1 | version="version oldpolymake.lib 4.0.0.0 Jun_2013 "; |
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2 | category="Tropical Geometry"; |
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3 | info=" |
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4 | LIBRARY: oldpolymake.lib Computations with polytopes and fans, |
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5 | interface to polymake and TOPCOM |
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6 | AUTHOR: Thomas Markwig, email: keilen@mathematik.uni-kl.de |
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7 | |
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8 | WARNING: |
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9 | Most procedures will not work unless polymake or topcom is installed and |
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10 | if so, they will only work with the operating system LINUX! |
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11 | For more detailed information see IMPORTANT NOTE respectively consult the |
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12 | help string of the procedures. |
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13 | |
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14 | The conventions used in this library for polytopes and fans, e.g. the |
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15 | length and labeling of their vertices resp. rays, differs from the conventions |
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16 | used in polymake and thus from the conventions used in the polymake |
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17 | extension polymake.so of Singular. We recommend to use the newer polymake.so |
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18 | whenever possible. |
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19 | |
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20 | IMPORTANT NOTE: |
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21 | Even though this is a Singular library for computing polytopes and fans |
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22 | such as the Newton polytope or the Groebner fan of a polynomial, most of |
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23 | the hard computations are NOT done by Singular but by the program |
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24 | @* - polymake by Ewgenij Gawrilow, TU Berlin and Michael Joswig, TU Darmstadt |
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25 | @* (see http://www.math.tu-berlin.de/polymake/), |
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26 | @* respectively (only in the procedure triangulations) by the program |
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27 | @* - topcom by Joerg Rambau, Universitaet Bayreuth (see |
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28 | @* http://www.uni-bayreuth.de/departments/wirtschaftsmathematik/rambau/TOPCOM); |
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29 | @* this library should rather be seen as an interface which allows to use a |
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30 | (very limited) number of options which polymake respectively topcom offers |
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31 | to compute with polytopes and fans and to make the results available in |
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32 | Singular for further computations; |
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33 | moreover, the user familiar with Singular does not have to learn the syntax |
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34 | of polymake or topcom, if the options offered here are sufficient for his |
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35 | purposes. |
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36 | @* Note, though, that the procedures concerned with planar polygons are |
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37 | independent of both, polymake and topcom. |
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38 | |
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39 | PROCEDURES USING POLYMAKE: |
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40 | polymakePolytope() computes the vertices of a polytope using polymake |
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41 | newtonPolytope() computes the Newton polytope of a polynomial |
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42 | newtonPolytopeLP() computes the lattice points of the Newton polytope |
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43 | normalFan() computes the normal fan of a polytope |
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44 | groebnerFan() computes the Groebner fan of a polynomial |
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45 | |
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46 | PROCEDURES USING TOPCOM: |
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47 | triangulations() computes all triangulations of a marked polytope |
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48 | secondaryPolytope() computes the secondary polytope of a marked polytope |
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49 | |
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50 | PROCEDURES USING POLYMAKE AND TOPCOM: |
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51 | secondaryFan() computes the secondary fan of a marked polytope |
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52 | |
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53 | PROCEDURES CONERNED WITH PLANAR POLYGONS: |
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54 | cycleLength() computes the cycleLength of cycle |
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55 | splitPolygon() splits a marked polygon into vertices, facets, interior points |
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56 | eta() computes the eta-vector of a triangulation |
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57 | findOrientedBoundary() computes the boundary of a convex hull |
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58 | cyclePoints() computes lattice points connected to some lattice point |
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59 | latticeArea() computes the lattice area of a polygon |
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60 | picksFormula() computes the ingrediants of Pick's formula for a polygon |
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61 | ellipticNF() computes the normal form of an elliptic polygon |
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62 | ellipticNFDB() displays the 16 normal forms of elliptic polygons |
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63 | |
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64 | KEYWORDS: polytope; fan; secondary fan; secondary polytope; polymake; |
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65 | Newton polytope; Groebner fan |
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66 | "; |
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67 | |
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68 | //////////////////////////////////////////////////////////////////////////////// |
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69 | /// Auxilary Static Procedures in this Library |
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70 | //////////////////////////////////////////////////////////////////////////////// |
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71 | /// - scalarproduct |
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72 | /// - intmatcoldelete |
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73 | /// - intmatconcat |
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74 | /// - sortlist |
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75 | /// - minInList |
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76 | /// - stringdelete |
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77 | /// - abs |
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78 | /// - commondenominator |
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79 | /// - maxPosInIntvec |
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80 | /// - maxPosInIntmat |
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81 | /// - sortintvec |
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82 | /// - matrixtointmat |
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83 | //////////////////////////////////////////////////////////////////////////////// |
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84 | |
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85 | //////////////////////////////////////////////////////////////////////////////// |
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86 | LIB "poly.lib"; |
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87 | LIB "linalg.lib"; |
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88 | LIB "random.lib"; |
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89 | //////////////////////////////////////////////////////////////////////////////// |
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90 | |
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91 | static proc mod_init () |
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92 | { |
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93 | LIB "gfanlib.so"; |
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94 | LIB "polymake.so"; |
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95 | } |
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96 | |
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97 | ///////////////////////////////////////////////////////////////////////////// |
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98 | /// PROCEDURES USING POLYMAKE |
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99 | ///////////////////////////////////////////////////////////////////////////// |
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100 | |
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101 | proc polymakePolytope (intmat points) |
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102 | "USAGE: polymakePolytope(points); polytope intmat |
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103 | ASSUME: each row of points gives the coordinates of a lattice point of a |
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104 | polytope with their affine coordinates as given by the output of |
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105 | secondaryPolytope |
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106 | PURPOSE: the procedure calls polymake to compute the vertices of the polytope |
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107 | as well as its dimension and information on its facets |
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108 | RETURN: list, L with four entries |
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109 | @* L[1] : an integer matrix whose rows are the coordinates of vertices |
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110 | of the polytope |
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111 | @* L[2] : the dimension of the polytope |
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112 | @* L[3] : a list whose ith entry explains to which vertices the |
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113 | ith vertex of the Newton polytope is connected |
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114 | -- i.e. L[3][i] is an integer vector and an entry k in |
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115 | there means that the vertex L[1][i] is connected to the |
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116 | vertex L[1][k] |
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117 | @* L[4] : an matrix of type bigintmat whose rows mulitplied by |
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118 | (1,var(1),...,var(nvar)) give a linear system of equations |
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119 | describing the affine hull of the polytope, |
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120 | i.e. the smallest affine space containing the polytope |
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121 | NOTE: - for its computations the procedure calls the program polymake by |
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122 | Ewgenij Gawrilow, TU Berlin and Michael Joswig, TU Darmstadt; |
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123 | it therefore is necessary that this program is installed in order |
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124 | to use this procedure; |
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125 | see http://www.math.tu-berlin.de/polymake/ |
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126 | @* - note that in the vertex edge graph we have changed the polymake |
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127 | convention which starts indexing its vertices by zero while we start |
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128 | with one ! |
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129 | EXAMPLE: example polymakePolytope; shows an example" |
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130 | { |
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131 | // add a first column to polytope as homogenising coordinate |
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132 | points=intmatAddFirstColumn(points,"points"); |
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133 | polytope polytop=polytopeViaPoints(points); |
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134 | list graph=vertexAdjacencyGraph(polytop)[2]; |
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135 | int i,j; |
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136 | for (i=1;i<=size(graph);i++) |
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137 | { |
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138 | for (j=1;j<=size(graph[i]);j++) |
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139 | { |
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140 | graph[i][j]=graph[i][j]+1; |
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141 | } |
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142 | } |
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143 | return(list(intmatcoldelete(vertices(polytop),1),dimension(polytop),graph,equations(polytop))); |
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144 | } |
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145 | example |
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146 | { |
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147 | "EXAMPLE:"; |
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148 | echo=2; |
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149 | // the lattice points of the unit square in the plane |
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150 | list points=intvec(0,0),intvec(0,1),intvec(1,0),intvec(1,1); |
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151 | // the secondary polytope of this lattice point configuration is computed |
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152 | intmat secpoly=secondaryPolytope(points)[1]; |
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153 | list np=polymakePolytope(secpoly); |
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154 | // the vertices of the secondary polytope are: |
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155 | np[1]; |
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156 | // its dimension is |
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157 | np[2]; |
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158 | // np[3] contains information how the vertices are connected to each other, |
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159 | // e.g. the first vertex (number 0) is connected to the second one |
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160 | np[3][1]; |
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161 | // the affine hull has the equation |
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162 | ring r=0,x(1..4),dp; |
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163 | matrix M[5][1]=1,x(1),x(2),x(3),x(4); |
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164 | intmat(np[4])*M; |
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165 | } |
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166 | |
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167 | ///////////////////////////////////////////////////////////////////////////// |
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168 | |
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169 | proc newtonPolytope (poly f) |
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170 | "USAGE: newtonPolytope(f); f poly |
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171 | RETURN: list, L with four entries |
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172 | @* L[1] : an integer matrix whose rows are the coordinates of vertices |
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173 | of the Newton polytope of f |
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174 | @* L[2] : the dimension of the Newton polytope of f |
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175 | @* L[3] : a list whose ith entry explains to which vertices the |
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176 | ith vertex of the Newton polytope is connected |
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177 | -- i.e. L[3][i] is an integer vector and an entry k in |
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178 | there means that the vertex L[1][i] is |
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179 | connected to the vertex L[1][k] |
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180 | @* L[4] : an matrix of type bigintmat whose rows mulitplied by |
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181 | (1,var(1),...,var(nvar)) give a linear system of equations |
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182 | describing the affine hull of the Newton polytope, i.e. the |
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183 | smallest affine space containing the Newton polytope |
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184 | NOTE: - if we replace the first column of L[4] by zeros, i.e. if we move |
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185 | the affine hull to the origin, then we get the equations for the |
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186 | orthogonal complement of the linearity space of the normal fan dual |
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187 | to the Newton polytope, i.e. we get the EQUATIONS that |
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188 | we need as input for polymake when computing the normal fan |
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189 | @* - the procedure calls for its computation polymake by Ewgenij Gawrilow, |
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190 | TU Berlin and Michael Joswig, so it only works if polymake is installed; |
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191 | see http://www.math.tu-berlin.de/polymake/ |
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192 | EXAMPLE: example newtonPolytope; shows an example" |
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193 | { |
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194 | int i,j; |
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195 | // compute the list of exponent vectors of the polynomial, |
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196 | // which are the lattice points |
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197 | // whose convex hull is the Newton polytope of f |
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198 | intmat exponents[size(f)][nvars(basering)]; |
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199 | while (f!=0) |
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200 | { |
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201 | i++; |
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202 | exponents[i,1..nvars(basering)]=leadexp(f); |
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203 | f=f-lead(f); |
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204 | } |
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205 | // call polymakePolytope with exponents |
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206 | return(polymakePolytope(exponents)); |
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207 | } |
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208 | example |
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209 | { |
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210 | "EXAMPLE:"; |
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211 | echo=2; |
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212 | ring r=0,(x,y,z),dp; |
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213 | matrix M[4][1]=1,x,y,z; |
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214 | poly f=y3+x2+xy+2xz+yz+z2+1; |
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215 | // the Newton polytope of f is |
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216 | list np=newtonPolytope(f); |
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217 | // the vertices of the Newton polytope are: |
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218 | np[1]; |
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219 | // its dimension is |
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220 | np[2]; |
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221 | // np[3] contains information how the vertices are connected to each other, |
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222 | // e.g. the first vertex (number 0) is connected to the second, third and |
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223 | // fourth vertex |
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224 | np[3][1]; |
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225 | ////////////////////////// |
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226 | f=x2-y3; |
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227 | // the Newton polytope of f is |
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228 | np=newtonPolytope(f); |
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229 | // the vertices of the Newton polytope are: |
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230 | np[1]; |
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231 | // its dimension is |
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232 | np[2]; |
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233 | // the Newton polytope is contained in the affine space given |
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234 | // by the equations |
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235 | intmat(np[4])*M; |
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236 | } |
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237 | |
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238 | ///////////////////////////////////////////////////////////////////////////// |
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239 | |
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240 | proc newtonPolytopeLP (poly f) |
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241 | "USAGE: newtonPolytopeLP(f); f poly |
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242 | RETURN: list, the exponent vectors of the monomials occuring in f, |
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243 | i.e. the lattice points of the Newton polytope of f |
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244 | EXAMPLE: example normalFan; shows an example" |
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245 | { |
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246 | list np; |
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247 | int i=1; |
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248 | while (f!=0) |
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249 | { |
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250 | np[i]=leadexp(f); |
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251 | f=f-lead(f); |
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252 | i++; |
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253 | } |
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254 | return(np); |
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255 | } |
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256 | example |
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257 | { |
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258 | "EXAMPLE:"; |
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259 | echo=2; |
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260 | ring r=0,(x,y,z),dp; |
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261 | poly f=y3+x2+xy+2xz+yz+z2+1; |
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262 | // the lattice points of the Newton polytope of f are |
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263 | newtonPolytopeLP(f); |
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264 | } |
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265 | |
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266 | ///////////////////////////////////////////////////////////////////////////// |
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267 | |
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268 | proc normalFan (intmat vertices,intmat affinehull,list graph,int er,list #) |
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269 | "USAGE: normalFan (vert,aff,graph,rays,[,#]); vert,aff intmat, graph list, rays int, # string |
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270 | ASSUME: - vert is an integer matrix whose rows are the coordinate of |
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271 | the vertices of a convex lattice polytope; |
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272 | @* - aff describes the affine hull of this polytope, i.e. |
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273 | the smallest affine space containing it, in the following sense: |
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274 | denote by n the number of columns of vert, then multiply aff by |
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275 | (1,x(1),...,x(n)) and set the resulting terms to zero in order to |
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276 | get the equations for the affine hull; |
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277 | @* - the ith entry of graph is an integer vector describing to which |
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278 | vertices the ith vertex is connected, i.e. a k as entry means that |
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279 | the vertex vert[i] is connected to vert[k]; |
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280 | @* - the integer rays is either one (if the extreme rays should be |
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281 | computed) or zero (otherwise) |
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282 | RETURN: list, the ith entry of L[1] contains information about the cone in the |
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283 | normal fan dual to the ith vertex of the polytope |
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284 | @* L[1][i][1] = integer matrix representing the inequalities which |
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285 | describe the cone dual to the ith vertex |
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286 | @* L[1][i][2] = a list which contains the inequalities represented |
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287 | by L[i][1] as a list of strings, where we use the |
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288 | variables x(1),...,x(n) |
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289 | @* L[1][i][3] = only present if 'er' is set to 1; in that case it is |
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290 | an interger matrix whose rows are the extreme rays |
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291 | of the cone |
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292 | @* L[2] = is an integer matrix whose rows span the linearity space |
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293 | of the fan, i.e. the linear space which is contained in |
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294 | each cone |
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295 | NOTE: - the procedure calls for its computation polymake by Ewgenij Gawrilow, |
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296 | TU Berlin and Michael Joswig, so it only works if polymake is |
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297 | installed; |
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298 | see http://www.math.tu-berlin.de/polymake/ |
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299 | @* - in the optional argument # it is possible to hand over other names |
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300 | for the variables to be used -- be careful, the format must be correct |
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301 | and that is not tested, e.g. if you want the variable names to be |
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302 | u00,u10,u01,u11 then you must hand over the string u11,u10,u01,u11 |
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303 | EXAMPLE: example normalFan; shows an example" |
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304 | { |
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305 | list ineq; // stores the inequalities of the cones |
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306 | int i,j,k; |
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307 | // we work over the following ring |
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308 | execute("ring ineqring=0,x(1.."+string(ncols(vertices))+"),lp;"); |
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309 | string greatersign=">"; |
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310 | // create the variable names |
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311 | if (size(#)>0) |
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312 | { |
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313 | if (typeof(#[1])=="string") |
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314 | { |
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315 | kill ineqring; |
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316 | execute("ring ineqring=0,("+#[1]+"),lp;"); |
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317 | } |
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318 | if (size(#)>1) |
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319 | { |
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320 | greatersign="<"; |
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321 | } |
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322 | } |
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323 | ////////////////////////////////////////////////////////////////// |
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324 | // Compute first the inequalities of the cones |
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325 | ////////////////////////////////////////////////////////////////// |
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326 | matrix VAR[1][ncols(vertices)]=maxideal(1); |
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327 | matrix EXP[ncols(vertices)][1]; |
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328 | poly p,pl,pr; |
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329 | // consider all vertices of the polytope |
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330 | for (i=1;i<=nrows(vertices);i++) |
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331 | { |
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332 | // first we produce for each vertex in the polytope |
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333 | // the inequalities describing the dual cone in the normal fan |
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334 | list pp; // contain strings representing the inequalities |
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335 | // describing the normal cone |
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336 | intmat ie[size(graph[i])][ncols(vertices)]; // contains the inequalities |
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337 | // as rows |
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338 | // consider all the vertices to which the ith vertex in the |
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339 | // polytope is connected by an edge |
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340 | for (j=1;j<=size(graph[i]);j++) |
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341 | { |
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342 | // produce the vector ie_j pointing from the jth vertex to the ith vertex; |
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343 | // this will be the jth inequality for the cone in the normal fan dual to |
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344 | // the ith vertex |
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345 | ie[j,1..ncols(vertices)]=vertices[i,1..ncols(vertices)]-vertices[graph[i][j],1..ncols(vertices)]; |
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346 | EXP=ie[j,1..ncols(vertices)]; |
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347 | // build a linear polynomial with the entries of ie_j as coefficients |
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348 | p=(VAR*EXP)[1,1]; |
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349 | pl,pr=0,0; |
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350 | // separate the terms with positive coefficients in p from |
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351 | // those with negative coefficients |
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352 | for (k=1;k<=size(p);k++) |
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353 | { |
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354 | if (leadcoef(p[k])<0) |
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355 | { |
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356 | pr=pr-p[k]; |
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357 | } |
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358 | else |
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359 | { |
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360 | pl=pl+p[k]; |
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361 | } |
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362 | } |
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363 | // build the string which represents the jth inequality |
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364 | // for the cone dual to the ith vertex |
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365 | // as polynomial inequality of type string, and store this |
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366 | // in the list pp as jth entry |
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367 | pp[j]=string(pl)+" "+greatersign+" "+string(pr); |
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368 | } |
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369 | // all inequalities for the ith vertex are stored in the list ineq |
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370 | ineq[i]=list(ie,pp); |
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371 | kill ie,pp; // kill certain lists |
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372 | } |
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373 | // remove the first column of affine hull to compute the linearity space |
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374 | intmat linearity[1][ncols(vertices)]; |
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375 | if (nrows(affinehull)>0) |
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376 | { |
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377 | linearity=intmatcoldelete(affinehull,1); |
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378 | } |
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379 | ////////////////////////////////////////////////////////////////// |
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380 | // Compute next the extreme rays of the cones |
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381 | ////////////////////////////////////////////////////////////////// |
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382 | if (er==1) |
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383 | { |
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384 | list extremerays; // keeps the result |
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385 | cone kegel; |
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386 | intmat linearspan=intmatAddFirstColumn(linearity,"rays"); |
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387 | intmat M; // the matrix keeping the inequalities |
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388 | for (i=1;i<=size(ineq);i++) |
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389 | { |
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390 | kegel=coneViaInequalities(intmatAddFirstColumn(ineq[i][1],"rays"),linearspan); |
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391 | extremerays[i]=intmatcoldelete(rays(kegel),1); |
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392 | } |
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393 | for (i=1;i<=size(ineq);i++) |
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394 | { |
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395 | ineq[i]=ineq[i]+list(extremerays[i]); |
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396 | } |
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397 | } |
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398 | // get the linearity space |
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399 | return(list(ineq,linearity)); |
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400 | } |
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401 | example |
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402 | { |
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403 | "EXAMPLE:"; |
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404 | echo=2; |
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405 | ring r=0,(x,y,z),dp; |
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406 | matrix M[4][1]=1,x,y,z; |
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407 | poly f=y3+x2+xy+2xz+yz+z2+1; |
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408 | // the Newton polytope of f is |
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409 | list np=newtonPolytope(f); |
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410 | // the Groebner fan of f, i.e. the normal fan of the Newton polytope |
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411 | list gf=normalFan(np[1],np[4],np[3],1,"x,y,z"); |
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412 | // the number of cones in the Groebner fan of f is: |
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413 | size(gf[1]); |
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414 | // the inequalities of the first cone as matrix are: |
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415 | print(gf[1][1][1]); |
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416 | // the inequalities of the first cone as string are: |
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417 | print(gf[1][1][2]); |
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418 | // the rows of the following matrix are the extreme rays of the first cone: |
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419 | print(gf[1][1][3]); |
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420 | // each cone contains the linearity space spanned by: |
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421 | print(gf[2]); |
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422 | } |
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423 | |
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424 | ///////////////////////////////////////////////////////////////////////////// |
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425 | |
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426 | proc groebnerFan (poly f) |
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427 | "USAGE: groebnerFan(f); f poly |
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428 | RETURN: list, the ith entry of L[1] contains information about the ith cone |
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429 | in the Groebner fan dual to the ith vertex in the Newton |
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430 | polytope of the f |
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431 | @* L[1][i][1] = integer matrix representing the inequalities |
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432 | which describe the cone |
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433 | @* L[1][i][2] = a list which contains the inequalities represented |
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434 | by L[1][i][1] as a list of strings |
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435 | @* L[1][i][3] = an interger matrix whose rows are the extreme rays |
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436 | of the cone |
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437 | @* L[2] = is an integer matrix whose rows span the linearity space |
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438 | of the fan, i.e. the linear space which is contained |
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439 | in each cone |
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440 | @* L[3] = the Newton polytope of f in the format of the procedure |
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441 | newtonPolytope |
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442 | @* L[4] = integer matrix where each row represents the exponent |
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443 | vector of one monomial occuring in the input polynomial |
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444 | NOTE: - if you have already computed the Newton polytope of f then you might want |
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445 | to use the procedure normalFan instead in order to avoid doing costly |
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446 | computation twice |
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447 | @* - the procedure calls for its computation polymake by Ewgenij Gawrilow, |
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448 | TU Berlin and Michael Joswig, so it only works if polymake is installed; |
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449 | see http://www.math.tu-berlin.de/polymake/ |
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450 | EXAMPLE: example groebnerFan; shows an example" |
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451 | { |
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452 | int i,j; |
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453 | // compute the list of exponent vectors of the polynomial, which are |
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454 | // the lattice points whose convex hull is the Newton polytope of f |
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455 | intmat exponents[size(f)][nvars(basering)]; |
---|
456 | while (f!=0) |
---|
457 | { |
---|
458 | i++; |
---|
459 | exponents[i,1..nvars(basering)]=leadexp(f); |
---|
460 | f=f-lead(f); |
---|
461 | } |
---|
462 | // call polymakePolytope with exponents |
---|
463 | list newtonp=polymakePolytope(exponents); |
---|
464 | // get the variables as string |
---|
465 | string variablen; |
---|
466 | for (i=1;i<=nvars(basering);i++) |
---|
467 | { |
---|
468 | variablen=variablen+string(var(i))+","; |
---|
469 | } |
---|
470 | variablen=variablen[1,size(variablen)-1]; |
---|
471 | // call normalFan in order to compute the Groebner fan |
---|
472 | list gf=normalFan(newtonp[1],newtonp[4],newtonp[3],1,variablen); |
---|
473 | // append newtonp to gf |
---|
474 | gf[3]=newtonp; |
---|
475 | // append the exponent vectors to gf |
---|
476 | gf[4]=exponents; |
---|
477 | return(gf); |
---|
478 | } |
---|
479 | example |
---|
480 | { |
---|
481 | "EXAMPLE:"; |
---|
482 | echo=2; |
---|
483 | ring r=0,(x,y,z),dp; |
---|
484 | matrix M[4][1]=1,x,y,z; |
---|
485 | poly f=y3+x2+xy+2xz+yz+z2+1; |
---|
486 | // the Newton polytope of f is |
---|
487 | list gf=groebnerFan(f); |
---|
488 | // the exponent vectors of f are ordered as follows |
---|
489 | gf[4]; |
---|
490 | // the first cone of the groebner fan has the inequalities |
---|
491 | gf[1][1][1]; |
---|
492 | // as a string they look like |
---|
493 | gf[1][1][2]; |
---|
494 | // and it has the extreme rays |
---|
495 | print(gf[1][1][3]); |
---|
496 | // the linearity space is spanned by |
---|
497 | print(gf[2]); |
---|
498 | // the vertices of the Newton polytope are: |
---|
499 | gf[3][1]; |
---|
500 | // its dimension is |
---|
501 | gf[3][2]; |
---|
502 | // np[3] contains information how the vertices are connected to each other, |
---|
503 | // e.g. the 1st vertex is connected to the 2nd, 3rd and 4th vertex |
---|
504 | gf[3][3][1]; |
---|
505 | } |
---|
506 | |
---|
507 | |
---|
508 | |
---|
509 | /////////////////////////////////////////////////////////////////////////////// |
---|
510 | /// PROCEDURES USING TOPCOM |
---|
511 | /////////////////////////////////////////////////////////////////////////////// |
---|
512 | |
---|
513 | proc triangulations (list polygon,list #) |
---|
514 | "USAGE: triangulations(polygon[,#]); list polygon, list # |
---|
515 | ASSUME: polygon is a list of integer vectors of the same size representing |
---|
516 | the affine coordinates of the lattice points |
---|
517 | PURPOSE: the procedure considers the marked polytope given as the convex hull of |
---|
518 | the lattice points and with these lattice points as markings; it then |
---|
519 | computes all possible triangulations of this marked polytope |
---|
520 | RETURN: list, each entry corresponds to one triangulation and the ith entry is |
---|
521 | itself a list of integer vectors of size three, where each integer |
---|
522 | vector defines one triangle in the triangulation by telling which |
---|
523 | points of the input are the vertices of the triangle |
---|
524 | NOTE:- the procedure calls for its computations the program points2triangs |
---|
525 | from the program topcom by Joerg Rambau, Universitaet Bayreuth; it |
---|
526 | therefore is necessary that this program is installed in order to use |
---|
527 | this procedure; see |
---|
528 | @* http://www.uni-bayreuth.de/departments/wirtschaftsmathematik/rambau/TOPCOM |
---|
529 | @* - if you only want to have the regular triangulations the procedure should |
---|
530 | be called with the string 'regular' as optional argument |
---|
531 | @* - the procedure creates the files /tmp/triangulationsinput and |
---|
532 | /tmp/triangulationsoutput; |
---|
533 | the former is used as input for points2triangs and the latter is its |
---|
534 | output containing the triangulations of corresponding to points in the |
---|
535 | format of points2triangs; if you wish to use this for further |
---|
536 | computations with topcom, you have to call the procedure with the |
---|
537 | string 'keepfiles' as optional argument |
---|
538 | @* - note that an integer i in an integer vector representing a triangle |
---|
539 | refers to the ith lattice point, i.e. polygon[i]; this convention is |
---|
540 | different from TOPCOM's convention, where i would refer to the i-1st |
---|
541 | lattice point |
---|
542 | EXAMPLE: example triangulations; shows an example" |
---|
543 | { |
---|
544 | int i,j; |
---|
545 | // check for optional arguments |
---|
546 | int regular,keepfiles; |
---|
547 | if (size(#)>0) |
---|
548 | { |
---|
549 | for (i=1;i<=size(#);i++) |
---|
550 | { |
---|
551 | if (typeof(#[i])=="string") |
---|
552 | { |
---|
553 | if (#[i]=="keepfiles") |
---|
554 | { |
---|
555 | keepfiles=1; |
---|
556 | } |
---|
557 | if (#[i]=="regular") |
---|
558 | { |
---|
559 | regular=1; |
---|
560 | } |
---|
561 | } |
---|
562 | } |
---|
563 | } |
---|
564 | // prepare the input for points2triangs by writing the input polygon in the |
---|
565 | // necessary format |
---|
566 | string spi="["; |
---|
567 | for (i=1;i<=size(polygon);i++) |
---|
568 | { |
---|
569 | polygon[i][size(polygon[i])+1]=1; |
---|
570 | spi=spi+"["+string(polygon[i])+"]"; |
---|
571 | if (i<size(polygon)) |
---|
572 | { |
---|
573 | spi=spi+","; |
---|
574 | } |
---|
575 | } |
---|
576 | spi=spi+"]"; |
---|
577 | write(":w /tmp/triangulationsinput",spi); |
---|
578 | // call points2triangs |
---|
579 | if (regular==1) // compute only regular triangulations |
---|
580 | { |
---|
581 | system("sh","cd /tmp; points2triangs --regular < triangulationsinput > triangulationsoutput"); |
---|
582 | } |
---|
583 | else // compute all triangulations |
---|
584 | { |
---|
585 | system("sh","cd /tmp; points2triangs < triangulationsinput > triangulationsoutput"); |
---|
586 | } |
---|
587 | string p2t=read("/tmp/triangulationsoutput"); // takes result of points2triangs |
---|
588 | // delete the tmp-files, if no second argument is given |
---|
589 | if (keepfiles==0) |
---|
590 | { |
---|
591 | system("sh","cd /tmp; rm -f triangulationsinput; rm -f triangulationsoutput"); |
---|
592 | } |
---|
593 | // preprocessing of p2t if points2triangs is version >= 0.15 |
---|
594 | // brings p2t to the format of version 0.14 |
---|
595 | string np2t; // takes the triangulations in Singular format |
---|
596 | for (i=1;i<=size(p2t)-2;i++) |
---|
597 | { |
---|
598 | if ((p2t[i]==":") and (p2t[i+1]=="=") and (p2t[i+2]=="[")) |
---|
599 | { |
---|
600 | np2t=np2t+p2t[i]+p2t[i+1]; |
---|
601 | i=i+3; |
---|
602 | while (p2t[i]!=":") |
---|
603 | { |
---|
604 | i=i+1; |
---|
605 | } |
---|
606 | } |
---|
607 | else |
---|
608 | { |
---|
609 | if ((p2t[i]=="]") and (p2t[i+1]==";")) |
---|
610 | { |
---|
611 | np2t=np2t+p2t[i+1]; |
---|
612 | i=i+1; |
---|
613 | } |
---|
614 | else |
---|
615 | { |
---|
616 | np2t=np2t+p2t[i]; |
---|
617 | } |
---|
618 | } |
---|
619 | } |
---|
620 | if (p2t[size(p2t)-1]=="]") |
---|
621 | { |
---|
622 | np2t=np2t+p2t[size(p2t)]; |
---|
623 | } |
---|
624 | else |
---|
625 | { |
---|
626 | if (np2t[size(np2t)]!=";") |
---|
627 | { |
---|
628 | np2t=np2t+p2t[size(p2t)-1]+p2t[size(p2t)]; |
---|
629 | } |
---|
630 | } |
---|
631 | p2t=np2t; |
---|
632 | np2t=""; |
---|
633 | // transform the points2triangs output of version 0.14 into Singular format |
---|
634 | for (i=1;i<=size(p2t);i++) |
---|
635 | { |
---|
636 | if (p2t[i]=="=") |
---|
637 | { |
---|
638 | np2t=np2t+p2t[i]+"list("; |
---|
639 | i++; |
---|
640 | } |
---|
641 | else |
---|
642 | { |
---|
643 | if (p2t[i]!=":") |
---|
644 | { |
---|
645 | if ((p2t[i]=="}") and (p2t[i+1]=="}")) |
---|
646 | { |
---|
647 | np2t=np2t+"))"; |
---|
648 | i++; |
---|
649 | } |
---|
650 | else |
---|
651 | { |
---|
652 | if (p2t[i]=="{") |
---|
653 | { |
---|
654 | np2t=np2t+"intvec("; |
---|
655 | } |
---|
656 | else |
---|
657 | { |
---|
658 | if (p2t[i]=="}") |
---|
659 | { |
---|
660 | np2t=np2t+")"; |
---|
661 | } |
---|
662 | else |
---|
663 | { |
---|
664 | if (p2t[i]=="[") |
---|
665 | { |
---|
666 | // in Topcom version 17.4 (and maybe also in earlier versions) the list |
---|
667 | // of triangulations is indexed starting with index 0, in Singular |
---|
668 | // we have to start with index 1 |
---|
669 | np2t=np2t+p2t[i]+"1+"; |
---|
670 | } |
---|
671 | else |
---|
672 | { |
---|
673 | np2t=np2t+p2t[i]; |
---|
674 | } |
---|
675 | } |
---|
676 | } |
---|
677 | } |
---|
678 | } |
---|
679 | } |
---|
680 | } |
---|
681 | list T; |
---|
682 | execute(np2t); |
---|
683 | // depending on the version of Topcom, the list T has or has not an entry T[1] |
---|
684 | // if it has none, the entry should be removed |
---|
685 | while (typeof(T[1])=="none") |
---|
686 | { |
---|
687 | T=delete(T,1); |
---|
688 | } |
---|
689 | // raise each index by one |
---|
690 | for (i=1;i<=size(T);i++) |
---|
691 | { |
---|
692 | for (j=1;j<=size(T[i]);j++) |
---|
693 | { |
---|
694 | T[i][j]=T[i][j]+1; |
---|
695 | } |
---|
696 | } |
---|
697 | return(T); |
---|
698 | } |
---|
699 | example |
---|
700 | { |
---|
701 | "EXAMPLE:"; |
---|
702 | echo=2; |
---|
703 | // the lattice points of the unit square in the plane |
---|
704 | list polygon=intvec(0,0),intvec(0,1),intvec(1,0),intvec(1,1); |
---|
705 | // the triangulations of this lattice point configuration are computed |
---|
706 | list triang=triangulations(polygon); |
---|
707 | triang; |
---|
708 | } |
---|
709 | |
---|
710 | ///////////////////////////////////////////////////////////////////////////// |
---|
711 | |
---|
712 | proc secondaryPolytope (list polygon,list #) |
---|
713 | "USAGE: secondaryPolytope(polygon[,#]); list polygon, list # |
---|
714 | ASSUME: - polygon is a list of integer vectors of the same size representing |
---|
715 | the affine coordinates of lattice points |
---|
716 | @* - if the triangulations of the corresponding polygon have already been |
---|
717 | computed with the procedure triangulations then these can be given as |
---|
718 | a second (optional) argument in order to avoid doing this computation |
---|
719 | again |
---|
720 | PURPOSE: the procedure considers the marked polytope given as the convex hull of |
---|
721 | the lattice points and with these lattice points as markings; it then |
---|
722 | computes the lattice points of the secondary polytope given by this |
---|
723 | marked polytope which correspond to the triangulations computed by |
---|
724 | the procedure triangulations |
---|
725 | RETURN: list, say L, such that: |
---|
726 | @* L[1] = intmat, each row gives the affine coordinates of a lattice |
---|
727 | point in the secondary polytope given by the marked |
---|
728 | polytope corresponding to polygon |
---|
729 | @* L[2] = the list of corresponding triangulations |
---|
730 | NOTE: if the triangluations are not handed over as optional argument the |
---|
731 | procedure calls for its computation of these triangulations the program |
---|
732 | points2triangs from the program topcom by Joerg Rambau, Universitaet |
---|
733 | Bayreuth; it therefore is necessary that this program is installed in |
---|
734 | order to use this procedure; see |
---|
735 | @* http://www.uni-bayreuth.de/departments/wirtschaftsmathematik/rambau/TOPCOM |
---|
736 | EXAMPLE: example secondaryPolytope; shows an example" |
---|
737 | { |
---|
738 | // compute the triangulations of the point configuration with points2triangs |
---|
739 | if (size(#)==0) |
---|
740 | { |
---|
741 | list triangs=triangulations(polygon); |
---|
742 | } |
---|
743 | else |
---|
744 | { |
---|
745 | list triangs=#; |
---|
746 | } |
---|
747 | int i,j,k,l; |
---|
748 | intmat N[2][2]; // is used to compute areas of triangles |
---|
749 | intvec vertex; // stores a point in the secondary polytope as |
---|
750 | // intermediate result |
---|
751 | int eintrag; |
---|
752 | int halt; |
---|
753 | intmat secpoly[size(triangs)][size(polygon)]; // stores all lattice points |
---|
754 | // of the secondary polytope |
---|
755 | // consider each triangulation and compute the corresponding point |
---|
756 | // in the secondary polytope |
---|
757 | for (i=1;i<=size(triangs);i++) |
---|
758 | { |
---|
759 | // for each triangulation we have to compute the coordinates |
---|
760 | // corresponding to each marked point |
---|
761 | for (j=1;j<=size(polygon);j++) |
---|
762 | { |
---|
763 | eintrag=0; |
---|
764 | // for each marked point we have to consider all triangles in the |
---|
765 | // triangulation which involve this particular point |
---|
766 | for (k=1;k<=size(triangs[i]);k++) |
---|
767 | { |
---|
768 | halt=0; |
---|
769 | for (l=1;(l<=3) and (halt==0);l++) |
---|
770 | { |
---|
771 | if (triangs[i][k][l]==j) |
---|
772 | { |
---|
773 | halt=1; |
---|
774 | N[1,1]=polygon[triangs[i][k][3]][1]-polygon[triangs[i][k][1]][1]; |
---|
775 | N[1,2]=polygon[triangs[i][k][2]][1]-polygon[triangs[i][k][1]][1]; |
---|
776 | N[2,1]=polygon[triangs[i][k][3]][2]-polygon[triangs[i][k][1]][2]; |
---|
777 | N[2,2]=polygon[triangs[i][k][2]][2]-polygon[triangs[i][k][1]][2]; |
---|
778 | eintrag=eintrag+abs(det(N)); |
---|
779 | } |
---|
780 | } |
---|
781 | } |
---|
782 | vertex[j]=eintrag; |
---|
783 | } |
---|
784 | secpoly[i,1..size(polygon)]=vertex; |
---|
785 | } |
---|
786 | return(list(secpoly,triangs)); |
---|
787 | } |
---|
788 | example |
---|
789 | { |
---|
790 | "EXAMPLE:"; |
---|
791 | echo=2; |
---|
792 | // the lattice points of the unit square in the plane |
---|
793 | list polygon=intvec(0,0),intvec(0,1),intvec(1,0),intvec(1,1); |
---|
794 | // the secondary polytope of this lattice point configuration is computed |
---|
795 | list secpoly=secondaryPolytope(polygon); |
---|
796 | // the points in the secondary polytope |
---|
797 | print(secpoly[1]); |
---|
798 | // the corresponding triangulations |
---|
799 | secpoly[2]; |
---|
800 | } |
---|
801 | |
---|
802 | /////////////////////////////////////////////////////////////////////////////// |
---|
803 | /// PROCEDURES USING POLYMAKE AND TOPCOM |
---|
804 | /////////////////////////////////////////////////////////////////////////////// |
---|
805 | |
---|
806 | proc secondaryFan (list polygon,list #) |
---|
807 | "USAGE: secondaryFan(polygon[,#]); list polygon, list # |
---|
808 | ASSUME: - polygon is a list of integer vectors of the same size representing |
---|
809 | the affine coordinates of lattice points |
---|
810 | @* - if the triangulations of the corresponding polygon have already been |
---|
811 | computed with the procedure triangulations then these can be given |
---|
812 | as a second (optional) argument in order to avoid doing this |
---|
813 | computation again |
---|
814 | PURPOSE: the procedure considers the marked polytope given as the convex hull of |
---|
815 | the lattice points and with these lattice points as markings; it then |
---|
816 | computes the lattice points of the secondary polytope given by this |
---|
817 | marked polytope which correspond to the triangulations computed by |
---|
818 | the procedure triangulations |
---|
819 | RETURN: list, the ith entry of L[1] contains information about the ith cone in |
---|
820 | the secondary fan of the polygon, i.e. the cone dual to the |
---|
821 | ith vertex of the secondary polytope |
---|
822 | @* L[1][i][1] = integer matrix representing the inequalities which |
---|
823 | describe the cone dual to the ith vertex |
---|
824 | @* L[1][i][2] = a list which contains the inequalities represented |
---|
825 | by L[1][i][1] as a list of strings, where we use the |
---|
826 | variables x(1),...,x(n) |
---|
827 | @* L[1][i][3] = only present if 'er' is set to 1; in that case it is |
---|
828 | an interger matrix whose rows are the extreme rays |
---|
829 | of the cone |
---|
830 | @* L[2] = is an integer matrix whose rows span the linearity space |
---|
831 | of the fan, i.e. the linear space which is contained in |
---|
832 | each cone |
---|
833 | @* L[3] = the secondary polytope in the format of the procedure |
---|
834 | polymakePolytope |
---|
835 | @* L[4] = the list of triangulations corresponding to the vertices |
---|
836 | of the secondary polytope |
---|
837 | NOTE:- the procedure calls for its computation polymake by Ewgenij Gawrilow, |
---|
838 | TU Berlin and Michael Joswig, so it only works if polymake is installed; |
---|
839 | see http://www.math.tu-berlin.de/polymake/ |
---|
840 | @* - in the optional argument # it is possible to hand over other names for |
---|
841 | the variables to be used -- be careful, the format must be correct and |
---|
842 | that is not tested, e.g. if you want the variable names to be |
---|
843 | u00,u10,u01,u11 then you must hand over the string 'u11,u10,u01,u11' |
---|
844 | @* - if the triangluations are not handed over as optional argument the |
---|
845 | procedure calls for its computation of these triangulations the program |
---|
846 | points2triangs from the program topcom by Joerg Rambau, Universitaet |
---|
847 | Bayreuth; it therefore is necessary that this program is installed in |
---|
848 | order to use this procedure; see |
---|
849 | @* http://www.uni-bayreuth.de/departments/wirtschaftsmathematik/rambau/TOPCOM |
---|
850 | EXAMPLE: example secondaryFan; shows an example" |
---|
851 | { |
---|
852 | if (size(#)==0) |
---|
853 | { |
---|
854 | list triang=triangulations(polygon); |
---|
855 | } |
---|
856 | else |
---|
857 | { |
---|
858 | list triang=#[1]; |
---|
859 | } |
---|
860 | list sp=secondaryPolytope(polygon,triang); |
---|
861 | list spp=polymakePolytope(sp[1]); |
---|
862 | list sf=normalFan(spp[1],spp[4],spp[3],1); |
---|
863 | return(list(sf[1],sf[2],spp,triang)); |
---|
864 | } |
---|
865 | example |
---|
866 | { |
---|
867 | "EXAMPLE:"; |
---|
868 | echo=2; |
---|
869 | // the lattice points of the unit square in the plane |
---|
870 | list polygon=intvec(0,0),intvec(0,1),intvec(1,0),intvec(1,1); |
---|
871 | // the secondary polytope of this lattice point configuration is computed |
---|
872 | list secfan=secondaryFan(polygon); |
---|
873 | // the number of cones in the secondary fan of the polygon |
---|
874 | size(secfan[1]); |
---|
875 | // the inequalities of the first cone as matrix are: |
---|
876 | print(secfan[1][1][1]); |
---|
877 | // the inequalities of the first cone as string are: |
---|
878 | print(secfan[1][1][2]); |
---|
879 | // the rows of the following matrix are the extreme rays of the first cone: |
---|
880 | print(secfan[1][1][3]); |
---|
881 | // each cone contains the linearity space spanned by: |
---|
882 | print(secfan[2]); |
---|
883 | // the points in the secondary polytope |
---|
884 | print(secfan[3][1]); |
---|
885 | // the corresponding triangulations |
---|
886 | secfan[4]; |
---|
887 | } |
---|
888 | |
---|
889 | |
---|
890 | //////////////////////////////////////////////////////////////////////////////// |
---|
891 | /// PROCEDURES CONCERNED WITH PLANAR POLYGONS |
---|
892 | //////////////////////////////////////////////////////////////////////////////// |
---|
893 | |
---|
894 | proc cycleLength (list boundary,intvec interior) |
---|
895 | "USAGE: cycleLength(boundary,interior); list boundary, intvec interior |
---|
896 | ASSUME: boundary is a list of integer vectors describing a cycle in some |
---|
897 | convex lattice polygon around the lattice point interior ordered |
---|
898 | clock wise |
---|
899 | RETURN: string, the cycle length of the corresponding cycle in the dual |
---|
900 | tropical curve |
---|
901 | EXAMPLE: example cycleLength; shows an example" |
---|
902 | { |
---|
903 | int j; |
---|
904 | // create a ring whose variables are indexed by the points in |
---|
905 | // boundary resp. by interior |
---|
906 | string rst="ring cyclering=0,(u"+string(interior[1])+string(interior[2]); |
---|
907 | for (j=1;j<=size(boundary);j++) |
---|
908 | { |
---|
909 | rst=rst+",u"+string(boundary[j][1])+string(boundary[j][2]); |
---|
910 | } |
---|
911 | rst=rst+"),lp;"; |
---|
912 | execute(rst); |
---|
913 | // add the first and second point at the end of boundary |
---|
914 | boundary[size(boundary)+1]=boundary[1]; |
---|
915 | boundary[size(boundary)+1]=boundary[2]; |
---|
916 | poly cl,summand; // takes the cycle length |
---|
917 | matrix N1[2][2]; // used to compute the area of a triangle |
---|
918 | matrix N2[2][2]; // used to compute the area of a triangle |
---|
919 | matrix N3[2][2]; // used to compute the area of a triangle |
---|
920 | // for each original point in boundary compute its contribution to the cycle |
---|
921 | for (j=2;j<=size(boundary)-1;j++) |
---|
922 | { |
---|
923 | N1=boundary[j-1]-interior,boundary[j]-interior; |
---|
924 | N2=boundary[j]-interior,boundary[j+1]-interior; |
---|
925 | N3=boundary[j+1]-interior,boundary[j-1]-interior; |
---|
926 | execute("summand=-u"+string(boundary[j][1])+string(boundary[j][2])+"+u"+string(interior[1])+string(interior[2])+";"); |
---|
927 | summand=summand*(det(N1)+det(N2)+det(N3))/(det(N1)*det(N2)); |
---|
928 | cl=cl+summand; |
---|
929 | } |
---|
930 | return(string(cl)); |
---|
931 | } |
---|
932 | example |
---|
933 | { |
---|
934 | "EXAMPLE:"; |
---|
935 | echo=2; |
---|
936 | // the integer vectors in boundary are lattice points on the boundary |
---|
937 | // of a convex lattice polygon in the plane |
---|
938 | list boundary=intvec(0,0),intvec(0,1),intvec(0,2),intvec(2,2), |
---|
939 | intvec(2,1),intvec(2,0); |
---|
940 | // interior is a lattice point in the interior of this lattice polygon |
---|
941 | intvec interior=1,1; |
---|
942 | // compute the general cycle length of a cycle of the corresponding cycle |
---|
943 | // in the dual tropical curve, note that (0,1) and (2,1) do not contribute |
---|
944 | cycleLength(boundary,interior); |
---|
945 | } |
---|
946 | |
---|
947 | ///////////////////////////////////////////////////////////////////////////// |
---|
948 | |
---|
949 | proc splitPolygon (list markings) |
---|
950 | "USAGE: splitPolygon (markings); markings list |
---|
951 | ASSUME: markings is a list of integer vectors representing lattice points in |
---|
952 | the plane which we consider as the marked points of the convex lattice |
---|
953 | polytope spanned by them |
---|
954 | PURPOSE: split the marked points in the vertices, the points on the facets |
---|
955 | which are not vertices, and the interior points |
---|
956 | RETURN: list, L consisting of three lists |
---|
957 | @* L[1] : represents the vertices the polygon ordered clockwise |
---|
958 | @* L[1][i][1] = intvec, the coordinates of the ith vertex |
---|
959 | @* L[1][i][2] = int, the position of L[1][i][1] in markings |
---|
960 | @* L[2][i] : represents the lattice points on the facet of the |
---|
961 | polygon with endpoints L[1][i] and L[1][i+1] |
---|
962 | (i considered modulo size(L[1])) |
---|
963 | @* L[2][i][j][1] = intvec, the coordinates of the jth |
---|
964 | lattice point on that facet |
---|
965 | @* L[2][i][j][2] = int, the position of L[2][i][j][1] |
---|
966 | in markings |
---|
967 | @* L[3] : represents the interior lattice points of the polygon |
---|
968 | @* L[3][i][1] = intvec, coordinates of ith interior point |
---|
969 | @* L[3][i][2] = int, the position of L[3][i][1] in markings |
---|
970 | EXAMPLE: example splitPolygon; shows an example" |
---|
971 | { |
---|
972 | list vert; // stores the result |
---|
973 | // compute the boundary of the polygon in an oriented way |
---|
974 | list pb=findOrientedBoundary(markings); |
---|
975 | // the vertices are just the second entry of pb |
---|
976 | vert[1]=pb[2]; |
---|
977 | int i,j,k; // indices |
---|
978 | list boundary; // stores the points on the facets of the |
---|
979 | // polygon which are not vertices |
---|
980 | // append to the boundary points as well as to the vertices |
---|
981 | // the first vertex a second time |
---|
982 | pb[1]=pb[1]+list(pb[1][1]); |
---|
983 | pb[2]=pb[2]+list(pb[2][1]); |
---|
984 | // for each vertex find all points on the facet of the polygon with this vertex |
---|
985 | // and the next vertex as endpoints |
---|
986 | int z=2; |
---|
987 | for (i=1;i<=size(vert[1]);i++) |
---|
988 | { |
---|
989 | j=1; |
---|
990 | list facet; // stores the points on this facet which are not vertices |
---|
991 | // while the next vertex is not reached, store the boundary lattice point |
---|
992 | while (pb[1][z]!=pb[2][i+1]) |
---|
993 | { |
---|
994 | facet[j]=pb[1][z]; |
---|
995 | j++; |
---|
996 | z++; |
---|
997 | } |
---|
998 | // store the points on the ith facet as boundary[i] |
---|
999 | boundary[i]=facet; |
---|
1000 | kill facet; |
---|
1001 | z++; |
---|
1002 | } |
---|
1003 | // store the information on the boundary in vert[2] |
---|
1004 | vert[2]=boundary; |
---|
1005 | // find the remaining points in the input which are not on |
---|
1006 | // the boundary by checking |
---|
1007 | // for each point in markings if it is contained in pb[1] |
---|
1008 | list interior=markings; |
---|
1009 | for (i=size(interior);i>=1;i--) |
---|
1010 | { |
---|
1011 | for (j=1;j<=size(pb[1])-1;j++) |
---|
1012 | { |
---|
1013 | if (interior[i]==pb[1][j]) |
---|
1014 | { |
---|
1015 | interior=delete(interior,i); |
---|
1016 | j=size(pb[1]); |
---|
1017 | } |
---|
1018 | } |
---|
1019 | } |
---|
1020 | // store the interior points in vert[3] |
---|
1021 | vert[3]=interior; |
---|
1022 | // add to each point in vert the index which it gets from |
---|
1023 | // its position in the input markings; |
---|
1024 | // do so for ver[1] |
---|
1025 | for (i=1;i<=size(vert[1]);i++) |
---|
1026 | { |
---|
1027 | j=1; |
---|
1028 | while (markings[j]!=vert[1][i]) |
---|
1029 | { |
---|
1030 | j++; |
---|
1031 | } |
---|
1032 | vert[1][i]=list(vert[1][i],j); |
---|
1033 | } |
---|
1034 | // do so for ver[2] |
---|
1035 | for (i=1;i<=size(vert[2]);i++) |
---|
1036 | { |
---|
1037 | for (k=1;k<=size(vert[2][i]);k++) |
---|
1038 | { |
---|
1039 | j=1; |
---|
1040 | while (markings[j]!=vert[2][i][k]) |
---|
1041 | { |
---|
1042 | j++; |
---|
1043 | } |
---|
1044 | vert[2][i][k]=list(vert[2][i][k],j); |
---|
1045 | } |
---|
1046 | } |
---|
1047 | // do so for ver[3] |
---|
1048 | for (i=1;i<=size(vert[3]);i++) |
---|
1049 | { |
---|
1050 | j=1; |
---|
1051 | while (markings[j]!=vert[3][i]) |
---|
1052 | { |
---|
1053 | j++; |
---|
1054 | } |
---|
1055 | vert[3][i]=list(vert[3][i],j); |
---|
1056 | } |
---|
1057 | return(vert); |
---|
1058 | } |
---|
1059 | example |
---|
1060 | { |
---|
1061 | "EXAMPLE:"; |
---|
1062 | echo=2; |
---|
1063 | // the lattice polygon spanned by the points (0,0), (3,0) and (0,3) |
---|
1064 | // with all integer points as markings |
---|
1065 | list polygon=intvec(1,1),intvec(3,0),intvec(2,0),intvec(1,0), |
---|
1066 | intvec(0,0),intvec(2,1),intvec(0,1),intvec(1,2), |
---|
1067 | intvec(0,2),intvec(0,3); |
---|
1068 | // split the polygon in its vertices, its facets and its interior points |
---|
1069 | list sp=splitPolygon(polygon); |
---|
1070 | // the vertices |
---|
1071 | sp[1]; |
---|
1072 | // the points on facets which are not vertices |
---|
1073 | sp[2]; |
---|
1074 | // the interior points |
---|
1075 | sp[3]; |
---|
1076 | } |
---|
1077 | |
---|
1078 | |
---|
1079 | ///////////////////////////////////////////////////////////////////////////// |
---|
1080 | |
---|
1081 | proc eta (list triang,list polygon) |
---|
1082 | "USAGE: eta(triang,polygon); triang, polygon list |
---|
1083 | ASSUME: polygon has the format of the output of splitPolygon, i.e. it is a |
---|
1084 | list with three entries describing a convex lattice polygon in the |
---|
1085 | following way: |
---|
1086 | @* polygon[1] : is a list of lists; for each i the entry polygon[1][i][1] |
---|
1087 | is a lattice point which is a vertex of the lattice |
---|
1088 | polygon, and polygon[1][i][2] is an integer assigned to |
---|
1089 | this lattice point as identifying index |
---|
1090 | @* polygon[2] : is a list of lists; for each vertex of the polygon, |
---|
1091 | i.e. for each entry in polygon[1], it contains a list |
---|
1092 | polygon[2][i], which contains the lattice points on the |
---|
1093 | facet with endpoints polygon[1][i] and polygon[1][i+1] |
---|
1094 | - i considered mod size(polygon[1]); |
---|
1095 | each such lattice point contributes an entry |
---|
1096 | polygon[2][i][j][1] which is an integer |
---|
1097 | vector giving the coordinate of the lattice point and an |
---|
1098 | entry polygon[2][i][j][2] which is the identifying index |
---|
1099 | @* polygon[3] : is a list of lists, where each entry corresponds to a |
---|
1100 | lattice point in the interior of the polygon, with |
---|
1101 | polygon[3][j][1] being the coordinates of the point |
---|
1102 | and polygon[3][j][2] being the identifying index; |
---|
1103 | @* triang is a list of integer vectors all of size three describing a |
---|
1104 | triangulation of the polygon described by polygon; if an entry of |
---|
1105 | triang is the vector (i,j,k) then the triangle is built by the vertices |
---|
1106 | with indices i, j and k |
---|
1107 | RETURN: intvec, the integer vector eta describing that vertex of the Newton |
---|
1108 | polytope discriminant of the polygone whose dual cone in the |
---|
1109 | Groebner fan contains the cone of the secondary fan of the |
---|
1110 | polygon corresponding to the given triangulation |
---|
1111 | NOTE: for a better description of eta see Gelfand, Kapranov, |
---|
1112 | Zelevinski: Discriminants, Resultants and multidimensional Determinants. |
---|
1113 | Chapter 10. |
---|
1114 | EXAMPLE: example eta; shows an example" |
---|
1115 | { |
---|
1116 | int i,j,k,l,m,n; // index variables |
---|
1117 | list ordpolygon; // stores the lattice points in the order |
---|
1118 | // used in the triangulation |
---|
1119 | list triangarea; // stores the areas of the triangulations |
---|
1120 | intmat N[2][2]; // used to compute triangle areas |
---|
1121 | // 1) store the lattice points in the order used in the triangulation |
---|
1122 | // go first through all vertices of the polytope |
---|
1123 | for (j=1;j<=size(polygon[1]);j++) |
---|
1124 | { |
---|
1125 | ordpolygon[polygon[1][j][2]]=polygon[1][j][1]; |
---|
1126 | } |
---|
1127 | // then consider all inner points |
---|
1128 | for (j=1;j<=size(polygon[3]);j++) |
---|
1129 | { |
---|
1130 | ordpolygon[polygon[3][j][2]]=polygon[3][j][1]; |
---|
1131 | } |
---|
1132 | // finally consider all lattice points on the boundary which are not vertices |
---|
1133 | for (j=1;j<=size(polygon[2]);j++) |
---|
1134 | { |
---|
1135 | for (i=1;i<=size(polygon[2][j]);i++) |
---|
1136 | { |
---|
1137 | ordpolygon[polygon[2][j][i][2]]=polygon[2][j][i][1]; |
---|
1138 | } |
---|
1139 | } |
---|
1140 | // 2) compute for each triangle in the triangulation the area of the triangle |
---|
1141 | for (i=1;i<=size(triang);i++) |
---|
1142 | { |
---|
1143 | // Note that the ith lattice point in orderedpolygon has the |
---|
1144 | // number i-1 in the triangulation! |
---|
1145 | N=ordpolygon[triang[i][1]]-ordpolygon[triang[i][2]],ordpolygon[triang[i][1]]-ordpolygon[triang[i][3]]; |
---|
1146 | triangarea[i]=abs(det(N)); |
---|
1147 | } |
---|
1148 | intvec ETA; // stores the eta_ij |
---|
1149 | int etaij; // stores the part of eta_ij during computations |
---|
1150 | // which comes from triangle areas |
---|
1151 | int seitenlaenge; // stores the part of eta_ij during computations |
---|
1152 | // which comes from boundary facets |
---|
1153 | list seiten; // stores the lattice points on facets of the polygon |
---|
1154 | intvec v; // used to compute a facet length |
---|
1155 | // 3) store first in seiten[i] all lattice points on the facet |
---|
1156 | // connecting the ith vertex, |
---|
1157 | // i.e. polygon[1][i], with the i+1st vertex, i.e. polygon[1][i+1], |
---|
1158 | // where we replace i+1 |
---|
1159 | // 1 if i=size(polygon[1]); |
---|
1160 | // then append the last entry of seiten once more at the very |
---|
1161 | // beginning of seiten, so |
---|
1162 | // that the index is shifted by one |
---|
1163 | for (i=1;i<=size(polygon[1]);i++) |
---|
1164 | { |
---|
1165 | if (i<size(polygon[1])) |
---|
1166 | { |
---|
1167 | seiten[i]=list(polygon[1][i])+polygon[2][i]+list(polygon[1][i+1]); |
---|
1168 | } |
---|
1169 | else |
---|
1170 | { |
---|
1171 | seiten[i]=list(polygon[1][i])+polygon[2][i]+list(polygon[1][1]); |
---|
1172 | } |
---|
1173 | } |
---|
1174 | seiten=insert(seiten,seiten[size(seiten)],0); |
---|
1175 | // 4) compute the eta_ij for all vertices of the polygon |
---|
1176 | for (j=1;j<=size(polygon[1]);j++) |
---|
1177 | { |
---|
1178 | // the vertex itself contributes a 1 |
---|
1179 | etaij=1; |
---|
1180 | // check for each triangle in the triangulation ... |
---|
1181 | for (k=1;k<=size(triang);k++) |
---|
1182 | { |
---|
1183 | // ... if the vertex is actually a vertex of the triangle ... |
---|
1184 | if ((polygon[1][j][2]==triang[k][1]) or (polygon[1][j][2]==triang[k][2]) or (polygon[1][j][2]==triang[k][3])) |
---|
1185 | { |
---|
1186 | // ... if so, add the area of the triangle to etaij |
---|
1187 | etaij=etaij+triangarea[k]; |
---|
1188 | // then check if that triangle has a facet which is contained |
---|
1189 | // in one of the |
---|
1190 | // two facets of the polygon which are adjecent to the given vertex ... |
---|
1191 | // these two facets are seiten[j] and seiten[j+1] |
---|
1192 | for (n=j;n<=j+1;n++) |
---|
1193 | { |
---|
1194 | // check for each lattice point in the facet of the polygon ... |
---|
1195 | for (l=1;l<=size(seiten[n]);l++) |
---|
1196 | { |
---|
1197 | // ... and for each lattice point in the triangle ... |
---|
1198 | for (m=1;m<=size(triang[k]);m++) |
---|
1199 | { |
---|
1200 | // ... if they coincide and are not the vertex itself ... |
---|
1201 | if ((seiten[n][l][2]==triang[k][m]) and (seiten[n][l][2]!=polygon[1][j][2])) |
---|
1202 | { |
---|
1203 | // if so, then compute the vector pointing from this |
---|
1204 | // lattice point to the vertex |
---|
1205 | v=polygon[1][j][1]-seiten[n][l][1]; |
---|
1206 | // and the lattice length of this vector has to be |
---|
1207 | // subtracted from etaij |
---|
1208 | etaij=etaij-abs(gcd(v[1],v[2])); |
---|
1209 | } |
---|
1210 | } |
---|
1211 | } |
---|
1212 | } |
---|
1213 | } |
---|
1214 | } |
---|
1215 | // store etaij in the list |
---|
1216 | ETA[polygon[1][j][2]]=etaij; |
---|
1217 | } |
---|
1218 | // 5) compute the eta_ij for all lattice points on the facets |
---|
1219 | // of the polygon which are not vertices, these are the |
---|
1220 | // lattice points in polygon[2][1] to polygon[2][size(polygon[1])] |
---|
1221 | for (i=1;i<=size(polygon[2]);i++) |
---|
1222 | { |
---|
1223 | for (j=1;j<=size(polygon[2][i]);j++) |
---|
1224 | { |
---|
1225 | // initialise etaij |
---|
1226 | etaij=0; |
---|
1227 | // initialise seitenlaenge |
---|
1228 | seitenlaenge=0; |
---|
1229 | // check for each triangle in the triangulation ... |
---|
1230 | for (k=1;k<=size(triang);k++) |
---|
1231 | { |
---|
1232 | // ... if the vertex is actually a vertex of the triangle ... |
---|
1233 | 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])) |
---|
1234 | { |
---|
1235 | // ... if so, add the area of the triangle to etaij |
---|
1236 | etaij=etaij+triangarea[k]; |
---|
1237 | // then check if that triangle has a facet which is contained in the |
---|
1238 | // facet of the polygon which contains the lattice point in question, |
---|
1239 | // this is the facet seiten[i+1]; |
---|
1240 | // check for each lattice point in the facet of the polygon ... |
---|
1241 | for (l=1;l<=size(seiten[i+1]);l++) |
---|
1242 | { |
---|
1243 | // ... and for each lattice point in the triangle ... |
---|
1244 | for (m=1;m<=size(triang[k]);m++) |
---|
1245 | { |
---|
1246 | // ... if they coincide and are not the vertex itself ... |
---|
1247 | if ((seiten[i+1][l][2]==triang[k][m]) and (seiten[i+1][l][2]!=polygon[2][i][j][2])) |
---|
1248 | { |
---|
1249 | // if so, then compute the vector pointing from |
---|
1250 | // this lattice point to the vertex |
---|
1251 | v=polygon[2][i][j][1]-seiten[i+1][l][1]; |
---|
1252 | // and the lattice length of this vector contributes |
---|
1253 | // to seitenlaenge |
---|
1254 | seitenlaenge=seitenlaenge+abs(gcd(v[1],v[2])); |
---|
1255 | } |
---|
1256 | } |
---|
1257 | } |
---|
1258 | } |
---|
1259 | } |
---|
1260 | // if the lattice point was a vertex of any triangle |
---|
1261 | // in the triangulation ... |
---|
1262 | if (etaij!=0) |
---|
1263 | { |
---|
1264 | // then eta_ij is the sum of the triangle areas minus seitenlaenge |
---|
1265 | ETA[polygon[2][i][j][2]]=etaij-seitenlaenge; |
---|
1266 | } |
---|
1267 | else |
---|
1268 | { |
---|
1269 | // otherwise it is just zero |
---|
1270 | ETA[polygon[2][i][j][2]]=0; |
---|
1271 | } |
---|
1272 | } |
---|
1273 | } |
---|
1274 | // 4) compute the eta_ij for all inner lattice points of the polygon |
---|
1275 | for (j=1;j<=size(polygon[3]);j++) |
---|
1276 | { |
---|
1277 | // initialise etaij |
---|
1278 | etaij=0; |
---|
1279 | // check for each triangle in the triangulation ... |
---|
1280 | for (k=1;k<=size(triang);k++) |
---|
1281 | { |
---|
1282 | // ... if the vertex is actually a vertex of the triangle ... |
---|
1283 | if ((polygon[3][j][2]==triang[k][1]) or (polygon[3][j][2]==triang[k][2]) or (polygon[3][j][2]==triang[k][3])) |
---|
1284 | { |
---|
1285 | // ... if so, add the area of the triangle to etaij |
---|
1286 | etaij=etaij+triangarea[k]; |
---|
1287 | } |
---|
1288 | } |
---|
1289 | // store etaij in ETA |
---|
1290 | ETA[polygon[3][j][2]]=etaij; |
---|
1291 | } |
---|
1292 | return(ETA); |
---|
1293 | } |
---|
1294 | example |
---|
1295 | { |
---|
1296 | "EXAMPLE:"; |
---|
1297 | echo=2; |
---|
1298 | // the lattice polygon spanned by the points (0,0), (3,0) and (0,3) |
---|
1299 | // with all integer points as markings |
---|
1300 | list polygon=intvec(1,1),intvec(3,0),intvec(2,0),intvec(1,0), |
---|
1301 | intvec(0,0),intvec(2,1),intvec(0,1),intvec(1,2), |
---|
1302 | intvec(0,2),intvec(0,3); |
---|
1303 | // split the polygon in its vertices, its facets and its interior points |
---|
1304 | list sp=splitPolygon(polygon); |
---|
1305 | // define a triangulation by connecting the only interior point |
---|
1306 | // with the vertices |
---|
1307 | list triang=intvec(1,2,5),intvec(1,5,10),intvec(1,5,10); |
---|
1308 | // compute the eta-vector of this triangulation |
---|
1309 | eta(triang,sp); |
---|
1310 | } |
---|
1311 | |
---|
1312 | ///////////////////////////////////////////////////////////////////////////// |
---|
1313 | |
---|
1314 | proc findOrientedBoundary (list polygon) |
---|
1315 | "USAGE: findOrientedBoundary(polygon); polygon list |
---|
1316 | ASSUME: polygon is a list of integer vectors defining integer lattice points |
---|
1317 | in the plane |
---|
1318 | RETURN: list l with the following interpretation |
---|
1319 | @* l[1] = list of integer vectors such that the polygonal path |
---|
1320 | defined by these is the boundary of the convex hull of |
---|
1321 | the lattice points in polygon |
---|
1322 | @* l[2] = list, the redundant points in l[1] have been removed |
---|
1323 | EXAMPLE: example findOrientedBoundary; shows an example" |
---|
1324 | { |
---|
1325 | // Order the vertices such that passing from one to the next we travel along |
---|
1326 | // the boundary of the convex hull of the vertices clock wise |
---|
1327 | int d,k,i,j; |
---|
1328 | intmat D[2][2]; |
---|
1329 | ///////////////////////////////////// |
---|
1330 | // Treat first the pathological cases that the polygon is not two-dimensional: |
---|
1331 | ///////////////////////////////////// |
---|
1332 | // if the polygon is empty or only one point or a line segment of two points |
---|
1333 | if (size(polygon)<=2) |
---|
1334 | { |
---|
1335 | return(list(polygon,polygon)); |
---|
1336 | } |
---|
1337 | // check is the polygon is only a line segment given by more than two points; |
---|
1338 | // for this first compute sum of the absolute values of the determinants |
---|
1339 | // of the matrices whose |
---|
1340 | // rows are the vectors pointing from the first to the second point |
---|
1341 | // and from the |
---|
1342 | // the first point to the ith point for i=3,...,size(polygon); |
---|
1343 | // if this sum is zero |
---|
1344 | // then the polygon is a line segment and we have to find its end points |
---|
1345 | d=0; |
---|
1346 | for (i=3;i<=size(polygon);i++) |
---|
1347 | { |
---|
1348 | D=polygon[2]-polygon[1],polygon[i]-polygon[1]; |
---|
1349 | d=d+abs(det(D)); |
---|
1350 | } |
---|
1351 | if (d==0) // then polygon is a line segment |
---|
1352 | { |
---|
1353 | intmat laenge[size(polygon)][size(polygon)]; |
---|
1354 | intvec mp; |
---|
1355 | // for this collect first all vectors pointing from one lattice |
---|
1356 | // point to the next, |
---|
1357 | // compute their pairwise angles and their lengths |
---|
1358 | for (i=1;i<=size(polygon)-1;i++) |
---|
1359 | { |
---|
1360 | for (j=i+1;j<=size(polygon);j++) |
---|
1361 | { |
---|
1362 | mp=polygon[i]-polygon[j]; |
---|
1363 | laenge[i,j]=abs(gcd(mp[1],mp[2])); |
---|
1364 | } |
---|
1365 | } |
---|
1366 | mp=maxPosInIntmat(laenge); |
---|
1367 | list endpoints=polygon[mp[1]],polygon[mp[2]]; |
---|
1368 | intvec abstand; |
---|
1369 | for (i=1;i<=size(polygon);i++) |
---|
1370 | { |
---|
1371 | abstand[i]=0; |
---|
1372 | if (i<mp[1]) |
---|
1373 | { |
---|
1374 | abstand[i]=laenge[i,mp[1]]; |
---|
1375 | } |
---|
1376 | if (i>mp[1]) |
---|
1377 | { |
---|
1378 | abstand[i]=laenge[mp[1],i]; |
---|
1379 | } |
---|
1380 | } |
---|
1381 | polygon=sortlistbyintvec(polygon,abstand); |
---|
1382 | return(list(polygon,endpoints)); |
---|
1383 | } |
---|
1384 | /////////////////////////////////////////////////////////////// |
---|
1385 | list orderedvertices; // stores the vertices in an ordered way |
---|
1386 | list minimisedorderedvertices; // stores the vertices in an ordered way; |
---|
1387 | // redundant ones removed |
---|
1388 | list comparevertices; // stores vertices which should be compared to |
---|
1389 | // the testvertex |
---|
1390 | orderedvertices[1]=polygon[1]; // set the starting vertex |
---|
1391 | minimisedorderedvertices[1]=polygon[1]; // set the starting vertex |
---|
1392 | intvec testvertex=polygon[1]; //vertex to which the others have to be compared |
---|
1393 | intvec startvertex=polygon[1]; // keep the starting vertex to test, |
---|
1394 | // when the end is reached |
---|
1395 | int endtest; // is set to one, when the end is reached |
---|
1396 | int startvertexfound;// is 1, once for some testvertex a candidate |
---|
1397 | // for the next vertex has been found |
---|
1398 | polygon=delete(polygon,1); // delete the testvertex |
---|
1399 | intvec v,w; |
---|
1400 | int l=1; // counts the vertices |
---|
1401 | // the basic idea is that a vertex can be |
---|
1402 | // the next one on the boundary if all other vertices |
---|
1403 | // lie to the right of the vector v pointing |
---|
1404 | // from the testvertex to this one; this can be tested |
---|
1405 | // by checking if the determinant of the 2x2-matrix |
---|
1406 | // with first column v and second column the vector w, |
---|
1407 | // pointing from the testvertex to the new vertex, |
---|
1408 | // is non-positive; if this is the case for all |
---|
1409 | // new vertices, then the one in consideration is |
---|
1410 | // a possible choice for the next vertex on the boundary |
---|
1411 | // and it is stored in naechste; we can then order |
---|
1412 | // the candidates according to their distance from |
---|
1413 | // the testvertex; then they occur on the boundary in that order! |
---|
1414 | while (endtest==0) |
---|
1415 | { |
---|
1416 | list naechste; // stores the possible choices for the next vertex |
---|
1417 | k=1; |
---|
1418 | for (i=1;i<=size(polygon);i++) |
---|
1419 | { |
---|
1420 | d=0; // stores the value of the determinant of (v,w) |
---|
1421 | v=polygon[i]-testvertex; // points from the testvertex to the ith vertex |
---|
1422 | comparevertices=delete(polygon,i); // we needn't compare v to itself |
---|
1423 | // we should compare v to the startvertex-testvertex; |
---|
1424 | // in the first calling of the loop |
---|
1425 | // this is irrelevant since the difference will be zero; |
---|
1426 | // however, later on it will |
---|
1427 | // be vital, since we delete the vertices |
---|
1428 | // which we have already tested from the list |
---|
1429 | // of all vertices, and when all vertices |
---|
1430 | // on the boundary have been found we would |
---|
1431 | // therefore find a vertex in the interior |
---|
1432 | // as candidate; but always testing against |
---|
1433 | // the starting vertex, this cannot happen |
---|
1434 | comparevertices[size(comparevertices)+1]=startvertex; |
---|
1435 | for (j=1;(j<=size(comparevertices)) and (d<=0);j++) |
---|
1436 | { |
---|
1437 | w=comparevertices[j]-testvertex; // points form the testvertex |
---|
1438 | // to the jth vertex |
---|
1439 | D=v,w; |
---|
1440 | d=det(D); |
---|
1441 | } |
---|
1442 | if (d<=0) // if all determinants are non-positive, |
---|
1443 | { // then the ith vertex is a candidate |
---|
1444 | naechste[k]=list(polygon[i],i,scalarproduct(v,v));// we store the vertex, |
---|
1445 | //its position, and its |
---|
1446 | k++; // distance from the testvertex |
---|
1447 | } |
---|
1448 | } |
---|
1449 | if (size(naechste)>0) // then a candidate for the next vertex has been found |
---|
1450 | { |
---|
1451 | startvertexfound=1; // at least once a candidate has been found |
---|
1452 | naechste=sortlist(naechste,3); // we order the candidates according |
---|
1453 | // to their distance from testvertex; |
---|
1454 | for (j=1;j<=size(naechste);j++) // then we store them in this |
---|
1455 | { // order in orderedvertices |
---|
1456 | l++; |
---|
1457 | orderedvertices[l]=naechste[j][1]; |
---|
1458 | } |
---|
1459 | testvertex=naechste[size(naechste)][1]; // we store the last one as |
---|
1460 | // next testvertex; |
---|
1461 | // store the next corner of NSD |
---|
1462 | minimisedorderedvertices[size(minimisedorderedvertices)+1]=testvertex; |
---|
1463 | naechste=sortlist(naechste,2); // then we reorder the vertices |
---|
1464 | // according to their position |
---|
1465 | for (j=size(naechste);j>=1;j--) // and we delete them from the vertices |
---|
1466 | { |
---|
1467 | polygon=delete(polygon,naechste[j][2]); |
---|
1468 | } |
---|
1469 | } |
---|
1470 | else // that means either that the vertex was inside the polygon, |
---|
1471 | { // or that we have reached the last vertex on the boundary |
---|
1472 | // of the polytope |
---|
1473 | if (startvertexfound==0) // the vertex was in the interior; |
---|
1474 | { // we delete it and start all over again |
---|
1475 | orderedvertices[1]=polygon[1]; |
---|
1476 | minimisedorderedvertices[1]=polygon[1]; |
---|
1477 | testvertex=polygon[1]; |
---|
1478 | startvertex=polygon[1]; |
---|
1479 | polygon=delete(polygon,1); |
---|
1480 | } |
---|
1481 | else // we have reached the last vertex on the boundary of |
---|
1482 | { // the polytope and can stop |
---|
1483 | endtest=1; |
---|
1484 | } |
---|
1485 | } |
---|
1486 | kill naechste; |
---|
1487 | } |
---|
1488 | // test if the first vertex in minimisedorderedvertices |
---|
1489 | // is on the same line with the second and |
---|
1490 | // the last, i.e. if we started our search in the |
---|
1491 | // middle of a face; if so, delete it |
---|
1492 | v=minimisedorderedvertices[2]-minimisedorderedvertices[1]; |
---|
1493 | w=minimisedorderedvertices[size(minimisedorderedvertices)]-minimisedorderedvertices[1]; |
---|
1494 | D=v,w; |
---|
1495 | if (det(D)==0) |
---|
1496 | { |
---|
1497 | minimisedorderedvertices=delete(minimisedorderedvertices,1); |
---|
1498 | } |
---|
1499 | // test if the first vertex in minimisedorderedvertices |
---|
1500 | // is on the same line with the two |
---|
1501 | // last ones, i.e. if we started our search at the end of a face; |
---|
1502 | // if so, delete it |
---|
1503 | v=minimisedorderedvertices[size(minimisedorderedvertices)-1]-minimisedorderedvertices[1]; |
---|
1504 | w=minimisedorderedvertices[size(minimisedorderedvertices)]-minimisedorderedvertices[1]; |
---|
1505 | D=v,w; |
---|
1506 | if (det(D)==0) |
---|
1507 | { |
---|
1508 | minimisedorderedvertices=delete(minimisedorderedvertices,size(minimisedorderedvertices)); |
---|
1509 | } |
---|
1510 | return(list(orderedvertices,minimisedorderedvertices)); |
---|
1511 | } |
---|
1512 | example |
---|
1513 | { |
---|
1514 | "EXAMPLE:"; |
---|
1515 | echo=2; |
---|
1516 | // the following lattice points in the plane define a polygon |
---|
1517 | list polygon=intvec(0,0),intvec(3,1),intvec(1,0),intvec(2,0), |
---|
1518 | intvec(1,1),intvec(3,2),intvec(1,2),intvec(2,3), |
---|
1519 | intvec(2,4); |
---|
1520 | // we compute its boundary |
---|
1521 | list boundarypolygon=findOrientedBoundary(polygon); |
---|
1522 | // the points on the boundary ordered clockwise are boundarypolygon[1] |
---|
1523 | boundarypolygon[1]; |
---|
1524 | // the vertices of the boundary are boundarypolygon[2] |
---|
1525 | boundarypolygon[2]; |
---|
1526 | } |
---|
1527 | |
---|
1528 | |
---|
1529 | ///////////////////////////////////////////////////////////////////////////// |
---|
1530 | |
---|
1531 | proc cyclePoints (list triang,list points,int pt) |
---|
1532 | "USAGE: cyclePoints(triang,points,pt) triang,points list, pt int |
---|
1533 | ASSUME: - points is a list of integer vectors describing the lattice |
---|
1534 | points of a marked polygon; |
---|
1535 | @* - triang is a list of integer vectors describing a triangulation |
---|
1536 | of the marked polygon in the sense that an integer vector of |
---|
1537 | the form (i,j,k) describes the triangle formed by polygon[i], |
---|
1538 | polygon[j] and polygon[k]; |
---|
1539 | @* - pt is an integer between 1 and size(points), singling out a |
---|
1540 | lattice point among the marked points |
---|
1541 | PURPOSE: consider the convex lattice polygon, say P, spanned by all lattice |
---|
1542 | points in points which in the triangulation triang are connected |
---|
1543 | to the point points[pt]; the procedure computes all marked points |
---|
1544 | in points which lie on the boundary of that polygon, ordered |
---|
1545 | clockwise |
---|
1546 | RETURN: list, of integer vectors which are the coordinates of the lattice |
---|
1547 | points on the boundary of the above mentioned polygon P, if |
---|
1548 | this polygon is not the empty set (that would be the case if |
---|
1549 | points[pt] is not a vertex of any triangle in the |
---|
1550 | triangulation); otherwise return the empty list |
---|
1551 | EXAMPLE: example cyclePoints; shows an example" |
---|
1552 | { |
---|
1553 | int i,j; // indices |
---|
1554 | list v; // saves the indices of lattice points connected to the |
---|
1555 | // interior point in the triangulation |
---|
1556 | // save all points in triangulations containing pt in v |
---|
1557 | for (i=1;i<=size(triang);i++) |
---|
1558 | { |
---|
1559 | if ((triang[i][1]==pt) or (triang[i][2]==pt) or (triang[i][3]==pt)) |
---|
1560 | { |
---|
1561 | j++; |
---|
1562 | v[3*j-2]=triang[i][1]; |
---|
1563 | v[3*j-1]=triang[i][2]; |
---|
1564 | v[3*j]=triang[i][3]; |
---|
1565 | } |
---|
1566 | } |
---|
1567 | if (size(v)==0) |
---|
1568 | { |
---|
1569 | return(list()); |
---|
1570 | } |
---|
1571 | // remove pt itself and redundancies in v |
---|
1572 | for (i=size(v);i>=1;i--) |
---|
1573 | { |
---|
1574 | j=1; |
---|
1575 | while ((j<i) and (v[i]!=v[j])) |
---|
1576 | { |
---|
1577 | j++; |
---|
1578 | } |
---|
1579 | if ((j<i) or (v[i]==pt)) |
---|
1580 | { |
---|
1581 | v=delete(v,i); |
---|
1582 | } |
---|
1583 | } |
---|
1584 | // save in pts the coordinates of the points with indices in v |
---|
1585 | list pts; |
---|
1586 | for (i=1;i<=size(v);i++) |
---|
1587 | { |
---|
1588 | pts[i]=points[v[i]]; |
---|
1589 | } |
---|
1590 | // consider the convex polytope spanned by the points in pts, |
---|
1591 | // find the points on the |
---|
1592 | // boundary and order them clockwise |
---|
1593 | return(findOrientedBoundary(pts)[1]); |
---|
1594 | } |
---|
1595 | example |
---|
1596 | { |
---|
1597 | "EXAMPLE:"; |
---|
1598 | echo=2; |
---|
1599 | // the lattice polygon spanned by the points (0,0), (3,0) and (0,3) |
---|
1600 | // with all integer points as markings |
---|
1601 | list points=intvec(1,1),intvec(3,0),intvec(2,0),intvec(1,0), |
---|
1602 | intvec(0,0),intvec(2,1),intvec(0,1),intvec(1,2), |
---|
1603 | intvec(0,2),intvec(0,3); |
---|
1604 | // define a triangulation |
---|
1605 | list triang=intvec(1,2,5),intvec(1,5,7),intvec(1,7,9),intvec(8,9,10), |
---|
1606 | intvec(1,8,9),intvec(1,2,8); |
---|
1607 | // compute the points connected to (1,1) in triang |
---|
1608 | cyclePoints(triang,points,1); |
---|
1609 | } |
---|
1610 | |
---|
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 |
---|
1617 | polygon, 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), |
---|
1637 | intvec(2,1),intvec(0,0); |
---|
1638 | latticeArea(polygon); |
---|
1639 | } |
---|
1640 | |
---|
1641 | ///////////////////////////////////////////////////////////////////////////// |
---|
1642 | |
---|
1643 | proc picksFormula (list polygon) |
---|
1644 | "USAGE: picksFormula(polygon); polygon list |
---|
1645 | ASSUME: polygon is a list of integer vectors in the plane and consider their |
---|
1646 | convex hull C |
---|
1647 | RETURN: list, L of three integersthe |
---|
1648 | @* L[1] : the lattice area of C, i.e. twice the Euclidean area |
---|
1649 | @* L[2] : the number of lattice points on the boundary of C |
---|
1650 | @* L[3] : the number of interior lattice points of C |
---|
1651 | NOTE: the integers in L are related by Pick's formula, namely: L[1]=L[2]+2*L[3]-2 |
---|
1652 | EXAMPLE: example picksFormula; shows an example" |
---|
1653 | { |
---|
1654 | list pg=findOrientedBoundary(polygon)[2]; |
---|
1655 | int area,bdpts,i; |
---|
1656 | intmat M[2][2]; |
---|
1657 | // compute the lattice area of the polygon, i.e. twice the Euclidean area |
---|
1658 | for (i=2;i<=size(pg)-1;i++) |
---|
1659 | { |
---|
1660 | M[1,1..2]=pg[i]-pg[1]; |
---|
1661 | M[2,1..2]=pg[i+1]-pg[1]; |
---|
1662 | area=area+abs(det(M)); |
---|
1663 | } |
---|
1664 | // compute the number of lattice points on the boundary |
---|
1665 | intvec edge; |
---|
1666 | pg[size(pg)+1]=pg[1]; |
---|
1667 | for (i=1;i<=size(pg)-1;i++) |
---|
1668 | { |
---|
1669 | edge=pg[i]-pg[i+1]; |
---|
1670 | bdpts=bdpts+abs(gcd(edge[1],edge[2])); |
---|
1671 | } |
---|
1672 | // Pick's formula says that the lattice area A, the number g of interior |
---|
1673 | // points and |
---|
1674 | // the number b of boundary points are connected by the formula: A=b+2g-2 |
---|
1675 | return(list(area,bdpts,(area-bdpts+2) div 2)); |
---|
1676 | } |
---|
1677 | example |
---|
1678 | { |
---|
1679 | "EXAMPLE:"; |
---|
1680 | echo=2; |
---|
1681 | // define a polygon with lattice area 5 |
---|
1682 | list polygon=intvec(1,2),intvec(1,0),intvec(2,0),intvec(1,1), |
---|
1683 | intvec(2,1),intvec(0,0); |
---|
1684 | list pick=picksFormula(polygon); |
---|
1685 | // the lattice area of the polygon is: |
---|
1686 | pick[1]; |
---|
1687 | // the number of lattice points on the boundary is: |
---|
1688 | pick[2]; |
---|
1689 | // the number of interior lattice points is: |
---|
1690 | pick[3]; |
---|
1691 | // the number's are related by Pick's formula: |
---|
1692 | pick[1]-pick[2]-2*pick[3]+2; |
---|
1693 | } |
---|
1694 | |
---|
1695 | ///////////////////////////////////////////////////////////////////////////// |
---|
1696 | |
---|
1697 | proc ellipticNF (list polygon) |
---|
1698 | "USAGE: ellipticNF(polygon); polygon list |
---|
1699 | ASSUME: polygon is a list of integer vectors in the plane such that their |
---|
1700 | convex hull C has precisely one interior lattice point; i.e. C is the |
---|
1701 | Newton polygon of an elliptic curve |
---|
1702 | PURPOSE: compute the normal form of the polygon with respect to the unimodular |
---|
1703 | affine transformations T=A*x+v; there are sixteen different normal forms |
---|
1704 | (see e.g. Bjorn Poonen, Fernando Rodriguez-Villegas: Lattice Polygons |
---|
1705 | and the number 12. Amer. Math. Monthly 107 (2000), no. 3, |
---|
1706 | 238--250.) |
---|
1707 | RETURN: list, L such that |
---|
1708 | @* L[1] : list whose entries are the vertices of the normal form of |
---|
1709 | the polygon |
---|
1710 | @* L[2] : the matrix A of the unimodular transformation |
---|
1711 | @* L[3] : the translation vector v of the unimodular transformation |
---|
1712 | @* L[4] : list such that the ith entry is the image of polygon[i] |
---|
1713 | under the unimodular transformation T |
---|
1714 | EXAMPLE: example ellipticNF; shows an example" |
---|
1715 | { |
---|
1716 | int i; // index |
---|
1717 | intvec edge; // stores the vector of an edge |
---|
1718 | intvec boundary; // stores lattice lengths of the edges of the Newton cycle |
---|
1719 | // find the vertices of the Newton cycle and order it clockwise |
---|
1720 | list pg=findOrientedBoundary(polygon)[2]; |
---|
1721 | // check if there is precisely one interior point in the Newton polygon |
---|
1722 | if (picksFormula(pg)[3]!=1) |
---|
1723 | { |
---|
1724 | ERROR("The polygon has not precisely one interior point!"); |
---|
1725 | } |
---|
1726 | // insert the first vertex at the end once again |
---|
1727 | pg[size(pg)+1]=pg[1]; |
---|
1728 | // compute the number of lattice points on each edge |
---|
1729 | for (i=1;i<=size(pg)-1;i++) |
---|
1730 | { |
---|
1731 | edge=pg[i]-pg[i+1]; |
---|
1732 | boundary[i]=1+abs(gcd(edge[1],edge[2])); |
---|
1733 | } |
---|
1734 | // store the values of boundary once more adding the first two at the end |
---|
1735 | intvec tboundary=boundary,boundary[1],boundary[2]; |
---|
1736 | // sort boundary in an asecending way |
---|
1737 | intvec sbd=sortintvec(boundary); |
---|
1738 | // find the first edge having the maximal number of lattice points |
---|
1739 | int max=maxPosInIntvec(boundary); |
---|
1740 | // some computations have to be done over the rationals |
---|
1741 | ring transformationring=0,x,lp; |
---|
1742 | intvec trans; // stores the vector by which we have to translate the polygon |
---|
1743 | intmat A[2][2]; // stores the matrix by which we have to transform the polygon |
---|
1744 | matrix M[3][3]; // stores the projective coordinates of the points |
---|
1745 | // which are to be transformed |
---|
1746 | matrix N[3][3]; // stores the projective coordinates of the points to |
---|
1747 | // which M is to be transformed |
---|
1748 | intmat T[3][3]; // stores the unimodular affine transformation in |
---|
1749 | // projective form |
---|
1750 | // add the second point of pg once again at the end |
---|
1751 | pg=insert(pg,pg[2],size(pg)); |
---|
1752 | // if there is only one edge which has the maximal number of lattice points, |
---|
1753 | // then M should be: |
---|
1754 | M=pg[max],1,pg[max+1],1,pg[max+2],1; |
---|
1755 | // consider the 16 different cases which can occur: |
---|
1756 | // Case 1: |
---|
1757 | if (sbd==intvec(2,2,2)) |
---|
1758 | { |
---|
1759 | N=0,1,1,1,2,1,2,0,1; |
---|
1760 | } |
---|
1761 | // Case 2: |
---|
1762 | if (sbd==intvec(2,2,3)) |
---|
1763 | { |
---|
1764 | N=2,0,1,0,0,1,1,2,1; |
---|
1765 | } |
---|
1766 | // Case 3: |
---|
1767 | if (sbd==intvec(2,3,4)) |
---|
1768 | { |
---|
1769 | // here the orientation of the Newton polygon is important ! |
---|
1770 | if (tboundary[max+1]==3) |
---|
1771 | { |
---|
1772 | N=3,0,1,0,0,1,0,2,1; |
---|
1773 | } |
---|
1774 | else |
---|
1775 | { |
---|
1776 | N=0,0,1,3,0,1,0,2,1; |
---|
1777 | } |
---|
1778 | } |
---|
1779 | // Case 4: |
---|
1780 | if (sbd==intvec(3,3,5)) |
---|
1781 | { |
---|
1782 | N=4,0,1,0,0,1,0,2,1; |
---|
1783 | } |
---|
1784 | // Case 5: |
---|
1785 | if (sbd==intvec(4,4,4)) |
---|
1786 | { |
---|
1787 | N=3,0,1,0,0,1,0,3,1; |
---|
1788 | } |
---|
1789 | // Case 6+7: |
---|
1790 | if (sbd==intvec(2,2,2,2)) |
---|
1791 | { |
---|
1792 | // there are two different polygons which has four edges all of length 2, |
---|
1793 | // but only one of them has two edges whose direction vectors form a matrix |
---|
1794 | // of determinant 3 |
---|
1795 | A=pg[1]-pg[2],pg[3]-pg[2]; |
---|
1796 | while ((max<4) and (det(A)!=3)) |
---|
1797 | { |
---|
1798 | max++; |
---|
1799 | A=pg[max]-pg[max+1],pg[max+2]-pg[max+1]; |
---|
1800 | } |
---|
1801 | // Case 6: |
---|
1802 | if (det(A)==3) |
---|
1803 | { |
---|
1804 | M=pg[max],1,pg[max+1],1,pg[max+2],1; |
---|
1805 | N=1,0,1,0,2,1,2,1,1; |
---|
1806 | } |
---|
1807 | // Case 7: |
---|
1808 | else |
---|
1809 | { |
---|
1810 | N=2,1,1,1,0,1,0,1,1; |
---|
1811 | } |
---|
1812 | } |
---|
1813 | // Case 8: |
---|
1814 | if (sbd==intvec(2,2,2,3)) |
---|
1815 | { |
---|
1816 | // the orientation of the polygon is important |
---|
1817 | A=pg[max]-pg[max+1],pg[max+2]-pg[max+1]; |
---|
1818 | if (det(A)==2) |
---|
1819 | { |
---|
1820 | N=2,0,1,0,0,1,0,1,1; |
---|
1821 | } |
---|
1822 | else |
---|
1823 | { |
---|
1824 | N=0,0,1,2,0,1,1,2,1; |
---|
1825 | } |
---|
1826 | } |
---|
1827 | // Case 9: |
---|
1828 | if (sbd==intvec(2,2,3,3)) |
---|
1829 | { |
---|
1830 | // if max==1, then the 5th entry in tboundary is the same as the first |
---|
1831 | if (max==1) |
---|
1832 | { |
---|
1833 | max=5; |
---|
1834 | } |
---|
1835 | // if boundary=3,2,2,3 then set max=4 |
---|
1836 | if (tboundary[max+1]!=3) |
---|
1837 | { |
---|
1838 | max=4; |
---|
1839 | } |
---|
1840 | M=pg[max],1,pg[max+1],1,pg[max+2],1; |
---|
1841 | // the orientation of the polygon matters |
---|
1842 | A=pg[max-1]-pg[max],pg[max+1]-pg[max]; |
---|
1843 | if (det(A)==4) |
---|
1844 | { |
---|
1845 | N=2,0,1,0,0,1,0,2,1; |
---|
1846 | } |
---|
1847 | else |
---|
1848 | { |
---|
1849 | N=0,2,1,0,0,1,2,0,1; |
---|
1850 | } |
---|
1851 | } |
---|
1852 | // Case 10: |
---|
1853 | if (sbd==intvec(2,2,3,4)) |
---|
1854 | { |
---|
1855 | // the orientation of the polygon matters |
---|
1856 | if (tboundary[max+1]==3) |
---|
1857 | { |
---|
1858 | N=3,0,1,0,0,1,0,2,1; |
---|
1859 | } |
---|
1860 | else |
---|
1861 | { |
---|
1862 | N=0,0,1,3,0,1,2,1,1; |
---|
1863 | } |
---|
1864 | } |
---|
1865 | // Case 11: |
---|
1866 | if (sbd==intvec(2,3,3,4)) |
---|
1867 | { |
---|
1868 | N=3,0,1,0,0,1,0,2,1; |
---|
1869 | } |
---|
1870 | // Case 12: |
---|
1871 | if (sbd==intvec(3,3,3,3)) |
---|
1872 | { |
---|
1873 | N=2,0,1,0,0,1,0,2,1; |
---|
1874 | } |
---|
1875 | // Case 13: |
---|
1876 | if (sbd==intvec(2,2,2,2,2)) |
---|
1877 | { |
---|
1878 | // compute the angles of the polygon vertices |
---|
1879 | intvec dt; |
---|
1880 | for (i=1;i<=5;i++) |
---|
1881 | { |
---|
1882 | A=pg[i]-pg[i+1],pg[i+2]-pg[i+1]; |
---|
1883 | dt[i]=det(A); |
---|
1884 | } |
---|
1885 | dt[6]=dt[1]; |
---|
1886 | // find the vertex to be mapped to (0,1) |
---|
1887 | max=1; |
---|
1888 | while ((dt[max]!=2) or (dt[max+1]!=2)) |
---|
1889 | { |
---|
1890 | max++; |
---|
1891 | } |
---|
1892 | M=pg[max],1,pg[max+1],1,pg[max+2],1; |
---|
1893 | N=0,1,1,1,2,1,2,1,1; |
---|
1894 | } |
---|
1895 | // Case 14: |
---|
1896 | if (sbd==intvec(2,2,2,2,3)) |
---|
1897 | { |
---|
1898 | N=2,0,1,0,0,1,0,1,1; |
---|
1899 | } |
---|
1900 | // Case 15: |
---|
1901 | if (sbd==intvec(2,2,2,3,3)) |
---|
1902 | { |
---|
1903 | // find the vertix to be mapped to (2,0) |
---|
1904 | if (tboundary[max+1]!=3) |
---|
1905 | { |
---|
1906 | max=5; |
---|
1907 | M=pg[max],1,pg[max+1],1,pg[max+2],1; |
---|
1908 | } |
---|
1909 | N=2,0,1,0,0,1,0,2,1; |
---|
1910 | } |
---|
1911 | // Case 16: |
---|
1912 | if (sbd==intvec(2,2,2,2,2,2)) |
---|
1913 | { |
---|
1914 | N=2,0,1,1,0,1,0,1,1; |
---|
1915 | } |
---|
1916 | // we have to transpose the matrices M and N |
---|
1917 | M=transpose(M); |
---|
1918 | N=transpose(N); |
---|
1919 | // compute the unimodular affine transformation, which is of the form |
---|
1920 | // A11 A12 | T1 |
---|
1921 | // A21 A22 | T2 |
---|
1922 | // 0 0 | 1 |
---|
1923 | T=matrixtointmat(N*inverse(M)); |
---|
1924 | // the upper-left 2x2-block is A |
---|
1925 | A=T[1..2,1..2]; |
---|
1926 | // the upper-right 2x1-block is the translation vector |
---|
1927 | trans=T[1,3],T[2,3]; |
---|
1928 | // transform now the lattice points of the polygon with respect to A and T |
---|
1929 | list nf; |
---|
1930 | for (i=1;i<=size(polygon);i++) |
---|
1931 | { |
---|
1932 | intmat V[2][1]=polygon[i]; |
---|
1933 | V=A*V; |
---|
1934 | nf[i]=intvec(V[1,1]+trans[1],V[2,1]+trans[2]); |
---|
1935 | kill V; |
---|
1936 | } |
---|
1937 | return(list(findOrientedBoundary(nf)[2],A,trans,nf)); |
---|
1938 | } |
---|
1939 | example |
---|
1940 | { |
---|
1941 | "EXAMPLE:"; |
---|
1942 | echo=2; |
---|
1943 | ring r=0,(x,y),dp; |
---|
1944 | // the Newton polygon of the following polynomial |
---|
1945 | // has precisely one interior point |
---|
1946 | poly f=x22y11+x19y10+x17y9+x16y9+x12y7+x9y6+x7y5+x2y3; |
---|
1947 | list polygon=newtonPolytopeLP(f); |
---|
1948 | // its lattice points are |
---|
1949 | polygon; |
---|
1950 | // find its normal form |
---|
1951 | list nf=ellipticNF(polygon); |
---|
1952 | // the vertices of the normal form are |
---|
1953 | nf[1]; |
---|
1954 | // it has been transformed by the unimodular affine transformation A*x+v |
---|
1955 | // with matrix A |
---|
1956 | nf[2]; |
---|
1957 | // and translation vector v |
---|
1958 | nf[3]; |
---|
1959 | // the 3rd lattice point ... |
---|
1960 | polygon[3]; |
---|
1961 | // ... has been transformed to |
---|
1962 | nf[4][3]; |
---|
1963 | } |
---|
1964 | |
---|
1965 | |
---|
1966 | ///////////////////////////////////////////////////////////////////////////// |
---|
1967 | |
---|
1968 | proc ellipticNFDB (int n,list #) |
---|
1969 | "USAGE: ellipticNFDB(n[,#]); n int, # list |
---|
1970 | ASSUME: n is an integer between 1 and 16 |
---|
1971 | PURPOSE: this is a database storing the 16 normal forms of planar polygons with |
---|
1972 | precisely one interior point up to unimodular affine transformations |
---|
1973 | @* (see e.g. Bjorn Poonen, Fernando Rodriguez-Villegas: Lattice Polygons |
---|
1974 | and the number 12. Amer. Math. Monthly 107 (2000), no. 3, |
---|
1975 | 238--250.) |
---|
1976 | RETURN: list, L such that |
---|
1977 | @* L[1] : list whose entries are the vertices of the nth normal form |
---|
1978 | @* L[2] : list whose entries are all the lattice points of the |
---|
1979 | nth normal form |
---|
1980 | @* L[3] : only present if the optional parameter # is present, and |
---|
1981 | then it is a polynomial in the variables (x,y) whose |
---|
1982 | Newton polygon is the nth normal form |
---|
1983 | NOTE: the optional parameter is only allowed if the basering has the |
---|
1984 | variables x and y |
---|
1985 | EXAMPLE: example ellipticNFDB; shows an example" |
---|
1986 | { |
---|
1987 | if ((n<1) or (n>16)) |
---|
1988 | { |
---|
1989 | ERROR("n is not between 1 and 16."); |
---|
1990 | } |
---|
1991 | if (size(#)>0) |
---|
1992 | { |
---|
1993 | if ((defined(x)==0) or (defined(y)==0)) |
---|
1994 | { |
---|
1995 | ERROR("The variables x and y are not defined."); |
---|
1996 | } |
---|
1997 | } |
---|
1998 | if ((defined(x)==0) or (defined(y)==0)) |
---|
1999 | { |
---|
2000 | ring nfring=0,(x,y),dp; |
---|
2001 | } |
---|
2002 | // store the normal forms as polynomials |
---|
2003 | 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, |
---|
2004 | 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, |
---|
2005 | 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; |
---|
2006 | list pg=newtonPolytopeLP(nf[n]); |
---|
2007 | if (size(#)==0) |
---|
2008 | { |
---|
2009 | return(list(findOrientedBoundary(pg)[2],pg)); |
---|
2010 | } |
---|
2011 | else |
---|
2012 | { |
---|
2013 | return(list(findOrientedBoundary(pg)[2],pg,nf[n])); |
---|
2014 | } |
---|
2015 | } |
---|
2016 | example |
---|
2017 | { |
---|
2018 | "EXAMPLE:"; |
---|
2019 | echo=2; |
---|
2020 | list nf=ellipticNFDB(5); |
---|
2021 | // the vertices of the 5th normal form are |
---|
2022 | nf[1]; |
---|
2023 | // its lattice points are |
---|
2024 | nf[2]; |
---|
2025 | } |
---|
2026 | |
---|
2027 | |
---|
2028 | ///////////////////////////////////////////////////////////////////////////////// |
---|
2029 | ///////////////////////////////////////////////////////////////////////////////// |
---|
2030 | /// AUXILARY PROCEDURES, WHICH ARE DECLARED STATIC |
---|
2031 | ///////////////////////////////////////////////////////////////////////////////// |
---|
2032 | ///////////////////////////////////////////////////////////////////////////////// |
---|
2033 | /// - scalarproduct |
---|
2034 | /// - intmatcoldelete |
---|
2035 | /// - intmatconcat |
---|
2036 | /// - sortlist |
---|
2037 | /// - minInList |
---|
2038 | /// - stringdelete |
---|
2039 | /// - abs |
---|
2040 | /// - commondenominator |
---|
2041 | /// - maxPosInIntvec |
---|
2042 | /// - maxPosInIntmat |
---|
2043 | /// - sortintvec |
---|
2044 | /// - matrixtointmat |
---|
2045 | ///////////////////////////////////////////////////////////////////////////////// |
---|
2046 | |
---|
2047 | static proc scalarproduct (intvec w,intvec v) |
---|
2048 | "USAGE: scalarproduct(w,v); w,v intvec |
---|
2049 | ASSUME: w and v are integer vectors of the same length |
---|
2050 | RETURN: int, the scalarproduct of v and w |
---|
2051 | NOTE: the procedure is called by findOrientedBoundary" |
---|
2052 | { |
---|
2053 | int sp; |
---|
2054 | for (int i=1;i<=size(w);i++) |
---|
2055 | { |
---|
2056 | sp=sp+v[i]*w[i]; |
---|
2057 | } |
---|
2058 | return(sp); |
---|
2059 | } |
---|
2060 | |
---|
2061 | static proc intmatcoldelete (intmat w,int i) |
---|
2062 | "USAGE: intmatcoldelete(w,i); w intmat, i int |
---|
2063 | RETURN: intmat, the integer matrix w with the ith comlumn deleted |
---|
2064 | NOTE: the procedure is called by intmatsort and normalFan" |
---|
2065 | { |
---|
2066 | if ((i<1) or (i>ncols(w)) or (ncols(w)==1)) |
---|
2067 | { |
---|
2068 | return(w); |
---|
2069 | } |
---|
2070 | if (i==1) |
---|
2071 | { |
---|
2072 | intmat M[nrows(w)][ncols(w)-1]=w[1..nrows(w),2..ncols(w)]; |
---|
2073 | return(M); |
---|
2074 | } |
---|
2075 | if (i==ncols(w)) |
---|
2076 | { |
---|
2077 | intmat M[nrows(w)][ncols(w)-1]=w[1..nrows(w),1..ncols(w)-1]; |
---|
2078 | return(M); |
---|
2079 | } |
---|
2080 | else |
---|
2081 | { |
---|
2082 | intmat M[nrows(w)][i-1]=w[1..nrows(w),1..i-1]; |
---|
2083 | intmat N[nrows(w)][ncols(w)-i]=w[1..nrows(w),i+1..ncols(w)]; |
---|
2084 | return(intmatconcat(M,N)); |
---|
2085 | } |
---|
2086 | } |
---|
2087 | |
---|
2088 | static proc intmatconcat (intmat M,intmat N) |
---|
2089 | "USAGE: intmatconcat(M,N); M,N intmat |
---|
2090 | RETURN: intmat, M and N concatenated |
---|
2091 | NOTE: the procedure is called by intmatcoldelete and sortintmat" |
---|
2092 | { |
---|
2093 | if (nrows(M)>=nrows(N)) |
---|
2094 | { |
---|
2095 | int m=nrows(M); |
---|
2096 | |
---|
2097 | } |
---|
2098 | else |
---|
2099 | { |
---|
2100 | int m=nrows(N); |
---|
2101 | } |
---|
2102 | intmat P[m][ncols(M)+ncols(N)]; |
---|
2103 | P[1..nrows(M),1..ncols(M)]=M[1..nrows(M),1..ncols(M)]; |
---|
2104 | P[1..nrows(N),ncols(M)+1..ncols(M)+ncols(N)]=N[1..nrows(N),1..ncols(N)]; |
---|
2105 | return(P); |
---|
2106 | } |
---|
2107 | |
---|
2108 | static proc sortlist (list v,int pos) |
---|
2109 | "USAGE: sortlist(v,pos); v list, pos int |
---|
2110 | RETURN: list, the list L ordered in an ascending way according to the pos-th entries |
---|
2111 | NOTE: called by tropicalCurve" |
---|
2112 | { |
---|
2113 | if(size(v)==1) |
---|
2114 | { |
---|
2115 | return(v); |
---|
2116 | } |
---|
2117 | list w=minInList(v,pos); |
---|
2118 | v=delete(v,w[2]); |
---|
2119 | v=sortlist(v,pos); |
---|
2120 | v=list(w[1])+v; |
---|
2121 | return(v); |
---|
2122 | } |
---|
2123 | |
---|
2124 | static proc minInList (list v,int pos) |
---|
2125 | "USAGE: minInList(v,pos); v list, pos int |
---|
2126 | RETURN: list, (v[i],i) such that v[i][pos] is minimal |
---|
2127 | NOTE: called by sortlist" |
---|
2128 | { |
---|
2129 | int min=v[1][pos]; |
---|
2130 | int minpos=1; |
---|
2131 | for (int i=2;i<=size(v);i++) |
---|
2132 | { |
---|
2133 | if (v[i][pos]<min) |
---|
2134 | { |
---|
2135 | min=v[i][pos]; |
---|
2136 | minpos=i; |
---|
2137 | } |
---|
2138 | } |
---|
2139 | return(list(v[minpos],minpos)); |
---|
2140 | } |
---|
2141 | |
---|
2142 | static proc stringdelete (string w,int i) |
---|
2143 | "USAGE: stringdelete(w,i); w string, i int |
---|
2144 | RETURN: string, the string w with the ith component deleted |
---|
2145 | NOTE: the procedure is called by texnumber and choosegfanvector" |
---|
2146 | { |
---|
2147 | if ((i>size(w)) or (i<=0)) |
---|
2148 | { |
---|
2149 | return(w); |
---|
2150 | } |
---|
2151 | if ((size(w)==1) and (i==1)) |
---|
2152 | { |
---|
2153 | return(""); |
---|
2154 | |
---|
2155 | } |
---|
2156 | if (i==1) |
---|
2157 | { |
---|
2158 | return(w[2..size(w)]); |
---|
2159 | } |
---|
2160 | if (i==size(w)) |
---|
2161 | { |
---|
2162 | return(w[1..size(w)-1]); |
---|
2163 | } |
---|
2164 | else |
---|
2165 | { |
---|
2166 | string erg=w[1..i-1],w[i+1..size(w)]; |
---|
2167 | return(erg); |
---|
2168 | } |
---|
2169 | } |
---|
2170 | |
---|
2171 | static proc abs (def n) |
---|
2172 | "USAGE: abs(n); n poly or int |
---|
2173 | RETURN: poly or int, the absolute value of n" |
---|
2174 | { |
---|
2175 | if (n>=0) |
---|
2176 | { |
---|
2177 | return(n); |
---|
2178 | } |
---|
2179 | else |
---|
2180 | { |
---|
2181 | return(-n); |
---|
2182 | } |
---|
2183 | } |
---|
2184 | |
---|
2185 | static proc commondenominator (matrix M) |
---|
2186 | "USAGE: commondenominator(M); M matrix |
---|
2187 | ASSUME: the base ring has characteristic zero |
---|
2188 | RETURN: int, the lowest common multiple of the denominators of the leading coefficients |
---|
2189 | of the entries in M |
---|
2190 | NOTE: the procedure is called from polymakeToIntmat" |
---|
2191 | { |
---|
2192 | int i,j; |
---|
2193 | int kgV=1; |
---|
2194 | // successively build the lowest common multiple of the denominators of the leading coefficients |
---|
2195 | // of the entries in M |
---|
2196 | for (i=1;i<=nrows(M);i++) |
---|
2197 | { |
---|
2198 | for (j=1;j<=ncols(M);j++) |
---|
2199 | { |
---|
2200 | kgV=lcm(kgV,int(denominator(leadcoef(M[i,j])))); |
---|
2201 | } |
---|
2202 | } |
---|
2203 | return(kgV); |
---|
2204 | } |
---|
2205 | |
---|
2206 | static proc maxPosInIntvec (intvec v) |
---|
2207 | "USAGE: maxPosInIntvec(v); v intvec |
---|
2208 | RETURN: int, the first position of a maximal entry in v |
---|
2209 | NOTE: called by sortintmat" |
---|
2210 | { |
---|
2211 | int max=v[1]; |
---|
2212 | int maxpos=1; |
---|
2213 | for (int i=2;i<=size(v);i++) |
---|
2214 | { |
---|
2215 | if (v[i]>max) |
---|
2216 | { |
---|
2217 | max=v[i]; |
---|
2218 | maxpos=i; |
---|
2219 | } |
---|
2220 | } |
---|
2221 | return(maxpos); |
---|
2222 | } |
---|
2223 | |
---|
2224 | static proc maxPosInIntmat (intmat v) |
---|
2225 | "USAGE: maxPosInIntmat(v); v intmat |
---|
2226 | ASSUME: v has a unique maximal entry |
---|
2227 | RETURN: intvec, the position (i,j) of the maximal entry in v |
---|
2228 | NOTE: called by findOrientedBoundary" |
---|
2229 | { |
---|
2230 | int max=v[1,1]; |
---|
2231 | intvec maxpos=1,1; |
---|
2232 | int i,j; |
---|
2233 | for (i=1;i<=nrows(v);i++) |
---|
2234 | { |
---|
2235 | for (j=1;j<=ncols(v);j++) |
---|
2236 | { |
---|
2237 | if (v[i,j]>max) |
---|
2238 | { |
---|
2239 | max=v[i,j]; |
---|
2240 | maxpos=i,j; |
---|
2241 | } |
---|
2242 | } |
---|
2243 | } |
---|
2244 | return(maxpos); |
---|
2245 | } |
---|
2246 | |
---|
2247 | static proc sortintvec (intvec w) |
---|
2248 | "USAGE: sortintvec(v); v intvec |
---|
2249 | RETURN: intvec, the entries of v are ordered in an ascending way |
---|
2250 | NOTE: called from ellipticNF" |
---|
2251 | { |
---|
2252 | int j,k,stop; |
---|
2253 | intvec v=w[1]; |
---|
2254 | for (j=2;j<=size(w);j++) |
---|
2255 | { |
---|
2256 | k=1; |
---|
2257 | stop=0; |
---|
2258 | while ((k<=size(v)) and (stop==0)) |
---|
2259 | { |
---|
2260 | if (v[k]<w[j]) |
---|
2261 | { |
---|
2262 | k++; |
---|
2263 | } |
---|
2264 | else |
---|
2265 | { |
---|
2266 | stop=1; |
---|
2267 | } |
---|
2268 | } |
---|
2269 | if (k==size(v)+1) |
---|
2270 | { |
---|
2271 | v=v,w[j]; |
---|
2272 | } |
---|
2273 | else |
---|
2274 | { |
---|
2275 | if (k==1) |
---|
2276 | { |
---|
2277 | v=w[j],v; |
---|
2278 | } |
---|
2279 | else |
---|
2280 | { |
---|
2281 | v=v[1..k-1],w[j],v[k..size(v)]; |
---|
2282 | } |
---|
2283 | } |
---|
2284 | } |
---|
2285 | return(v); |
---|
2286 | } |
---|
2287 | |
---|
2288 | static proc sortlistbyintvec (list L,intvec w) |
---|
2289 | "USAGE: sortlistbyintvec(L,w); L list, w intvec |
---|
2290 | RETURN: list, the entries of L are ordered such that the corresponding reordering of |
---|
2291 | w would order w in an ascending way |
---|
2292 | NOTE: called from ellipticNF" |
---|
2293 | { |
---|
2294 | int j,k,stop; |
---|
2295 | intvec v=w[1]; |
---|
2296 | list LL=L[1]; |
---|
2297 | for (j=2;j<=size(w);j++) |
---|
2298 | { |
---|
2299 | k=1; |
---|
2300 | stop=0; |
---|
2301 | while ((k<=size(v)) and (stop==0)) |
---|
2302 | { |
---|
2303 | if (v[k]<w[j]) |
---|
2304 | { |
---|
2305 | k++; |
---|
2306 | } |
---|
2307 | else |
---|
2308 | { |
---|
2309 | stop=1; |
---|
2310 | } |
---|
2311 | } |
---|
2312 | if (k==size(v)+1) |
---|
2313 | { |
---|
2314 | v=v,w[j]; |
---|
2315 | LL=insert(LL,L[j],size(LL)); |
---|
2316 | } |
---|
2317 | else |
---|
2318 | { |
---|
2319 | if (k==1) |
---|
2320 | { |
---|
2321 | v=w[j],v; |
---|
2322 | LL=insert(LL,L[j]); |
---|
2323 | } |
---|
2324 | else |
---|
2325 | { |
---|
2326 | v=v[1..k-1],w[j],v[k..size(v)]; |
---|
2327 | LL=insert(LL,L[j],k-1); |
---|
2328 | } |
---|
2329 | } |
---|
2330 | } |
---|
2331 | return(LL); |
---|
2332 | } |
---|
2333 | |
---|
2334 | static proc matrixtointmat (matrix MM) |
---|
2335 | "USAGE: matrixtointmat(v); MM matrix |
---|
2336 | ASSUME: MM is a matrix with only integers as entries |
---|
2337 | RETURN: intmat, the matrix MM has been transformed to type intmat |
---|
2338 | NOTE: called from ellipticNF" |
---|
2339 | { |
---|
2340 | intmat M[nrows(MM)][ncols(MM)]=M; |
---|
2341 | int i,j; |
---|
2342 | for (i=1;i<=nrows(M);i++) |
---|
2343 | { |
---|
2344 | for (j=1;j<=ncols(M);j++) |
---|
2345 | { |
---|
2346 | execute("M["+string(i)+","+string(j)+"]="+string(MM[i,j])+";"); |
---|
2347 | } |
---|
2348 | } |
---|
2349 | return(M); |
---|
2350 | } |
---|
2351 | |
---|
2352 | ////////////////////////////////////////////////////////////////////////////// |
---|
2353 | |
---|
2354 | static proc polygonToCoordinates (list points) |
---|
2355 | "USAGE: polygonToCoordinates(points); points list |
---|
2356 | ASSUME: points is a list of integer vectors each of size two describing the |
---|
2357 | marked points of a convex lattice polygon like the output of |
---|
2358 | polygonDB |
---|
2359 | RETURN: list, the first entry is a string representing the coordinates |
---|
2360 | corresponding to the latticpoints seperated by commata |
---|
2361 | the second entry is a list where the ith entry is a string |
---|
2362 | representing the coordinate of corresponding to the ith |
---|
2363 | lattice point the third entry is the latex format of the |
---|
2364 | first entry |
---|
2365 | NOTE: the procedure is called by fan" |
---|
2366 | { |
---|
2367 | string coord; |
---|
2368 | list coords; |
---|
2369 | string latex; |
---|
2370 | for (int i=1;i<=size(points);i++) |
---|
2371 | { |
---|
2372 | coords[i]="u"+string(points[i][1])+string(points[i][2]); |
---|
2373 | coord=coord+coords[i]+","; |
---|
2374 | latex=latex+"u_{"+string(points[i][1])+string(points[i][2])+"},"; |
---|
2375 | } |
---|
2376 | coord=coord[1,size(coord)-1]; |
---|
2377 | latex=latex[1,size(latex)-1]; |
---|
2378 | return(list(coord,coords,latex)); |
---|
2379 | } |
---|
2380 | |
---|
2381 | static proc intmatAddFirstColumn (intmat M,string art) |
---|
2382 | "USAGE: intmatAddFirstColumn(M,art); M intmat, art string |
---|
2383 | ASSUME: - M is an integer matrix where a first column of 0's or 1's should be added |
---|
2384 | @* - art is one of the following strings: |
---|
2385 | @* + 'rays' : indicating that a first column of 0's should be added |
---|
2386 | @* + 'points' : indicating that a first column of 1's should be added |
---|
2387 | RETURN: intmat, a first column has been added to the matrix" |
---|
2388 | { |
---|
2389 | intmat N[nrows(M)][ncols(M)+1]; |
---|
2390 | int i,j; |
---|
2391 | for (i=1;i<=nrows(M);i++) |
---|
2392 | { |
---|
2393 | if (art=="rays") |
---|
2394 | { |
---|
2395 | N[i,1]=0; |
---|
2396 | } |
---|
2397 | else |
---|
2398 | { |
---|
2399 | N[i,1]=1; |
---|
2400 | } |
---|
2401 | for (j=1;j<=ncols(M);j++) |
---|
2402 | { |
---|
2403 | N[i,j+1]=M[i,j]; |
---|
2404 | } |
---|
2405 | } |
---|
2406 | return(N); |
---|
2407 | } |
---|
2408 | |
---|
2409 | |
---|
2410 | |
---|