1 | version="$Id: polymake.lib,v 1.4 2008-08-06 13:14:56 keilen Exp $"; |
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2 | category="Tropical Geometry"; |
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3 | info=" |
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4 | LIBRARY: polymake.lib |
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5 | |
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6 | AUTHORS: Thomas Markwig, email: keilen@mathematik.uni-kl.de |
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7 | |
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8 | KEYWORDS: polytope, fan, secondary fan, secondary polytope, polymake, |
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9 | Newton polytope, Groebner fan |
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10 | |
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11 | WARNING: |
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12 | Most procedures will not work unless polymake or topcom is installed and |
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13 | if so, they will only work with the operating system LINUX! |
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14 | For more detailed information see IMPORTANT NOTE respectively consult the |
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15 | help string of the procedures. |
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16 | |
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17 | IMPORTANT NOTE: |
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18 | Even though this is a Singular library for computing polytopes and fans such |
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19 | as the Newton polytope or the Groebner fan of a polynomial, most of the hard |
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20 | computations are NOT done by Singular but by the program |
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21 | - polymake by Ewgenij Gawrilow, TU Berlin and Michael Joswig, TU Darmstadt |
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22 | (see http://www.math.tu-berlin.de/polymake/), |
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23 | respectively (only in the procedure triangularions) by the program |
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24 | - topcom by Joerg Rambau, Universitaet Bayreuth |
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25 | (see http://www.uni-bayreuth.de/departments/wirtschaftsmathematik/rambau/TOPCOM); |
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26 | this library should rather be seen as an interface which allows to use a (very limited) |
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27 | number of options which polymake respectively topcom offers to compute with polytopes |
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28 | and fans and to make the results available in Singular for further computations; |
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29 | moreover, the user familiar with Singular does not have to learn the syntax of polymake |
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30 | or topcom, if the options offered here are sufficient for his purposes. |
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31 | Note, though, that the procedures concerned with planar polygons are independent of |
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32 | both, polymake and topcom. |
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33 | |
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34 | PROCEDURES USING POLYMAKE: |
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35 | polymakePolytope(list) computes the vertices of a polytope using polymake |
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36 | newtonPolytope(poly) computes the Newton polytope of the polynomial |
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37 | newtonPolytopeLP(poly) computes the lattice points of the Newton polytope of the polynomial |
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38 | normalFan(intmat,intmat,list) computes the normal fan of a polytope |
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39 | groebnerFan(poly) computes the Groebner fan of a polynomial |
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40 | intmatToPolymake(intmat,string) transforms an integer matrix into polymake format |
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41 | polymakeToIntmat(string,string) transforms polymake output into an integer matrix |
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42 | |
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43 | PROCEDURES USING TOPCOM: |
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44 | triangulations(list) computes all triangulations of a marked polytope |
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45 | secondaryPolytope(list) computes the secondary polytope of a marked polytope |
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46 | |
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47 | PROCEDURES CONERNED WITH PLANAR POLYGONS: |
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48 | cycleLength(list,intvec) computes the cycleLength of cycle dual to list with interior point intvec |
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49 | splitPolygon(list) splits a marked polygon into vertices, facets and interior points |
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50 | eta(list,list) computes the eta-vector of a triangulation |
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51 | findOrientedBoundary(list) computes the boundary of the convex hull of a list of lattice points |
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52 | cyclePoints(list,list,int) computes lattice points connected to a lattice point in a triangulation |
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53 | latticeArea(list) computes the lattice area of a polygon |
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54 | picksFormula(list) computes the ingrediants of Pick's formula for a polygon |
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55 | ellipticNF(list) computes the normal form of an elliptic polygon |
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56 | ellipticNFDB(int) displays the 16 normal forms of elliptic polygons |
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57 | "; |
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58 | |
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59 | //////////////////////////////////////////////////////////////////////////////////// |
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60 | /// Auxilary Static Procedures in this Library |
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61 | //////////////////////////////////////////////////////////////////////////////////// |
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62 | /// - scalarproduct |
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63 | /// - intmatcoldelete |
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64 | /// - intmatconcat |
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65 | /// - sortlist |
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66 | /// - minInList |
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67 | /// - stringdelete |
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68 | /// - abs |
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69 | /// - commondenominator |
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70 | /// - maxPosInIntvec |
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71 | /// - maxPosInIntmat |
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72 | /// - sortintvec |
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73 | /// - matrixtointmat |
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74 | ///////////////////////////////////////////////////////////////////////////////////// |
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75 | |
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76 | //////////////////////////////////////////////////////////////////////////////////// |
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77 | LIB "poly.lib"; |
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78 | LIB "linalg.lib"; |
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79 | LIB "random.lib"; |
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80 | //////////////////////////////////////////////////////////////////////////////////// |
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81 | |
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82 | //////////////////////////////////////////////////////////////////////////////////////// |
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83 | /// PROCEDURES USING POLYMAKE |
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84 | //////////////////////////////////////////////////////////////////////////////////////// |
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85 | |
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86 | proc polymakePolytope (intmat polytope,list #) |
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87 | "USAGE: polymakePolytope(polytope[,#]); polytope list, # string |
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88 | ASSUME: each row of polytope gives the coordinates of a lattice point of a polytope |
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89 | with their affine coordinates as given by the output of secondaryPolytope |
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90 | PURPOSE: the procedure calls polymake to compute the vertices of the polytope as well |
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91 | as its dimension and information on its facets |
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92 | RETURN: list, L with four entries |
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93 | L[1] : an integer matrix whose rows are the coordinates of vertices |
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94 | of the polytope |
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95 | L[2] : the dimension of the polytope |
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96 | L[3] : a list whose ith entry explains to which vertices the ith vertex |
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97 | of the Newton polytope is connected -- i.e. L[3][i] is an integer |
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98 | vector and an entry k in there means that the vertex L[1][i] is |
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99 | connected to the vertex L[1][k] |
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100 | L[4] : an integer matrix whose rows mulitplied by (1,var(1),...,var(nvar)) give |
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101 | a linear system of equations describing the affine hull of the polytope, |
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102 | i.e. the smallest affine space containing the polytope |
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103 | NOTE: - for its computations the procedure calls the program polymake by Ewgenij Gawrilow, |
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104 | TU Berlin and Michael Joswig, TU Darmstadt; it therefore is necessary that |
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105 | this program is installed in order to use this procedure; |
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106 | see http://www.math.tu-berlin.de/polymake/ |
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107 | - note that in the vertex edge graph we have changed the polymake convention which |
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108 | starts indexing its vertices by zero while we start with one ! |
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109 | - the procedure creates the file /tmp/polytope.polymake which contains the polytope |
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110 | in polymake format and which can be used for further computations with polymake |
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111 | - moreover, the procedure creates the file /tmp/polytope.output which it deletes |
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112 | again before ending |
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113 | - it is possible to give as an optional second argument as string which then will be |
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114 | used instead of 'polytope' in the name of the polymake output file |
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115 | EXAMPLE: example polymakePolytope; shows an example" |
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116 | { |
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117 | // the header for the file secendarypolytope.polymake |
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118 | string sp="_application polytope |
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119 | _version 2.2 |
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120 | _type RationalPolytope |
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121 | |
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122 | POINTS |
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123 | "; |
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124 | int i,j; |
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125 | // set the name for the polymake output file |
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126 | if (size(#)>0) |
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127 | { |
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128 | if (typeof(#[1])=="string") |
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129 | { |
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130 | string dateiname=#[1]; |
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131 | } |
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132 | else |
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133 | { |
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134 | string dateiname="polytope"; |
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135 | } |
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136 | } |
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137 | else |
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138 | { |
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139 | string dateiname="polytope"; |
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140 | } |
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141 | // create the lattice point list for polymake |
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142 | sp=sp+intmatToPolymake(polytope,"points"); |
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143 | // initialise dateiname.polymake and compute the vertices |
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144 | write(":w /tmp/"+dateiname+".polymake",sp); |
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145 | system("sh","cd /tmp; polymake "+dateiname+".polymake VERTICES > "+dateiname+".output"); |
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146 | string vertices=read("/tmp/"+dateiname+".output"); |
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147 | system("sh","/bin/rm /tmp/"+dateiname+".output"); |
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148 | intmat np=polymakeToIntmat(vertices,"affine"); |
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149 | // compute the dimension |
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150 | system("sh","cd /tmp; polymake "+dateiname+".polymake DIM > "+dateiname+".output"); |
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151 | string pdim=read("/tmp/"+dateiname+".output"); |
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152 | system("sh","/bin/rm /tmp/"+dateiname+".output"); |
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153 | pdim=pdim[5,size(pdim)-6]; |
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154 | execute("int nd="+pdim+";"); |
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155 | // compute the vertex-edge graph |
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156 | system("sh","cd /tmp; polymake "+dateiname+".polymake GRAPH > "+dateiname+".output"); |
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157 | string vertexedgegraph=read("/tmp/"+dateiname+".output"); |
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158 | system("sh","/bin/rm /tmp/"+dateiname+".output"); |
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159 | vertexedgegraph=vertexedgegraph[7,size(vertexedgegraph)-8]; |
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160 | string newveg; |
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161 | for (i=1;i<=size(vertexedgegraph);i++) |
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162 | { |
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163 | if (vertexedgegraph[i]=="{") |
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164 | { |
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165 | newveg=newveg+"intvec("; |
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166 | } |
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167 | else |
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168 | { |
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169 | if (vertexedgegraph[i]=="}") |
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170 | { |
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171 | newveg=newveg+"),"; |
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172 | } |
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173 | else |
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174 | { |
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175 | if (vertexedgegraph[i]==" ") |
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176 | { |
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177 | newveg=newveg+","; |
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178 | } |
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179 | else |
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180 | { |
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181 | newveg=newveg+vertexedgegraph[i]; |
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182 | } |
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183 | } |
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184 | } |
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185 | } |
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186 | newveg=newveg[1,size(newveg)-1]; |
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187 | execute("list nveg="+newveg+";"); |
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188 | // raise each entry in nveg by one |
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189 | for (i=1;i<=size(nveg);i++) |
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190 | { |
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191 | for (j=1;j<=size(nveg[i]);j++) |
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192 | { |
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193 | nveg[i][j]=nveg[i][j]+1; |
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194 | } |
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195 | } |
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196 | // compute the affine hull |
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197 | system("sh","cd /tmp; polymake "+dateiname+".polymake AFFINE_HULL > "+dateiname+".output"); |
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198 | string equations=read("/tmp/"+dateiname+".output"); |
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199 | system("sh","/bin/rm /tmp/"+dateiname+".output"); |
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200 | if (size(equations)>14) |
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201 | { |
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202 | intmat neq=polymakeToIntmat(equations,"cleardenom"); |
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203 | } |
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204 | else |
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205 | { |
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206 | intmat neq[1][ncols(polytope)+1]; |
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207 | } |
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208 | return(list(np,nd,nveg,neq)); |
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209 | } |
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210 | example |
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211 | { |
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212 | "EXAMPLE:"; |
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213 | echo=2; |
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214 | // the lattice points of the unit square in the plane |
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215 | list points=intvec(0,0),intvec(0,1),intvec(1,0),intvec(1,1); |
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216 | // the secondary polytope of this lattice point configuration is computed |
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217 | intmat secpoly=secondaryPolytope(points)[1]; |
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218 | list np=polymakePolytope(secpoly); |
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219 | // the vertices of the Newton polytope are: |
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220 | np[1]; |
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221 | // its dimension is |
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222 | np[2]; |
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223 | // np[3] contains information how the vertices are connected to each other, |
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224 | // e.g. the first vertex (number 0) is connected to the second one |
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225 | np[3][1]; |
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226 | // the affine hull has the equation |
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227 | ring r=0,x(1..4),dp; |
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228 | matrix M[5][1]=1,x(1),x(2),x(3),x(4); |
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229 | np[4]*M; |
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230 | } |
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231 | |
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232 | proc newtonPolytope (poly f,list #) |
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233 | "USAGE: newtonPolytope(f[,#]); f poly, # string |
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234 | RETURN: list, L with four entries |
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235 | L[1] : an integer matrix whose rows are the coordinates of vertices |
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236 | of the Newton polytope of f |
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237 | L[2] : the dimension of the Newton polytope of f |
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238 | L[3] : a list whose ith entry explains to which vertices the ith vertex |
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239 | of the Newton polytope is connected -- i.e. L[3][i] is an integer |
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240 | vector and an entry k in there means that the vertex L[1][i] is |
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241 | connected to the vertex L[1][k] |
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242 | L[4] : an integer matrix whose rows mulitplied by (1,var(1),...,var(nvar)) give |
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243 | a linear system of equations describing the affine hull of the Newton |
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244 | polytope, i.e. the smallest affine space containing the Newton polytope |
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245 | 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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246 | the origin, then we get the equations for the orthogonal comploment of the linearity |
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247 | space of the normal fan dual to the Newton polytope, i.e. we get the EQUATIONS that |
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248 | we need as input for polymake when computing the normal fan |
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249 | - the procedure calls for its computation polymake by Ewgenij Gawrilow, |
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250 | TU Berlin and Michael Joswig, so it only works if polymake is installed; |
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251 | see http://www.math.tu-berlin.de/polymake/ |
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252 | - the procedure creates the file /tmp/newtonPolytope.polymake which contains the polytope |
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253 | in polymake format and which can be used for further computations with polymake |
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254 | - moreover, the procedure creates the file /tmp/newtonPolytope.output which it deletes |
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255 | again before ending |
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256 | - it is possible to give as an optional second argument as string which then will be |
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257 | used instead of 'newtonPolytope' in the name of the polymake output file |
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258 | EXAMPLE: example newtonPolytope; shows an example" |
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259 | { |
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260 | int i,j; |
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261 | // compute the list of exponent vectors of the polynomial, which are the lattice points |
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262 | // whose convex hull is the Newton polytope of f |
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263 | intmat exponents[size(f)][nvars(basering)]; |
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264 | while (f!=0) |
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265 | { |
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266 | i++; |
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267 | exponents[i,1..nvars(basering)]=leadexp(f); |
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268 | f=f-lead(f); |
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269 | } |
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270 | if (size(#)==0) |
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271 | { |
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272 | #[1]="newtonPolytope"; |
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273 | } |
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274 | // call polymakePolytope with exponents |
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275 | return(polymakePolytope(exponents,#)); |
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276 | } |
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277 | example |
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278 | { |
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279 | "EXAMPLE:"; |
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280 | echo=2; |
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281 | ring r=0,(x,y,z),dp; |
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282 | matrix M[4][1]=1,x,y,z; |
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283 | poly f=y3+x2+xy+2xz+yz+z2+1; |
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284 | // the Newton polytope of f is |
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285 | list np=newtonPolytope(f); |
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286 | // the vertices of the Newton polytope are: |
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287 | np[1]; |
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288 | // its dimension is |
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289 | np[2]; |
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290 | // np[3] contains information how the vertices are connected to each other, |
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291 | // e.g. the first vertex (number 0) is connected to the second, third and |
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292 | // fourth vertex |
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293 | np[3][1]; |
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294 | ////////////////////////// |
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295 | f=x2-y3; |
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296 | // the Newton polytope of f is |
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297 | np=newtonPolytope(f); |
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298 | // the vertices of the Newton polytope are: |
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299 | np[1]; |
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300 | // its dimension is |
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301 | np[2]; |
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302 | // the Newton polytope is contained in the affine space given by the equations |
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303 | np[4]*M; |
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304 | } |
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305 | |
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306 | proc newtonPolytopeLP (poly f) |
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307 | "USAGE: newtonPolytopeLP(f); f poly |
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308 | RETURN: list, the exponent vectors of the monomials occuring in f, i.e. the lattice |
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309 | points of the Newton polytope of f |
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310 | EXAMPLE: example normalFan; shows an example" |
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311 | { |
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312 | list np; |
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313 | int i=1; |
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314 | while (f!=0) |
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315 | { |
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316 | np[i]=leadexp(f); |
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317 | f=f-lead(f); |
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318 | i++; |
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319 | } |
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320 | return(np); |
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321 | } |
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322 | example |
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323 | { |
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324 | "EXAMPLE:"; |
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325 | echo=2; |
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326 | ring r=0,(x,y,z),dp; |
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327 | poly f=y3+x2+xy+2xz+yz+z2+1; |
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328 | // the lattice points of the Newton polytope of f are |
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329 | newtonPolytopeLP(f); |
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330 | } |
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331 | |
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332 | proc normalFan (intmat vertices,intmat affinehull,list graph,int er,list #) |
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333 | "USAGE: normalFan (vert,aff,graph,rays,[,#]); vert,aff intmat, graph list, rays int, # string |
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334 | ASSUME: - vert is an integer matrix whose rows are the coordinate of the vertices of |
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335 | a convex lattice polygon; |
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336 | - aff describes the affine hull of this polytope, i.e. |
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337 | the smallest affine space containing it, in the following sense: |
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338 | denote by n the number of columns of vert, then multiply aff by (1,x(1),...,x(n)) |
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339 | and set the resulting terms to zero in order to get the equations for the affine hull; |
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340 | - the ith entry of graph is an integer vector describing to which vertices |
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341 | the ith vertex is connected, i.e. a k as entry means that the vertex vert[i] is |
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342 | connected to vert[k]; |
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343 | - the integer rays is either one (if the extreme rays should be computed) or zero (otherwise) |
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344 | RETURN: list, the ith entry of L[1] contains information about the cone in the normal fan |
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345 | dual to the ith vertex of the polytope |
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346 | L[1][i][1] = integer matrix representing the inequalities which describe the |
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347 | cone dual to the ith vertex |
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348 | L[1][i][2] = a list which contains the inequalities represented by L[i][1] |
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349 | as a list of strings, where we use the variables x(1),...,x(n) |
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350 | L[1][i][3] = only present if 'er' is set to 1; in that case it is an interger matrix |
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351 | whose rows are the extreme rays of the cone |
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352 | L[2] = is an integer matrix whose rows span the linearity space of the fan, |
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353 | i.e. the linear space which is contained in each cone |
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354 | NOTE: - the procedure calls for its computation polymake by Ewgenij Gawrilow, |
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355 | TU Berlin and Michael Joswig, so it only works if polymake is installed; |
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356 | see http://www.math.tu-berlin.de/polymake/ |
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357 | - in the optional argument # it is possible to hand over other names for the |
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358 | variables to be used -- be carful, the format must be correct and that is |
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359 | not tested, e.g. if you want the variable names to be u00,u10,u01,u11 |
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360 | then you must hand over the string u11,u10,u01,u11 |
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361 | EXAMPLE: example normalFan; shows an example" |
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362 | { |
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363 | list ineq; // stores the inequalities of the cones |
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364 | int i,j,k; |
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365 | // we work over the following ring |
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366 | execute("ring ineqring=0,x(1.."+string(ncols(vertices))+"),lp;"); |
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367 | string greatersign=">"; |
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368 | // create the variable names |
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369 | if (size(#)>0) |
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370 | { |
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371 | if (typeof(#[1])=="string") |
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372 | { |
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373 | kill ineqring; |
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374 | execute("ring ineqring=0,("+#[1]+"),lp;"); |
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375 | } |
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376 | if (size(#)>1) |
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377 | { |
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378 | greatersign="<"; |
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379 | } |
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380 | } |
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381 | ////////////////////////////////////////////////////////////////// |
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382 | // Compute first the inequalities of the cones |
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383 | ////////////////////////////////////////////////////////////////// |
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384 | matrix VAR[1][ncols(vertices)]=maxideal(1); |
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385 | matrix EXP[ncols(vertices)][1]; |
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386 | poly p,pl,pr; |
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387 | // consider all vertices of the polytope |
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388 | for (i=1;i<=nrows(vertices);i++) |
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389 | { |
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390 | // first we produce for each vertex in the polytope |
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391 | // the inequalities describing the dual cone in the normal fan |
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392 | list pp; // contain strings representing the inequalities describing the normal cone |
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393 | intmat ie[size(graph[i])][ncols(vertices)]; // contains the inequalities as rows |
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394 | // consider all the vertices to which the ith vertex in the polytope is connected by an edge |
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395 | for (j=1;j<=size(graph[i]);j++) |
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396 | { |
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397 | // produce the vector ie_j pointing from the jth vertex to the ith vertex; |
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398 | // this will be the jth inequality for the cone in the normal fan dual to the ith vertex |
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399 | ie[j,1..ncols(vertices)]=vertices[i,1..ncols(vertices)]-vertices[graph[i][j],1..ncols(vertices)]; |
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400 | EXP=ie[j,1..ncols(vertices)]; |
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401 | // build a linear polynomial with the entries of ie_j as coefficients |
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402 | p=(VAR*EXP)[1,1]; |
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403 | pl,pr=0,0; |
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404 | // separate the terms with positive coefficients in p from those with negative coefficients |
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405 | for (k=1;k<=size(p);k++) |
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406 | { |
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407 | if (leadcoef(p[k])<0) |
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408 | { |
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409 | pr=pr-p[k]; |
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410 | } |
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411 | else |
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412 | { |
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413 | pl=pl+p[k]; |
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414 | } |
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415 | } |
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416 | // build the string which represents the jth inequality for the cone dual to the ith vertex |
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417 | // as polynomial inequality of type string, and store this in the list pp as jth entry |
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418 | pp[j]=string(pl)+" "+greatersign+" "+string(pr); |
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419 | } |
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420 | // all inequalities for the ith vertex are stored in the list ineq |
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421 | ineq[i]=list(ie,pp); |
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422 | kill ie,pp; // kill certain lists |
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423 | } |
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424 | // remove the first column of affine hull to compute the linearity space |
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425 | intmat linearity=intmatcoldelete(affinehull,1); |
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426 | ////////////////////////////////////////////////////////////////// |
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427 | // Compute next the extreme rays of the cones |
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428 | ////////////////////////////////////////////////////////////////// |
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429 | if (er==1) |
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430 | { |
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431 | list extremerays; // keeps the result |
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432 | string polymake; // keeps polymake output |
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433 | // the header for ineq.polymake |
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434 | string head="_application polytope |
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435 | _version 2.2 |
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436 | _type RationalPolytope |
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437 | |
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438 | INEQUALITIES |
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439 | "; |
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440 | // the tail for both polymake files |
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441 | string tail=" |
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442 | EQUATIONS |
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443 | "; |
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444 | tail=tail+intmatToPolymake(linearity,"rays"); |
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445 | string ungleichungen; // keeps the inequalities for the polymake code |
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446 | intmat M; // the matrix keeping the inequalities |
---|
447 | // create the file ineq.output |
---|
448 | write(":w /tmp/ineq.output",""); |
---|
449 | int dimension; // keeps the dimension of the intersection the bad cones with the u11tobeseencone |
---|
450 | for (i=1;i<=size(ineq);i++) |
---|
451 | { |
---|
452 | i,". Cone of ",nrows(vertices); // indicate how many vertices have been dealt with |
---|
453 | ungleichungen=intmatToPolymake(ineq[i][1],"rays"); |
---|
454 | // write the inequalities to ineq.polymake and call polymake |
---|
455 | write(":w /tmp/ineq.polymake",head+ungleichungen+tail); |
---|
456 | ungleichungen=""; // clear ungleichungen |
---|
457 | system("sh","cd /tmp; /bin/rm ineq.output; polymake ineq.polymake VERTICES > ineq.output"); |
---|
458 | // read the result of polymake |
---|
459 | polymake=read("/tmp/ineq.output"); |
---|
460 | intmat VERT=polymakeToIntmat(polymake,"affine"); |
---|
461 | extremerays[i]=VERT; |
---|
462 | kill VERT; |
---|
463 | } |
---|
464 | for (i=1;i<=size(ineq);i++) |
---|
465 | { |
---|
466 | ineq[i]=ineq[i]+list(extremerays[i]); |
---|
467 | } |
---|
468 | } |
---|
469 | // get the linearity space |
---|
470 | return(list(ineq,linearity)); |
---|
471 | } |
---|
472 | example |
---|
473 | { |
---|
474 | "EXAMPLE:"; |
---|
475 | echo=2; |
---|
476 | ring r=0,(x,y,z),dp; |
---|
477 | matrix M[4][1]=1,x,y,z; |
---|
478 | poly f=y3+x2+xy+2xz+yz+z2+1; |
---|
479 | // the Newton polytope of f is |
---|
480 | list np=newtonPolytope(f); |
---|
481 | // the Groebner fan of f, i.e. the normal fan of the Newton polytope |
---|
482 | list gf=normalFan(np[1],np[4],np[3],1,"x,y,z"); |
---|
483 | // the number of cones in the Groebner fan of f is: |
---|
484 | size(gf[1]); |
---|
485 | // the inequalities of the first cone as matrix are: |
---|
486 | print(gf[1][1][1]); |
---|
487 | // the inequalities of the first cone as string are: |
---|
488 | print(gf[1][1][2]); |
---|
489 | // the rows of the following matrix are the extreme rays of the first cone: |
---|
490 | print(gf[1][1][3]); |
---|
491 | // each cone contains the linearity space spanned by: |
---|
492 | print(gf[2]); |
---|
493 | } |
---|
494 | |
---|
495 | proc groebnerFan (poly f,list #) |
---|
496 | "USAGE: groebnerFan(f[,#]); f poly, # string |
---|
497 | RETURN: list, the ith entry of L[1] contains information about the ith cone in the Groebner fan |
---|
498 | dual to the ith vertex in the Newton polytope of the f |
---|
499 | L[1][i][1] = integer matrix representing the inequalities which describe the cone |
---|
500 | L[1][i][2] = a list which contains the inequalities represented by L[i][1] |
---|
501 | as a list of strings |
---|
502 | L[1][i][3] = an interger matrix whose rows are the extreme rays of the cone |
---|
503 | L[2] = is an integer matrix whose rows span the linearity space of the fan, |
---|
504 | i.e. the linear space which is contained in each cone |
---|
505 | L[3] = the Newton polytope of f in the format of the procedure newtonPolytope |
---|
506 | NOTE: - if you have alread computed the Newton polytope of f then you might want |
---|
507 | to use the procedure normalFan instead in order to avoid doing costly computation |
---|
508 | twice |
---|
509 | - the procedure calls for its computation polymake by Ewgenij Gawrilow, |
---|
510 | TU Berlin and Michael Joswig, so it only works if polymake is installed; |
---|
511 | see http://www.math.tu-berlin.de/polymake/ |
---|
512 | - the procedure creates the file /tmp/newtonPolytope.polymake which contains the |
---|
513 | Newton polytope of f in polymake format and which can be used for further |
---|
514 | computations with polymake |
---|
515 | - it is possible to give as an optional second argument as string which then will be |
---|
516 | used instead of 'newtonPolytope' in the name of the polymake output file |
---|
517 | EXAMPLE: example groebnerFan; shows an example" |
---|
518 | { |
---|
519 | int i,j; |
---|
520 | // compute the list of exponent vectors of the polynomial, which are the lattice points |
---|
521 | // whose convex hull is the Newton polytope of f |
---|
522 | intmat exponents[size(f)][nvars(basering)]; |
---|
523 | while (f!=0) |
---|
524 | { |
---|
525 | i++; |
---|
526 | exponents[i,1..nvars(basering)]=leadexp(f); |
---|
527 | f=f-lead(f); |
---|
528 | } |
---|
529 | if (size(#)==0) |
---|
530 | { |
---|
531 | #[1]="newtonPolytope"; |
---|
532 | } |
---|
533 | // call polymakePolytope with exponents |
---|
534 | list newtonp=polymakePolytope(exponents,"newtonPolytope"); |
---|
535 | // get the variables as string |
---|
536 | string variablen; |
---|
537 | for (i=1;i<=nvars(basering);i++) |
---|
538 | { |
---|
539 | variablen=variablen+string(var(i))+","; |
---|
540 | } |
---|
541 | variablen=variablen[1,size(variablen)-1]; |
---|
542 | // call normalFan in order to compute the Groebner fan |
---|
543 | list gf=normalFan(newtonp[1],newtonp[4],newtonp[3],1,variablen); |
---|
544 | // append newtonp to gf |
---|
545 | gf[3]=newtonp; |
---|
546 | // append newtonp the exponent vectors to gt |
---|
547 | gf[4]=exponents; |
---|
548 | return(gf); |
---|
549 | } |
---|
550 | example |
---|
551 | { |
---|
552 | "EXAMPLE:"; |
---|
553 | echo=2; |
---|
554 | ring r=0,(x,y,z),dp; |
---|
555 | matrix M[4][1]=1,x,y,z; |
---|
556 | poly f=y3+x2+xy+2xz+yz+z2+1; |
---|
557 | // the Newton polytope of f is |
---|
558 | list gf=groebnerFan(f); |
---|
559 | // the exponent vectors of f are ordered as follows |
---|
560 | gf[4]; |
---|
561 | // the first cone of the groebner fan has the inequalities |
---|
562 | gf[1][1][1]; |
---|
563 | // as a string they look like |
---|
564 | gf[1][1][2]; |
---|
565 | // and it has the extreme rays |
---|
566 | print(gf[1][1][3]); |
---|
567 | // the linearity space is spanned by |
---|
568 | print(gf[2]); |
---|
569 | // the vertices of the Newton polytope are: |
---|
570 | gf[3][1]; |
---|
571 | // its dimension is |
---|
572 | gf[3][2]; |
---|
573 | // np[3] contains information how the vertices are connected to each other, |
---|
574 | // e.g. the first vertex is connected to the second, third and fourth vertex |
---|
575 | gf[3][3][1]; |
---|
576 | } |
---|
577 | |
---|
578 | |
---|
579 | |
---|
580 | proc intmatToPolymake (intmat M,string art) |
---|
581 | "USAGE: intmatToPolymake(M,art); M intmat, art string |
---|
582 | ASSUME: - M is an integer matrix which should be transformed into polymake format; |
---|
583 | - art is one of the following strings |
---|
584 | - 'rays' : indicating that a first column of 0's should be added |
---|
585 | - 'points' : indicating that a first column of 1's should be added |
---|
586 | RETURN: string, the matrix is transformed in a string and a first column has been added |
---|
587 | EXAMPLE: example intmatToPolymake; shows an example" |
---|
588 | { |
---|
589 | if (art=="rays") |
---|
590 | { |
---|
591 | string anf="0 "; |
---|
592 | } |
---|
593 | else |
---|
594 | { |
---|
595 | string anf="1 "; |
---|
596 | } |
---|
597 | string sp; |
---|
598 | int i,j; |
---|
599 | // create the lattice point list for polymake |
---|
600 | for (i=1;i<=nrows(M);i++) |
---|
601 | { |
---|
602 | sp=sp+anf; |
---|
603 | for (j=1;j<=ncols(M);j++) |
---|
604 | { |
---|
605 | sp=sp+string(M[i,j])+" "; |
---|
606 | if (j==ncols(M)) |
---|
607 | { |
---|
608 | sp=sp+" |
---|
609 | "; |
---|
610 | } |
---|
611 | } |
---|
612 | } |
---|
613 | return(sp); |
---|
614 | } |
---|
615 | example |
---|
616 | { |
---|
617 | "EXAMPLE:"; |
---|
618 | echo=2; |
---|
619 | intmat M[3][4]=1,2,3,4,5,6,7,8,9,10,11,12; |
---|
620 | intmatToPolymake(M,"rays"); |
---|
621 | intmatToPolymake(M,"points"); |
---|
622 | } |
---|
623 | |
---|
624 | proc polymakeToIntmat (string pm,string art) |
---|
625 | "USAGE: polymakeToIntmat(pm,art); pm, art string |
---|
626 | ASSUME: pm is the result of calling polymake with one 'argument' like VERTICES, AFFINE_HULL, etc., |
---|
627 | so that the first row of the string is the name of the corresponding 'argument' and |
---|
628 | the further rows contain the result which consists of vectors either over the integers |
---|
629 | or over the rationals |
---|
630 | RETURN: intmat, the rows of the matrix are basically the vectors in pm from the second row on |
---|
631 | where each row has been multiplied with the lowest common multiple of the |
---|
632 | denominators of its entries so as to be an integer matrix; moreover, |
---|
633 | if art=='affine', then the first column is omitted since we only want affine |
---|
634 | coordinates |
---|
635 | EXAMPLE: example polymakeToIntmat; shows an example" |
---|
636 | { |
---|
637 | // we need a line break |
---|
638 | string zeilenumbruch=" |
---|
639 | "; |
---|
640 | // remove the 'argment' name, i.e. the first row of pm |
---|
641 | while (pm[1]!=zeilenumbruch) |
---|
642 | { |
---|
643 | pm=stringdelete(pm,1); |
---|
644 | } |
---|
645 | pm=stringdelete(pm,1); |
---|
646 | // find out how many entries each vector has, namely one more than 'spaces' in a row |
---|
647 | int i=1; |
---|
648 | int s=1; |
---|
649 | int z=1; |
---|
650 | while (pm[i]!=zeilenumbruch) |
---|
651 | { |
---|
652 | if (pm[i]==" ") |
---|
653 | { |
---|
654 | s++; |
---|
655 | } |
---|
656 | i++; |
---|
657 | } |
---|
658 | // if we want to have affine coordinates |
---|
659 | if (art=="affine") |
---|
660 | { |
---|
661 | s--; // then there is one column less |
---|
662 | // and the entry of the first column (in the first row) has to be removed |
---|
663 | while (pm[1]!=" ") |
---|
664 | { |
---|
665 | pm=stringdelete(pm,1); |
---|
666 | } |
---|
667 | pm=stringdelete(pm,1); |
---|
668 | } |
---|
669 | // we add two line breaks at the end in order to have this as a stopping criterion |
---|
670 | pm=pm+zeilenumbruch+zeilenumbruch; |
---|
671 | // we now have to work through each row |
---|
672 | for (i=1;i<=size(pm);i++) |
---|
673 | { |
---|
674 | // if there are two consecutive line breaks we are done |
---|
675 | if ((pm[i]==zeilenumbruch) and (pm[i+1]==zeilenumbruch)) |
---|
676 | { |
---|
677 | i=size(pm)+1; |
---|
678 | } |
---|
679 | else |
---|
680 | { |
---|
681 | // a line break has to be replaced by a comma |
---|
682 | if (pm[i]==zeilenumbruch) |
---|
683 | { |
---|
684 | z++; |
---|
685 | pm[i]=","; |
---|
686 | // if we want to have affine coordinates, then we have to delete the first entry in each row |
---|
687 | if (art=="affine") |
---|
688 | { |
---|
689 | while (pm[i+1]!=" ") |
---|
690 | { |
---|
691 | pm=stringdelete(pm,i+1); |
---|
692 | } |
---|
693 | pm=stringdelete(pm,i+1); |
---|
694 | } |
---|
695 | } |
---|
696 | // a space has to be replaced by a comma |
---|
697 | if (pm[i]==" ") |
---|
698 | { |
---|
699 | pm[i]=","; |
---|
700 | } |
---|
701 | } |
---|
702 | } |
---|
703 | // if we have introduced superflous commata at the end, they should be removed |
---|
704 | while (pm[size(pm)]==",") |
---|
705 | { |
---|
706 | pm=stringdelete(pm,size(pm)); |
---|
707 | } |
---|
708 | // since the matrix could be over the rationals, we need a ring with rational coefficients |
---|
709 | ring zwischering=0,x,lp; |
---|
710 | // create the matrix with the elements of pm as entries |
---|
711 | execute("matrix mm["+string(z)+"]["+string(s)+"]="+pm+";"); |
---|
712 | // transform this into an integer matrix |
---|
713 | matrix M[1][ncols(mm)]; // takes a row of mm |
---|
714 | int cm; // takes a lowest common multiple |
---|
715 | // multiply each row by an integer such that its entries are integers |
---|
716 | for (int j=1;j<=nrows(mm);j++) |
---|
717 | { |
---|
718 | M=mm[j,1..ncols(mm)]; |
---|
719 | cm=commondenominator(M); |
---|
720 | for (i=1;i<=ncols(mm);i++) |
---|
721 | { |
---|
722 | mm[j,i]=cm*mm[j,i]; |
---|
723 | } |
---|
724 | } |
---|
725 | // transform the matrix mm into an integer matrix |
---|
726 | execute("intmat im["+string(z)+"]["+string(s)+"]="+string(mm)+";"); |
---|
727 | return(im); |
---|
728 | } |
---|
729 | example |
---|
730 | { |
---|
731 | "EXAMPLE:"; |
---|
732 | echo=2; |
---|
733 | // this is the usual output of some polymake computation |
---|
734 | string pm="VERTICES |
---|
735 | 0 1 3 5/3 1/3 -1 -23/3 -1/3 5/3 1/3 1 |
---|
736 | 0 1 3 -23/3 5/3 1 5/3 1/3 1/3 -1/3 -1 |
---|
737 | 0 1 1 1/3 -1/3 -1 5/3 1/3 -23/3 5/3 3 |
---|
738 | 0 1 1 5/3 -23/3 3 1/3 5/3 -1/3 1/3 -1 |
---|
739 | 0 1 -1 1/3 5/3 3 -1/3 -23/3 1/3 5/3 1 |
---|
740 | 0 1 -1 -1/3 1/3 1 1/3 5/3 5/3 -23/3 3 |
---|
741 | 0 1 -1 1 3 -5 -1 3 -1 1 -1 |
---|
742 | 0 1 -1 -1 -1 -1 1 1 3 3 -5 |
---|
743 | 0 1 -5 3 1 -1 3 -1 1 -1 -1 |
---|
744 | |
---|
745 | "; |
---|
746 | intmat PM=polymakeToIntmat(pm,"affine"); |
---|
747 | // note that the first column has been removed, since we asked for |
---|
748 | // affine coordinates, and the denominators have been cleared |
---|
749 | print(PM); |
---|
750 | } |
---|
751 | |
---|
752 | //////////////////////////////////////////////////////////////////////////////////////// |
---|
753 | /// PROCEDURES USING TOPCOM |
---|
754 | //////////////////////////////////////////////////////////////////////////////////////// |
---|
755 | |
---|
756 | proc triangulations (list polygon) |
---|
757 | "USAGE: triangulations(polygon); list polygon |
---|
758 | ASSUME: polygon is a list of integer vectors of the same size representing the affine |
---|
759 | coordinates of the lattice points |
---|
760 | PURPOSE: the procedure considers the marked polytope given as the convex hull of |
---|
761 | the lattice points and with these lattice points as markings; it then |
---|
762 | computes all possible triangulations of this marked polytope |
---|
763 | RETURN: list, each entry corresponds to one triangulation and the ith entry is |
---|
764 | itself a list of integer vectors of size three, where each integer |
---|
765 | vector defines one triangle in the triangulation by telling which |
---|
766 | points of the input are the vertices of the triangle |
---|
767 | NOTE: - the procedure calls for its computations the program points2triangs |
---|
768 | from the program topcom by Joerg Rambau, Universitaet Bayreuth; it |
---|
769 | therefore is necessary that this program is installed in order to use this |
---|
770 | procedure; |
---|
771 | see http://www.uni-bayreuth.de/departments/wirtschaftsmathematik/rambau/TOPCOM |
---|
772 | - the procedure creates the files /tmp/triangulationsinput and /tmp/triangulationsoutput; |
---|
773 | the former is used as input for points2triangs and the latter is its output |
---|
774 | containing the triangulations of corresponding to points in the format |
---|
775 | of points2triangs; this can be use for further computations with topcom |
---|
776 | - note that an integer i in an integer vector representing a triangle refers to |
---|
777 | the ith lattice point, i.e. polygon[i]; this convention is different from |
---|
778 | TOPCOM's convention, where i would refer to the i-1st lattice point |
---|
779 | EXAMPLE: example triangulations; shows an example" |
---|
780 | { |
---|
781 | int i,j; |
---|
782 | // prepare the input for points2triangs by writing the input polygon in the |
---|
783 | // necessary format |
---|
784 | string spi="["; |
---|
785 | for (i=1;i<=size(polygon);i++) |
---|
786 | { |
---|
787 | polygon[i][size(polygon[i])+1]=1; |
---|
788 | spi=spi+"["+string(polygon[i])+"]"; |
---|
789 | if (i<size(polygon)) |
---|
790 | { |
---|
791 | spi=spi+","; |
---|
792 | } |
---|
793 | } |
---|
794 | spi=spi+"]"; |
---|
795 | write(":w /tmp/triangulationsinput",spi); |
---|
796 | // call points2triangs |
---|
797 | system("sh","cd /tmp; points2triangs < triangulationsinput > triangulationsoutput"); |
---|
798 | string p2t=read("/tmp/triangulationsoutput"); // takes the result of points2triangs |
---|
799 | // preprocessing of p2t if points2triangs is version >= 0.15 brings p2t to the format of version 0.14 |
---|
800 | string np2t; // takes the triangulations in Singular format |
---|
801 | for (i=1;i<=size(p2t)-2;i++) |
---|
802 | { |
---|
803 | if ((p2t[i]==":") and (p2t[i+1]=="=") and (p2t[i+2]=="[")) |
---|
804 | { |
---|
805 | np2t=np2t+p2t[i]+p2t[i+1]; |
---|
806 | i=i+3; |
---|
807 | while (p2t[i]!=":") |
---|
808 | { |
---|
809 | i=i+1; |
---|
810 | } |
---|
811 | } |
---|
812 | else |
---|
813 | { |
---|
814 | if ((p2t[i]=="]") and (p2t[i+1]==";")) |
---|
815 | { |
---|
816 | np2t=np2t+p2t[i+1]; |
---|
817 | i=i+1; |
---|
818 | } |
---|
819 | else |
---|
820 | { |
---|
821 | np2t=np2t+p2t[i]; |
---|
822 | } |
---|
823 | } |
---|
824 | } |
---|
825 | if (p2t[size(p2t)-1]=="]") |
---|
826 | { |
---|
827 | np2t=np2t+p2t[size(p2t)]; |
---|
828 | } |
---|
829 | else |
---|
830 | { |
---|
831 | if (np2t[size(np2t)]!=";") |
---|
832 | { |
---|
833 | np2t=np2t+p2t[size(p2t)-1]+p2t[size(p2t)]; |
---|
834 | } |
---|
835 | } |
---|
836 | p2t=np2t; |
---|
837 | np2t=""; |
---|
838 | // transform the points2triangs output of version 0.14 into Singular format |
---|
839 | for (i=1;i<=size(p2t);i++) |
---|
840 | { |
---|
841 | if (p2t[i]=="=") |
---|
842 | { |
---|
843 | np2t=np2t+p2t[i]+"list("; |
---|
844 | i++; |
---|
845 | } |
---|
846 | else |
---|
847 | { |
---|
848 | if (p2t[i]!=":") |
---|
849 | { |
---|
850 | if ((p2t[i]=="}") and (p2t[i+1]=="}")) |
---|
851 | { |
---|
852 | np2t=np2t+"))"; |
---|
853 | i++; |
---|
854 | } |
---|
855 | else |
---|
856 | { |
---|
857 | if (p2t[i]=="{") |
---|
858 | { |
---|
859 | np2t=np2t+"intvec("; |
---|
860 | } |
---|
861 | else |
---|
862 | { |
---|
863 | if (p2t[i]=="}") |
---|
864 | { |
---|
865 | np2t=np2t+")"; |
---|
866 | } |
---|
867 | else |
---|
868 | { |
---|
869 | np2t=np2t+p2t[i]; |
---|
870 | } |
---|
871 | } |
---|
872 | } |
---|
873 | } |
---|
874 | } |
---|
875 | } |
---|
876 | list T; |
---|
877 | execute(np2t); |
---|
878 | // raise each index by one |
---|
879 | for (i=1;i<=size(T);i++) |
---|
880 | { |
---|
881 | for (j=1;j<=size(T[i]);j++) |
---|
882 | { |
---|
883 | T[i][j]=T[i][j]+1; |
---|
884 | } |
---|
885 | } |
---|
886 | return(T); |
---|
887 | } |
---|
888 | example |
---|
889 | { |
---|
890 | "EXAMPLE:"; |
---|
891 | echo=2; |
---|
892 | // the lattice points of the unit square in the plane |
---|
893 | list polygon=intvec(0,0),intvec(0,1),intvec(1,0),intvec(1,1); |
---|
894 | // the triangulations of this lattice point configuration are computed |
---|
895 | list triang=triangulations(polygon); |
---|
896 | triang; |
---|
897 | } |
---|
898 | |
---|
899 | proc secondaryPolytope (list polygon,list #) |
---|
900 | "USAGE: secondaryPolytope(polygon[,#]); list polygon, list # |
---|
901 | ASSUME: - polygon is a list of integer vectors of the same size representing the affine |
---|
902 | coordinates of lattice points |
---|
903 | - if the triangulations of the corresponding polygon have already been computed |
---|
904 | with the procedure triangulations then these can be given as a second (optional) |
---|
905 | argument in order to avoid doing this computation again |
---|
906 | PURPOSE: the procedure considers the marked polytope given as the convex hull of |
---|
907 | the lattice points and with these lattice points as markings; it then |
---|
908 | computes the lattice points of the secondary polytope given by this |
---|
909 | marked polytope which correspond to the triangulations computed by |
---|
910 | the procedure triangulations |
---|
911 | RETURN: list, say L, such that: |
---|
912 | L[1] = intmat, each row gives the affine coordinates of a lattice point |
---|
913 | in the secondary polytope given by the marked |
---|
914 | polytope corresponding to polygon |
---|
915 | L[2] = the list of corresponding triangulations |
---|
916 | NOTE: if the triangluations are not handed over as optional argument the procedure calls |
---|
917 | for its computation of these triangulations the program points2triangs |
---|
918 | from the program topcom by Joerg Rambau, Universitaet Bayreuth; it |
---|
919 | therefore is necessary that this program is installed in order to use this |
---|
920 | procedure; |
---|
921 | see http://www.uni-bayreuth.de/departments/wirtschaftsmathematik/rambau/TOPCOM |
---|
922 | EXAMPLE: example secondaryPolytope; shows an example" |
---|
923 | { |
---|
924 | // compute the triangulations of the point configuration with points2triangs |
---|
925 | if (size(#)==0) |
---|
926 | { |
---|
927 | list triangs=triangulations(polygon); |
---|
928 | } |
---|
929 | else |
---|
930 | { |
---|
931 | list triangs=#; |
---|
932 | } |
---|
933 | int i,j,k,l; |
---|
934 | intmat N[2][2]; // is used to compute areas of triangles |
---|
935 | intvec vertex; // stores a point in the secondary polytope as intermediate result |
---|
936 | int eintrag; |
---|
937 | int halt; |
---|
938 | intmat secpoly[size(triangs)][size(polygon)]; // stores all lattice points of the secondary polytope |
---|
939 | // consider each triangulation and compute the corresponding point in the secondary polytope |
---|
940 | for (i=1;i<=size(triangs);i++) |
---|
941 | { |
---|
942 | // for each triangulation we have to compute the coordinates corresponding to each marked point |
---|
943 | for (j=1;j<=size(polygon);j++) |
---|
944 | { |
---|
945 | eintrag=0; |
---|
946 | // for each marked point we have to consider all triangles in the triangulation |
---|
947 | // which involve this particular point |
---|
948 | for (k=1;k<=size(triangs[i]);k++) |
---|
949 | { |
---|
950 | halt=0; |
---|
951 | for (l=1;(l<=3) and (halt==0);l++) |
---|
952 | { |
---|
953 | if (triangs[i][k][l]==j) |
---|
954 | { |
---|
955 | halt=1; |
---|
956 | N[1,1]=polygon[triangs[i][k][3]][1]-polygon[triangs[i][k][1]][1]; |
---|
957 | N[1,2]=polygon[triangs[i][k][2]][1]-polygon[triangs[i][k][1]][1]; |
---|
958 | N[2,1]=polygon[triangs[i][k][3]][2]-polygon[triangs[i][k][1]][2]; |
---|
959 | N[2,2]=polygon[triangs[i][k][2]][2]-polygon[triangs[i][k][1]][2]; |
---|
960 | eintrag=eintrag+abs(det(N)); |
---|
961 | } |
---|
962 | } |
---|
963 | } |
---|
964 | vertex[j]=eintrag; |
---|
965 | } |
---|
966 | secpoly[i,1..size(polygon)]=vertex; |
---|
967 | } |
---|
968 | return(list(secpoly,triangs)); |
---|
969 | } |
---|
970 | example |
---|
971 | { |
---|
972 | "EXAMPLE:"; |
---|
973 | echo=2; |
---|
974 | // the lattice points of the unit square in the plane |
---|
975 | list polygon=intvec(0,0),intvec(0,1),intvec(1,0),intvec(1,1); |
---|
976 | // the secondary polytope of this lattice point configuration is computed |
---|
977 | list secpoly=secondaryPolytope(polygon); |
---|
978 | // the points in the secondary polytope |
---|
979 | print(secpoly[1]); |
---|
980 | // the corresponding triangulations |
---|
981 | secpoly[2]; |
---|
982 | } |
---|
983 | |
---|
984 | //////////////////////////////////////////////////////////////////////////////////////// |
---|
985 | /// PROCEDURES CONCERNED WITH PLANAR POLYGONS |
---|
986 | //////////////////////////////////////////////////////////////////////////////////////// |
---|
987 | |
---|
988 | proc cycleLength (list boundary,intvec interior) |
---|
989 | "USAGE: cycleLength(boundary,interior); list boundary, intvec interior |
---|
990 | ASSUME: boundary is a list of integer vectors describing a cycle in some convex lattice |
---|
991 | polygon around the lattice point interior ordered clock wise |
---|
992 | RETURN: string, the cycle length of the corresponding cycle in the dual tropical curve |
---|
993 | EXAMPLE: example cycleLength; shows an example" |
---|
994 | { |
---|
995 | int j; |
---|
996 | // create a ring whose variables are indexed by the points in boundary resp. by interior |
---|
997 | string rst="ring cyclering=0,(u"+string(interior[1])+string(interior[2]); |
---|
998 | for (j=1;j<=size(boundary);j++) |
---|
999 | { |
---|
1000 | rst=rst+",u"+string(boundary[j][1])+string(boundary[j][2]); |
---|
1001 | } |
---|
1002 | rst=rst+"),lp;"; |
---|
1003 | execute(rst); |
---|
1004 | // add the first and second point at the end of boundary |
---|
1005 | boundary[size(boundary)+1]=boundary[1]; |
---|
1006 | boundary[size(boundary)+1]=boundary[2]; |
---|
1007 | poly cl,summand; // takes the cycle length |
---|
1008 | matrix N1[2][2]; // used to compute the area of a triangle |
---|
1009 | matrix N2[2][2]; // used to compute the area of a triangle |
---|
1010 | matrix N3[2][2]; // used to compute the area of a triangle |
---|
1011 | // for each original point in the boundary compute its contribution to the cycle |
---|
1012 | for (j=2;j<=size(boundary)-1;j++) |
---|
1013 | { |
---|
1014 | N1=boundary[j-1]-interior,boundary[j]-interior; |
---|
1015 | N2=boundary[j]-interior,boundary[j+1]-interior; |
---|
1016 | N3=boundary[j+1]-interior,boundary[j-1]-interior; |
---|
1017 | execute("summand=-u"+string(boundary[j][1])+string(boundary[j][2])+"+u"+string(interior[1])+string(interior[2])+";"); |
---|
1018 | summand=summand*(det(N1)+det(N2)+det(N3))/(det(N1)*det(N2)); |
---|
1019 | cl=cl+summand; |
---|
1020 | } |
---|
1021 | return(string(cl)); |
---|
1022 | } |
---|
1023 | example |
---|
1024 | { |
---|
1025 | "EXAMPLE:"; |
---|
1026 | echo=2; |
---|
1027 | // the integer vectors in boundary are lattice points on the boundary |
---|
1028 | // of a convex lattice polygon in the plane |
---|
1029 | list boundary=intvec(0,0),intvec(0,1),intvec(0,2),intvec(2,2),intvec(2,1),intvec(2,0); |
---|
1030 | // interior is a lattice point in the interior of this lattice polygon |
---|
1031 | intvec interior=1,1; |
---|
1032 | // compute the general cycle length of a cycle of the corresponding cycle |
---|
1033 | // in the dual tropical curve -- note that (0,1) and (2,1) do not contribute |
---|
1034 | cycleLength(boundary,interior); |
---|
1035 | } |
---|
1036 | |
---|
1037 | proc splitPolygon (list markings) |
---|
1038 | "USAGE: splitPolygon (markings); markings list |
---|
1039 | ASSUME: markings is a list of integer vectors representing lattice points in the plane |
---|
1040 | which we consider as the marked points of the convex lattice polytope spanned by them |
---|
1041 | PURPOSE: split the marked points in the vertices, the points on the facets which are not vertices, |
---|
1042 | and the interior points |
---|
1043 | RETURN: list, L consisting of three lists |
---|
1044 | L[1] : represents the vertices the polygon ordered clockwise |
---|
1045 | L[1][i][1] = intvec, the coordinates of the ith vertex |
---|
1046 | L[1][i][2] = int, the position of L[1][i][1] in markings |
---|
1047 | L[2][i] : represents the lattice points on the facet of the polygon with |
---|
1048 | endpoints L[1][i] and L[1][i+1] (i considered modulo size(L[1])) |
---|
1049 | L[2][i][j][1] = intvec, the coordinates of the jth lattice point on that facet |
---|
1050 | L[2][i][j][2] = int, the position of L[2][i][j][1] in markings |
---|
1051 | L[3] : represents the interior lattice points of the polygon |
---|
1052 | L[3][i][1] = intvec, the coordinates of the ith interior point |
---|
1053 | L[3][i][2] = int, the position of L[3][i][1] in markings |
---|
1054 | EXAMPLE: example splitPolygon; shows an example" |
---|
1055 | { |
---|
1056 | list vert; // stores the result |
---|
1057 | // compute the boundary of the polygon in an oriented way |
---|
1058 | list pb=findOrientedBoundary(markings); |
---|
1059 | // the vertices are just the second entry of pb |
---|
1060 | vert[1]=pb[2]; |
---|
1061 | int i,j,k; // indices |
---|
1062 | list boundary; // stores the points on the facets of the polygon which are not vertices |
---|
1063 | // append to the boundary points as well as to the vertices the first vertex a second time |
---|
1064 | pb[1]=pb[1]+list(pb[1][1]); |
---|
1065 | pb[2]=pb[2]+list(pb[2][1]); |
---|
1066 | // for each vertex find all points on the facet of the polygon with this vertex |
---|
1067 | // and the next vertex as endpoints |
---|
1068 | int z=2; |
---|
1069 | for (i=1;i<=size(vert[1]);i++) |
---|
1070 | { |
---|
1071 | j=1; |
---|
1072 | list facet; // stores the points on this facet which are not vertices |
---|
1073 | // while the next vertex is not reached, store the boundary lattice point |
---|
1074 | while (pb[1][z]!=pb[2][i+1]) |
---|
1075 | { |
---|
1076 | facet[j]=pb[1][z]; |
---|
1077 | j++; |
---|
1078 | z++; |
---|
1079 | } |
---|
1080 | // store the points on the ith facet as boundary[i] |
---|
1081 | boundary[i]=facet; |
---|
1082 | kill facet; |
---|
1083 | z++; |
---|
1084 | } |
---|
1085 | // store the information on the boundary in vert[2] |
---|
1086 | vert[2]=boundary; |
---|
1087 | // find the remaining points in the input which are not on the boundary by checking |
---|
1088 | // for each point in markings if it is contained in pb[1] |
---|
1089 | list interior=markings; |
---|
1090 | for (i=size(interior);i>=1;i--) |
---|
1091 | { |
---|
1092 | for (j=1;j<=size(pb[1])-1;j++) |
---|
1093 | { |
---|
1094 | if (interior[i]==pb[1][j]) |
---|
1095 | { |
---|
1096 | interior=delete(interior,i); |
---|
1097 | j=size(pb[1]); |
---|
1098 | } |
---|
1099 | } |
---|
1100 | } |
---|
1101 | // store the interior points in vert[3] |
---|
1102 | vert[3]=interior; |
---|
1103 | // add to each point in vert the index which it gets from its position in the input markings; |
---|
1104 | // do so for ver[1] |
---|
1105 | for (i=1;i<=size(vert[1]);i++) |
---|
1106 | { |
---|
1107 | j=1; |
---|
1108 | while (markings[j]!=vert[1][i]) |
---|
1109 | { |
---|
1110 | j++; |
---|
1111 | } |
---|
1112 | vert[1][i]=list(vert[1][i],j); |
---|
1113 | } |
---|
1114 | // do so for ver[2] |
---|
1115 | for (i=1;i<=size(vert[2]);i++) |
---|
1116 | { |
---|
1117 | for (k=1;k<=size(vert[2][i]);k++) |
---|
1118 | { |
---|
1119 | j=1; |
---|
1120 | while (markings[j]!=vert[2][i][k]) |
---|
1121 | { |
---|
1122 | j++; |
---|
1123 | } |
---|
1124 | vert[2][i][k]=list(vert[2][i][k],j); |
---|
1125 | } |
---|
1126 | } |
---|
1127 | // do so for ver[3] |
---|
1128 | for (i=1;i<=size(vert[3]);i++) |
---|
1129 | { |
---|
1130 | j=1; |
---|
1131 | while (markings[j]!=vert[3][i]) |
---|
1132 | { |
---|
1133 | j++; |
---|
1134 | } |
---|
1135 | vert[3][i]=list(vert[3][i],j); |
---|
1136 | } |
---|
1137 | return(vert); |
---|
1138 | } |
---|
1139 | example |
---|
1140 | { |
---|
1141 | "EXAMPLE:"; |
---|
1142 | echo=2; |
---|
1143 | // the lattice polygon spanned by the points (0,0), (3,0) and (0,3) |
---|
1144 | // with all integer points as markings |
---|
1145 | list polygon=intvec(1,1),intvec(3,0),intvec(2,0),intvec(1,0),intvec(0,0),intvec(2,1),intvec(0,1),intvec(1,2),intvec(0,2),intvec(0,3); |
---|
1146 | // split the polygon in its vertices, its facets and its interior points |
---|
1147 | list sp=splitPolygon(polygon); |
---|
1148 | // the vertices |
---|
1149 | sp[1]; |
---|
1150 | // the points on facets which are not vertices |
---|
1151 | sp[2]; |
---|
1152 | // the interior points |
---|
1153 | sp[3]; |
---|
1154 | } |
---|
1155 | |
---|
1156 | |
---|
1157 | proc eta (list triang,list polygon) |
---|
1158 | "USAGE: eta(triang,polygon); triang, polygon list |
---|
1159 | ASSUME: polygon has the format of the output of splitPolygon, i.e. it is a list with three |
---|
1160 | entries describing a convex lattice polygon in the following way: |
---|
1161 | polygon[1] : is a list of lists; for each i the entry polygon[1][i][1] is a lattice point which is |
---|
1162 | a vertex of the lattice polygon, and polygon[1][i][2] is an integer assigned to |
---|
1163 | this lattice point as identifying index |
---|
1164 | polygon[2] : is a list of lists; for each vertex of the polygon, i.e. for each entry in polygon[1], |
---|
1165 | it contains a list polygon[2][i], which contains the lattice points on the facet |
---|
1166 | with endpoints polygon[1][i] and polygon[1][i+1] - i considered mod size(polygon[1]); |
---|
1167 | each such lattice point contributes an entry polygon[2][i][j][1] which is an integer |
---|
1168 | vector giving the coordinate of the lattice point and an entry polygon[2][i][j][2] |
---|
1169 | which is the identifying index |
---|
1170 | polygon[3] : is a list of lists, where each entry corresponds to a lattice point in the |
---|
1171 | interior of the polygon, with polygon[3][j][1] being the coordinates of the point |
---|
1172 | and polygon[3][j][2] being the identifying index; |
---|
1173 | triang is a list of integer vectors all of size three describing a triangulation of the |
---|
1174 | polygon described by polygon; if an entry of triang is the vector (i,j,k) then the triangle |
---|
1175 | is build by the vertices with indices i, j and k |
---|
1176 | RETURN: intvec, the integer vector eta describing that vertex of the Newton polytope discriminant |
---|
1177 | of the polygone whose dual cone in the Groebner fan contains the cone of the |
---|
1178 | secondary fan of the polygon corresponding to the given triangulation |
---|
1179 | NOTE: for a better description of eta see either Gelfand, Kapranov, Zelevinski: Discriminants, |
---|
1180 | Resultants and multidimensional Determinants. Chapter 10. |
---|
1181 | EXAMPLE: example eta; shows an example" |
---|
1182 | { |
---|
1183 | int i,j,k,l,m,n; // index variables |
---|
1184 | list ordpolygon; // stores the lattice points in the order used in the triangulation |
---|
1185 | list triangarea; // stores the areas of the triangulations |
---|
1186 | intmat N[2][2]; // used to compute triangle areas |
---|
1187 | // 1) store the lattice points in the order used in the triangulation |
---|
1188 | // go first through all vertices of the polytope |
---|
1189 | for (j=1;j<=size(polygon[1]);j++) |
---|
1190 | { |
---|
1191 | ordpolygon[polygon[1][j][2]]=polygon[1][j][1]; |
---|
1192 | } |
---|
1193 | // then consider all inner points |
---|
1194 | for (j=1;j<=size(polygon[3]);j++) |
---|
1195 | { |
---|
1196 | ordpolygon[polygon[3][j][2]]=polygon[3][j][1]; |
---|
1197 | } |
---|
1198 | // finally consider all lattice points on the boundary which are not vertices |
---|
1199 | for (j=1;j<=size(polygon[2]);j++) |
---|
1200 | { |
---|
1201 | for (i=1;i<=size(polygon[2][j]);i++) |
---|
1202 | { |
---|
1203 | ordpolygon[polygon[2][j][i][2]]=polygon[2][j][i][1]; |
---|
1204 | } |
---|
1205 | } |
---|
1206 | // 2) compute for each triangle in the triangulation the area of the triangle |
---|
1207 | for (i=1;i<=size(triang);i++) |
---|
1208 | { |
---|
1209 | // Note that the ith lattice point in orderedpolygon has the number i-1 in the triangulation! |
---|
1210 | N=ordpolygon[triang[i][1]]-ordpolygon[triang[i][2]],ordpolygon[triang[i][1]]-ordpolygon[triang[i][3]]; |
---|
1211 | triangarea[i]=abs(det(N)); |
---|
1212 | } |
---|
1213 | intvec ETA; // stores the eta_ij |
---|
1214 | int etaij; // stores the part of eta_ij during computations which comes from triangle areas |
---|
1215 | int seitenlaenge; // stores the part of eta_ij during computations which comes from boundary facets |
---|
1216 | list seiten; // stores the lattice points on facets of the polygon |
---|
1217 | intvec v; // used to compute a facet length |
---|
1218 | // 3) store first in seiten[i] all lattice points on the facet connecting the ith vertex, |
---|
1219 | // i.e. polygon[1][i], with the i+1st vertex, i.e. polygon[1][i+1], where we replace i+1 |
---|
1220 | // 1 if i=size(polygon[1]); |
---|
1221 | // then append the last entry of seiten once more at the very beginning of seiten, so |
---|
1222 | // that the index is shifted by one |
---|
1223 | for (i=1;i<=size(polygon[1]);i++) |
---|
1224 | { |
---|
1225 | if (i<size(polygon[1])) |
---|
1226 | { |
---|
1227 | seiten[i]=list(polygon[1][i])+polygon[2][i]+list(polygon[1][i+1]); |
---|
1228 | } |
---|
1229 | else |
---|
1230 | { |
---|
1231 | seiten[i]=list(polygon[1][i])+polygon[2][i]+list(polygon[1][1]); |
---|
1232 | } |
---|
1233 | } |
---|
1234 | seiten=insert(seiten,seiten[size(seiten)],0); |
---|
1235 | // 4) compute the eta_ij for all vertices of the polygon |
---|
1236 | for (j=1;j<=size(polygon[1]);j++) |
---|
1237 | { |
---|
1238 | // the vertex itself contributes a 1 |
---|
1239 | etaij=1; |
---|
1240 | // check for each triangle in the triangulation ... |
---|
1241 | for (k=1;k<=size(triang);k++) |
---|
1242 | { |
---|
1243 | // ... if the vertex is actually a vertex of the triangle ... |
---|
1244 | if ((polygon[1][j][2]==triang[k][1]) or (polygon[1][j][2]==triang[k][2]) or (polygon[1][j][2]==triang[k][3])) |
---|
1245 | { |
---|
1246 | // ... if so, add the area of the triangle to etaij |
---|
1247 | etaij=etaij+triangarea[k]; |
---|
1248 | // then check if that triangle has a facet which is contained in one of the |
---|
1249 | // two facets of the polygon which are adjecent to the given vertex ... |
---|
1250 | // these two facets are seiten[j] and seiten[j+1] |
---|
1251 | for (n=j;n<=j+1;n++) |
---|
1252 | { |
---|
1253 | // check for each lattice point in the facet of the polygon ... |
---|
1254 | for (l=1;l<=size(seiten[n]);l++) |
---|
1255 | { |
---|
1256 | // ... and for each lattice point in the triangle ... |
---|
1257 | for (m=1;m<=size(triang[k]);m++) |
---|
1258 | { |
---|
1259 | // ... if they coincide and are not the vertex itself ... |
---|
1260 | if ((seiten[n][l][2]==triang[k][m]) and (seiten[n][l][2]!=polygon[1][j][2])) |
---|
1261 | { |
---|
1262 | // if so, then compute the vector pointing from this lattice point to the vertex |
---|
1263 | v=polygon[1][j][1]-seiten[n][l][1]; |
---|
1264 | // and the lattice length of this vector has to be subtracted from etaij |
---|
1265 | etaij=etaij-abs(gcd(v[1],v[2])); |
---|
1266 | } |
---|
1267 | } |
---|
1268 | } |
---|
1269 | } |
---|
1270 | } |
---|
1271 | } |
---|
1272 | // store etaij in the list |
---|
1273 | ETA[polygon[1][j][2]]=etaij; |
---|
1274 | } |
---|
1275 | // 5) compute the eta_ij for all lattice points on the facets of the polygon which are not vertices, |
---|
1276 | // these are the lattice points in polygon[2][1] to polygon[2][size(polygon[1])] |
---|
1277 | for (i=1;i<=size(polygon[2]);i++) |
---|
1278 | { |
---|
1279 | for (j=1;j<=size(polygon[2][i]);j++) |
---|
1280 | { |
---|
1281 | // initialise etaij |
---|
1282 | etaij=0; |
---|
1283 | // initialise seitenlaenge |
---|
1284 | seitenlaenge=0; |
---|
1285 | // check for each triangle in the triangulation ... |
---|
1286 | for (k=1;k<=size(triang);k++) |
---|
1287 | { |
---|
1288 | // ... if the vertex is actually a vertex of the triangle ... |
---|
1289 | 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])) |
---|
1290 | { |
---|
1291 | // ... if so, add the area of the triangle to etaij |
---|
1292 | etaij=etaij+triangarea[k]; |
---|
1293 | // then check if that triangle has a facet which is contained in the |
---|
1294 | // facet of the polygon which contains the lattice point in question, |
---|
1295 | // this is the facet seiten[i+1]; |
---|
1296 | // check for each lattice point in the facet of the polygon ... |
---|
1297 | for (l=1;l<=size(seiten[i+1]);l++) |
---|
1298 | { |
---|
1299 | // ... and for each lattice point in the triangle ... |
---|
1300 | for (m=1;m<=size(triang[k]);m++) |
---|
1301 | { |
---|
1302 | // ... if they coincide and are not the vertex itself ... |
---|
1303 | if ((seiten[i+1][l][2]==triang[k][m]) and (seiten[i+1][l][2]!=polygon[2][i][j][2])) |
---|
1304 | { |
---|
1305 | // if so, then compute the vector pointing from this lattice point to the vertex |
---|
1306 | v=polygon[2][i][j][1]-seiten[i+1][l][1]; |
---|
1307 | // and the lattice length of this vector contributes to seitenlaenge |
---|
1308 | seitenlaenge=seitenlaenge+abs(gcd(v[1],v[2])); |
---|
1309 | } |
---|
1310 | } |
---|
1311 | } |
---|
1312 | } |
---|
1313 | } |
---|
1314 | // if the lattice point was a vertex of any triangle in the triangulation ... |
---|
1315 | if (etaij!=0) |
---|
1316 | { |
---|
1317 | // then eta_ij is the sum of the triangle areas minus seitenlaenge |
---|
1318 | ETA[polygon[2][i][j][2]]=etaij-seitenlaenge; |
---|
1319 | } |
---|
1320 | else |
---|
1321 | { |
---|
1322 | // otherwise it is just zero |
---|
1323 | ETA[polygon[2][i][j][2]]=0; |
---|
1324 | } |
---|
1325 | } |
---|
1326 | } |
---|
1327 | // 4) compute the eta_ij for all inner lattice points of the polygon |
---|
1328 | for (j=1;j<=size(polygon[3]);j++) |
---|
1329 | { |
---|
1330 | // initialise etaij |
---|
1331 | etaij=0; |
---|
1332 | // check for each triangle in the triangulation ... |
---|
1333 | for (k=1;k<=size(triang);k++) |
---|
1334 | { |
---|
1335 | // ... if the vertex is actually a vertex of the triangle ... |
---|
1336 | if ((polygon[3][j][2]==triang[k][1]) or (polygon[3][j][2]==triang[k][2]) or (polygon[3][j][2]==triang[k][3])) |
---|
1337 | { |
---|
1338 | // ... if so, add the area of the triangle to etaij |
---|
1339 | etaij=etaij+triangarea[k]; |
---|
1340 | } |
---|
1341 | } |
---|
1342 | // store etaij in ETA |
---|
1343 | ETA[polygon[3][j][2]]=etaij; |
---|
1344 | } |
---|
1345 | return(ETA); |
---|
1346 | } |
---|
1347 | example |
---|
1348 | { |
---|
1349 | "EXAMPLE:"; |
---|
1350 | echo=2; |
---|
1351 | // the lattice polygon spanned by the points (0,0), (3,0) and (0,3) |
---|
1352 | // with all integer points as markings |
---|
1353 | list polygon=intvec(1,1),intvec(3,0),intvec(2,0),intvec(1,0),intvec(0,0),intvec(2,1),intvec(0,1),intvec(1,2),intvec(0,2),intvec(0,3); |
---|
1354 | // split the polygon in its vertices, its facets and its interior points |
---|
1355 | list sp=splitPolygon(polygon); |
---|
1356 | // define a triangulation by connecting the only interior point with the vertices |
---|
1357 | list triang=intvec(1,2,5),intvec(1,5,10),intvec(1,5,10); |
---|
1358 | // compute the eta-vector of this triangulation |
---|
1359 | eta(triang,sp); |
---|
1360 | } |
---|
1361 | |
---|
1362 | proc findOrientedBoundary (list polygon) |
---|
1363 | "USAGE: findOrientedBoundary(polygon); polygon list |
---|
1364 | ASSUME: polygon is a list of integer vectors defining integer lattice points in the plane |
---|
1365 | RETURN: list, l[1] = list of integer vectors such that the polygonal path defined by |
---|
1366 | these is the boundary of the convex hull of the lattice points in polygon |
---|
1367 | l[2] = list, the redundant points in l[1] have been removed |
---|
1368 | EXAMPLE: example findOrientedBoundary; shows an example" |
---|
1369 | { |
---|
1370 | // Order the vertices such that passing from one to the next we travel along |
---|
1371 | // the boundary of the convex hull of the vertices clock wise |
---|
1372 | int d,k,i,j; |
---|
1373 | intmat D[2][2]; |
---|
1374 | ///////////////////////////////////// |
---|
1375 | // Treat first the pathological cases that the polygon is not two-dimensional: |
---|
1376 | ///////////////////////////////////// |
---|
1377 | // if the polygon is empty or only one point or a line segment of two points |
---|
1378 | if (size(polygon)<=2) |
---|
1379 | { |
---|
1380 | return(list(polygon,polygon)); |
---|
1381 | } |
---|
1382 | // check is the polygon is only a line segment given by more than two points; |
---|
1383 | // for this first compute sum of the absolute values of the determinants of the matrices whose |
---|
1384 | // rows are the vectors pointing from the first to the second point and from the |
---|
1385 | // the first point to the ith point for i=3,...,size(polygon); if this sum is zero |
---|
1386 | // then the polygon is a line segment and we have to find its end points |
---|
1387 | d=0; |
---|
1388 | for (i=3;i<=size(polygon);i++) |
---|
1389 | { |
---|
1390 | D=polygon[2]-polygon[1],polygon[i]-polygon[1]; |
---|
1391 | d=d+abs(det(D)); |
---|
1392 | } |
---|
1393 | if (d==0) // then polygon is a line segment |
---|
1394 | { |
---|
1395 | intmat laenge[size(polygon)][size(polygon)]; |
---|
1396 | intvec mp; |
---|
1397 | // for this collect first all vectors pointing from one lattice point to the next, |
---|
1398 | // compute their pairwise angles and their lengths |
---|
1399 | for (i=1;i<=size(polygon)-1;i++) |
---|
1400 | { |
---|
1401 | for (j=i+1;j<=size(polygon);j++) |
---|
1402 | { |
---|
1403 | mp=polygon[i]-polygon[j]; |
---|
1404 | laenge[i,j]=abs(gcd(mp[1],mp[2])); |
---|
1405 | } |
---|
1406 | } |
---|
1407 | mp=maxPosInIntmat(laenge); |
---|
1408 | list endpoints=polygon[mp[1]],polygon[mp[2]]; |
---|
1409 | intvec abstand; |
---|
1410 | for (i=1;i<=size(polygon);i++) |
---|
1411 | { |
---|
1412 | abstand[i]=0; |
---|
1413 | if (i<mp[1]) |
---|
1414 | { |
---|
1415 | abstand[i]=laenge[i,mp[1]]; |
---|
1416 | } |
---|
1417 | if (i>mp[1]) |
---|
1418 | { |
---|
1419 | abstand[i]=laenge[mp[1],i]; |
---|
1420 | } |
---|
1421 | } |
---|
1422 | polygon=sortlistbyintvec(polygon,abstand); |
---|
1423 | return(list(polygon,endpoints)); |
---|
1424 | } |
---|
1425 | /////////////////////////////////////////////////////////////// |
---|
1426 | list orderedvertices; // stores the vertices in an ordered way |
---|
1427 | list minimisedorderedvertices; // stores the vertices in an ordered way; redundant ones removed |
---|
1428 | list comparevertices; // stores vertices which should be compared to the testvertex |
---|
1429 | orderedvertices[1]=polygon[1]; // set the starting vertex |
---|
1430 | minimisedorderedvertices[1]=polygon[1]; // set the starting vertex |
---|
1431 | intvec testvertex=polygon[1]; // the vertex to which the others have to be compared |
---|
1432 | intvec startvertex=polygon[1]; // keep the starting vertex to test, when the end is reached |
---|
1433 | int endtest; // is set to one, when the end is reached |
---|
1434 | int startvertexfound;// is 1, once for some testvertex a candidate for the next vertex has been found |
---|
1435 | polygon=delete(polygon,1); // delete the testvertex |
---|
1436 | intvec v,w; |
---|
1437 | int l=1; // counts the vertices |
---|
1438 | // the basic idea is that a vertex can be the next one on the boundary if all other vertices |
---|
1439 | // ly to the right of the vector v pointing from the testvertex to this one; this can be tested |
---|
1440 | // by checking if the determinant of the 2x2-matrix with first column v and second column the vector w, |
---|
1441 | // pointing from the testvertex to the new vertex, is non-positive; if this is the case for all |
---|
1442 | // new vertices, then the one in consideration is a possible choice for the next vertex on the boundary |
---|
1443 | // and it is stored in naechste; we can then order the candidates according to their distance from |
---|
1444 | // the testvertex; then they occur on the boundary in that order! |
---|
1445 | while (endtest==0) |
---|
1446 | { |
---|
1447 | list naechste; // stores the possible choices for the next vertex |
---|
1448 | k=1; |
---|
1449 | for (i=1;i<=size(polygon);i++) |
---|
1450 | { |
---|
1451 | d=0; // stores the value of the determinant of (v,w) |
---|
1452 | v=polygon[i]-testvertex; // points from the testvertex to the ith vertex |
---|
1453 | comparevertices=delete(polygon,i); // we needn't compare v to itself |
---|
1454 | // we should compare v to the startvertex-testvertex; in the first calling of the loop |
---|
1455 | // this is irrelevant since the difference will be zero; however, later on it will |
---|
1456 | // be vital, since we delete the vertices which we have already tested from the list |
---|
1457 | // of all vertices, and when all vertices on the boundary have been found we would |
---|
1458 | // therefore find a vertex in the interior as candidate; but always testing against |
---|
1459 | // the starting vertex, this can not happen |
---|
1460 | comparevertices[size(comparevertices)+1]=startvertex; |
---|
1461 | for (j=1;(j<=size(comparevertices)) and (d<=0);j++) |
---|
1462 | { |
---|
1463 | w=comparevertices[j]-testvertex; // points form the testvertex to the jth vertex |
---|
1464 | D=v,w; |
---|
1465 | d=det(D); |
---|
1466 | } |
---|
1467 | if (d<=0) // if all determinants are non-positive, then the ith vertex is a candidate |
---|
1468 | { |
---|
1469 | naechste[k]=list(polygon[i],i,scalarproduct(v,v)); // we store the vertex, its position, and its |
---|
1470 | k++; // distance from the testvertex |
---|
1471 | } |
---|
1472 | } |
---|
1473 | if (size(naechste)>0) // then a candidate for the next vertex has been found |
---|
1474 | { |
---|
1475 | startvertexfound=1; // at least once a candidate has been found |
---|
1476 | naechste=sortlist(naechste,3); //we order the candidates according to their distance from testvertex; |
---|
1477 | for (j=1;j<=size(naechste);j++) // then we store them in this order in orderedvertices |
---|
1478 | { |
---|
1479 | l++; |
---|
1480 | orderedvertices[l]=naechste[j][1]; |
---|
1481 | } |
---|
1482 | testvertex=naechste[size(naechste)][1]; // we store the last one as next testvertex; |
---|
1483 | minimisedorderedvertices[size(minimisedorderedvertices)+1]=testvertex; // store the next corner of NSD |
---|
1484 | naechste=sortlist(naechste,2); // then we reorder the vertices according to their position |
---|
1485 | for (j=size(naechste);j>=1;j--) // and we delete them from the vertices |
---|
1486 | { |
---|
1487 | polygon=delete(polygon,naechste[j][2]); |
---|
1488 | } |
---|
1489 | } |
---|
1490 | else // that means either that the vertex was inside the polygon, |
---|
1491 | { // or that we have reached the last vertex on the boundary of the polytope |
---|
1492 | if (startvertexfound==0) // the vertex was in the interior; we delete it and start all over again |
---|
1493 | { |
---|
1494 | orderedvertices[1]=polygon[1]; |
---|
1495 | minimisedorderedvertices[1]=polygon[1]; |
---|
1496 | testvertex=polygon[1]; |
---|
1497 | startvertex=polygon[1]; |
---|
1498 | polygon=delete(polygon,1); |
---|
1499 | } |
---|
1500 | else // we have reached the last vertex on the boundary of the polytope and can stop |
---|
1501 | { |
---|
1502 | endtest=1; |
---|
1503 | } |
---|
1504 | } |
---|
1505 | kill naechste; |
---|
1506 | } |
---|
1507 | // test if the first vertex in minimisedorderedvertices is on the same line with the second and |
---|
1508 | // the last, i.e. if we started our search in the middle of a face; if so, delete it |
---|
1509 | v=minimisedorderedvertices[2]-minimisedorderedvertices[1]; |
---|
1510 | w=minimisedorderedvertices[size(minimisedorderedvertices)]-minimisedorderedvertices[1]; |
---|
1511 | D=v,w; |
---|
1512 | if (det(D)==0) |
---|
1513 | { |
---|
1514 | minimisedorderedvertices=delete(minimisedorderedvertices,1); |
---|
1515 | } |
---|
1516 | // test if the first vertex in minimisedorderedvertices is on the same line with the two |
---|
1517 | // last ones, i.e. if we started our search at the end of a face; if so, delete it |
---|
1518 | v=minimisedorderedvertices[size(minimisedorderedvertices)-1]-minimisedorderedvertices[1]; |
---|
1519 | w=minimisedorderedvertices[size(minimisedorderedvertices)]-minimisedorderedvertices[1]; |
---|
1520 | D=v,w; |
---|
1521 | if (det(D)==0) |
---|
1522 | { |
---|
1523 | minimisedorderedvertices=delete(minimisedorderedvertices,size(minimisedorderedvertices)); |
---|
1524 | } |
---|
1525 | return(list(orderedvertices,minimisedorderedvertices)); |
---|
1526 | } |
---|
1527 | example |
---|
1528 | { |
---|
1529 | "EXAMPLE:"; |
---|
1530 | echo=2; |
---|
1531 | // the following lattice points in the plane define a polygon |
---|
1532 | list polygon=intvec(0,0),intvec(3,1),intvec(1,0),intvec(2,0),intvec(1,1),intvec(3,2),intvec(1,2),intvec(2,3),intvec(2,4); |
---|
1533 | // we compute its boundary |
---|
1534 | list boundarypolygon=findOrientedBoundary(polygon); |
---|
1535 | // the points on the boundary ordered clockwise are boundarypolygon[1] |
---|
1536 | boundarypolygon[1]; |
---|
1537 | // the vertices of the boundary are boundarypolygon[2] |
---|
1538 | boundarypolygon[2]; |
---|
1539 | } |
---|
1540 | |
---|
1541 | |
---|
1542 | proc cyclePoints (list triang,list points,int pt) |
---|
1543 | "USAGE: cyclePoints(triang,points,pt) triang,points list, pt int |
---|
1544 | ASSUME: points is a list of integer vectors describing the lattice points of a marked polygon; |
---|
1545 | triang is a list of integer vectors describing a triangulation of the marked polygon |
---|
1546 | in the sense that an integer vector of the form (i,j,k) describes the triangle formed |
---|
1547 | by polygon[i], polygon[j] and polygon[k]; |
---|
1548 | pt is an integer between 1 and size(points), singling out a lattice point among |
---|
1549 | the marked points |
---|
1550 | PURPOSE: consider the convex lattice polygon, say P, spanned by all lattice points in points which |
---|
1551 | in the triangulation triang are connected to the point points[pt]; the procedure |
---|
1552 | computes all marked points in points which ly on the boundary of that polygon, ordered |
---|
1553 | clockwise |
---|
1554 | RETURN: list, of integer vectors which are the coordinates of the lattice points on |
---|
1555 | the boundary of the above mentioned polygon P, if this polygon is not the |
---|
1556 | empty set (that would be the case if points[pt] is not a vertex of any |
---|
1557 | triangle in the triangulation); otherwise return the empty list |
---|
1558 | EXAMPLE: example cyclePoints; shows an example" |
---|
1559 | { |
---|
1560 | int i,j; // indices |
---|
1561 | list v; // saves the indices of lattice points connected to the interior point in the triangulation |
---|
1562 | // save all points in triangulations containing pt in v |
---|
1563 | for (i=1;i<=size(triang);i++) |
---|
1564 | { |
---|
1565 | if ((triang[i][1]==pt) or (triang[i][2]==pt) or (triang[i][3]==pt)) |
---|
1566 | { |
---|
1567 | j++; |
---|
1568 | v[3*j-2]=triang[i][1]; |
---|
1569 | v[3*j-1]=triang[i][2]; |
---|
1570 | v[3*j]=triang[i][3]; |
---|
1571 | } |
---|
1572 | } |
---|
1573 | if (size(v)==0) |
---|
1574 | { |
---|
1575 | return(list()); |
---|
1576 | } |
---|
1577 | // remove pt itself and redundancies in v |
---|
1578 | for (i=size(v);i>=1;i--) |
---|
1579 | { |
---|
1580 | j=1; |
---|
1581 | while ((j<i) and (v[i]!=v[j])) |
---|
1582 | { |
---|
1583 | j++; |
---|
1584 | } |
---|
1585 | if ((j<i) or (v[i]==pt)) |
---|
1586 | { |
---|
1587 | v=delete(v,i); |
---|
1588 | } |
---|
1589 | } |
---|
1590 | // save in pts the coordinates of the points with indices in v |
---|
1591 | list pts; |
---|
1592 | for (i=1;i<=size(v);i++) |
---|
1593 | { |
---|
1594 | pts[i]=points[v[i]]; |
---|
1595 | } |
---|
1596 | // consider the convex polytope spanned by the points in pts, find the points on the |
---|
1597 | // boundary and order them clockwise |
---|
1598 | return(findOrientedBoundary(pts)[1]); |
---|
1599 | } |
---|
1600 | example |
---|
1601 | { |
---|
1602 | "EXAMPLE:"; |
---|
1603 | echo=2; |
---|
1604 | // the lattice polygon spanned by the points (0,0), (3,0) and (0,3) |
---|
1605 | // with all integer points as markings |
---|
1606 | list points=intvec(1,1),intvec(3,0),intvec(2,0),intvec(1,0),intvec(0,0),intvec(2,1),intvec(0,1),intvec(1,2),intvec(0,2),intvec(0,3); |
---|
1607 | // define a triangulation |
---|
1608 | list triang=intvec(1,2,5),intvec(1,5,7),intvec(1,7,9),intvec(8,9,10),intvec(1,8,9),intvec(1,2,8); |
---|
1609 | // compute the points connected to (1,1) in triang |
---|
1610 | cyclePoints(triang,points,1); |
---|
1611 | } |
---|
1612 | |
---|
1613 | proc latticeArea (list polygon) |
---|
1614 | "USAGE: latticeArea(polygon); polygon list |
---|
1615 | ASSUME: polygon is a list of integer vectors in the plane |
---|
1616 | RETURN: int, the lattice area of the convex hull of the lattice points in polygon, |
---|
1617 | i.e. twice the Euclidean area |
---|
1618 | EXAMPLE: example polygonlatticeArea; shows an example" |
---|
1619 | { |
---|
1620 | list pg=findOrientedBoundary(polygon)[2]; |
---|
1621 | int area; |
---|
1622 | intmat M[2][2]; |
---|
1623 | for (int i=2;i<=size(pg)-1;i++) |
---|
1624 | { |
---|
1625 | M[1,1..2]=pg[i]-pg[1]; |
---|
1626 | M[2,1..2]=pg[i+1]-pg[1]; |
---|
1627 | area=area+abs(det(M)); |
---|
1628 | } |
---|
1629 | return(area); |
---|
1630 | } |
---|
1631 | example |
---|
1632 | { |
---|
1633 | "EXAMPLE:"; |
---|
1634 | echo=2; |
---|
1635 | // define a polygon with lattice area 5 |
---|
1636 | list polygon=intvec(1,2),intvec(1,0),intvec(2,0),intvec(1,1),intvec(2,1),intvec(0,0); |
---|
1637 | latticeArea(polygon); |
---|
1638 | } |
---|
1639 | |
---|
1640 | proc picksFormula (list polygon) |
---|
1641 | "USAGE: picksFormula(polygon); polygon list |
---|
1642 | ASSUME: polygon is a list of integer vectors in the plane and consider their convex hull C |
---|
1643 | RETURN: list, L of three integersthe |
---|
1644 | L[1] : the lattice area of C, i.e. twice the Euclidean area |
---|
1645 | L[2] : the number of lattice points on the boundary of C |
---|
1646 | L[3] : the number of interior lattice points of C |
---|
1647 | NOTE: the integers in L are related by Pick's formula, namely: L[1]=L[2]+2*L[3]-2 |
---|
1648 | EXAMPLE: example picksFormula; shows an example" |
---|
1649 | { |
---|
1650 | list pg=findOrientedBoundary(polygon)[2]; |
---|
1651 | int area,bdpts,i; |
---|
1652 | intmat M[2][2]; |
---|
1653 | // compute the lattice area of the polygon, i.e. twice the Euclidean area |
---|
1654 | for (i=2;i<=size(pg)-1;i++) |
---|
1655 | { |
---|
1656 | M[1,1..2]=pg[i]-pg[1]; |
---|
1657 | M[2,1..2]=pg[i+1]-pg[1]; |
---|
1658 | area=area+abs(det(M)); |
---|
1659 | } |
---|
1660 | // compute the number of lattice points on the boundary |
---|
1661 | intvec edge; |
---|
1662 | pg[size(pg)+1]=pg[1]; |
---|
1663 | for (i=1;i<=size(pg)-1;i++) |
---|
1664 | { |
---|
1665 | edge=pg[i]-pg[i+1]; |
---|
1666 | bdpts=bdpts+abs(gcd(edge[1],edge[2])); |
---|
1667 | } |
---|
1668 | // Pick's formula says that the lattice area A, the number g of interior points and |
---|
1669 | // the number b of boundary points are connected by the formula: A=b+2g-2 |
---|
1670 | return(list(area,bdpts,(area-bdpts+2)/2)); |
---|
1671 | } |
---|
1672 | example |
---|
1673 | { |
---|
1674 | "EXAMPLE:"; |
---|
1675 | echo=2; |
---|
1676 | // define a polygon with lattice area 5 |
---|
1677 | list polygon=intvec(1,2),intvec(1,0),intvec(2,0),intvec(1,1),intvec(2,1),intvec(0,0); |
---|
1678 | list pick=picksFormula(polygon); |
---|
1679 | // the lattice area of the polygon is: |
---|
1680 | pick[1]; |
---|
1681 | // the number of lattice points on the boundary is: |
---|
1682 | pick[2]; |
---|
1683 | // the number of interior lattice points is: |
---|
1684 | pick[3]; |
---|
1685 | // the number's are related by Pick's formula: |
---|
1686 | pick[1]-pick[2]-2*pick[3]+2; |
---|
1687 | } |
---|
1688 | |
---|
1689 | proc ellipticNF (list polygon) |
---|
1690 | "USAGE: ellipticNF(polygon); polygon list |
---|
1691 | ASSUME: polygon is a list of integer vectors in the plane such that their convex hull C |
---|
1692 | has precisely one interior lattice point; i.e. C is the Newton polygon of an |
---|
1693 | elliptic curve |
---|
1694 | PURPOSE: compute the normal form of the polygon with respect to the unimodular affine |
---|
1695 | transformations T=A*x+v; there are sixteen different normal forms |
---|
1696 | (see e.g. Bjorn Poonen, Fernando Rodriguez-Villegas: Lattice Polygons and |
---|
1697 | the number 12. Amer. Math. Monthly 107 (2000), no. 3, 238--250.) |
---|
1698 | RETURN: list, L such that |
---|
1699 | L[1] : list whose entries are the vertices of the normal form of the polygon |
---|
1700 | L[2] : the matrix A of the unimodular transformation |
---|
1701 | L[3] : the translation vector v of the unimodular transformation |
---|
1702 | L[4] : list such that the ith entry is the image of polygon[i] under the |
---|
1703 | unimodular transformation T |
---|
1704 | EXAMPLE: example ellipticNF; shows an example" |
---|
1705 | { |
---|
1706 | int i; // index |
---|
1707 | intvec edge; // stores the vector of an edge |
---|
1708 | intvec boundary; // stores the lattice lengths of the edges of the Newton cylce |
---|
1709 | // find the vertices of the Newton cycle and order it clockwise |
---|
1710 | list pg=findOrientedBoundary(polygon)[2]; |
---|
1711 | // check if there is precisely one interior point in the Newton polygon |
---|
1712 | if (picksFormula(pg)[3]!=1) |
---|
1713 | { |
---|
1714 | ERROR("The polygon has not precisely one interior point!"); |
---|
1715 | } |
---|
1716 | // insert the first vertex at the end once again |
---|
1717 | pg[size(pg)+1]=pg[1]; |
---|
1718 | // compute the number of lattice points on each edge |
---|
1719 | for (i=1;i<=size(pg)-1;i++) |
---|
1720 | { |
---|
1721 | edge=pg[i]-pg[i+1]; |
---|
1722 | boundary[i]=1+abs(gcd(edge[1],edge[2])); |
---|
1723 | } |
---|
1724 | // store the values of boundary once more adding the first two at the end |
---|
1725 | intvec tboundary=boundary,boundary[1],boundary[2]; |
---|
1726 | // sort boundary in an asecending way |
---|
1727 | intvec sbd=sortintvec(boundary); |
---|
1728 | // find the first edge having the maximal number of lattice points |
---|
1729 | int max=maxPosInIntvec(boundary); |
---|
1730 | // some computations have to be done over the rationals |
---|
1731 | ring transformationring=0,x,lp; |
---|
1732 | intvec trans; // stores the vector by which we have to translate the polygon |
---|
1733 | intmat A[2][2]; // stores the matrix by which we have to transform the polygon |
---|
1734 | matrix M[3][3]; // stores the projective coordinates of the points which are to be transformed |
---|
1735 | matrix N[3][3]; // stores the projective coordinates of the points to which M is to be transformed |
---|
1736 | intmat T[3][3]; // stores the unimodular affine transformation in projective form |
---|
1737 | // add the second point of pg once again at the end |
---|
1738 | pg=insert(pg,pg[2],size(pg)); |
---|
1739 | // if there is only one edge which has the maximal number of lattice points, then M should be: |
---|
1740 | M=pg[max],1,pg[max+1],1,pg[max+2],1; |
---|
1741 | // consider the 16 different cases which can occur: |
---|
1742 | // Case 1: |
---|
1743 | if (sbd==intvec(2,2,2)) |
---|
1744 | { |
---|
1745 | N=0,1,1,1,2,1,2,0,1; |
---|
1746 | } |
---|
1747 | // Case 2: |
---|
1748 | if (sbd==intvec(2,2,3)) |
---|
1749 | { |
---|
1750 | N=2,0,1,0,0,1,1,2,1; |
---|
1751 | } |
---|
1752 | // Case 3: |
---|
1753 | if (sbd==intvec(2,3,4)) |
---|
1754 | { |
---|
1755 | // here the orientation of the Newton polygon is important ! |
---|
1756 | if (tboundary[max+1]==3) |
---|
1757 | { |
---|
1758 | N=3,0,1,0,0,1,0,2,1; |
---|
1759 | } |
---|
1760 | else |
---|
1761 | { |
---|
1762 | N=0,0,1,3,0,1,0,2,1; |
---|
1763 | } |
---|
1764 | } |
---|
1765 | // Case 4: |
---|
1766 | if (sbd==intvec(3,3,5)) |
---|
1767 | { |
---|
1768 | N=4,0,1,0,0,1,0,2,1; |
---|
1769 | } |
---|
1770 | // Case 5: |
---|
1771 | if (sbd==intvec(4,4,4)) |
---|
1772 | { |
---|
1773 | N=3,0,1,0,0,1,0,3,1; |
---|
1774 | } |
---|
1775 | // Case 6+7: |
---|
1776 | if (sbd==intvec(2,2,2,2)) |
---|
1777 | { |
---|
1778 | // there are two different polygons which has four edges all of length 2, |
---|
1779 | // but only one of them has two edges whose direction vectors form a matrix |
---|
1780 | // of determinant 3 |
---|
1781 | A=pg[1]-pg[2],pg[3]-pg[2]; |
---|
1782 | while ((max<4) and (det(A)!=3)) |
---|
1783 | { |
---|
1784 | max++; |
---|
1785 | A=pg[max]-pg[max+1],pg[max+2]-pg[max+1]; |
---|
1786 | } |
---|
1787 | // Case 6: |
---|
1788 | if (det(A)==3) |
---|
1789 | { |
---|
1790 | M=pg[max],1,pg[max+1],1,pg[max+2],1; |
---|
1791 | N=1,0,1,0,2,1,2,1,1; |
---|
1792 | } |
---|
1793 | // Case 7: |
---|
1794 | else |
---|
1795 | { |
---|
1796 | N=2,1,1,1,0,1,0,1,1; |
---|
1797 | } |
---|
1798 | } |
---|
1799 | // Case 8: |
---|
1800 | if (sbd==intvec(2,2,2,3)) |
---|
1801 | { |
---|
1802 | // the orientation of the polygon is important |
---|
1803 | A=pg[max]-pg[max+1],pg[max+2]-pg[max+1]; |
---|
1804 | if (det(A)==2) |
---|
1805 | { |
---|
1806 | N=2,0,1,0,0,1,0,1,1; |
---|
1807 | } |
---|
1808 | else |
---|
1809 | { |
---|
1810 | N=0,0,1,2,0,1,1,2,1; |
---|
1811 | } |
---|
1812 | } |
---|
1813 | // Case 9: |
---|
1814 | if (sbd==intvec(2,2,3,3)) |
---|
1815 | { |
---|
1816 | // if max==1, then the 5th entry in tboundary is the same as the first |
---|
1817 | if (max==1) |
---|
1818 | { |
---|
1819 | max=5; |
---|
1820 | } |
---|
1821 | // if boundary=3,2,2,3 then set max=4 |
---|
1822 | if (tboundary[max+1]!=3) |
---|
1823 | { |
---|
1824 | max=4; |
---|
1825 | } |
---|
1826 | M=pg[max],1,pg[max+1],1,pg[max+2],1; |
---|
1827 | // the orientation of the polygon matters |
---|
1828 | A=pg[max-1]-pg[max],pg[max+1]-pg[max]; |
---|
1829 | if (det(A)==4) |
---|
1830 | { |
---|
1831 | N=2,0,1,0,0,1,0,2,1; |
---|
1832 | } |
---|
1833 | else |
---|
1834 | { |
---|
1835 | N=0,2,1,0,0,1,2,0,1; |
---|
1836 | } |
---|
1837 | } |
---|
1838 | // Case 10: |
---|
1839 | if (sbd==intvec(2,2,3,4)) |
---|
1840 | { |
---|
1841 | // the orientation of the polygon matters |
---|
1842 | if (tboundary[max+1]==3) |
---|
1843 | { |
---|
1844 | N=3,0,1,0,0,1,0,2,1; |
---|
1845 | } |
---|
1846 | else |
---|
1847 | { |
---|
1848 | N=0,0,1,3,0,1,2,1,1; |
---|
1849 | } |
---|
1850 | } |
---|
1851 | // Case 11: |
---|
1852 | if (sbd==intvec(2,3,3,4)) |
---|
1853 | { |
---|
1854 | N=3,0,1,0,0,1,0,2,1; |
---|
1855 | } |
---|
1856 | // Case 12: |
---|
1857 | if (sbd==intvec(3,3,3,3)) |
---|
1858 | { |
---|
1859 | N=2,0,1,0,0,1,0,2,1; |
---|
1860 | } |
---|
1861 | // Case 13: |
---|
1862 | if (sbd==intvec(2,2,2,2,2)) |
---|
1863 | { |
---|
1864 | // compute the angles of the polygon vertices |
---|
1865 | intvec dt; |
---|
1866 | for (i=1;i<=5;i++) |
---|
1867 | { |
---|
1868 | A=pg[i]-pg[i+1],pg[i+2]-pg[i+1]; |
---|
1869 | dt[i]=det(A); |
---|
1870 | } |
---|
1871 | dt[6]=dt[1]; |
---|
1872 | // find the vertex to be mapped to (0,1) |
---|
1873 | max=1; |
---|
1874 | while ((dt[max]!=2) or (dt[max+1]!=2)) |
---|
1875 | { |
---|
1876 | max++; |
---|
1877 | } |
---|
1878 | M=pg[max],1,pg[max+1],1,pg[max+2],1; |
---|
1879 | N=0,1,1,1,2,1,2,1,1; |
---|
1880 | } |
---|
1881 | // Case 14: |
---|
1882 | if (sbd==intvec(2,2,2,2,3)) |
---|
1883 | { |
---|
1884 | N=2,0,1,0,0,1,0,1,1; |
---|
1885 | } |
---|
1886 | // Case 15: |
---|
1887 | if (sbd==intvec(2,2,2,3,3)) |
---|
1888 | { |
---|
1889 | // find the vertix to be mapped to (2,0) |
---|
1890 | if (tboundary[max+1]!=3) |
---|
1891 | { |
---|
1892 | max=5; |
---|
1893 | M=pg[max],1,pg[max+1],1,pg[max+2],1; |
---|
1894 | } |
---|
1895 | N=2,0,1,0,0,1,0,2,1; |
---|
1896 | } |
---|
1897 | // Case 16: |
---|
1898 | if (sbd==intvec(2,2,2,2,2,2)) |
---|
1899 | { |
---|
1900 | N=2,0,1,1,0,1,0,1,1; |
---|
1901 | } |
---|
1902 | // we have to transpose the matrices M and N |
---|
1903 | M=transpose(M); |
---|
1904 | N=transpose(N); |
---|
1905 | // compute the unimodular affine transformation, which is of the form |
---|
1906 | // A11 A12 | T1 |
---|
1907 | // A21 A22 | T2 |
---|
1908 | // 0 0 | 1 |
---|
1909 | T=matrixtointmat(N*inverse(M)); |
---|
1910 | // the upper-left 2x2-block is A |
---|
1911 | A=T[1..2,1..2]; |
---|
1912 | // the upper-right 2x1-block is the translation vector |
---|
1913 | trans=T[1,3],T[2,3]; |
---|
1914 | // transform now the lattice points of the polygon with respect to A and T |
---|
1915 | list nf; |
---|
1916 | for (i=1;i<=size(polygon);i++) |
---|
1917 | { |
---|
1918 | intmat V[2][1]=polygon[i]; |
---|
1919 | V=A*V; |
---|
1920 | nf[i]=intvec(V[1,1]+trans[1],V[2,1]+trans[2]); |
---|
1921 | kill V; |
---|
1922 | } |
---|
1923 | return(list(findOrientedBoundary(nf)[2],A,trans,nf)); |
---|
1924 | } |
---|
1925 | example |
---|
1926 | { |
---|
1927 | "EXAMPLE:"; |
---|
1928 | echo=2; |
---|
1929 | ring r=0,(x,y),dp; |
---|
1930 | // the Newton polygon of the following polynomial has precisely one interior point |
---|
1931 | poly f=x22y11+x19y10+x17y9+x16y9+x12y7+x9y6+x7y5+x2y3; |
---|
1932 | list polygon=newtonPolytopeLP(f); |
---|
1933 | // its lattice points are |
---|
1934 | polygon; |
---|
1935 | // find its normal form |
---|
1936 | list nf=ellipticNF(polygon); |
---|
1937 | // the vertices of the normal form are |
---|
1938 | nf[1]; |
---|
1939 | // it has been transformed by the unimodular affine transformation A*x+v |
---|
1940 | // with matrix A |
---|
1941 | nf[2]; |
---|
1942 | // and translation vector v |
---|
1943 | nf[3]; |
---|
1944 | // the 3rd lattice point ... |
---|
1945 | polygon[3]; |
---|
1946 | // ... has been transformed to |
---|
1947 | nf[4][3]; |
---|
1948 | } |
---|
1949 | |
---|
1950 | |
---|
1951 | proc ellipticNFDB (int n,list #) |
---|
1952 | "USAGE: ellipticNFDB(n[,#]); n int, # list |
---|
1953 | ASSUME: n is an intger between 1 and 16 |
---|
1954 | PURPOSE: this is a database storing the 16 normal forms of planar polygons with |
---|
1955 | precisely one interior point up to unimodular affine transformations |
---|
1956 | (see e.g. Bjorn Poonen, Fernando Rodriguez-Villegas: Lattice Polygons and |
---|
1957 | the number 12. Amer. Math. Monthly 107 (2000), no. 3, 238--250.) |
---|
1958 | RETURN: list, L such that |
---|
1959 | L[1] : list whose entries are the vertices of the nth normal form |
---|
1960 | L[2] : list whose entries are all the lattice points of the nth normal form |
---|
1961 | L[3] : only present if the optional parameter # is present, and then |
---|
1962 | it is a polynomial in the variables (x,y) whose Newton polygon |
---|
1963 | is the nth normal form |
---|
1964 | NOTE: the optional parameter is only allowed if the basering has the variables x and y |
---|
1965 | EXAMPLE: example ellipticNFDB; shows an example" |
---|
1966 | { |
---|
1967 | if ((n<1) or (n>16)) |
---|
1968 | { |
---|
1969 | ERROR("n is not between 1 and 16."); |
---|
1970 | } |
---|
1971 | if (size(#)>0) |
---|
1972 | { |
---|
1973 | if ((defined(x)==0) or (defined(y)==0)) |
---|
1974 | { |
---|
1975 | ERROR("The variables x and y are not defined."); |
---|
1976 | } |
---|
1977 | } |
---|
1978 | if ((defined(x)==0) or (defined(y)==0)) |
---|
1979 | { |
---|
1980 | ring nfring=0,(x,y),dp; |
---|
1981 | } |
---|
1982 | // store the normal forms as polynomials |
---|
1983 | 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, |
---|
1984 | 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, |
---|
1985 | 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; |
---|
1986 | list pg=newtonPolytopeLP(nf[n]); |
---|
1987 | if (size(#)==0) |
---|
1988 | { |
---|
1989 | return(list(findOrientedBoundary(pg)[2],pg)); |
---|
1990 | } |
---|
1991 | else |
---|
1992 | { |
---|
1993 | return(list(findOrientedBoundary(pg)[2],pg,nf[n])); |
---|
1994 | } |
---|
1995 | } |
---|
1996 | example |
---|
1997 | { |
---|
1998 | "EXAMPLE:"; |
---|
1999 | echo=2; |
---|
2000 | list nf=ellipticNFDB(5); |
---|
2001 | // the vertices of the 5th normal form are |
---|
2002 | nf[1]; |
---|
2003 | // its lattice points are |
---|
2004 | nf[2]; |
---|
2005 | } |
---|
2006 | |
---|
2007 | ///////////////////////////////////////////////////////////////////////////////// |
---|
2008 | ///////////////////////////////////////////////////////////////////////////////// |
---|
2009 | /// AUXILARY PROCEDURES, WHICH ARE DECLARED STATIC |
---|
2010 | ///////////////////////////////////////////////////////////////////////////////// |
---|
2011 | ///////////////////////////////////////////////////////////////////////////////// |
---|
2012 | /// - scalarproduct |
---|
2013 | /// - intmatcoldelete |
---|
2014 | /// - intmatconcat |
---|
2015 | /// - sortlist |
---|
2016 | /// - minInList |
---|
2017 | /// - stringdelete |
---|
2018 | /// - abs |
---|
2019 | /// - commondenominator |
---|
2020 | /// - maxPosInIntvec |
---|
2021 | /// - maxPosInIntmat |
---|
2022 | /// - sortintvec |
---|
2023 | /// - matrixtointmat |
---|
2024 | ///////////////////////////////////////////////////////////////////////////////// |
---|
2025 | |
---|
2026 | static proc scalarproduct (intvec w,intvec v) |
---|
2027 | "USAGE: scalarproduct(w,v); w,v intvec |
---|
2028 | ASSUME: w and v are integer vectors of the same length |
---|
2029 | RETURN: int, the scalarproduct of v and w |
---|
2030 | NOTE: the procedure is called by findOrientedBoundary" |
---|
2031 | { |
---|
2032 | int sp; |
---|
2033 | for (int i=1;i<=size(w);i++) |
---|
2034 | { |
---|
2035 | sp=sp+v[i]*w[i]; |
---|
2036 | } |
---|
2037 | return(sp); |
---|
2038 | } |
---|
2039 | |
---|
2040 | static proc intmatcoldelete (intmat w,int i) |
---|
2041 | "USAGE: intmatcoldelete(w,i); w intmat, i int |
---|
2042 | RETURN: intmat, the integer matrix w with the ith comlumn deleted |
---|
2043 | NOTE: the procedure is called by intmatsort and normalFan" |
---|
2044 | { |
---|
2045 | if ((i<1) or (i>ncols(w)) or (ncols(w)==1)) |
---|
2046 | { |
---|
2047 | return(w); |
---|
2048 | } |
---|
2049 | if (i==1) |
---|
2050 | { |
---|
2051 | intmat M[nrows(w)][ncols(w)-1]=w[1..nrows(w),2..ncols(w)]; |
---|
2052 | return(M); |
---|
2053 | } |
---|
2054 | if (i==ncols(w)) |
---|
2055 | { |
---|
2056 | intmat M[nrows(w)][ncols(w)-1]=w[1..nrows(w),1..ncols(w)-1]; |
---|
2057 | return(M); |
---|
2058 | } |
---|
2059 | else |
---|
2060 | { |
---|
2061 | intmat M[nrows(w)][i-1]=w[1..nrows(w),1..i-1]; |
---|
2062 | intmat N[nrows(w)][ncols(w)-i]=w[1..nrows(w),i+1..ncols(w)]; |
---|
2063 | return(intmatconcat(M,N)); |
---|
2064 | } |
---|
2065 | } |
---|
2066 | |
---|
2067 | static proc intmatconcat (intmat M,intmat N) |
---|
2068 | "USAGE: intmatconcat(M,N); M,N intmat |
---|
2069 | RETURN: intmat, M and N concatenated |
---|
2070 | NOTE: the procedure is called by intmatcoldelete and sortintmat" |
---|
2071 | { |
---|
2072 | if (nrows(M)>=nrows(N)) |
---|
2073 | { |
---|
2074 | int m=nrows(M); |
---|
2075 | |
---|
2076 | } |
---|
2077 | else |
---|
2078 | { |
---|
2079 | int m=nrows(N); |
---|
2080 | } |
---|
2081 | intmat P[m][ncols(M)+ncols(N)]; |
---|
2082 | P[1..nrows(M),1..ncols(M)]=M[1..nrows(M),1..ncols(M)]; |
---|
2083 | P[1..nrows(N),ncols(M)+1..ncols(M)+ncols(N)]=N[1..nrows(N),1..ncols(N)]; |
---|
2084 | return(P); |
---|
2085 | } |
---|
2086 | |
---|
2087 | static proc sortlist (list v,int pos) |
---|
2088 | "USAGE: sortlist(v,pos); v list, pos int |
---|
2089 | RETURN: list, the list L ordered in an ascending way according to the pos-th entries |
---|
2090 | NOTE: called by tropicalCurve" |
---|
2091 | { |
---|
2092 | if(size(v)==1) |
---|
2093 | { |
---|
2094 | return(v); |
---|
2095 | } |
---|
2096 | list w=minInList(v,pos); |
---|
2097 | v=delete(v,w[2]); |
---|
2098 | v=sortlist(v,pos); |
---|
2099 | v=list(w[1])+v; |
---|
2100 | return(v); |
---|
2101 | } |
---|
2102 | |
---|
2103 | static proc minInList (list v,int pos) |
---|
2104 | "USAGE: minInList(v,pos); v list, pos int |
---|
2105 | RETURN: list, (v[i],i) such that v[i][pos] is minimal |
---|
2106 | NOTE: called by sortlist" |
---|
2107 | { |
---|
2108 | int min=v[1][pos]; |
---|
2109 | int minpos=1; |
---|
2110 | for (int i=2;i<=size(v);i++) |
---|
2111 | { |
---|
2112 | if (v[i][pos]<min) |
---|
2113 | { |
---|
2114 | min=v[i][pos]; |
---|
2115 | minpos=i; |
---|
2116 | } |
---|
2117 | } |
---|
2118 | return(list(v[minpos],minpos)); |
---|
2119 | } |
---|
2120 | |
---|
2121 | static proc stringdelete (string w,int i) |
---|
2122 | "USAGE: stringdelete(w,i); w string, i int |
---|
2123 | RETURN: string, the string w with the ith component deleted |
---|
2124 | NOTE: the procedure is called by texnumber and choosegfanvector" |
---|
2125 | { |
---|
2126 | if ((i>size(w)) or (i<=0)) |
---|
2127 | { |
---|
2128 | return(w); |
---|
2129 | } |
---|
2130 | if ((size(w)==1) and (i==1)) |
---|
2131 | { |
---|
2132 | return(""); |
---|
2133 | |
---|
2134 | } |
---|
2135 | if (i==1) |
---|
2136 | { |
---|
2137 | return(w[2..size(w)]); |
---|
2138 | } |
---|
2139 | if (i==size(w)) |
---|
2140 | { |
---|
2141 | return(w[1..size(w)-1]); |
---|
2142 | } |
---|
2143 | else |
---|
2144 | { |
---|
2145 | string erg=w[1..i-1],w[i+1..size(w)]; |
---|
2146 | return(erg); |
---|
2147 | } |
---|
2148 | } |
---|
2149 | |
---|
2150 | static proc abs (def n) |
---|
2151 | "USAGE: abs(n); n poly or int |
---|
2152 | RETURN: poly or int, the absolute value of n" |
---|
2153 | { |
---|
2154 | if (n>=0) |
---|
2155 | { |
---|
2156 | return(n); |
---|
2157 | } |
---|
2158 | else |
---|
2159 | { |
---|
2160 | return(-n); |
---|
2161 | } |
---|
2162 | } |
---|
2163 | |
---|
2164 | static proc commondenominator (matrix M) |
---|
2165 | "USAGE: commondenominator(M); M matrix |
---|
2166 | ASSUME: the base ring has characteristic zero |
---|
2167 | RETURN: int, the lowest common multiple of the denominators of the leading coefficients |
---|
2168 | of the entries in M |
---|
2169 | NOTE: the procedure is called from polymakeToIntmat" |
---|
2170 | { |
---|
2171 | int i,j; |
---|
2172 | int kgV=1; |
---|
2173 | // successively build the lowest common multiple of the denominators of the leading coefficients |
---|
2174 | // of the entries in M |
---|
2175 | for (i=1;i<=nrows(M);i++) |
---|
2176 | { |
---|
2177 | for (j=1;j<=ncols(M);j++) |
---|
2178 | { |
---|
2179 | kgV=lcm(kgV,int(denominator(leadcoef(M[i,j])))); |
---|
2180 | } |
---|
2181 | } |
---|
2182 | return(kgV); |
---|
2183 | } |
---|
2184 | |
---|
2185 | static proc maxPosInIntvec (intvec v) |
---|
2186 | "USAGE: maxPosInIntvec(v); v intvec |
---|
2187 | RETURN: int, the first position of a maximal entry in v |
---|
2188 | NOTE: called by sortintmat" |
---|
2189 | { |
---|
2190 | int max=v[1]; |
---|
2191 | int maxpos=1; |
---|
2192 | for (int i=2;i<=size(v);i++) |
---|
2193 | { |
---|
2194 | if (v[i]>max) |
---|
2195 | { |
---|
2196 | max=v[i]; |
---|
2197 | maxpos=i; |
---|
2198 | } |
---|
2199 | } |
---|
2200 | return(maxpos); |
---|
2201 | } |
---|
2202 | |
---|
2203 | static proc maxPosInIntmat (intmat v) |
---|
2204 | "USAGE: maxPosInIntmat(v); v intmat |
---|
2205 | ASSUME: v has a unique maximal entry |
---|
2206 | RETURN: intvec, the position (i,j) of the maximal entry in v |
---|
2207 | NOTE: called by findOrientedBoundary" |
---|
2208 | { |
---|
2209 | int max=v[1,1]; |
---|
2210 | intvec maxpos=1,1; |
---|
2211 | int i,j; |
---|
2212 | for (i=1;i<=nrows(v);i++) |
---|
2213 | { |
---|
2214 | for (j=1;j<=ncols(v);j++) |
---|
2215 | { |
---|
2216 | if (v[i,j]>max) |
---|
2217 | { |
---|
2218 | max=v[i,j]; |
---|
2219 | maxpos=i,j; |
---|
2220 | } |
---|
2221 | } |
---|
2222 | } |
---|
2223 | return(maxpos); |
---|
2224 | } |
---|
2225 | |
---|
2226 | static proc sortintvec (intvec w) |
---|
2227 | "USAGE: sortintvec(v); v intvec |
---|
2228 | RETURN: intvec, the entries of v are ordered in an ascending way |
---|
2229 | NOTE: called from ellipticNF" |
---|
2230 | { |
---|
2231 | int j,k,stop; |
---|
2232 | intvec v=w[1]; |
---|
2233 | for (j=2;j<=size(w);j++) |
---|
2234 | { |
---|
2235 | k=1; |
---|
2236 | stop=0; |
---|
2237 | while ((k<=size(v)) and (stop==0)) |
---|
2238 | { |
---|
2239 | if (v[k]<w[j]) |
---|
2240 | { |
---|
2241 | k++; |
---|
2242 | } |
---|
2243 | else |
---|
2244 | { |
---|
2245 | stop=1; |
---|
2246 | } |
---|
2247 | } |
---|
2248 | if (k==size(v)+1) |
---|
2249 | { |
---|
2250 | v=v,w[j]; |
---|
2251 | } |
---|
2252 | else |
---|
2253 | { |
---|
2254 | if (k==1) |
---|
2255 | { |
---|
2256 | v=w[j],v; |
---|
2257 | } |
---|
2258 | else |
---|
2259 | { |
---|
2260 | v=v[1..k-1],w[j],v[k..size(v)]; |
---|
2261 | } |
---|
2262 | } |
---|
2263 | } |
---|
2264 | return(v); |
---|
2265 | } |
---|
2266 | |
---|
2267 | static proc sortlistbyintvec (list L,intvec w) |
---|
2268 | "USAGE: sortlistbyintvec(L,w); L list, w intvec |
---|
2269 | RETURN: list, the entries of L are ordered such that the corresponding reordering of |
---|
2270 | w would order w in an ascending way |
---|
2271 | NOTE: called from ellipticNF" |
---|
2272 | { |
---|
2273 | int j,k,stop; |
---|
2274 | intvec v=w[1]; |
---|
2275 | list LL=L[1]; |
---|
2276 | for (j=2;j<=size(w);j++) |
---|
2277 | { |
---|
2278 | k=1; |
---|
2279 | stop=0; |
---|
2280 | while ((k<=size(v)) and (stop==0)) |
---|
2281 | { |
---|
2282 | if (v[k]<w[j]) |
---|
2283 | { |
---|
2284 | k++; |
---|
2285 | } |
---|
2286 | else |
---|
2287 | { |
---|
2288 | stop=1; |
---|
2289 | } |
---|
2290 | } |
---|
2291 | if (k==size(v)+1) |
---|
2292 | { |
---|
2293 | v=v,w[j]; |
---|
2294 | LL=insert(LL,L[j],size(LL)); |
---|
2295 | } |
---|
2296 | else |
---|
2297 | { |
---|
2298 | if (k==1) |
---|
2299 | { |
---|
2300 | v=w[j],v; |
---|
2301 | LL=insert(LL,L[j]); |
---|
2302 | } |
---|
2303 | else |
---|
2304 | { |
---|
2305 | v=v[1..k-1],w[j],v[k..size(v)]; |
---|
2306 | LL=insert(LL,L[j],k-1); |
---|
2307 | } |
---|
2308 | } |
---|
2309 | } |
---|
2310 | return(LL); |
---|
2311 | } |
---|
2312 | |
---|
2313 | static proc matrixtointmat (matrix MM) |
---|
2314 | "USAGE: matrixtointmat(v); MM matrix |
---|
2315 | ASSUME: MM is a matrix with only integers as entries |
---|
2316 | RETURN: intmat, the matrix MM has been transformed to type intmat |
---|
2317 | NOTE: called from ellipticNF" |
---|
2318 | { |
---|
2319 | intmat M[nrows(MM)][ncols(MM)]=M; |
---|
2320 | int i,j; |
---|
2321 | for (i=1;i<=nrows(M);i++) |
---|
2322 | { |
---|
2323 | for (j=1;j<=ncols(M);j++) |
---|
2324 | { |
---|
2325 | execute("M["+string(i)+","+string(j)+"]="+string(MM[i,j])+";"); |
---|
2326 | } |
---|
2327 | } |
---|
2328 | return(M); |
---|
2329 | } |
---|
2330 | |
---|
2331 | ////////////////////////////////////////////////////////////////////////////// |
---|
2332 | |
---|
2333 | static proc polygonToCoordinates (list points) |
---|
2334 | "USAGE: polygonToCoordinates(points); points list |
---|
2335 | ASSUME: points is a list of integer vectors each of size two describing the |
---|
2336 | marked points of a convex lattice polygon like the output of polygonDB |
---|
2337 | RETURN: list, the first entry is a string representing the coordinates corresponding |
---|
2338 | to the latticpoints seperated by commata |
---|
2339 | the second entry is a list where the ith entry is a string representing |
---|
2340 | the coordinate of corresponding to the ith lattice point |
---|
2341 | the third entry is the latex format of the first entry |
---|
2342 | NOTE: the procedure is called by fan" |
---|
2343 | { |
---|
2344 | string coord; |
---|
2345 | list coords; |
---|
2346 | string latex; |
---|
2347 | for (int i=1;i<=size(points);i++) |
---|
2348 | { |
---|
2349 | coords[i]="u"+string(points[i][1])+string(points[i][2]); |
---|
2350 | coord=coord+coords[i]+","; |
---|
2351 | latex=latex+"u_{"+string(points[i][1])+string(points[i][2])+"},"; |
---|
2352 | } |
---|
2353 | coord=coord[1,size(coord)-1]; |
---|
2354 | latex=latex[1,size(latex)-1]; |
---|
2355 | return(list(coord,coords,latex)); |
---|
2356 | } |
---|
2357 | |
---|
2358 | |
---|
2359 | /* |
---|
2360 | proc ADeterminant (list polygon,list #) |
---|
2361 | { |
---|
2362 | list triangs=triangulations(polygon); |
---|
2363 | list sppg=splitPolygon(polygon); |
---|
2364 | list etavectors; |
---|
2365 | int i,j; |
---|
2366 | int stop; |
---|
2367 | for (i=1;i<=size(triangs);i++) |
---|
2368 | { |
---|
2369 | etavectors[i]=eta(triangs[i],sppg); |
---|
2370 | } |
---|
2371 | size(etavectors); |
---|
2372 | |
---|
2373 | for (i=size(etavectors);i>=2;i--) |
---|
2374 | { |
---|
2375 | stop=0; |
---|
2376 | for (j=1;(j<i) and (stop==0);j++) |
---|
2377 | { |
---|
2378 | if (etavectors[i]==etavectors[j]) |
---|
2379 | { |
---|
2380 | etavectors=delete(etavectors,i); |
---|
2381 | stop=1; |
---|
2382 | } |
---|
2383 | } |
---|
2384 | } |
---|
2385 | size(etavectors); |
---|
2386 | if (size(#)>0) |
---|
2387 | { |
---|
2388 | execute("ring ADring=(0,a(1.."+string(size(etavectors))+")),("+polygonToCoordinates(polygon)[1]+"),lp;"); |
---|
2389 | list terme; |
---|
2390 | poly ad,term; |
---|
2391 | matrix XE[1][1]; |
---|
2392 | for (i=1;i<=size(etavectors);i++) |
---|
2393 | { |
---|
2394 | term=1; |
---|
2395 | for (j=1;j<=nvars(basering);j++) |
---|
2396 | { |
---|
2397 | term=term*var(j)^etavectors[i][j]; |
---|
2398 | } |
---|
2399 | terme[i]=term; |
---|
2400 | ad=ad+a(i)*term; |
---|
2401 | } |
---|
2402 | matrix M[size(etavectors)][nvars(basering)]; |
---|
2403 | for (i=1;i<=size(etavectors);i++) |
---|
2404 | { |
---|
2405 | } |
---|
2406 | return(list(etavectors,string(ad))); |
---|
2407 | |
---|
2408 | |
---|
2409 | } |
---|
2410 | |
---|
2411 | return(etavectors); |
---|
2412 | } |
---|
2413 | |
---|
2414 | proc adsub () |
---|
2415 | { |
---|
2416 | ring r=0,(x,y,u00,u10,u20,u01,u11,u02,a(1..5)),dp; |
---|
2417 | poly f1=(3x-y+1)*(2x+y+1); |
---|
2418 | poly f2=(7x+2y-1)*(x+y); |
---|
2419 | poly f3=(x-y-2)*(x+y+3); |
---|
2420 | poly f4=(17x-11y+3)*(x+7y-2); |
---|
2421 | poly f5=(x+2y-7)*(3x+3y-1); |
---|
2422 | poly f6=(2x+12y-17)*(33x-3y-1); |
---|
2423 | matrix M1=coeffs(f1,ideal(1,x,x2,y,xy,y2)); |
---|
2424 | matrix M2=coeffs(f2,ideal(1,x,x2,y,xy,y2)); |
---|
2425 | matrix M3=coeffs(f3,ideal(1,x,x2,y,xy,y2)); |
---|
2426 | matrix M4=coeffs(f4,ideal(1,x,x2,y,xy,y2)); |
---|
2427 | matrix M5=coeffs(f5,ideal(1,x,x2,y,xy,y2)); |
---|
2428 | matrix M6=coeffs(f6,ideal(1,x,x2,y,xy,y2)); |
---|
2429 | 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; |
---|
2430 | 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]); |
---|
2431 | 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]); |
---|
2432 | 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]); |
---|
2433 | 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]); |
---|
2434 | 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]); |
---|
2435 | 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]); |
---|
2436 | ideal i=g1,g2,g3,g4,g5,g6; |
---|
2437 | option(redSB); |
---|
2438 | ideal j=std(i); |
---|
2439 | poly ff=substitute(f,a(5),4,a(4),-1,a(3),-1,a(2),1,a(1),-1); |
---|
2440 | return(string(ff)); |
---|
2441 | } |
---|
2442 | |
---|
2443 | */ |
---|