1 | //////////////////////////////////////////////////////////////////// |
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2 | version="version gitfan.lib 4.0.0.0 Jun_2013 "; |
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3 | category="Algebraic Geometry"; |
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4 | info=" |
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5 | LIBRARY: gitfan.lib Compute GIT-fans. |
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6 | |
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7 | AUTHORS: Janko Boehm boehm@mathematik.uni-kl.de |
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8 | @* Simon Keicher keicher@mail.mathematik.uni-tuebingen.de |
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9 | @* Yue Ren ren@mathematik.uni-kl.de |
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10 | |
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11 | OVERVIEW: |
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12 | This library computes GIT-fans, torus orbits and GKZ-fans. |
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13 | It uses the package 'gfanlib' by Anders N. Jensen |
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14 | and some algorithms have been outsourced to C++ to improve the performance. |
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15 | Check https://github.com/skeicher/gitfan_singular for updates. |
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16 | |
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17 | KEYWORDS: library; gitfan; GIT; geometric invariant theory; quotients |
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18 | |
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19 | PROCEDURES: |
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20 | afaces(ideal); Returns a list of intvecs that correspond to all a-faces |
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21 | gitCone(ideal,bigintmat,bigintmat); Returns the GIT-cone around the given weight vector w |
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22 | gitFan(ideal,bigintmat); Returns the GIT-fan of the H-action defined by Q on V(a) |
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23 | gkzFan(bigintmat); Returns the GKZ-fan of the matrix Q |
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24 | isAface(ideal,intvec); Checks whether intvec corresponds to an ideal-face |
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25 | orbitCones(ideal,bigintmat); Returns the list of all projected a-faces |
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26 | "; |
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27 | |
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28 | LIB "parallel.lib"; // for parallelWaitAll |
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29 | |
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30 | //////////////////////////////////////////////////// |
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31 | proc mod_init() |
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32 | { |
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33 | LIB"gfanlib.so"; |
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34 | } |
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35 | |
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36 | static proc int2face(int n, int r) |
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37 | { |
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38 | int k = r-1; |
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39 | intvec v; |
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40 | int n0 = n; |
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41 | |
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42 | while(n0 > 0){ |
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43 | while(2^k > n0){ |
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44 | k--; |
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45 | //v[size(v)+1] = 0; |
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46 | } |
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47 | |
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48 | v = k+1,v; |
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49 | n0 = n0 - 2^k; |
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50 | k--; |
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51 | } |
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52 | v = v[1..size(v)-1]; |
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53 | return(v); |
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54 | } |
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55 | |
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56 | ///////////////////////////////// |
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57 | |
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58 | proc isAface(ideal a, intvec gam0) |
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59 | "USAGE: isAface(a,gam0); a: ideal, gam0:intvec |
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60 | PURPOSE: Checks whether the face of the positive orthant, |
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61 | that is spanned by all i-th unit vectors, |
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62 | where i ranges amongst the entries of gam0, |
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63 | is an a-face. |
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64 | RETURN: int |
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65 | EXAMPLE: example isaface; shows an example |
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66 | " |
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67 | { |
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68 | poly pz; |
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69 | |
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70 | // special case: gam0 is the zero-cone: |
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71 | if (size(gam0) == 1 and gam0[1] == 0){ |
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72 | ideal G; |
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73 | |
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74 | // is an a-face if and only if RL0(0,...,0) = const |
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75 | // set all entries to 0: |
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76 | int i; |
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77 | for (int k = 1; k <= size(a); k++) { |
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78 | pz = subst(a[k], var(1), 0); |
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79 | for (i = 2; i <= nvars(basering); i++) { |
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80 | pz = subst(pz, var(i), 0); |
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81 | } |
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82 | G = G, pz; |
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83 | } |
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84 | |
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85 | G = std(G); |
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86 | |
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87 | // monomial inside?: |
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88 | if(1 == G){ |
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89 | return(0); |
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90 | } |
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91 | |
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92 | return(1); |
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93 | } |
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94 | |
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95 | |
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96 | // ring is too big: Switch to KK[T_i; e_i\in gam0] instead: |
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97 | string initNewRing = "ring Rgam0 = 0,("; |
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98 | |
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99 | for (int i=1; i<size(gam0); i++){ |
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100 | initNewRing = initNewRing + string(var(gam0[i])) + ","; |
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101 | } |
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102 | |
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103 | initNewRing = initNewRing + string(var(gam0[size(gam0)])) + "),dp;"; |
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104 | def R = basering; |
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105 | execute(initNewRing); |
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106 | |
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107 | ideal agam0 = imap(R,a); |
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108 | |
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109 | poly p = var(1); // first entry of g; p = prod T_i with i element of g |
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110 | for ( i = 2; i <= nvars(basering); i++ ) { |
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111 | p = p * var(i); |
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112 | } |
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113 | // p is now the product over all T_i, with e_i in gam0 |
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114 | |
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115 | agam0 = agam0, p - 1; // rad-membership |
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116 | ideal G = std(agam0); |
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117 | |
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118 | // does G contain 1?, i.e. is G = 1? |
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119 | if(G <> 1) { |
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120 | return(1); // true |
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121 | } |
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122 | |
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123 | return(0); // false |
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124 | } |
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125 | example |
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126 | { |
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127 | echo = 2; |
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128 | |
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129 | ring R = 0,(T(1..4)),dp; |
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130 | ideal I = T(1)*T(2)-T(4); |
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131 | |
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132 | intvec w = 1,4; |
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133 | intvec v = 1,2,4; |
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134 | |
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135 | isAface(I,w); // should be 0 |
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136 | "-----------"; |
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137 | isAface(I,v); // should be 1 |
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138 | } |
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139 | |
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140 | //////////////////////////////////////////////////// |
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141 | |
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142 | proc afacesPart(ideal a, int d, int start, int end, int r){ |
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143 | intvec gam0; |
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144 | int i; |
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145 | list AF; |
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146 | |
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147 | for(i = start; i <= end; i++){ |
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148 | if(i < 2^r){ |
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149 | gam0 = int2face(i,r); |
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150 | |
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151 | // take gam0 only if it has |
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152 | // at least d rays: |
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153 | if(size(gam0) >= d){ |
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154 | if (isAface(a,gam0)){ |
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155 | AF[size(AF) + 1] = gam0; |
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156 | } |
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157 | } |
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158 | } |
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159 | } |
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160 | |
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161 | return(AF); |
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162 | } |
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163 | |
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164 | //////////////////////////////////////////////////// |
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165 | |
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166 | proc afaces(ideal a, list #) |
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167 | "USAGE: afaces(a, b, c); a: ideal, d: int, c: int |
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168 | PURPOSE: Returns a list of all a-faces (represented by intvecs). |
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169 | Moreover, it is possible to specify a dimensional bound b, |
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170 | upon which only cones of that dimension and above are returned. |
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171 | Lastly, as the computation is parallizable, one can specify c, |
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172 | the number of cores to be used by the computation. |
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173 | RETURN: a list of intvecs |
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174 | EXAMPLE: example afaces; shows an example |
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175 | " |
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176 | { |
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177 | int d = 1; |
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178 | int ncores = 1; |
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179 | |
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180 | if ((size(#) > 0) and (typeof(#[1]) == "int")){ |
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181 | d = #[1]; |
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182 | } |
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183 | |
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184 | if ((size(#) > 1) and (typeof(#[2]) == "int")){ |
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185 | ncores = #[2]; |
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186 | } |
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187 | |
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188 | list AF; |
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189 | intvec gam0; |
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190 | int r = nvars(basering); |
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191 | |
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192 | // check if 0 is an a-face: |
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193 | gam0 = 0; |
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194 | if (isAface(a,gam0)){ |
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195 | AF[size(AF) + 1] = gam0; |
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196 | } |
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197 | |
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198 | // check for other a-faces: |
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199 | // make ncores processes: |
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200 | int step = 2^r div ncores; |
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201 | int i; |
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202 | |
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203 | list args; |
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204 | for(int k = 0; k < ncores; k++){ |
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205 | args[size(args) + 1] = list(a, d, k * step + 1, (k+1) * step, r); |
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206 | } |
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207 | |
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208 | string command = "afacesPart"; |
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209 | list out = parallelWaitAll(command, args); |
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210 | |
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211 | // do remaining ones: |
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212 | for(i = ncores * step +1; i < 2^r; i++){ |
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213 | "another one needed"; |
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214 | gam0 = int2face(i,r); |
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215 | |
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216 | // take gam0 only if it has |
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217 | // at least d rays: |
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218 | if(size(gam0) >= d){ |
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219 | if (isAface(a,gam0)){ |
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220 | AF[size(AF) + 1] = gam0; |
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221 | } |
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222 | } |
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223 | } |
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224 | |
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225 | // read out afaces of out into AF: |
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226 | for(i = 1; i <= size(out); i++){ |
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227 | AF = AF + out[i]; |
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228 | } |
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229 | |
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230 | return(AF); |
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231 | } |
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232 | example |
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233 | { |
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234 | |
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235 | echo = 2; |
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236 | ring R = 0,T(1..3),dp; |
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237 | ideal a = T(1)+T(2)+T(3); |
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238 | |
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239 | list F = afaces(a,3,4); |
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240 | print(F); |
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241 | print(size(F)); |
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242 | |
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243 | // 2nd ex // |
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244 | ring R2 = 0,T(1..3),dp; |
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245 | ideal a2 = T(2)^2*T(3)^2+T(1)*T(3); |
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246 | |
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247 | list F2 = afaces(a2,3,4); |
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248 | print(F2); |
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249 | print(size(F2)); |
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250 | |
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251 | // 3rd ex // |
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252 | ring R3 = 0,T(1..3),dp; |
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253 | ideal a3 = 0; |
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254 | |
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255 | list F3 = afaces(a3,3,4); |
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256 | print(F3); |
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257 | print(size(F3)); |
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258 | |
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259 | // bigger example // |
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260 | ring R = 0,T(1..15),dp; |
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261 | ideal a = |
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262 | T(1)*T(10)-T(2)*T(7)+T(3)*T(6), |
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263 | T(1)*T(11)-T(2)*T(8)+T(4)*T(6), |
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264 | T(1)*T(12)-T(2)*T(9)+T(5)*T(6), |
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265 | T(1)*T(13)-T(3)*T(8)+T(4)*T(7), |
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266 | T(1)*T(14)-T(3)*T(9)+T(5)*T(7), |
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267 | T(1)*T(15)-T(4)*T(9)+T(5)*T(8), |
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268 | T(2)*T(13)-T(3)*T(11)+T(4)*T(10), |
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269 | T(2)*T(14)-T(3)*T(12)+T(5)*T(10), |
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270 | T(2)*T(15)-T(4)*T(12)+T(5)*T(11), |
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271 | T(3)*T(15)-T(4)*T(14)+T(5)*T(13), |
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272 | T(6)*T(13)-T(7)*T(11)+T(8)*T(10), |
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273 | T(6)*T(14)-T(7)*T(12)+T(9)*T(10), |
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274 | T(6)*T(15)-T(8)*T(12)+T(9)*T(11), |
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275 | T(7)*T(15)-T(8)*T(14)+T(9)*T(13), |
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276 | T(10)*T(15)-T(11)*T(14)+T(12)*T(13); |
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277 | |
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278 | int t = timer; |
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279 | list F4 = afaces(a,0,2); |
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280 | print(size(F4)); |
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281 | timer - t; |
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282 | |
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283 | int t = timer; |
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284 | list F4 = afaces(a,0); |
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285 | print(size(F4)); |
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286 | timer - t; |
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287 | |
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288 | } |
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289 | |
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290 | /////////////////////////////////////// |
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291 | |
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292 | proc orbitCones(ideal a, bigintmat Q, list #) |
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293 | "USAGE: orbitCones(a, Q, b, c); a: ideal, Q: bigintmat, b: int, c: int |
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294 | PURPOSE: Returns a list consisting of all cones Q(gam0) where gam0 is an a-face. |
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295 | Moreover, it is possible to specify a dimensional bound b, |
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296 | upon which only cones of that dimension and above are returned. |
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297 | Lastly, as the computation is parallizable, one can specify c, |
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298 | the number of cores to be used by the computation. |
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299 | RETURN: a list of cones |
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300 | EXAMPLE: example orbitCones; shows an example |
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301 | " |
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302 | { |
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303 | list AF; |
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304 | |
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305 | if((size(#) > 1) and (typeof(#[2]) == "int")){ |
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306 | AF = afaces(a, nrows(Q), #[2]); |
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307 | } else |
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308 | { |
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309 | AF = afaces(a, nrows(Q)); |
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310 | } |
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311 | |
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312 | int dimensionBound = 0; |
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313 | if((size(#) > 0) and (typeof(#[1]) == "int")){ |
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314 | dimensionBound = #[1]; |
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315 | } |
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316 | |
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317 | list OC; |
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318 | intvec gam0; |
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319 | int j; |
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320 | |
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321 | for(int i = 1; i <= size(AF); i++){ |
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322 | gam0 = AF[i]; |
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323 | |
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324 | if(gam0 == 0){ |
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325 | bigintmat M[1][nrows(Q)]; |
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326 | } else { |
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327 | bigintmat M[size(gam0)][nrows(Q)]; |
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328 | for (j = 1; j <= size(gam0); j++){ |
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329 | M[j,1..ncols(M)] = Q[1..nrows(Q),gam0[j]]; |
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330 | } |
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331 | } |
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332 | cone c = coneViaPoints(M); |
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333 | |
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334 | if((dimension(c) >= dimensionBound) and (!(listContainsCone(OC, c)))){ |
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335 | OC[size(OC)+1] = c; |
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336 | } |
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337 | |
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338 | kill M, c; |
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339 | } |
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340 | |
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341 | return(OC); |
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342 | } |
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343 | example |
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344 | { |
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345 | echo=2; |
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346 | intmat Q[3][4] = |
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347 | 1,0,1,0, |
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348 | 0,1,0,1, |
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349 | 0,0,1,1; |
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350 | |
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351 | ring R = 0,T(1..4),dp; |
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352 | ideal a = 0; |
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353 | |
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354 | orbitCones(a, Q); |
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355 | } |
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356 | |
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357 | /////////////////////////////////////// |
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358 | |
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359 | proc gitCone(ideal a, bigintmat Q, bigintmat w) |
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360 | "USAGE: gitCone(a, Q, w); a: ideal, Q:bigintmat, w:bigintmat |
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361 | PURPOSE: Returns the GIT-cone lambda(w), i.e. the intersection of all |
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362 | orbit cones containing the vector w. |
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363 | NOTE: call this only if you are interested in a single GIT-cone. |
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364 | RETURN: a cone. |
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365 | EXAMPLE: example gitCone; shows an example |
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366 | " |
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367 | { |
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368 | list OC = orbitCones(a, Q); |
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369 | cone lambda = nrows(Q); |
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370 | |
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371 | for(int i = 1; i <= size(OC); i++){ |
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372 | cone c = OC[i]; |
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373 | |
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374 | if(containsInSupport(c, w)){ |
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375 | lambda = convexIntersection(lambda, c); |
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376 | } |
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377 | |
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378 | kill c; |
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379 | } |
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380 | |
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381 | return(lambda); |
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382 | } |
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383 | example |
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384 | { |
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385 | echo=2; |
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386 | intmat Q[3][4] = |
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387 | 1,0,1,0, |
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388 | 0,1,0,1, |
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389 | 0,0,1,1; |
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390 | |
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391 | ring R = 0,T(1..4),dp; |
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392 | ideal a = 0; |
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393 | |
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394 | bigintmat w[1][3] = 3,3,1; |
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395 | cone lambda = gitCone(a, Q, w); |
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396 | rays(lambda); |
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397 | |
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398 | bigintmat w2[1][3] = 1,1,1; |
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399 | cone lambda2 = gitCone(a, Q, w2); |
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400 | rays(lambda2); |
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401 | } |
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402 | |
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403 | ///////////////////////////////////// |
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404 | |
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405 | proc gitFan(ideal a, bigintmat Q, list #) |
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406 | "USAGE: gitFan(a, Q); a: ideal, Q:bigintmat |
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407 | PURPOSE: Returns the GIT-fan of the H-action defined by Q on V(a). |
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408 | An optional third parameter of type 'int' is interpreted as the |
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409 | number of CPU-cores you would like to use. |
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410 | Note that 'system("cpu");' returns the number of cpu available |
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411 | in your system. |
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412 | RETURN: a fan. |
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413 | EXAMPLE: example gitFan; shows an example |
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414 | " |
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415 | { |
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416 | list OC = orbitCones(a, Q, #); |
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417 | |
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418 | fan f = refineCones(OC, Q); |
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419 | return(f); |
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420 | } |
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421 | example |
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422 | { |
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423 | echo=2; |
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424 | intmat Q[3][4] = |
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425 | 1,0,1,0, |
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426 | 0,1,0,1, |
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427 | 0,0,1,1; |
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428 | |
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429 | ring R = 0,T(1..4),dp; |
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430 | ideal a = 0; |
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431 | |
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432 | gitFan(a, Q); |
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433 | |
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434 | // 2nd example // |
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435 | kill Q; |
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436 | intmat Q[3][6] = |
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437 | 1,1,0,0,-1,-1, |
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438 | 0,1,1,-1,-1,0, |
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439 | 1,1,1,1,1,1; |
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440 | |
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441 | ring R = 0,T(1..6),dp; |
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442 | ideal a = T(1)*T(6) + T(2)*T(5) + T(3)*T(4); |
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443 | |
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444 | int t = rtimer; |
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445 | fan F = gitFan(a, Q); |
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446 | t = rtimer - t; |
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447 | |
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448 | int tt = rtimer; |
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449 | fan F = gitFan(a, Q, 4); |
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450 | tt = rtimer - tt; |
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451 | |
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452 | t; |
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453 | tt; |
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454 | "--------"; |
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455 | kill R, Q, t, tt; |
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456 | // next example // |
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457 | ring R = 0,T(1..10),dp; |
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458 | ideal a = T(5)*T(10)-T(6)*T(9)+T(7)*T(8), |
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459 | T(1)*T(9)-T(2)*T(7)+T(4)*T(5), |
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460 | T(1)*T(8)-T(2)*T(6)+T(3)*T(5), |
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461 | T(1)*T(10)-T(3)*T(7)+T(4)*T(6), |
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462 | T(2)*T(10)-T(3)*T(9)+T(4)*T(8); |
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463 | |
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464 | bigintmat Q[4][10] = |
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465 | 1,0,0,0,1,1,1,0,0,0, |
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466 | 0,1,0,0,1,0,0,1,1,0, |
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467 | 0,0,1,0,0,1,0,1,0,1, |
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468 | 0,0,0,1,0,0,1,0,1,1; |
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469 | |
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470 | int t = rtimer; |
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471 | fan F = gitFan(a, Q); |
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472 | t = rtimer - t; |
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473 | |
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474 | int tt = rtimer; |
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475 | fan F = gitFan(a, Q, 4); |
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476 | tt = rtimer - tt; |
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477 | |
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478 | t; |
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479 | tt; |
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480 | |
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481 | "--------"; |
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482 | kill R, Q, t, tt; |
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483 | // next example // |
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484 | ring R = 0,T(1..15),dp; |
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485 | ideal a = |
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486 | T(1)*T(10)-T(2)*T(7)+T(3)*T(6), |
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487 | T(1)*T(11)-T(2)*T(8)+T(4)*T(6), |
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488 | T(1)*T(12)-T(2)*T(9)+T(5)*T(6), |
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489 | T(1)*T(13)-T(3)*T(8)+T(4)*T(7), |
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490 | T(1)*T(14)-T(3)*T(9)+T(5)*T(7), |
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491 | T(1)*T(15)-T(4)*T(9)+T(5)*T(8), |
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492 | T(2)*T(13)-T(3)*T(11)+T(4)*T(10), |
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493 | T(2)*T(14)-T(3)*T(12)+T(5)*T(10); |
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494 | |
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495 | bigintmat Q[5][15] = |
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496 | 1,0,0,0,0,1,1,1,1,0,0,0,0,0,0, |
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497 | 0,1,0,0,0,1,0,0,0,1,1,1,0,0,0, |
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498 | 0,0,1,0,0,0,1,0,0,1,0,0,1,1,0, |
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499 | 0,0,0,1,0,0,0,1,0,0,1,0,1,0,1, |
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500 | 0,0,0,0,1,0,0,0,1,0,0,1,0,1,1; |
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501 | |
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502 | int t = rtimer; |
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503 | fan F = gitFan(a, Q); |
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504 | t = rtimer - t; |
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505 | |
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506 | int tt = rtimer; |
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507 | fan F = gitFan(a, Q, 4); |
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508 | tt = rtimer - tt; |
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509 | |
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510 | t; |
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511 | tt; |
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512 | |
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513 | } |
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514 | |
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515 | ///////////////////////////////////// |
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516 | // Computes all simplicial orbit cones |
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517 | // w.r.t. the 0-ideal: |
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518 | |
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519 | static proc simplicialToricOrbitCones(bigintmat Q){ |
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520 | intvec gam0; |
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521 | list OC; |
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522 | cone c; |
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523 | int r = ncols(Q); |
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524 | int j; |
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525 | |
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526 | for(int i = 1; i < 2^r; i++ ){ |
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527 | gam0 = int2face(i,r); |
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528 | |
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529 | // each simplicial cone is generated by |
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530 | // exactly nrows(Q) many columns of Q: |
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531 | if(size(gam0) == nrows(Q)){ |
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532 | bigintmat M[size(gam0)][nrows(Q)]; |
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533 | |
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534 | for(j = 1; j <= size(gam0); j++){ |
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535 | M[j,1..ncols(M)] = Q[1..nrows(Q),gam0[j]]; |
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536 | } |
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537 | |
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538 | c = coneViaPoints(M); |
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539 | |
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540 | if((dimension(c) == nrows(Q)) and (!(listContainsCone(OC, c)))){ |
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541 | OC[size(OC)+1] = c; |
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542 | } |
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543 | |
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544 | kill M; |
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545 | } |
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546 | } |
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547 | |
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548 | return(OC); |
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549 | } |
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550 | |
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551 | ///////////////////////////////////// |
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552 | |
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553 | proc gkzFan(bigintmat Q) |
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554 | "USAGE: gkzFan(Q); a: ideal, Q:bigintmat |
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555 | PURPOSE: Returns the GKZ-fan of the matrix Q. |
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556 | RETURN: a fan. |
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557 | EXAMPLE: example gkzFan; shows an example |
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558 | " |
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559 | { |
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560 | // only difference to gitFan: |
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561 | // it suffices to consider all faces |
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562 | // that are simplicial: |
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563 | list OC = simplicialToricOrbitCones(Q); |
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564 | |
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565 | fan f = refineCones(OC, Q); |
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566 | return(f); |
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567 | } |
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568 | example |
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569 | { |
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570 | echo=2; |
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571 | intmat Q[3][4] = |
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572 | 1,0,1,0, |
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573 | 0,1,0,1, |
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574 | 0,0,1,1; |
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575 | |
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576 | gkzFan(Q); |
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577 | } |
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