeffs(f6,ideal(1,x,x2,y,xy,y2)); |
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2587 | poly f=(a(5))*u00*u20*u02+(a(3))*u00*u11^2+(a(4))*u10^2*u02+(a(2))*u10*u01*u11+(a(1))*u20*u01^2; |
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2588 | poly g1=substitute(f,u00,M1[1,1],u10,M1[2,1],u20,M1[3,1],u01,M1[4,1],u11,M1[5,1],u02,M1[6,1]); |
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2589 | poly g2=substitute(f,u00,M2[1,1],u10,M2[2,1],u20,M2[3,1],u01,M2[4,1],u11,M2[5,1],u02,M2[6,1]); |
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2590 | poly g3=substitute(f,u00,M3[1,1],u10,M3[2,1],u20,M3[3,1],u01,M3[4,1],u11,M3[5,1],u02,M3[6,1]); |
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2591 | poly g4=substitute(f,u00,M4[1,1],u10,M4[2,1],u20,M4[3,1],u01,M4[4,1],u11,M4[5,1],u02,M4[6,1]); |
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2592 | poly g5=substitute(f,u00,M5[1,1],u10,M5[2,1],u20,M5[3,1],u01,M5[4,1],u11,M5[5,1],u02,M5[6,1]); |
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2593 | poly g6=substitute(f,u00,M6[1,1],u10,M6[2,1],u20,M6[3,1],u01,M6[4,1],u11,M6[5,1],u02,M6[6,1]); |
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2594 | ideal i=g1,g2,g3,g4,g5,g6; |
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2595 | option(redSB); |
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2596 | ideal j=std(i); |
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2597 | poly ff=substitute(f,a(5),4,a(4),-1,a(3),-1,a(2),1,a(1),-1); |
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2598 | return(string(ff)); |
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2599 | } |
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2600 | |
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2601 | */ |
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