1 | /////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id: finvar.lib,v 1.50 2006-07-18 15:48:13 Singular Exp $" |
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3 | category="Invariant theory"; |
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4 | info=" |
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5 | LIBRARY: finvar.lib Invariant Rings of Finite Groups |
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6 | AUTHOR: Agnes E. Heydtmann, email: agnes@math.uni-sb.de; |
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7 | Simon A. King, email: king@mfo.de |
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8 | OVERVIEW: |
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9 | A library for computing polynomial invariants of finite matrix groups and |
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10 | generators of related varieties. The algorithms are based on B. Sturmfels, |
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11 | G. Kemper and W. Decker et al.. |
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12 | |
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13 | MAIN PROCEDURES: |
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14 | invariant_ring() generators of the invariant ring (i.r.) |
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15 | invariant_ring_random() generators of the i.r., randomized alg. |
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16 | primary_invariants() primary invariants (p.i.) |
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17 | primary_invariants_random() primary invariants, randomized alg. |
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18 | |
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19 | AUXILIARY PROCEDURES: |
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20 | cyclotomic() cyclotomic polynomial |
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21 | group_reynolds() finite group and Reynolds operator (R.o.) |
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22 | molien() Molien series (M.s.) |
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23 | reynolds_molien() Reynolds operator and Molien series |
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24 | partial_molien() partial expansion of Molien series |
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25 | evaluate_reynolds() image under the Reynolds operator |
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26 | invariant_basis() basis of homogeneous invariants of a degree |
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27 | invariant_basis_reynolds() as invariant_basis(), with R.o. |
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28 | primary_char0() primary invariants (p.i.) in char 0 |
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29 | primary_charp() primary invariants in char p |
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30 | primary_char0_no_molien() p.i., char 0, without Molien series |
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31 | primary_charp_no_molien() p.i., char p, without Molien series |
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32 | primary_charp_without() p.i., char p, without R.o. or Molien series |
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33 | primary_char0_random() primary invariants in char 0, randomized |
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34 | primary_charp_random() primary invariants in char p, randomized |
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35 | primary_char0_no_molien_random() p.i., char 0, without M.s., randomized |
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36 | primary_charp_no_molien_random() p.i., char p, without M.s., randomized |
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37 | primary_charp_without_random() p.i., char p, without R.o. or M.s., random. |
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38 | power_products() exponents for power products |
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39 | secondary_char0() secondary invariants (s.i.) in char 0 |
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40 | irred_secondary_char0() irreducible secondary invariants in char 0 |
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41 | secondary_charp() secondary invariants in char p |
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42 | secondary_no_molien() secondary invariants, without Molien series |
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43 | secondary_and_irreducibles_no_molien() s.i. & irreducible s.i., without M.s. |
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44 | secondary_not_cohen_macaulay() s.i. when invariant ring not Cohen-Macaulay |
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45 | orbit_variety() ideal of the orbit variety |
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46 | rel_orbit_variety() ideal of a relative orbit variety (new version) |
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47 | relative_orbit_variety() ideal of a relative orbit variety (old version) |
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48 | image_of_variety() ideal of the image of a variety |
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49 | "; |
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50 | /////////////////////////////////////////////////////////////////////////////// |
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51 | // perhaps useful procedures (no help provided): |
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52 | // unique() is a matrix among other matrices? |
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53 | // exponent() gives the exponent of a number |
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54 | // sort_of_invariant_basis() lin. ind. invariants of a degree mod p.i. |
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55 | // next_vector lists all of Z^n with first nonzero entry 1 |
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56 | // int_number_map integers 1..q are maped to q field elements |
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57 | // search searches a number of p.i., char 0 |
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58 | // p_search searches a number of p.i., char p |
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59 | // search_random searches a # of p.i., char 0, randomized |
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60 | // p_search_random searches a # of p.i., char p, randomized |
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61 | // concat_intmat concatenates two integer matrices |
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62 | /////////////////////////////////////////////////////////////////////////////// |
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63 | |
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64 | LIB "matrix.lib"; |
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65 | LIB "elim.lib"; |
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66 | LIB "general.lib"; |
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67 | LIB "algebra.lib"; |
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68 | |
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69 | /////////////////////////////////////////////////////////////////////////////// |
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70 | // Checks whether the last parameter, being a matrix, is among the previous |
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71 | // parameters, also being matrices |
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72 | /////////////////////////////////////////////////////////////////////////////// |
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73 | proc unique (list #) |
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74 | { int s=size(#); def m=#[s]; |
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75 | for (int i=1;i<s;i++) |
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76 | { if (#[i]==m) |
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77 | { return(0); } |
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78 | } |
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79 | return(1); |
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80 | } |
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81 | /////////////////////////////////////////////////////////////////////////////// |
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82 | |
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83 | proc cyclotomic (int i) |
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84 | "USAGE: cyclotomic(i); i integer > 0 |
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85 | RETURNS: the i-th cyclotomic polynomial (type <poly>) as one in the first ring |
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86 | variable |
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87 | THEORY: x^i-1 is divided by the j-th cyclotomic polynomial where j takes on |
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88 | the value of proper divisors of i |
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89 | EXAMPLE: example cyclotomic; shows an example |
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90 | " |
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91 | { if (i<=0) |
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92 | { "ERROR: the input should be > 0."; |
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93 | return(); |
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94 | } |
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95 | poly v1=var(1); |
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96 | if (i==1) |
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97 | { return(v1-1); // 1-st cyclotomic polynomial |
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98 | } |
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99 | poly min=v1^i-1; |
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100 | matrix s[1][2]; |
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101 | min=min/(v1-1); // dividing by the 1-st cyclotomic |
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102 | // polynomial |
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103 | int j=2; |
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104 | int n; |
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105 | poly c; |
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106 | int flag=1; |
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107 | while(2*j<=i) // there are no proper divisors of i |
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108 | { if ((i%j)==0) // greater than i/2 |
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109 | { if (flag==1) |
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110 | { n=j; // n stores the first proper divisor of |
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111 | } // i > 1 |
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112 | flag=0; |
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113 | c=cyclotomic(j); // recursive computation |
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114 | s=min,c; |
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115 | s=matrix(syz(ideal(s))); // dividing |
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116 | min=s[2,1]; |
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117 | } |
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118 | if (n*j==i) // the earliest possible point to break |
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119 | { break; |
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120 | } |
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121 | j++; |
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122 | } |
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123 | min=min/leadcoef(min); // making sure that the leading |
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124 | return(min); // coefficient is 1 |
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125 | } |
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126 | example |
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127 | { "EXAMPLE:"; echo=2; |
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128 | ring R=0,(x,y,z),dp; |
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129 | print(cyclotomic(25)); |
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130 | } |
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131 | |
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132 | proc group_reynolds (list #) |
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133 | "USAGE: group_reynolds(G1,G2,...[,v]); |
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134 | G1,G2,...: nxn <matrices> generating a finite matrix group, v: an |
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135 | optional <int> |
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136 | ASSUME: n is the number of variables of the basering, g the number of group |
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137 | elements |
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138 | RETURN: a <list>, the first list element will be a gxn <matrix> representing |
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139 | the Reynolds operator if we are in the non-modular case; if the |
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140 | characteristic is >0, minpoly==0 and the finite group non-cyclic the |
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141 | second list element is an <int> giving the lowest common multiple of |
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142 | the matrix group elements' order (used in molien); in general all |
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143 | other list elements are nxn <matrices> listing all elements of the |
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144 | finite group |
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145 | DISPLAY: information if v does not equal 0 |
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146 | THEORY: The entire matrix group is generated by getting all left products of |
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147 | generators with the new elements from the last run through the loop |
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148 | (or the generators themselves during the first run). All the ones that |
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149 | have been generated before are thrown out and the program terminates |
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150 | when no new elements found in one run. Additionally each time a new |
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151 | group element is found the corresponding ring mapping of which the |
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152 | Reynolds operator is made up is generated. They are stored in the rows |
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153 | of the first return value. |
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154 | EXAMPLE: example group_reynolds; shows an example |
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155 | " |
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156 | { int ch=char(basering); // the existance of the Reynolds operator |
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157 | // is dependent on the characteristic of |
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158 | // the base field |
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159 | int gen_num; // number of generators |
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160 | //------------------------ making sure the input is okay --------------------- |
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161 | if (typeof(#[size(#)])=="int") |
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162 | { if (size(#)==1) |
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163 | { "ERROR: there are no matrices given among the parameters"; |
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164 | return(); |
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165 | } |
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166 | int v=#[size(#)]; |
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167 | gen_num=size(#)-1; |
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168 | } |
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169 | else // last parameter is not <int> |
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170 | { int v=0; // no information is default |
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171 | gen_num=size(#); |
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172 | } |
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173 | if (typeof(#[1])<>"matrix") |
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174 | { "ERROR: The parameters must be a list of matrices and maybe an <int>"; |
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175 | return(); |
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176 | } |
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177 | int n=nrows(#[1]); |
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178 | if (n<>nvars(basering)) |
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179 | { "ERROR: the number of variables of the basering needs to be the same"; |
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180 | " as the dimension of the matrices"; |
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181 | return(); |
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182 | } |
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183 | if (n<>ncols(#[1])) |
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184 | { "ERROR: matrices need to be square and of the same dimensions"; |
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185 | return(); |
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186 | } |
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187 | matrix vars=matrix(maxideal(1)); // creating an nx1-matrix containing the |
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188 | vars=transpose(vars); // variables of the ring - |
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189 | matrix REY=#[1]*vars; // calculating the first ring mapping - |
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190 | // REY will contain the Reynolds |
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191 | // operator - |
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192 | matrix G(1)=#[1]; // G(k) are elements of the group - |
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193 | if (ch<>0 && minpoly==0 && gen_num<>1) // finding out of which order the |
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194 | { matrix I=diag(1,n); // group element is |
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195 | matrix TEST=G(1); |
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196 | int o1=1; |
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197 | int o2; |
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198 | while (TEST<>I) |
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199 | { TEST=TEST*G(1); |
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200 | o1++; |
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201 | } |
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202 | } |
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203 | int i=1; |
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204 | // -------------- doubles among the generators should be avoided ------------- |
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205 | for (int j=2;j<=gen_num;j++) // this loop adds the parameters to the |
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206 | { // group, leaving out doubles and |
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207 | // checking whether the parameters are |
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208 | // compatible with the task of the |
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209 | // procedure |
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210 | if (not(typeof(#[j])=="matrix")) |
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211 | { "ERROR: The parameters must be a list of matrices and maybe an <int>"; |
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212 | return(); |
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213 | } |
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214 | if ((n!=nrows(#[j])) or (n!=ncols(#[j]))) |
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215 | { "ERROR: matrices need to be square and of the same dimensions"; |
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216 | return(); |
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217 | } |
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218 | if (unique(G(1..i),#[j])) |
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219 | { i++; |
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220 | matrix G(i)=#[j]; |
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221 | if (ch<>0 && minpoly==0) // finding out of which order the group |
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222 | { TEST=G(i); // element is |
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223 | o2=1; |
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224 | while (TEST<>I) |
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225 | { TEST=TEST*G(i); |
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226 | o2++; |
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227 | } |
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228 | o1=o1*o2/gcd(o1,o2); // lowest common multiple of the element |
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229 | } // orders - |
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230 | REY=concat(REY,#[j]*vars); // adding ring homomorphisms to REY |
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231 | } |
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232 | } |
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233 | int g=i; // G(1)..G(i) are generators without |
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234 | // doubles - g generally is the number |
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235 | // of elements in the group so far - |
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236 | j=i; // j is the number of new elements that |
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237 | // we use as factors |
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238 | int k, m, l; |
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239 | if (v) |
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240 | { ""; |
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241 | " Generating the entire matrix group and the Reynolds operator..."; |
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242 | ""; |
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243 | } |
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244 | // -------------- main loop that finds all the group elements ---------------- |
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245 | while (1) |
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246 | { l=0; // l is the number of products we get in |
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247 | // one going |
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248 | for (m=g-j+1;m<=g;m++) |
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249 | { for (k=1;k<=i;k++) |
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250 | { l=l+1; |
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251 | matrix P(l)=G(k)*G(m); // possible new element |
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252 | } |
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253 | } |
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254 | j=0; |
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255 | for (k=1;k<=l;k++) |
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256 | { if (unique(G(1..g),P(k))) |
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257 | { j++; // a new factor for next run |
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258 | g++; |
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259 | matrix G(g)=P(k); // a new group element - |
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260 | if (ch<>0 && minpoly==0 && i<>1) // finding out of which order the |
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261 | { TEST=G(g); //group element is |
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262 | o2=1; |
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263 | while (TEST<>I) |
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264 | { TEST=TEST*G(g); |
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265 | o2++; |
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266 | } |
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267 | o1=o1*o2/gcd(o1,o2); // lowest common multiple of the element |
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268 | } // orders - |
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269 | REY=concat(REY,P(k)*vars); // adding new mapping to REY |
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270 | if (v) |
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271 | { " Group element "+string(g)+" has been found."; |
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272 | } |
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273 | } |
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274 | kill P(k); |
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275 | } |
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276 | if (j==0) // when we didn't add any new elements |
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277 | { break; // in one run through the while loop |
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278 | } // we are done |
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279 | } |
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280 | if (v) |
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281 | { if (g<=i) |
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282 | { " There are only "+string(g)+" group elements."; |
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283 | } |
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284 | ""; |
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285 | } |
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286 | REY=transpose(REY); // when we evaluate the Reynolds operator |
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287 | // later on, we actually want 1xn |
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288 | // matrices |
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289 | if (ch<>0) |
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290 | { if ((g%ch)==0) |
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291 | { if (voice==2) |
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292 | { "WARNING: The characteristic of the coefficient field divides the group order."; |
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293 | " Proceed without the Reynolds operator!"; |
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294 | } |
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295 | else |
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296 | { if (v) |
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297 | { " The characteristic of the base field divides the group order."; |
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298 | " We have to continue without Reynolds operator..."; |
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299 | ""; |
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300 | } |
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301 | } |
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302 | kill REY; |
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303 | matrix REY[1][1]=0; |
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304 | return(REY,G(1..g)); |
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305 | } |
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306 | if (minpoly==0) |
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307 | { if (i>1) |
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308 | { return(REY,o1,G(1..g)); |
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309 | } |
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310 | return(REY,G(1..g)); |
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311 | } |
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312 | } |
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313 | if (v) |
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314 | { " Done generating the group and the Reynolds operator."; |
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315 | ""; |
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316 | } |
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317 | return(REY,G(1..g)); |
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318 | } |
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319 | example |
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320 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
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321 | ring R=0,(x,y,z),dp; |
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322 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
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323 | list L=group_reynolds(A); |
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324 | print(L[1]); |
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325 | print(L[2..size(L)]); |
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326 | } |
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327 | |
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328 | /////////////////////////////////////////////////////////////////////////////// |
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329 | // Returns i such that root^i==n, i.e. it heavily relies on the right input. |
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330 | /////////////////////////////////////////////////////////////////////////////// |
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331 | proc exponent(number n, number root) |
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332 | { int i=0; |
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333 | while((n/root^i)<>1) |
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334 | { i++; |
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335 | } |
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336 | return(i); |
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337 | } |
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338 | /////////////////////////////////////////////////////////////////////////////// |
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339 | |
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340 | proc molien (list #) |
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341 | "USAGE: molien(G1,G2,...[,ringname,lcm,flags]); |
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342 | G1,G2,...: nxn <matrices>, all elements of a finite matrix group, |
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343 | ringname: a <string> giving a name for a new ring of characteristic 0 |
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344 | for the Molien series in case of prime characteristic, lcm: an <int> |
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345 | giving the lowest common multiple of the elements' orders in case of |
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346 | prime characteristic, minpoly==0 and a non-cyclic group, flags: an |
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347 | optional <intvec> with three components: if the first element is not |
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348 | equal to 0 characteristic 0 is simulated, i.e. the Molien series is |
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349 | computed as if the base field were characteristic 0 (the user must |
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350 | choose a field of large prime characteristic, e.g. 32003), the second |
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351 | component should give the size of intervals between canceling common |
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352 | factors in the expansion of the Molien series, 0 (the default) means |
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353 | only once after generating all terms, in prime characteristic also a |
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354 | negative number can be given to indicate that common factors should |
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355 | always be canceled when the expansion is simple (the root of the |
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356 | extension field does not occur among the coefficients) |
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357 | ASSUME: n is the number of variables of the basering, G1,G2... are the group |
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358 | elements generated by group_reynolds(), lcm is the second return value |
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359 | of group_reynolds() |
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360 | RETURN: in case of characteristic 0 a 1x2 <matrix> giving enumerator and |
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361 | denominator of Molien series; in case of prime characteristic a ring |
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362 | with the name `ringname` of characteristic 0 is created where the same |
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363 | Molien series (named M) is stored |
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364 | DISPLAY: information if the third component of flags does not equal 0 |
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365 | THEORY: In characteristic 0 the terms 1/det(1-xE) for all group elements of |
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366 | the Molien series are computed in a straight forward way. In prime |
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367 | characteristic a Brauer lift is involved. The returned matrix gives |
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368 | enumerator and denominator of the expanded version where common |
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369 | factors have been canceled. |
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370 | EXAMPLE: example molien; shows an example |
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371 | " |
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372 | { def br=basering; // the Molien series depends on the |
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373 | int ch=char(br); // characteristic of the coefficient |
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374 | // field - |
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375 | int g; // size of the group |
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376 | //---------------------- making sure the input is okay ----------------------- |
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377 | if (typeof(#[size(#)])=="intvec") |
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378 | { if (size(#[size(#)])==3) |
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379 | { int mol_flag=#[size(#)][1]; |
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380 | if (#[size(#)][2]<0 && (ch==0 or (ch<>0 && mol_flag<>0))) |
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381 | { "ERROR: the second component of <intvec> should be >=0" |
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382 | return(); |
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383 | } |
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384 | int interval=#[size(#)][2]; |
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385 | int v=#[size(#)][3]; |
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386 | } |
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387 | else |
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388 | { "ERROR: <intvec> should have three components"; |
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389 | return(); |
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390 | } |
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391 | if (ch<>0) |
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392 | { if (typeof(#[size(#)-1])=="int") |
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393 | { int r=#[size(#)-1]; |
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394 | if (typeof(#[size(#)-2])<>"string") |
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395 | { "ERROR: In characteristic p>0 a <string> must be given for the name of a new"; |
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396 | " ring where the Molien series can be stored"; |
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397 | return(); |
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398 | } |
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399 | else |
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400 | { if (#[size(#)-2]=="") |
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401 | { "ERROR: <string> may not be empty"; |
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402 | return(); |
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403 | } |
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404 | string newring=#[size(#)-2]; |
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405 | g=size(#)-3; |
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406 | } |
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407 | } |
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408 | else |
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409 | { if (typeof(#[size(#)-1])<>"string") |
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410 | { "ERROR: In characteristic p>0 a <string> must be given for the name of a new"; |
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411 | " ring where the Molien series can be stored"; |
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412 | return(); |
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413 | } |
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414 | else |
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415 | { if (#[size(#)-1]=="") |
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416 | { "ERROR: <string> may not be empty"; |
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417 | return(); |
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418 | } |
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419 | string newring=#[size(#)-1]; |
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420 | g=size(#)-2; |
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421 | int r=g; |
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422 | } |
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423 | } |
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424 | } |
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425 | else // then <string> ist not needed |
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426 | { g=size(#)-1; |
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427 | } |
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428 | } |
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429 | else // last parameter is not <intvec> |
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430 | { int v=0; // no information is default |
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431 | int mol_flag=0; // computing of Molien series is default |
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432 | int interval=0; |
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433 | if (ch<>0) |
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434 | { if (typeof(#[size(#)])=="int") |
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435 | { int r=#[size(#)]; |
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436 | if (typeof(#[size(#)-1])<>"string") |
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437 | { "ERROR: in characteristic p>0 a <string> must be given for the name of a new"; |
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438 | " ring where the Molien series can be stored"; |
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439 | return(); |
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440 | } |
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441 | else |
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442 | { if (#[size(#)-1]=="") |
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443 | { "ERROR: <string> may not be empty"; |
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444 | return(); |
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445 | } |
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446 | string newring=#[size(#)-1]; |
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447 | g=size(#)-2; |
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448 | } |
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449 | } |
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450 | else |
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451 | { if (typeof(#[size(#)])<>"string") |
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452 | { "ERROR: in characteristic p>0 a <string> must be given for the name of a new"; |
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453 | " ring where the Molien series can be stored"; |
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454 | return(); |
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455 | } |
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456 | else |
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457 | { if (#[size(#)]=="") |
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458 | { "ERROR: <string> may not be empty"; |
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459 | return(); |
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460 | } |
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461 | string newring=#[size(#)]; |
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462 | g=size(#)-1; |
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463 | int r=g; |
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464 | } |
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465 | } |
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466 | } |
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467 | else |
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468 | { g=size(#); |
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469 | } |
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470 | } |
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471 | if (ch<>0) |
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472 | { if ((g/r)*r<>g) |
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473 | { "ERROR: <int> should divide the group order." |
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474 | return(); |
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475 | } |
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476 | } |
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477 | if (ch<>0) |
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478 | { if ((g%ch)==0) |
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479 | { if (voice==2) |
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480 | { "WARNING: The characteristic of the coefficient field divides the group"; |
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481 | " order. Proceed without the Molien series!"; |
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482 | } |
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483 | else |
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484 | { if (v) |
---|
485 | { " The characteristic of the base field divides the group order."; |
---|
486 | " We have to continue without Molien series..."; |
---|
487 | ""; |
---|
488 | } |
---|
489 | } |
---|
490 | } |
---|
491 | if (minpoly<>0 && mol_flag==0) |
---|
492 | { if (voice==2) |
---|
493 | { "WARNING: It is impossible for this program to calculate the Molien series"; |
---|
494 | " for finite groups over extension fields of prime characteristic."; |
---|
495 | } |
---|
496 | else |
---|
497 | { if (v) |
---|
498 | { " Since it is impossible for this program to calculate the Molien series for"; |
---|
499 | " invariant rings over extension fields of prime characteristic, we have to"; |
---|
500 | " continue without it."; |
---|
501 | ""; |
---|
502 | } |
---|
503 | } |
---|
504 | return(); |
---|
505 | } |
---|
506 | } |
---|
507 | //---------------------------------------------------------------------------- |
---|
508 | if (not(typeof(#[1])=="matrix")) |
---|
509 | { "ERROR: the parameters must be a list of matrices and maybe an <intvec>"; |
---|
510 | return(); |
---|
511 | } |
---|
512 | int n=nrows(#[1]); |
---|
513 | if (n<>nvars(br)) |
---|
514 | { "ERROR: the number of variables of the basering needs to be the same"; |
---|
515 | " as the dimension of the square matrices"; |
---|
516 | return(); |
---|
517 | } |
---|
518 | if (v && voice<>2) |
---|
519 | { ""; |
---|
520 | " Generating the Molien series..."; |
---|
521 | ""; |
---|
522 | } |
---|
523 | if (v && voice==2) |
---|
524 | { ""; |
---|
525 | } |
---|
526 | //------------- calculating Molien series in characteristic 0 ---------------- |
---|
527 | if (ch==0) // when ch==0 we can calculate the Molien |
---|
528 | { matrix I=diag(1,n); // series in any case - |
---|
529 | poly v1=maxideal(1)[1]; // the Molien series will be in terms of |
---|
530 | // the first variable of the current |
---|
531 | // ring - |
---|
532 | matrix M[1][2]; // M will contain the Molien series - |
---|
533 | M[1,1]=0; // M[1,1] will be the numerator - |
---|
534 | M[1,2]=1; // M[1,2] will be the denominator - |
---|
535 | matrix s; // will help us canceling in the |
---|
536 | // fraction |
---|
537 | poly p; // will contain the denominator of the |
---|
538 | // new term of the Molien series |
---|
539 | //------------ computing 1/det(1+xE) for all E in the group ------------------ |
---|
540 | for (int j=1;j<=g;j++) |
---|
541 | { if (not(typeof(#[j])=="matrix")) |
---|
542 | { "ERROR: the parameters must be a list of matrices and maybe an <intvec>"; |
---|
543 | return(); |
---|
544 | } |
---|
545 | if ((n<>nrows(#[j])) or (n<>ncols(#[j]))) |
---|
546 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
547 | return(); |
---|
548 | } |
---|
549 | p=det(I-v1*#[j]); // denominator of new term - |
---|
550 | M[1,1]=M[1,1]*p+M[1,2]; // expanding M[1,1]/M[1,2] + 1/p |
---|
551 | M[1,2]=M[1,2]*p; |
---|
552 | if (interval<>0) // canceling common terms of denominator |
---|
553 | { if ((j/interval)*interval==j or j==g) // and enumerator - |
---|
554 | { s=matrix(syz(ideal(M))); // once gcd() is faster than syz() these |
---|
555 | M[1,1]=-s[2,1]; // three lines should be replaced by the |
---|
556 | M[1,2]=s[1,1]; // following three |
---|
557 | // p=gcd(M[1,1],M[1,2]); |
---|
558 | // M[1,1]=M[1,1]/p; |
---|
559 | // M[1,2]=M[1,2]/p; |
---|
560 | } |
---|
561 | } |
---|
562 | if (v) |
---|
563 | { " Term "+string(j)+" of the Molien series has been computed."; |
---|
564 | } |
---|
565 | } |
---|
566 | if (interval==0) // canceling common terms of denominator |
---|
567 | { // and enumerator - |
---|
568 | s=matrix(syz(ideal(M))); // once gcd() is faster than syz() these |
---|
569 | M[1,1]=-s[2,1]; // three lines should be replaced by the |
---|
570 | M[1,2]=s[1,1]; // following three |
---|
571 | // p=gcd(M[1,1],M[1,2]); |
---|
572 | // M[1,1]=M[1,1]/p; |
---|
573 | // M[1,2]=M[1,2]/p; |
---|
574 | } |
---|
575 | map slead=br,ideal(0); |
---|
576 | s=slead(M); |
---|
577 | M[1,1]=1/s[1,1]*M[1,1]; // numerator and denominator have to have |
---|
578 | M[1,2]=1/s[1,2]*M[1,2]; // a constant term of 1 |
---|
579 | if (v) |
---|
580 | { ""; |
---|
581 | " We are done calculating the Molien series."; |
---|
582 | ""; |
---|
583 | } |
---|
584 | return(M); |
---|
585 | } |
---|
586 | //---- calculating Molien series in prime characteristic with Brauer lift ---- |
---|
587 | if (ch<>0 && mol_flag==0) |
---|
588 | { if (g<>1) |
---|
589 | { matrix G(1..g)=#[1..g]; |
---|
590 | if (interval<0) |
---|
591 | { string Mstring; |
---|
592 | } |
---|
593 | //------ preparing everything for Brauer lifts into characteristic 0 --------- |
---|
594 | ring Q=0,x,dp; // we want to extend our ring as well as |
---|
595 | // the ring of rational numbers Q to |
---|
596 | // contain r-th primitive roots of unity |
---|
597 | // in order to factor characteristic |
---|
598 | // polynomials of group elements into |
---|
599 | // linear factors and lift eigenvalues to |
---|
600 | // characteristic 0 - |
---|
601 | poly minq=cyclotomic(r); // minq now contains the size-of-group-th |
---|
602 | // cyclotomic polynomial of Q, it is |
---|
603 | // irreducible there |
---|
604 | ring `newring`=(0,e),x,dp; |
---|
605 | map f=Q,ideal(e); |
---|
606 | minpoly=number(f(minq)); // e is now a r-th primitive root of |
---|
607 | // unity - |
---|
608 | kill Q, f; // no longer needed - |
---|
609 | poly p=1; // used to build the denominator of the |
---|
610 | // new term in the Molien series |
---|
611 | matrix s[1][2]; // used for canceling - |
---|
612 | matrix M[1][2]=0,1; // will contain Molien series - |
---|
613 | ring v1br=char(br),x,dp; // we calculate the r-th cyclotomic |
---|
614 | poly minp=cyclotomic(r); // polynomial of the base field and pick |
---|
615 | minp=factorize(minp)[1][2]; // an irreducible factor of it - |
---|
616 | if (deg(minp)==1) // in this case the base field contains |
---|
617 | { ring bre=char(br),x,dp; // r-th roots of unity already |
---|
618 | map f1=v1br,ideal(0); |
---|
619 | number e=-number((f1(minp))); // e is a r-th primitive root of unity |
---|
620 | } |
---|
621 | else |
---|
622 | { ring bre=(char(br),e),x,dp; |
---|
623 | map f1=v1br,ideal(e); |
---|
624 | minpoly=number(f1(minp)); // e is a r-th primitive root of unity |
---|
625 | } |
---|
626 | map f2=br,ideal(0); // we need f2 to map our group elements |
---|
627 | // to this new extension field bre |
---|
628 | matrix xI=diag(x,n); |
---|
629 | poly p; // used for the characteristic polynomial |
---|
630 | // to factor - |
---|
631 | list L; // will contain the linear factors of the |
---|
632 | ideal F; // characteristic polynomial of the group |
---|
633 | intvec C; // elements and their powers |
---|
634 | int i, j, k; |
---|
635 | // -------------- finding all the terms of the Molien series ----------------- |
---|
636 | for (i=1;i<=g;i++) |
---|
637 | { setring br; |
---|
638 | if (not(typeof(#[i])=="matrix")) |
---|
639 | { "ERROR: the parameters must be a list of matrices and maybe an <intvec>"; |
---|
640 | return(); |
---|
641 | } |
---|
642 | if ((n<>nrows(#[i])) or (n<>ncols(#[i]))) |
---|
643 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
644 | return(); |
---|
645 | } |
---|
646 | setring bre; |
---|
647 | p=det(xI-f2(G(i))); // characteristic polynomial of G(i) |
---|
648 | L=factorize(p); |
---|
649 | F=L[1]; |
---|
650 | C=L[2]; |
---|
651 | for (j=2;j<=ncols(F);j++) |
---|
652 | { F[j]=-1*(F[j]-x); // F[j] is now an eigenvalue of G(i), |
---|
653 | // it is a power of a primitive r-th root |
---|
654 | // of unity - |
---|
655 | k=exponent(number(F[j]),e); // F[j]==e^k |
---|
656 | setring `newring`; |
---|
657 | p=p*(1-x*(e^k))^C[j]; // building the denominator of the new |
---|
658 | setring bre; // term |
---|
659 | } |
---|
660 | // ----------- |
---|
661 | // k=0; |
---|
662 | // while(k<r) |
---|
663 | // { map f3=basering,ideal(e^k); |
---|
664 | // while (f3(p)==0) |
---|
665 | // { p=p/(x-e^k); |
---|
666 | // setring `newring`; |
---|
667 | // p=p*(1-x*(e^k)); // building the denominator of the new |
---|
668 | // setring bre; |
---|
669 | // } |
---|
670 | // kill f3; |
---|
671 | // if (p==1) |
---|
672 | // { break; |
---|
673 | // } |
---|
674 | // k=k+1; |
---|
675 | // } |
---|
676 | setring `newring`; |
---|
677 | M[1,1]=M[1,1]*p+M[1,2]; // expanding M[1,1]/M[1,2] + 1/p |
---|
678 | M[1,2]=M[1,2]*p; |
---|
679 | if (interval<0) |
---|
680 | { if (i<>g) |
---|
681 | { Mstring=string(M); |
---|
682 | for (j=1;j<=size(Mstring);j++) |
---|
683 | { if (Mstring[j]=="e") |
---|
684 | { interval=0; |
---|
685 | break; |
---|
686 | } |
---|
687 | } |
---|
688 | } |
---|
689 | if (interval<>0) |
---|
690 | { s=matrix(syz(ideal(M))); // once gcd() is faster than syz() |
---|
691 | M[1,1]=-s[2,1]; // these three lines should be |
---|
692 | M[1,2]=s[1,1]; // replaced by the following three |
---|
693 | // p=gcd(M[1,1],M[1,2]); |
---|
694 | // M[1,1]=M[1,1]/p; |
---|
695 | // M[1,2]=M[1,2]/p; |
---|
696 | } |
---|
697 | else |
---|
698 | { interval=-1; |
---|
699 | } |
---|
700 | } |
---|
701 | else |
---|
702 | { if (interval<>0) // canceling common terms of denominator |
---|
703 | { if ((i/interval)*interval==i or i==g) // and enumerator |
---|
704 | { s=matrix(syz(ideal(M))); // once gcd() is faster than syz() |
---|
705 | M[1,1]=-s[2,1]; // these three lines should be |
---|
706 | M[1,2]=s[1,1]; // replaced by the following three |
---|
707 | // p=gcd(M[1,1],M[1,2]); |
---|
708 | // M[1,1]=M[1,1]/p; |
---|
709 | // M[1,2]=M[1,2]/p; |
---|
710 | } |
---|
711 | } |
---|
712 | } |
---|
713 | p=1; |
---|
714 | setring bre; |
---|
715 | if (v) |
---|
716 | { " Term "+string(i)+" of the Molien series has been computed."; |
---|
717 | } |
---|
718 | } |
---|
719 | if (v) |
---|
720 | { ""; |
---|
721 | } |
---|
722 | setring `newring`; |
---|
723 | if (interval==0) // canceling common terms of denominator |
---|
724 | { // and enumerator - |
---|
725 | s=matrix(syz(ideal(M))); // once gcd() is faster than syz() these |
---|
726 | M[1,1]=-s[2,1]; // three lines should be replaced by the |
---|
727 | M[1,2]=s[1,1]; // following three |
---|
728 | // p=gcd(M[1,1],M[1,2]); |
---|
729 | // M[1,1]=M[1,1]/p; |
---|
730 | // M[1,2]=M[1,2]/p; |
---|
731 | } |
---|
732 | map slead=`newring`,ideal(0); |
---|
733 | s=slead(M); // forcing the constant term of numerator |
---|
734 | M[1,1]=1/s[1,1]*M[1,1]; // and denominator to be 1 |
---|
735 | M[1,2]=1/s[1,2]*M[1,2]; |
---|
736 | kill slead; |
---|
737 | kill s; |
---|
738 | kill p; |
---|
739 | } |
---|
740 | else // if the group only contains an identity |
---|
741 | { ring `newring`=0,x,dp; // element, it is very easy to calculate |
---|
742 | matrix M[1][2]=1,(1-x)^n; // the Molien series |
---|
743 | } |
---|
744 | exportto(Top,`newring`); // we keep the ring where we computed the |
---|
745 | export M; // Molien series in such that we can |
---|
746 | setring br; // keep it |
---|
747 | if (v) |
---|
748 | { " We are done calculating the Molien series."; |
---|
749 | ""; |
---|
750 | } |
---|
751 | } |
---|
752 | else // i.e. char<>0 and mol_flag<>0, the user |
---|
753 | { // has specified that we are dealing with |
---|
754 | // a ring of large characteristic which |
---|
755 | // can be treated like a ring of |
---|
756 | // characteristic 0; we'll avoid the |
---|
757 | // Brauer lifts |
---|
758 | //----------------------- simulating characteristic 0 ------------------------ |
---|
759 | string chst=charstr(br); |
---|
760 | for (int i=1;i<=size(chst);i++) |
---|
761 | { if (chst[i]==",") |
---|
762 | { break; |
---|
763 | } |
---|
764 | } |
---|
765 | //----------------- generating ring of characteristic 0 ---------------------- |
---|
766 | if (minpoly==0) |
---|
767 | { if (i>size(chst)) |
---|
768 | { execute("ring "+newring+"=0,("+varstr(br)+"),("+ordstr(br)+")"); |
---|
769 | } |
---|
770 | else |
---|
771 | { chst=chst[i..size(chst)]; |
---|
772 | execute |
---|
773 | ("ring "+newring+"=(0"+chst+"),("+varstr(br)+"),("+ordstr(br)+")"); |
---|
774 | } |
---|
775 | } |
---|
776 | else |
---|
777 | { string minp=string(minpoly); |
---|
778 | minp=minp[2..size(minp)-1]; |
---|
779 | chst=chst[i..size(chst)]; |
---|
780 | execute("ring "+newring+"=(0"+chst+"),("+varstr(br)+"),("+ordstr(br)+")"); |
---|
781 | execute("minpoly="+minp); |
---|
782 | } |
---|
783 | matrix I=diag(1,n); |
---|
784 | poly v1=maxideal(1)[1]; // the Molien series will be in terms of |
---|
785 | // the first variable of the current |
---|
786 | // ring - |
---|
787 | matrix M[1][2]; // M will contain the Molien series - |
---|
788 | M[1,1]=0; // M[1,1] will be the numerator - |
---|
789 | M[1,2]=1; // M[1,2] will be the denominator - |
---|
790 | matrix s; // will help us canceling in the |
---|
791 | // fraction |
---|
792 | poly p; // will contain the denominator of the |
---|
793 | // new term of the Molien series |
---|
794 | int j; |
---|
795 | string links, rechts; |
---|
796 | //----------------- finding all terms of the Molien series ------------------- |
---|
797 | for (i=1;i<=g;i++) |
---|
798 | { setring br; |
---|
799 | if (not(typeof(#[i])=="matrix")) |
---|
800 | { "ERROR: the parameters must be a list of matrices and maybe an <intvec>"; |
---|
801 | return(); |
---|
802 | } |
---|
803 | if ((n<>nrows(#[i])) or (n<>ncols(#[i]))) |
---|
804 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
805 | return(); |
---|
806 | } |
---|
807 | string stM(i)=string(#[i]); |
---|
808 | for (j=1;j<=size(stM(i));j++) |
---|
809 | { if (stM(i)[j]==" |
---|
810 | ") |
---|
811 | { links=stM(i)[1..j-1]; |
---|
812 | rechts=stM(i)[j+1..size(stM(i))]; |
---|
813 | stM(i)=links+rechts; |
---|
814 | } |
---|
815 | } |
---|
816 | setring `newring`; |
---|
817 | execute("matrix G(i)["+string(n)+"]["+string(n)+"]="+stM(i)); |
---|
818 | p=det(I-v1*G(i)); // denominator of new term - |
---|
819 | M[1,1]=M[1,1]*p+M[1,2]; // expanding M[1,1]/M[1,2] + 1/p |
---|
820 | M[1,2]=M[1,2]*p; |
---|
821 | if (interval<>0) // canceling common terms of denominator |
---|
822 | { if ((i/interval)*interval==i or i==g) // and enumerator |
---|
823 | { |
---|
824 | s=matrix(syz(ideal(M))); // once gcd() is faster than syz() these |
---|
825 | M[1,1]=-s[2,1]; // three lines should be replaced by the |
---|
826 | M[1,2]=s[1,1]; // following three |
---|
827 | // p=gcd(M[1,1],M[1,2]); |
---|
828 | // M[1,1]=M[1,1]/p; |
---|
829 | // M[1,2]=M[1,2]/p; |
---|
830 | } |
---|
831 | } |
---|
832 | if (v) |
---|
833 | { " Term "+string(i)+" of the Molien series has been computed."; |
---|
834 | } |
---|
835 | } |
---|
836 | if (interval==0) // canceling common terms of denominator |
---|
837 | { // and enumerator - |
---|
838 | s=matrix(syz(ideal(M))); // once gcd() is faster than syz() these |
---|
839 | M[1,1]=-s[2,1]; // three lines should be replaced by the |
---|
840 | M[1,2]=s[1,1]; // following three |
---|
841 | // p=gcd(M[1,1],M[1,2]); |
---|
842 | // M[1,1]=M[1,1]/p; |
---|
843 | // M[1,2]=M[1,2]/p; |
---|
844 | } |
---|
845 | map slead=`newring`,ideal(0); |
---|
846 | s=slead(M); |
---|
847 | M[1,1]=1/s[1,1]*M[1,1]; // numerator and denominator have to have |
---|
848 | M[1,2]=1/s[1,2]*M[1,2]; // a constant term of 1 |
---|
849 | if (v) |
---|
850 | { ""; |
---|
851 | " We are done calculating the Molien series."; |
---|
852 | ""; |
---|
853 | } |
---|
854 | kill G(1..g), s, slead, p, v1, I; |
---|
855 | export `newring`; // we keep the ring where we computed the |
---|
856 | export M; // the Molien series such that we can |
---|
857 | setring br; // keep it |
---|
858 | } |
---|
859 | } |
---|
860 | example |
---|
861 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
862 | " note the case of prime characteristic"; echo=2; |
---|
863 | ring R=0,(x,y,z),dp; |
---|
864 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
865 | list L=group_reynolds(A); |
---|
866 | matrix M=molien(L[2..size(L)]); |
---|
867 | print(M); |
---|
868 | ring S=3,(x,y,z),dp; |
---|
869 | string newring="alksdfjlaskdjf"; |
---|
870 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
871 | list L=group_reynolds(A); |
---|
872 | molien(L[2..size(L)],newring); |
---|
873 | setring alksdfjlaskdjf; |
---|
874 | print(M); |
---|
875 | setring S; |
---|
876 | kill alksdfjlaskdjf; |
---|
877 | } |
---|
878 | /////////////////////////////////////////////////////////////////////////////// |
---|
879 | |
---|
880 | proc reynolds_molien (list #) |
---|
881 | "USAGE: reynolds_molien(G1,G2,...[,ringname,flags]); |
---|
882 | G1,G2,...: nxn <matrices> generating a finite matrix group, ringname: |
---|
883 | a <string> giving a name for a new ring of characteristic 0 for the |
---|
884 | Molien series in case of prime characteristic, flags: an optional |
---|
885 | <intvec> with three components: if the first element is not equal to 0 |
---|
886 | characteristic 0 is simulated, i.e. the Molien series is computed as |
---|
887 | if the base field were characteristic 0 (the user must choose a field |
---|
888 | of large prime characteristic, e.g. 32003) the second component should |
---|
889 | give the size of intervals between canceling common factors in the |
---|
890 | expansion of the Molien series, 0 (the default) means only once after |
---|
891 | generating all terms, in prime characteristic also a negative number |
---|
892 | can be given to indicate that common factors should always be canceled |
---|
893 | when the expansion is simple (the root of the extension field does not |
---|
894 | occur among the coefficients) |
---|
895 | ASSUME: n is the number of variables of the basering, G1,G2... are the group |
---|
896 | elements generated by group_reynolds(), g is the size of the group |
---|
897 | RETURN: a gxn <matrix> representing the Reynolds operator is the first return |
---|
898 | value and in case of characteristic 0 a 1x2 <matrix> giving enumerator |
---|
899 | and denominator of Molien series is the second one; in case of prime |
---|
900 | characteristic a ring with the name `ringname` of characteristic 0 is |
---|
901 | created where the same Molien series (named M) is stored |
---|
902 | DISPLAY: information if the third component of flags does not equal 0 |
---|
903 | THEORY: The entire matrix group is generated by getting all left products of |
---|
904 | the generators with new elements from the last run through the loop |
---|
905 | (or the generators themselves during the first run). All the ones that |
---|
906 | have been generated before are thrown out and the program terminates |
---|
907 | when are no new elements found in one run. Additionally each time a |
---|
908 | new group element is found the corresponding ring mapping of which the |
---|
909 | Reynolds operator is made up is generated. They are stored in the rows |
---|
910 | of the first return value. In characteristic 0 the terms 1/det(1-xE) |
---|
911 | is computed whenever a new element E is found. In prime characteristic |
---|
912 | a Brauer lift is involved and the terms are only computed after the |
---|
913 | entire matrix group is generated (to avoid the modular case). The |
---|
914 | returned matrix gives enumerator and denominator of the expanded |
---|
915 | version where common factors have been canceled. |
---|
916 | EXAMPLE: example reynolds_molien; shows an example |
---|
917 | " |
---|
918 | { def br=basering; // the Molien series depends on the |
---|
919 | int ch=char(br); // characteristic of the coefficient |
---|
920 | // field |
---|
921 | int gen_num; |
---|
922 | //------------------- making sure the input is okay -------------------------- |
---|
923 | if (typeof(#[size(#)])=="intvec") |
---|
924 | { if (size(#[size(#)])==3) |
---|
925 | { int mol_flag=#[size(#)][1]; |
---|
926 | if (#[size(#)][2]<0 && (ch==0 or (ch<>0 && mol_flag<>0))) |
---|
927 | { "ERROR: the second component of the <intvec> should be >=0"; |
---|
928 | return(); |
---|
929 | } |
---|
930 | int interval=#[size(#)][2]; |
---|
931 | int v=#[size(#)][3]; |
---|
932 | } |
---|
933 | else |
---|
934 | { "ERROR: <intvec> should have three components"; |
---|
935 | return(); |
---|
936 | } |
---|
937 | if (ch<>0) |
---|
938 | { if (typeof(#[size(#)-1])<>"string") |
---|
939 | { "ERROR: in characteristic p a <string> must be given for the name"; |
---|
940 | " of a new ring where the Molien series can be stored"; |
---|
941 | return(); |
---|
942 | } |
---|
943 | else |
---|
944 | { if (#[size(#)-1]=="") |
---|
945 | { "ERROR: <string> may not be empty"; |
---|
946 | return(); |
---|
947 | } |
---|
948 | string newring=#[size(#)-1]; |
---|
949 | gen_num=size(#)-2; |
---|
950 | } |
---|
951 | } |
---|
952 | else // then <string> ist not needed |
---|
953 | { gen_num=size(#)-1; |
---|
954 | } |
---|
955 | } |
---|
956 | else // last parameter is not <intvec> |
---|
957 | { int v=0; // no information is default |
---|
958 | int interval; |
---|
959 | int mol_flag=0; // computing of Molien series is default |
---|
960 | if (ch<>0) |
---|
961 | { if (typeof(#[size(#)])<>"string") |
---|
962 | { "ERROR: in characteristic p a <string> must be given for the name"; |
---|
963 | " of a new ring where the Molien series can be stored"; |
---|
964 | return(); |
---|
965 | } |
---|
966 | else |
---|
967 | { if (#[size(#)]=="") |
---|
968 | { "ERROR: <string> may not be empty"; |
---|
969 | return(); |
---|
970 | } |
---|
971 | string newring=#[size(#)]; |
---|
972 | gen_num=size(#)-1; |
---|
973 | } |
---|
974 | } |
---|
975 | else |
---|
976 | { gen_num=size(#); |
---|
977 | } |
---|
978 | } |
---|
979 | // ----------------- computing the terms with Brauer lift -------------------- |
---|
980 | if (ch<>0 && mol_flag==0) |
---|
981 | { list L=group_reynolds(#[1..gen_num],v); |
---|
982 | if (L[1]==0) |
---|
983 | { if (voice==2) |
---|
984 | { "WARNING: The characteristic of the coefficient field divides the group order."; |
---|
985 | " Proceed without the Reynolds operator or the Molien series!"; |
---|
986 | return(); |
---|
987 | } |
---|
988 | if (v) |
---|
989 | { " The characteristic of the base field divides the group order."; |
---|
990 | " We have to continue without Reynolds operator or the Molien series..."; |
---|
991 | return(); |
---|
992 | } |
---|
993 | } |
---|
994 | if (minpoly<>0) |
---|
995 | { if (voice==2) |
---|
996 | { "WARNING: It is impossible for this program to calculate the Molien series"; |
---|
997 | " for finite groups over extension fields of prime characteristic."; |
---|
998 | return(L[1]); |
---|
999 | } |
---|
1000 | else |
---|
1001 | { if (v) |
---|
1002 | { " Since it is impossible for this program to calculate the Molien series for"; |
---|
1003 | " invariant rings over extension fields of prime characteristic, we have to"; |
---|
1004 | " continue without it."; |
---|
1005 | return(L[1]); |
---|
1006 | } |
---|
1007 | } |
---|
1008 | } |
---|
1009 | if (typeof(L[2])=="int") |
---|
1010 | { molien(L[3..size(L)],newring,L[2],intvec(mol_flag,interval,v)); |
---|
1011 | } |
---|
1012 | else |
---|
1013 | { molien(L[2..size(L)],newring,intvec(mol_flag,interval,v)); |
---|
1014 | } |
---|
1015 | return(L[1]); |
---|
1016 | } |
---|
1017 | //----------- computing Molien series in the straight forward way ------------ |
---|
1018 | if (ch==0) |
---|
1019 | { if (typeof(#[1])<>"matrix") |
---|
1020 | { "ERROR: the parameters must be a list of matrices and maybe an <intvec>"; |
---|
1021 | return(); |
---|
1022 | } |
---|
1023 | int n=nrows(#[1]); |
---|
1024 | if (n<>nvars(br)) |
---|
1025 | { "ERROR: the number of variables of the basering needs to be the same"; |
---|
1026 | " as the dimension of the matrices"; |
---|
1027 | return(); |
---|
1028 | } |
---|
1029 | if (n<>ncols(#[1])) |
---|
1030 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
1031 | return(); |
---|
1032 | } |
---|
1033 | matrix vars=matrix(maxideal(1)); // creating an nx1-matrix containing the |
---|
1034 | vars=transpose(vars); // variables of the ring - |
---|
1035 | matrix A(1)=#[1]*vars; // calculating the first ring mapping - |
---|
1036 | // A(1) will contain the Reynolds |
---|
1037 | // operator - |
---|
1038 | poly v1=vars[1,1]; // the Molien series will be in terms of |
---|
1039 | // the first variable of the current |
---|
1040 | // ring |
---|
1041 | matrix I=diag(1,n); |
---|
1042 | matrix A(2)[1][2]; // A(2) will contain the Molien series - |
---|
1043 | A(2)[1,1]=1; // A(2)[1,1] will be the numerator |
---|
1044 | matrix G(1)=#[1]; // G(k) are elements of the group - |
---|
1045 | A(2)[1,2]=det(I-v1*(G(1))); // A(2)[1,2] will be the denominator - |
---|
1046 | matrix s; // will help us canceling in the |
---|
1047 | // fraction |
---|
1048 | poly p; // will contain the denominator of the |
---|
1049 | // new term of the Molien series |
---|
1050 | int i=1; |
---|
1051 | for (int j=2;j<=gen_num;j++) // this loop adds the parameters to the |
---|
1052 | { // group, leaving out doubles and |
---|
1053 | // checking whether the parameters are |
---|
1054 | // compatible with the task of the |
---|
1055 | // procedure |
---|
1056 | if (not(typeof(#[j])=="matrix")) |
---|
1057 | { "ERROR: the parameters must be a list of matrices and maybe an <intvec>"; |
---|
1058 | return(); |
---|
1059 | } |
---|
1060 | if ((n!=nrows(#[j])) or (n!=ncols(#[j]))) |
---|
1061 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
1062 | return(); |
---|
1063 | } |
---|
1064 | if (unique(G(1..i),#[j])) |
---|
1065 | { i++; |
---|
1066 | matrix G(i)=#[j]; |
---|
1067 | A(1)=concat(A(1),#[j]*vars); // adding ring homomorphisms to A(1) - |
---|
1068 | p=det(I-v1*#[j]); // denominator of new term - |
---|
1069 | A(2)[1,1]=A(2)[1,1]*p+A(2)[1,2]; // expanding A(2)[1,1]/A(2)[1,2] +1/p |
---|
1070 | A(2)[1,2]=A(2)[1,2]*p; |
---|
1071 | if (interval<>0) // canceling common terms of denominator |
---|
1072 | { if ((i/interval)*interval==i) // and enumerator |
---|
1073 | { |
---|
1074 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() these |
---|
1075 | A(2)[1,1]=-s[2,1]; // three lines should be replaced by the |
---|
1076 | A(2)[1,2]=s[1,1]; // following three |
---|
1077 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
1078 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
1079 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
1080 | } |
---|
1081 | } |
---|
1082 | } |
---|
1083 | } |
---|
1084 | int g=i; // G(1)..G(i) are generators without |
---|
1085 | // doubles - g generally is the number |
---|
1086 | // of elements in the group so far - |
---|
1087 | j=i; // j is the number of new elements that |
---|
1088 | // we use as factors |
---|
1089 | int k, m, l; |
---|
1090 | if (v) |
---|
1091 | { ""; |
---|
1092 | " Generating the entire matrix group. Whenever a new group element is found,"; |
---|
1093 | " the corresponding ring homomorphism of the Reynolds operator and the"; |
---|
1094 | " corresponding term of the Molien series is generated."; |
---|
1095 | ""; |
---|
1096 | } |
---|
1097 | //----------- computing 1/det(I-xE) whenever a new element E is found -------- |
---|
1098 | while (1) |
---|
1099 | { l=0; // l is the number of products we get in |
---|
1100 | // one going |
---|
1101 | for (m=g-j+1;m<=g;m=m+1) |
---|
1102 | { for (k=1;k<=i;k++) |
---|
1103 | { l++; |
---|
1104 | matrix P(l)=G(k)*G(m); // possible new element |
---|
1105 | } |
---|
1106 | } |
---|
1107 | j=0; |
---|
1108 | for (k=1;k<=l;k++) |
---|
1109 | { if (unique(G(1..g),P(k))) |
---|
1110 | { j++; // a new factor for next run |
---|
1111 | g++; |
---|
1112 | matrix G(g)=P(k); // a new group element - |
---|
1113 | A(1)=concat(A(1),P(k)*vars); // adding new mapping to A(1) |
---|
1114 | p=det(I-v1*P(k)); // denominator of new term |
---|
1115 | A(2)[1,1]=A(2)[1,1]*p+A(2)[1,2]; |
---|
1116 | A(2)[1,2]=A(2)[1,2]*p; // expanding A(2)[1,1]/A(2)[1,2] + 1/p - |
---|
1117 | if (interval<>0) // canceling common terms of denominator |
---|
1118 | { if ((g/interval)*interval==g) // and enumerator |
---|
1119 | { |
---|
1120 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() |
---|
1121 | A(2)[1,1]=-s[2,1]; // these three lines should be replaced |
---|
1122 | A(2)[1,2]=s[1,1]; // by the following three |
---|
1123 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
1124 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
1125 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
1126 | } |
---|
1127 | } |
---|
1128 | if (v) |
---|
1129 | { " Group element "+string(g)+" has been found."; |
---|
1130 | } |
---|
1131 | } |
---|
1132 | kill P(k); |
---|
1133 | } |
---|
1134 | if (j==0) // when we didn't add any new elements |
---|
1135 | { break; // in one run through the while loop |
---|
1136 | } // we are done |
---|
1137 | } |
---|
1138 | if (v) |
---|
1139 | { if (g<=i) |
---|
1140 | { " There are only "+string(g)+" group elements."; |
---|
1141 | } |
---|
1142 | ""; |
---|
1143 | } |
---|
1144 | A(1)=transpose(A(1)); // when we evaluate the Reynolds operator |
---|
1145 | // later on, we actually want 1xn |
---|
1146 | // matrices |
---|
1147 | if (interval==0) // canceling common terms of denominator |
---|
1148 | { // and enumerator - |
---|
1149 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() |
---|
1150 | A(2)[1,1]=-s[2,1]; // these three lines should be replaced |
---|
1151 | A(2)[1,2]=s[1,1]; // by the following three |
---|
1152 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
1153 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
1154 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
1155 | } |
---|
1156 | if (interval<>0) // canceling common terms of denominator |
---|
1157 | { if ((g/interval)*interval<>g) // and enumerator |
---|
1158 | { |
---|
1159 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() |
---|
1160 | A(2)[1,1]=-s[2,1]; // these three lines should be replaced |
---|
1161 | A(2)[1,2]=s[1,1]; // by the following three |
---|
1162 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
1163 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
1164 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
1165 | } |
---|
1166 | } |
---|
1167 | map slead=br,ideal(0); |
---|
1168 | s=slead(A(2)); |
---|
1169 | A(2)[1,1]=1/s[1,1]*A(2)[1,1]; // numerator and denominator have to have |
---|
1170 | A(2)[1,2]=1/s[1,2]*A(2)[1,2]; // a constant term of 1 |
---|
1171 | if (v) |
---|
1172 | { " Now we are done calculating Molien series and Reynolds operator."; |
---|
1173 | ""; |
---|
1174 | } |
---|
1175 | return(A(1..2)); |
---|
1176 | } |
---|
1177 | //------------------------ simulating characteristic 0 ----------------------- |
---|
1178 | else // if ch<>0 and mol_flag<>0 |
---|
1179 | { if (typeof(#[1])<>"matrix") |
---|
1180 | { "ERROR: the parameters must be a list of matrices and maybe an <intvec>"; |
---|
1181 | return(); |
---|
1182 | } |
---|
1183 | int n=nrows(#[1]); |
---|
1184 | if (n<>nvars(br)) |
---|
1185 | { "ERROR: the number of variables of the basering needs to be the same"; |
---|
1186 | " as the dimension of the matrices"; |
---|
1187 | return(); |
---|
1188 | } |
---|
1189 | if (n<>ncols(#[1])) |
---|
1190 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
1191 | return(); |
---|
1192 | } |
---|
1193 | matrix vars=matrix(maxideal(1)); // creating an nx1-matrix containing the |
---|
1194 | vars=transpose(vars); // variables of the ring - |
---|
1195 | matrix A(1)=#[1]*vars; // calculating the first ring mapping - |
---|
1196 | // A(1) will contain the Reynolds |
---|
1197 | // operator |
---|
1198 | string chst=charstr(br); |
---|
1199 | for (int i=1;i<=size(chst);i++) |
---|
1200 | { if (chst[i]==",") |
---|
1201 | { break; |
---|
1202 | } |
---|
1203 | } |
---|
1204 | if (minpoly==0) |
---|
1205 | { if (i>size(chst)) |
---|
1206 | { execute("ring "+newring+"=0,("+varstr(br)+"),("+ordstr(br)+")"); |
---|
1207 | } |
---|
1208 | else |
---|
1209 | { chst=chst[i..size(chst)]; |
---|
1210 | execute |
---|
1211 | ("ring "+newring+"=(0"+chst+"),("+varstr(br)+"),("+ordstr(br)+")"); |
---|
1212 | } |
---|
1213 | } |
---|
1214 | else |
---|
1215 | { string minp=string(minpoly); |
---|
1216 | minp=minp[2..size(minp)-1]; |
---|
1217 | chst=chst[i..size(chst)]; |
---|
1218 | execute("ring "+newring+"=(0"+chst+"),("+varstr(br)+"),("+ordstr(br)+")"); |
---|
1219 | execute("minpoly="+minp); |
---|
1220 | } |
---|
1221 | poly v1=var(1); // the Molien series will be in terms of |
---|
1222 | // the first variable of the current |
---|
1223 | // ring |
---|
1224 | matrix I=diag(1,n); |
---|
1225 | int o; |
---|
1226 | setring br; |
---|
1227 | matrix G(1)=#[1]; |
---|
1228 | string links, rechts; |
---|
1229 | string stM(1)=string(#[1]); |
---|
1230 | for (o=1;o<=size(stM(1));o++) |
---|
1231 | { if (stM(1)[o]==" |
---|
1232 | ") |
---|
1233 | { links=stM(1)[1..o-1]; |
---|
1234 | rechts=stM(1)[o+1..size(stM(1))]; |
---|
1235 | stM(1)=links+rechts; |
---|
1236 | } |
---|
1237 | } |
---|
1238 | setring `newring`; |
---|
1239 | execute("matrix G(1)["+string(n)+"]["+string(n)+"]="+stM(1)); |
---|
1240 | matrix A(2)[1][2]; // A(2) will contain the Molien series - |
---|
1241 | A(2)[1,1]=1; // A(2)[1,1] will be the numerator |
---|
1242 | A(2)[1,2]=det(I-v1*(G(1))); // A(2)[1,2] will be the denominator - |
---|
1243 | matrix s; // will help us canceling in the |
---|
1244 | // fraction |
---|
1245 | poly p; // will contain the denominator of the |
---|
1246 | // new term of the Molien series |
---|
1247 | i=1; |
---|
1248 | for (int j=2;j<=gen_num;j++) // this loop adds the parameters to the |
---|
1249 | { // group, leaving out doubles and |
---|
1250 | // checking whether the parameters are |
---|
1251 | // compatible with the task of the |
---|
1252 | // procedure |
---|
1253 | setring br; |
---|
1254 | if (not(typeof(#[j])=="matrix")) |
---|
1255 | { "ERROR: the parameters must be a list of matrices and maybe an <intvec>"; |
---|
1256 | return(); |
---|
1257 | } |
---|
1258 | if ((n!=nrows(#[j])) or (n!=ncols(#[j]))) |
---|
1259 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
1260 | return(); |
---|
1261 | } |
---|
1262 | if (unique(G(1..i),#[j])) |
---|
1263 | { i++; |
---|
1264 | matrix G(i)=#[j]; |
---|
1265 | A(1)=concat(A(1),G(i)*vars); // adding ring homomorphisms to A(1) |
---|
1266 | string stM(i)=string(G(i)); |
---|
1267 | for (o=1;o<=size(stM(i));o++) |
---|
1268 | { if (stM(i)[o]==" |
---|
1269 | ") |
---|
1270 | { links=stM(i)[1..o-1]; |
---|
1271 | rechts=stM(i)[o+1..size(stM(i))]; |
---|
1272 | stM(i)=links+rechts; |
---|
1273 | } |
---|
1274 | } |
---|
1275 | setring `newring`; |
---|
1276 | execute("matrix G(i)["+string(n)+"]["+string(n)+"]="+stM(i)); |
---|
1277 | p=det(I-v1*G(i)); // denominator of new term - |
---|
1278 | A(2)[1,1]=A(2)[1,1]*p+A(2)[1,2]; // expanding A(2)[1,1]/A(2)[1,2] +1/p |
---|
1279 | A(2)[1,2]=A(2)[1,2]*p; |
---|
1280 | if (interval<>0) // canceling common terms of denominator |
---|
1281 | { if ((i/interval)*interval==i) // and enumerator |
---|
1282 | { |
---|
1283 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() these |
---|
1284 | A(2)[1,1]=-s[2,1]; // three lines should be replaced by the |
---|
1285 | A(2)[1,2]=s[1,1]; // following three |
---|
1286 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
1287 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
1288 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
1289 | } |
---|
1290 | } |
---|
1291 | setring br; |
---|
1292 | } |
---|
1293 | } |
---|
1294 | int g=i; // G(1)..G(i) are generators without |
---|
1295 | // doubles - g generally is the number |
---|
1296 | // of elements in the group so far - |
---|
1297 | j=i; // j is the number of new elements that |
---|
1298 | // we use as factors |
---|
1299 | int k, m, l; |
---|
1300 | if (v) |
---|
1301 | { ""; |
---|
1302 | " Generating the entire matrix group. Whenever a new group element is found,"; |
---|
1303 | " the corresponding ring homomorphism of the Reynolds operator and the"; |
---|
1304 | " corresponding term of the Molien series is generated."; |
---|
1305 | ""; |
---|
1306 | } |
---|
1307 | // taking all elements in a ring of characteristic 0 and computing the terms |
---|
1308 | // of the Molien series there |
---|
1309 | while (1) |
---|
1310 | { l=0; // l is the number of products we get in |
---|
1311 | // one going |
---|
1312 | for (m=g-j+1;m<=g;m++) |
---|
1313 | { for (k=1;k<=i;k++) |
---|
1314 | { l++; |
---|
1315 | matrix P(l)=G(k)*G(m); // possible new element |
---|
1316 | } |
---|
1317 | } |
---|
1318 | j=0; |
---|
1319 | for (k=1;k<=l;k++) |
---|
1320 | { if (unique(G(1..g),P(k))) |
---|
1321 | { j++; // a new factor for next run |
---|
1322 | g++; |
---|
1323 | matrix G(g)=P(k); // a new group element - |
---|
1324 | A(1)=concat(A(1),G(g)*vars); // adding new mapping to A(1) |
---|
1325 | string stM(g)=string(G(g)); |
---|
1326 | for (o=1;o<=size(stM(g));o++) |
---|
1327 | { if (stM(g)[o]==" |
---|
1328 | ") |
---|
1329 | { links=stM(g)[1..o-1]; |
---|
1330 | rechts=stM(g)[o+1..size(stM(g))]; |
---|
1331 | stM(g)=links+rechts; |
---|
1332 | } |
---|
1333 | } |
---|
1334 | setring `newring`; |
---|
1335 | execute("matrix G(g)["+string(n)+"]["+string(n)+"]="+stM(g)); |
---|
1336 | p=det(I-v1*G(g)); // denominator of new term |
---|
1337 | A(2)[1,1]=A(2)[1,1]*p+A(2)[1,2]; |
---|
1338 | A(2)[1,2]=A(2)[1,2]*p; // expanding A(2)[1,1]/A(2)[1,2] + 1/p - |
---|
1339 | if (interval<>0) // canceling common terms of denominator |
---|
1340 | { if ((g/interval)*interval==g) // and enumerator |
---|
1341 | { |
---|
1342 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() |
---|
1343 | A(2)[1,1]=-s[2,1]; // these three lines should be replaced |
---|
1344 | A(2)[1,2]=s[1,1]; // by the following three |
---|
1345 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
1346 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
1347 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
1348 | } |
---|
1349 | } |
---|
1350 | if (v) |
---|
1351 | { " Group element "+string(g)+" has been found."; |
---|
1352 | } |
---|
1353 | setring br; |
---|
1354 | } |
---|
1355 | kill P(k); |
---|
1356 | } |
---|
1357 | if (j==0) // when we didn't add any new elements |
---|
1358 | { break; // in one run through the while loop |
---|
1359 | } // we are done |
---|
1360 | } |
---|
1361 | if (v) |
---|
1362 | { if (g<=i) |
---|
1363 | { " There are only "+string(g)+" group elements."; |
---|
1364 | } |
---|
1365 | ""; |
---|
1366 | } |
---|
1367 | A(1)=transpose(A(1)); // when we evaluate the Reynolds operator |
---|
1368 | // later on, we actually want 1xn |
---|
1369 | // matrices |
---|
1370 | setring `newring`; |
---|
1371 | if (interval==0) // canceling common terms of denominator |
---|
1372 | { // and enumerator - |
---|
1373 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() |
---|
1374 | A(2)[1,1]=-s[2,1]; // these three lines should be replaced |
---|
1375 | A(2)[1,2]=s[1,1]; // by the following three |
---|
1376 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
1377 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
1378 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
1379 | } |
---|
1380 | if (interval<>0) // canceling common terms of denominator |
---|
1381 | { if ((g/interval)*interval<>g) // and enumerator |
---|
1382 | { |
---|
1383 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() |
---|
1384 | A(2)[1,1]=-s[2,1]; // these three lines should be replaced |
---|
1385 | A(2)[1,2]=s[1,1]; // by the following three |
---|
1386 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
1387 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
1388 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
1389 | } |
---|
1390 | } |
---|
1391 | map slead=`newring`,ideal(0); |
---|
1392 | s=slead(A(2)); |
---|
1393 | A(2)[1,1]=1/s[1,1]*A(2)[1,1]; // numerator and denominator have to have |
---|
1394 | A(2)[1,2]=1/s[1,2]*A(2)[1,2]; // a constant term of 1 |
---|
1395 | if (v) |
---|
1396 | { " Now we are done calculating Molien series and Reynolds operator."; |
---|
1397 | ""; |
---|
1398 | } |
---|
1399 | matrix M=A(2); |
---|
1400 | kill G(1..g), s, slead, p, v1, I, A(2); |
---|
1401 | export `newring`; // we keep the ring where we computed the |
---|
1402 | export M; // the Molien series such that we can |
---|
1403 | setring br; // keep it |
---|
1404 | return(A(1)); |
---|
1405 | } |
---|
1406 | } |
---|
1407 | example |
---|
1408 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
1409 | " note the case of prime characteristic"; echo=2; |
---|
1410 | ring R=0,(x,y,z),dp; |
---|
1411 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
1412 | matrix REY,M=reynolds_molien(A); |
---|
1413 | print(REY); |
---|
1414 | print(M); |
---|
1415 | ring S=3,(x,y,z),dp; |
---|
1416 | string newring="Qadjoint"; |
---|
1417 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
1418 | matrix REY=reynolds_molien(A,newring); |
---|
1419 | print(REY); |
---|
1420 | setring Qadjoint; |
---|
1421 | print(M); |
---|
1422 | setring S; |
---|
1423 | kill Qadjoint; |
---|
1424 | } |
---|
1425 | /////////////////////////////////////////////////////////////////////////////// |
---|
1426 | |
---|
1427 | proc partial_molien (matrix M, int n, list #) |
---|
1428 | "USAGE: partial_molien(M,n[,p]); |
---|
1429 | M: a 1x2 <matrix>, n: an <int> indicating number of terms in the |
---|
1430 | expansion, p: an optional <poly> |
---|
1431 | ASSUME: M is the return value of molien or the second return value of |
---|
1432 | reynolds_molien, p ought to be the second return value of a previous |
---|
1433 | run of partial_molien and avoids recalculating known terms |
---|
1434 | RETURN: n terms (type <poly>) of the partial expansion of the Molien series |
---|
1435 | (first n if there is no third parameter given, otherwise the next n |
---|
1436 | terms depending on a previous calculation) and an intermediate result |
---|
1437 | (type <poly>) of the calculation to be used as third parameter in a |
---|
1438 | next run of partial_molien |
---|
1439 | THEORY: The following calculation is implemented: |
---|
1440 | @format |
---|
1441 | (1+a1x+a2x^2+...+anx^n)/(1+b1x+b2x^2+...+bmx^m)=(1+(a1-b1)x+... |
---|
1442 | (1+b1x+b2x^2+...+bmx^m) |
---|
1443 | ----------------------- |
---|
1444 | (a1-b1)x+(a2-b2)x^2+... |
---|
1445 | (a1-b1)x+b1(a1-b1)x^2+... |
---|
1446 | @end format |
---|
1447 | EXAMPLE: example partial_molien; shows an example |
---|
1448 | " |
---|
1449 | { poly A(2); // A(2) will contain the return value of |
---|
1450 | // the intermediate result |
---|
1451 | if (char(basering)<>0) |
---|
1452 | { "ERROR: you have to change to a basering of characteristic 0, one in"; |
---|
1453 | " which the Molien series is defined"; |
---|
1454 | } |
---|
1455 | if (ncols(M)==2 && nrows(M)==1 && n>0 && size(#)<2) |
---|
1456 | { def br=basering; // keeping track of the old ring |
---|
1457 | map slead=br,ideal(0); |
---|
1458 | matrix s=slead(M); |
---|
1459 | if (s[1,1]<>1 || s[1,2]<>1) |
---|
1460 | { "ERROR: the constant terms of enumerator and denominator are not 1"; |
---|
1461 | return(); |
---|
1462 | } |
---|
1463 | |
---|
1464 | if (size(#)==0) |
---|
1465 | { A(2)=M[1,1]; // if a third parameter is not given, the |
---|
1466 | // intermediate result from the last run |
---|
1467 | // corresponds to the numerator - we need |
---|
1468 | } // its smallest term |
---|
1469 | else |
---|
1470 | { if (typeof(#[1])=="poly") |
---|
1471 | { A(2)=#[1]; // if a third term is given we 'start' |
---|
1472 | } // with its smallest term |
---|
1473 | else |
---|
1474 | { "ERROR: <poly> as third parameter expected"; |
---|
1475 | return(); |
---|
1476 | } |
---|
1477 | } |
---|
1478 | poly A(1)=M[1,2]; // denominator of Molien series (for now) |
---|
1479 | string mp=string(minpoly); |
---|
1480 | execute("ring R=("+charstr(br)+"),("+varstr(br)+"),ds;"); |
---|
1481 | execute("minpoly=number("+mp+");"); |
---|
1482 | poly A(1)=0; // A(1) will contain the sum of n terms - |
---|
1483 | poly min; // min will be our smallest term - |
---|
1484 | poly A(2)=fetch(br,A(2)); // fetching A(2) and M[1,2] into R |
---|
1485 | poly den=fetch(br,A(1)); |
---|
1486 | for (int i=1; i<=n; i++) // getting n terms and adding them up |
---|
1487 | { min=lead(A(2)); |
---|
1488 | A(1)=A(1)+min; |
---|
1489 | A(2)=A(2)-min*den; |
---|
1490 | } |
---|
1491 | setring br; // moving A(1) and A(2) back in the |
---|
1492 | A(1)=fetch(R,A(1)); // actual ring for output |
---|
1493 | A(2)=fetch(R,A(2)); |
---|
1494 | return(A(1..2)); |
---|
1495 | } |
---|
1496 | else |
---|
1497 | { "ERROR: the first parameter has to be a 1x2-matrix, i.e. the matrix"; |
---|
1498 | " returned by the procedure 'reynolds_molien', the second one"; |
---|
1499 | " should be > 0 and there should be no more than 3 parameters;" |
---|
1500 | return(); |
---|
1501 | } |
---|
1502 | } |
---|
1503 | example |
---|
1504 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
1505 | ring R=0,(x,y,z),dp; |
---|
1506 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
1507 | matrix REY,M=reynolds_molien(A); |
---|
1508 | poly p(1..2); |
---|
1509 | p(1..2)=partial_molien(M,5); |
---|
1510 | p(1); |
---|
1511 | p(1..2)=partial_molien(M,5,p(2)); |
---|
1512 | p(1); |
---|
1513 | } |
---|
1514 | /////////////////////////////////////////////////////////////////////////////// |
---|
1515 | |
---|
1516 | proc evaluate_reynolds (matrix REY, ideal I) |
---|
1517 | "USAGE: evaluate_reynolds(REY,I); |
---|
1518 | REY: a <matrix> representing the Reynolds operator, I: an arbitrary |
---|
1519 | <ideal> |
---|
1520 | ASSUME: REY is the first return value of group_reynolds() or reynolds_molien() |
---|
1521 | RETURNS: image of the polynomials defining I under the Reynolds operator |
---|
1522 | (type <ideal>) |
---|
1523 | NOTE: the characteristic of the coefficient field of the polynomial ring |
---|
1524 | should not divide the order of the finite matrix group |
---|
1525 | THEORY: REY has been constructed in such a way that each row serves as a ring |
---|
1526 | mapping of which the Reynolds operator is made up. |
---|
1527 | EXAMPLE: example evaluate_reynolds; shows an example |
---|
1528 | " |
---|
1529 | { def br=basering; |
---|
1530 | int n=nvars(br); |
---|
1531 | if (ncols(REY)==n) |
---|
1532 | { int m=nrows(REY); // we need m to 'cut' the ring |
---|
1533 | // homomorphisms 'out' of REY and to |
---|
1534 | // divide by the group order in the end |
---|
1535 | int num_poly=ncols(I); |
---|
1536 | matrix MI=matrix(I); |
---|
1537 | matrix MiI[1][num_poly]; |
---|
1538 | map pREY; |
---|
1539 | matrix rowREY[1][n]; |
---|
1540 | for (int i=1;i<=m;i++) |
---|
1541 | { rowREY=REY[i,1..n]; |
---|
1542 | pREY=br,ideal(rowREY); // f is now the i-th ring homomorphism |
---|
1543 | MiI=pREY(MI)+MiI; |
---|
1544 | } |
---|
1545 | MiI=(1/number(m))*MiI; |
---|
1546 | return(ideal(MiI)); |
---|
1547 | } |
---|
1548 | else |
---|
1549 | { "ERROR: the number of columns in the <matrix> should be the same as the"; |
---|
1550 | " number of variables in the basering; in fact it should be first"; |
---|
1551 | " return value of group_reynolds() or reynolds_molien()."; |
---|
1552 | return(); |
---|
1553 | } |
---|
1554 | } |
---|
1555 | example |
---|
1556 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
1557 | ring R=0,(x,y,z),dp; |
---|
1558 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
1559 | list L=group_reynolds(A); |
---|
1560 | ideal I=x2,y2,z2; |
---|
1561 | print(evaluate_reynolds(L[1],I)); |
---|
1562 | } |
---|
1563 | /////////////////////////////////////////////////////////////////////////////// |
---|
1564 | |
---|
1565 | proc invariant_basis (int g,list #) |
---|
1566 | "USAGE: invariant_basis(g,G1,G2,...); |
---|
1567 | g: an <int> indicating of which degree (>0) the homogeneous basis |
---|
1568 | shoud be, G1,G2,...: <matrices> generating a finite matrix group |
---|
1569 | RETURNS: the basis (type <ideal>) of the space of invariants of degree g |
---|
1570 | THEORY: A general polynomial of degree g is generated and the generators of |
---|
1571 | the matrix group applied. The difference ought to be 0 and this way a |
---|
1572 | system of linear equations is created. It is solved by computing |
---|
1573 | syzygies. |
---|
1574 | EXAMPLE: example invariant_basis; shows an example |
---|
1575 | " |
---|
1576 | { if (g<=0) |
---|
1577 | { "ERROR: the first parameter should be > 0"; |
---|
1578 | return(); |
---|
1579 | } |
---|
1580 | def br=basering; |
---|
1581 | ideal mon=sort(maxideal(g))[1]; // needed for constructing a general |
---|
1582 | int m=ncols(mon); // homogeneous polynomial of degree g |
---|
1583 | mon=sort(mon,intvec(m..1))[1]; |
---|
1584 | int a=size(#); |
---|
1585 | int i; |
---|
1586 | int n=nvars(br); |
---|
1587 | //---------------------- checking that the input is ok ----------------------- |
---|
1588 | for (i=1;i<=a;i++) |
---|
1589 | { if (typeof(#[i])=="matrix") |
---|
1590 | { if (nrows(#[i])==n && ncols(#[i])==n) |
---|
1591 | { matrix G(i)=#[i]; |
---|
1592 | } |
---|
1593 | else |
---|
1594 | { "ERROR: the number of variables of the base ring needs to be the same"; |
---|
1595 | " as the dimension of the square matrices"; |
---|
1596 | return(); |
---|
1597 | } |
---|
1598 | } |
---|
1599 | else |
---|
1600 | { "ERROR: the last parameters should be a list of matrices"; |
---|
1601 | return(); |
---|
1602 | } |
---|
1603 | } |
---|
1604 | //---------------------------------------------------------------------------- |
---|
1605 | execute("ring T=("+charstr(br)+"),("+varstr(br)+",p(1..m)),lp;"); |
---|
1606 | // p(1..m) are the general coefficients of the general polynomial of degree g |
---|
1607 | execute("ideal vars="+varstr(br)+";"); |
---|
1608 | map f; |
---|
1609 | ideal mon=imap(br,mon); |
---|
1610 | poly P=0; |
---|
1611 | for (i=m;i>=1;i--) |
---|
1612 | { P=P+p(i)*mon[i]; // P is the general polynomial |
---|
1613 | } |
---|
1614 | ideal I; // will help substituting variables in P |
---|
1615 | // by linear combinations of variables - |
---|
1616 | poly Pnew,temp; // Pnew is P with substitutions - |
---|
1617 | matrix S[m*a][m]; // will contain system of linear |
---|
1618 | // equations |
---|
1619 | int j,k; |
---|
1620 | //------------------- building the system of linear equations ---------------- |
---|
1621 | for (i=1;i<=a;i++) |
---|
1622 | { I=ideal(matrix(vars)*transpose(imap(br,G(i)))); |
---|
1623 | I=I,p(1..m); |
---|
1624 | f=T,I; |
---|
1625 | Pnew=f(P); |
---|
1626 | for (j=1;j<=m;j++) |
---|
1627 | { temp=P/mon[j]-Pnew/mon[j]; |
---|
1628 | for (k=1;k<=m;k++) |
---|
1629 | { S[m*(i-1)+j,k]=temp/p(k); |
---|
1630 | } |
---|
1631 | } |
---|
1632 | } |
---|
1633 | //---------------------------------------------------------------------------- |
---|
1634 | setring br; |
---|
1635 | map f=T,ideal(0); |
---|
1636 | matrix S=f(S); |
---|
1637 | matrix s=matrix(syz(S)); // s contains a basis of the space of |
---|
1638 | // solutions - |
---|
1639 | ideal I=ideal(matrix(mon)*s); // I contains a basis of homogeneous |
---|
1640 | if (I[1]<>0) // invariants of degree d |
---|
1641 | { for (i=1;i<=ncols(I);i++) |
---|
1642 | { I[i]=I[i]/leadcoef(I[i]); // setting leading coefficients to 1 |
---|
1643 | } |
---|
1644 | } |
---|
1645 | return(I); |
---|
1646 | } |
---|
1647 | example |
---|
1648 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
1649 | ring R=0,(x,y,z),dp; |
---|
1650 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
1651 | print(invariant_basis(2,A)); |
---|
1652 | } |
---|
1653 | /////////////////////////////////////////////////////////////////////////////// |
---|
1654 | |
---|
1655 | proc invariant_basis_reynolds (matrix REY,int d,list #) |
---|
1656 | "USAGE: invariant_basis_reynolds(REY,d[,flags]); |
---|
1657 | REY: a <matrix> representing the Reynolds operator, d: an <int> |
---|
1658 | indicating of which degree (>0) the homogeneous basis shoud be, flags: |
---|
1659 | an optional <intvec> with two entries: its first component gives the |
---|
1660 | dimension of the space (default <0 meaning unknown) and its second |
---|
1661 | component is used as the number of polynomials that should be mapped |
---|
1662 | to invariants during one call of evaluate_reynolds if the dimension of |
---|
1663 | the space is unknown or the number such that number x dimension |
---|
1664 | polynomials are mapped to invariants during one call of |
---|
1665 | evaluate_reynolds |
---|
1666 | ASSUME: REY is the first return value of group_reynolds() or reynolds_molien() |
---|
1667 | and flags[1] given by partial_molien |
---|
1668 | RETURN: the basis (type <ideal>) of the space of invariants of degree d |
---|
1669 | THEORY: Monomials of degree d are mapped to invariants with the Reynolds |
---|
1670 | operator. A linearly independent set is generated with the help of |
---|
1671 | minbase. |
---|
1672 | EXAMPLE: example invariant_basis_reynolds; shows an example |
---|
1673 | " |
---|
1674 | { |
---|
1675 | //---------------------- checking that the input is ok ----------------------- |
---|
1676 | if (d<=0) |
---|
1677 | { " ERROR: the second parameter should be > 0"; |
---|
1678 | return(); |
---|
1679 | } |
---|
1680 | if (size(#)>1) |
---|
1681 | { " ERROR: there should be at most three parameters"; |
---|
1682 | return(); |
---|
1683 | } |
---|
1684 | if (size(#)==1) |
---|
1685 | { if (typeof(#[1])<>"intvec") |
---|
1686 | { " ERROR: the third parameter should be of type <intvec>"; |
---|
1687 | return(); |
---|
1688 | } |
---|
1689 | if (size(#[1])<>2) |
---|
1690 | { " ERROR: there should be two components in <intvec>"; |
---|
1691 | return(); |
---|
1692 | } |
---|
1693 | else |
---|
1694 | { int cd=#[1][1]; |
---|
1695 | int step_fac=#[1][2]; |
---|
1696 | } |
---|
1697 | if (step_fac<=0) |
---|
1698 | { " ERROR: the second component of <intvec> should be > 0"; |
---|
1699 | return(); |
---|
1700 | } |
---|
1701 | if (cd==0) |
---|
1702 | { return(ideal(0)); |
---|
1703 | } |
---|
1704 | } |
---|
1705 | else |
---|
1706 | { int step_fac=1; |
---|
1707 | int cd=-1; |
---|
1708 | } |
---|
1709 | if (ncols(REY)<>nvars(basering)) |
---|
1710 | { "ERROR: the number of columns in the <matrix> should be the same as the"; |
---|
1711 | " number of variables in the basering; in fact it should be first"; |
---|
1712 | " return value of group_reynolds() or reynolds_molien()."; |
---|
1713 | return(); |
---|
1714 | } |
---|
1715 | //---------------------------------------------------------------------------- |
---|
1716 | ideal mon=sort(maxideal(d))[1]; |
---|
1717 | int DEGB = degBound; |
---|
1718 | degBound=d; |
---|
1719 | int j=ncols(mon); |
---|
1720 | mon=sort(mon,intvec(j..1))[1]; |
---|
1721 | ideal B; // will contain the basis |
---|
1722 | if (cd<0) |
---|
1723 | { if (step_fac>j) // all of mon will be mapped to |
---|
1724 | { B=evaluate_reynolds(REY,mon); // invariants at once |
---|
1725 | B=minbase(B); |
---|
1726 | degBound=DEGB; |
---|
1727 | return(B); |
---|
1728 | } |
---|
1729 | } |
---|
1730 | else |
---|
1731 | { if (step_fac*cd>j) // all of mon will be mapped to |
---|
1732 | { B=evaluate_reynolds(REY,mon); // invariants at once |
---|
1733 | B=minbase(B); |
---|
1734 | degBound=DEGB; |
---|
1735 | return(B); |
---|
1736 | } |
---|
1737 | } |
---|
1738 | int i,k; |
---|
1739 | int upper_bound=0; |
---|
1740 | int lower_bound=0; |
---|
1741 | ideal part_mon; // a part of mon of size step_fac*cd |
---|
1742 | while (1) |
---|
1743 | { lower_bound=upper_bound+1; |
---|
1744 | if (cd<0) |
---|
1745 | { upper_bound=upper_bound+step_fac; |
---|
1746 | } |
---|
1747 | else |
---|
1748 | { upper_bound=upper_bound+step_fac*cd; |
---|
1749 | } |
---|
1750 | if (upper_bound>j) |
---|
1751 | { upper_bound=j; |
---|
1752 | } |
---|
1753 | part_mon=mon[lower_bound..upper_bound]; |
---|
1754 | B=minbase(B+evaluate_reynolds(REY,part_mon)); |
---|
1755 | if ((ncols(B)==cd and B[1]<>0) or upper_bound==j) |
---|
1756 | { degBound=DEGB; |
---|
1757 | return(B); |
---|
1758 | } |
---|
1759 | } |
---|
1760 | } |
---|
1761 | example |
---|
1762 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
1763 | ring R=0,(x,y,z),dp; |
---|
1764 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
1765 | intvec flags=0,1,0; |
---|
1766 | matrix REY,M=reynolds_molien(A,flags); |
---|
1767 | flags=8,6; |
---|
1768 | print(invariant_basis_reynolds(REY,6,flags)); |
---|
1769 | } |
---|
1770 | |
---|
1771 | /////////////////////////////////////////////////////////////////////////////// |
---|
1772 | // This procedure generates linearly independent invariant polynomials of |
---|
1773 | // degree d that do not reduce to 0 modulo the primary invariants. It does this |
---|
1774 | // by applying the Reynolds operator to the monomials returned by kbase(sP,d). |
---|
1775 | // The result is used when computing secondary invariants. |
---|
1776 | /////////////////////////////////////////////////////////////////////////////// |
---|
1777 | proc sort_of_invariant_basis (ideal sP,matrix REY,int d,int step_fac) |
---|
1778 | { ideal mon=kbase(sP,d); |
---|
1779 | int DEGB=degBound; |
---|
1780 | degBound=d; |
---|
1781 | int j=ncols(mon); |
---|
1782 | int i; |
---|
1783 | mon=sort(mon,intvec(j..1))[1]; |
---|
1784 | ideal B; // will contain the "sort of basis" |
---|
1785 | if (step_fac>j) |
---|
1786 | { B=compress(evaluate_reynolds(REY,mon)); |
---|
1787 | for (i=1;i<=ncols(B);i++) // those are taken our that are o mod sP |
---|
1788 | { if (reduce(B[i],sP)==0) |
---|
1789 | { B[i]=0; |
---|
1790 | } |
---|
1791 | } |
---|
1792 | B=minbase(B); // here are the linearly independent ones |
---|
1793 | degBound=DEGB; |
---|
1794 | return(B); |
---|
1795 | } |
---|
1796 | int upper_bound=0; |
---|
1797 | int lower_bound=0; |
---|
1798 | ideal part_mon; // parts of mon |
---|
1799 | while (1) |
---|
1800 | { lower_bound=upper_bound+1; |
---|
1801 | upper_bound=upper_bound+step_fac; |
---|
1802 | if (upper_bound>j) |
---|
1803 | { upper_bound=j; |
---|
1804 | } |
---|
1805 | part_mon=mon[lower_bound..upper_bound]; |
---|
1806 | part_mon=compress(evaluate_reynolds(REY,part_mon)); |
---|
1807 | for (i=1;i<=ncols(part_mon);i++) |
---|
1808 | { if (reduce(part_mon[i],sP)==0) |
---|
1809 | { part_mon[i]=0; |
---|
1810 | } |
---|
1811 | } |
---|
1812 | B=minbase(B+part_mon); // here are the linearly independent ones |
---|
1813 | if (upper_bound==j) |
---|
1814 | { degBound=DEGB; |
---|
1815 | return(B); |
---|
1816 | } |
---|
1817 | } |
---|
1818 | } |
---|
1819 | |
---|
1820 | /////////////////////////////////////////////////////////////////////////////// |
---|
1821 | // Procedure returning the succeeding vector after vec. It is used to list |
---|
1822 | // all the vectors of Z^n with first nonzero entry 1. They are listed by |
---|
1823 | // increasing sum of the absolute value of their entries. |
---|
1824 | /////////////////////////////////////////////////////////////////////////////// |
---|
1825 | proc next_vector(intmat vec) |
---|
1826 | { int n=ncols(vec); // p: >0, n: <0, p0: >=0, n0: <=0 |
---|
1827 | for (int i=1;i<=n;i++) // finding out which is the first |
---|
1828 | { if (vec[1,i]<>0) // component <>0 |
---|
1829 | { break; |
---|
1830 | } |
---|
1831 | } |
---|
1832 | intmat new[1][n]; |
---|
1833 | if (i>n) // 0,...,0 --> 1,0....,0 |
---|
1834 | { new[1,1]=1; |
---|
1835 | return(new); |
---|
1836 | } |
---|
1837 | if (i==n) // 0,...,1 --> 1,1,0,...,0 |
---|
1838 | { new[1,1..2]=1,1; |
---|
1839 | return(new); |
---|
1840 | } |
---|
1841 | if (i==n-1) |
---|
1842 | { if (vec[1,n]==0) // 0,...,0,1,0 --> 0,...,0,1 |
---|
1843 | { new[1,n]=1; |
---|
1844 | return(new); |
---|
1845 | } |
---|
1846 | if (vec[1,n]>0) // 0,..,0,1,p --> 0,...,0,1,-p |
---|
1847 | { new[1,1..n]=vec[1,1..n-1],-vec[1,n]; |
---|
1848 | return(new); |
---|
1849 | } |
---|
1850 | new[1,1..2]=1,1-vec[1,n]; // 0,..,0,1,n --> 1,1-n,0,..,0 |
---|
1851 | return(new); |
---|
1852 | } |
---|
1853 | if (i>1) |
---|
1854 | { intmat temp[1][n-i+1]=vec[1,i..n]; // 0,...,0,1,*,...,* --> 1,*,...,* |
---|
1855 | temp=next_vector(temp); |
---|
1856 | new[1,i..n]=temp[1,1..n-i+1]; |
---|
1857 | return(new); |
---|
1858 | } // case left: 1,*,...,* |
---|
1859 | for (i=2;i<=n;i++) |
---|
1860 | { if (vec[1,i]>0) // make first positive negative and |
---|
1861 | { vec[1,i]=-vec[1,i]; // return |
---|
1862 | return(vec); |
---|
1863 | } |
---|
1864 | else |
---|
1865 | { vec[1,i]=-vec[1,i]; // make all negatives before positives |
---|
1866 | } // positive |
---|
1867 | } |
---|
1868 | for (i=2;i<=n-1;i++) // case: 1,p,...,p after 1,n,...,n |
---|
1869 | { if (vec[1,i]>0) |
---|
1870 | { vec[1,2]=vec[1,i]-1; // shuffleing things around... |
---|
1871 | if (i>2) // same sum of absolute values of entries |
---|
1872 | { vec[1,i]=0; |
---|
1873 | } |
---|
1874 | vec[1,i+1]=vec[1,i+1]+1; |
---|
1875 | return(vec); |
---|
1876 | } |
---|
1877 | } // case left: 1,0,...,0 --> 1,1,0,...,0 |
---|
1878 | new[1,2..3]=1,vec[1,n]; // and: 1,0,...,0,1 --> 0,1,1,0,...,0 |
---|
1879 | return(new); |
---|
1880 | } |
---|
1881 | |
---|
1882 | /////////////////////////////////////////////////////////////////////////////// |
---|
1883 | // Maps integers to elements of the base field. It is only called if the base |
---|
1884 | // field is of prime characteristic. If the base field has q elements |
---|
1885 | // (depending on minpoly) 1..q is mapped to those q elements. |
---|
1886 | /////////////////////////////////////////////////////////////////////////////// |
---|
1887 | proc int_number_map (int i) |
---|
1888 | { int p=char(basering); |
---|
1889 | if (minpoly==0) // if no minpoly is given, we have p |
---|
1890 | { i=i%p; // elements in the field |
---|
1891 | return(number(i)); |
---|
1892 | } |
---|
1893 | int d=pardeg(minpoly); |
---|
1894 | if (i<0) |
---|
1895 | { int bool=1; |
---|
1896 | i=(-1)*i; |
---|
1897 | } |
---|
1898 | i=i%p^d; // base field has p^d elements - |
---|
1899 | number a=par(1); // a is the root of the minpoly - we have |
---|
1900 | number out=0; // to construct a linear combination of |
---|
1901 | int j=1; // a^k |
---|
1902 | int k; |
---|
1903 | while (1) |
---|
1904 | { if (i<p^j) // finding an upper bound on i |
---|
1905 | { for (k=0;k<j-1;k++) |
---|
1906 | { out=out+((i/p^k)%p)*a^k; // finding how often p^k is contained in |
---|
1907 | } // i |
---|
1908 | out=out+(i/p^(j-1))*a^(j-1); |
---|
1909 | if (defined(bool)==voice) |
---|
1910 | { return((-1)*out); |
---|
1911 | } |
---|
1912 | return(out); |
---|
1913 | } |
---|
1914 | j++; |
---|
1915 | } |
---|
1916 | } |
---|
1917 | |
---|
1918 | /////////////////////////////////////////////////////////////////////////////// |
---|
1919 | // This procedure finds dif primary invariants in degree d. It returns all |
---|
1920 | // primary invariants found so far. The coefficients lie in a field of |
---|
1921 | // characteristic 0. |
---|
1922 | /////////////////////////////////////////////////////////////////////////////// |
---|
1923 | proc search (int n,int d,ideal B,int cd,ideal P,ideal sP,int i,int dif,int dB,ideal CI) |
---|
1924 | { intmat vec[1][cd]; // the coefficients for the next |
---|
1925 | // combination - |
---|
1926 | degBound=0; |
---|
1927 | poly test_poly; // the linear combination to test |
---|
1928 | int test_dim; |
---|
1929 | intvec h; // Hilbert series |
---|
1930 | int j=i+1; |
---|
1931 | matrix tB=transpose(B); |
---|
1932 | ideal TEST; |
---|
1933 | while(j<=i+dif) |
---|
1934 | { CI=CI+ideal(var(j)^d); // homogeneous polynomial of the same |
---|
1935 | // degree as the one we're looking for is |
---|
1936 | // added |
---|
1937 | // h=hilb(std(CI),1); |
---|
1938 | dB=dB+d-1; // used as degBound |
---|
1939 | while(1) |
---|
1940 | { vec=next_vector(vec); // next vector |
---|
1941 | test_poly=(vec*tB)[1,1]; |
---|
1942 | // degBound=dB; |
---|
1943 | TEST=sP+ideal(test_poly); |
---|
1944 | attrib(TEST,"isSB",1); |
---|
1945 | test_dim=dim(TEST); |
---|
1946 | // degBound=0; |
---|
1947 | if (n-test_dim==j) // the dimension has been lowered by one |
---|
1948 | { sP=TEST; |
---|
1949 | break; |
---|
1950 | } |
---|
1951 | // degBound=dB; |
---|
1952 | //TEST=std(sP+ideal(test_poly)); // should soon be replaced by next line |
---|
1953 | TEST=std(sP,test_poly); // or, better: |
---|
1954 | //TEST=std(sP,test_poly,h); // Hilbert driven std-calculation |
---|
1955 | test_dim=dim(TEST); |
---|
1956 | // degBound=0; |
---|
1957 | if (n-test_dim==j) // the dimension has been lowered by one |
---|
1958 | { sP=TEST; |
---|
1959 | break; |
---|
1960 | } |
---|
1961 | } |
---|
1962 | P[j]=test_poly; // test_poly ist added to primary |
---|
1963 | j++; // invariants |
---|
1964 | } |
---|
1965 | return(P,sP,CI,dB); |
---|
1966 | } |
---|
1967 | |
---|
1968 | /////////////////////////////////////////////////////////////////////////////// |
---|
1969 | // This procedure finds at most dif primary invariants in degree d. It returns |
---|
1970 | // all primary invariants found so far. The coefficients lie in the field of |
---|
1971 | // characteristic p>0. |
---|
1972 | /////////////////////////////////////////////////////////////////////////////// |
---|
1973 | proc p_search (int n,int d,ideal B,int cd,ideal P,ideal sP,int i,int dif,int dB,ideal CI) |
---|
1974 | { def br=basering; |
---|
1975 | degBound=0; |
---|
1976 | matrix vec(1)[1][cd]; // starting with 0-vector - |
---|
1977 | intmat new[1][cd]; // the coefficients for the next |
---|
1978 | // combination - |
---|
1979 | matrix pnew[1][cd]; // new needs to be mapped into br - |
---|
1980 | int counter=1; // counts the vectors |
---|
1981 | int j; |
---|
1982 | int p=char(br); |
---|
1983 | if (minpoly<>0) |
---|
1984 | { int ext_deg=pardeg(minpoly); // field has p^d elements |
---|
1985 | } |
---|
1986 | else |
---|
1987 | { int ext_deg=1; // field has p^d elements |
---|
1988 | } |
---|
1989 | poly test_poly; // the linear combination to test |
---|
1990 | int test_dim; |
---|
1991 | ring R=0,x,dp; // just to calculate next variable |
---|
1992 | // bound - |
---|
1993 | number bound=(number(p)^(ext_deg*cd)-1)/(number(p)^ext_deg-1)+1; |
---|
1994 | // this is how many linearly independent |
---|
1995 | // vectors of size cd exist having |
---|
1996 | // entries in the base field of br |
---|
1997 | setring br; |
---|
1998 | intvec h; // Hilbert series |
---|
1999 | int k=i+1; |
---|
2000 | if (ncols(B)<cd) { B[cd]=0; } |
---|
2001 | matrix tB=transpose(B); |
---|
2002 | ideal TEST; |
---|
2003 | while (k<=i+dif) |
---|
2004 | { CI=CI+ideal(var(k)^d); // homogeneous polynomial of the same |
---|
2005 | //degree as the one we're looking for is |
---|
2006 | // added |
---|
2007 | // h=hilb(std(CI),1); |
---|
2008 | dB=dB+d-1; // used as degBound |
---|
2009 | setring R; |
---|
2010 | while (number(counter)<>bound) // otherwise, we are done |
---|
2011 | { setring br; |
---|
2012 | new=next_vector(new); |
---|
2013 | for (j=1;j<=cd;j++) |
---|
2014 | { pnew[1,j]=int_number_map(new[1,j]); // mapping an integer into br |
---|
2015 | } |
---|
2016 | if (unique(vec(1..counter),pnew)) //checking whether we tried pnew before |
---|
2017 | { counter++; |
---|
2018 | matrix vec(counter)=pnew; // keeping track of the ones we tried - |
---|
2019 | test_poly=(vec(counter)*tB)[1,1]; // linear combination - |
---|
2020 | // degBound=dB; |
---|
2021 | TEST=sP+ideal(test_poly); |
---|
2022 | attrib(TEST,"isSB",1); |
---|
2023 | test_dim=dim(TEST); |
---|
2024 | // degBound=0; |
---|
2025 | if (n-test_dim==k) // the dimension has been lowered by one |
---|
2026 | { sP=TEST; |
---|
2027 | setring R; |
---|
2028 | break; |
---|
2029 | } |
---|
2030 | // degBound=dB; |
---|
2031 | //TEST=std(sP+ideal(test_poly)); // should soon to be replaced by next |
---|
2032 | // line |
---|
2033 | TEST=std(sP,test_poly); // or, better: |
---|
2034 | // TEST=std(sP,test_poly,h); // Hilbert driven std-calculation |
---|
2035 | test_dim=dim(TEST); |
---|
2036 | // degBound=0; |
---|
2037 | if (n-test_dim==k) // the dimension has been lowered by one |
---|
2038 | { sP=TEST; |
---|
2039 | setring R; |
---|
2040 | break; |
---|
2041 | } |
---|
2042 | } |
---|
2043 | setring R; |
---|
2044 | } |
---|
2045 | if (number(counter)<=bound) |
---|
2046 | { setring br; |
---|
2047 | P[k]=test_poly; // test_poly ist added to primary |
---|
2048 | } // invariants |
---|
2049 | else |
---|
2050 | { setring br; |
---|
2051 | CI=CI[1..size(CI)-1]; |
---|
2052 | return(P,sP,CI,dB-d+1); |
---|
2053 | } |
---|
2054 | k++; |
---|
2055 | } |
---|
2056 | return(P,sP,CI,dB); |
---|
2057 | } |
---|
2058 | /////////////////////////////////////////////////////////////////////////////// |
---|
2059 | |
---|
2060 | proc primary_char0 (matrix REY,matrix M,list #) |
---|
2061 | "USAGE: primary_char0(REY,M[,v]); |
---|
2062 | REY: a <matrix> representing the Reynolds operator, M: a 1x2 <matrix> |
---|
2063 | representing the Molien series, v: an optional <int> |
---|
2064 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien and |
---|
2065 | M the one of molien or the second one of reynolds_molien |
---|
2066 | DISPLAY: information about the various stages of the programme if v does not |
---|
2067 | equal 0 |
---|
2068 | RETURN: primary invariants (type <matrix>) of the invariant ring |
---|
2069 | THEORY: Bases of homogeneous invariants are generated successively and those |
---|
2070 | are chosen as primary invariants that lower the dimension of the ideal |
---|
2071 | generated by the previously found invariants (see paper \"Generating a |
---|
2072 | Noetherian Normalization of the Invariant Ring of a Finite Group\" by |
---|
2073 | Decker, Heydtmann, Schreyer (1998)). |
---|
2074 | EXAMPLE: example primary_char0; shows an example |
---|
2075 | " |
---|
2076 | { degBound=0; |
---|
2077 | if (char(basering)<>0) |
---|
2078 | { "ERROR: primary_char0 should only be used with rings of characteristic 0."; |
---|
2079 | return(); |
---|
2080 | } |
---|
2081 | //----------------- checking input and setting verbose mode ------------------ |
---|
2082 | if (size(#)>1) |
---|
2083 | { "ERROR: primary_char0 can only have three parameters."; |
---|
2084 | return(); |
---|
2085 | } |
---|
2086 | if (size(#)==1) |
---|
2087 | { if (typeof(#[1])<>"int") |
---|
2088 | { "ERROR: The third parameter should be of type <int>."; |
---|
2089 | return(); |
---|
2090 | } |
---|
2091 | else |
---|
2092 | { int v=#[1]; |
---|
2093 | } |
---|
2094 | } |
---|
2095 | else |
---|
2096 | { int v=0; |
---|
2097 | } |
---|
2098 | int n=nvars(basering); // n is the number of variables, as well |
---|
2099 | // as the size of the matrices, as well |
---|
2100 | // as the number of primary invariants, |
---|
2101 | // we should get |
---|
2102 | if (ncols(REY)<>n) |
---|
2103 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
2104 | return(); |
---|
2105 | } |
---|
2106 | if (ncols(M)<>2 or nrows(M)<>1) |
---|
2107 | { "ERROR: Second parameter ought to be the Molien series." |
---|
2108 | return(); |
---|
2109 | } |
---|
2110 | //---------------------------------------------------------------------------- |
---|
2111 | if (v && voice<>2) |
---|
2112 | { " We can start looking for primary invariants..."; |
---|
2113 | ""; |
---|
2114 | } |
---|
2115 | if (v && voice==2) |
---|
2116 | { ""; |
---|
2117 | } |
---|
2118 | //------------------------- initializing variables --------------------------- |
---|
2119 | int dB; |
---|
2120 | poly p(1..2); // p(1) will be used for single terms of |
---|
2121 | // the partial expansion, p(2) to store |
---|
2122 | p(1..2)=partial_molien(M,1); // the intermediate result - |
---|
2123 | poly v1=var(1); // we need v1 to split off coefficients |
---|
2124 | // in the partial expansion of M (which |
---|
2125 | // is in terms of the first variable) - |
---|
2126 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
2127 | // space of invariants of degree d, |
---|
2128 | // newdim: dimension the ideal generated |
---|
2129 | // the primary invariants plus basis |
---|
2130 | // elements, dif=n-i-newdim, i.e. the |
---|
2131 | // number of new primary invairants that |
---|
2132 | // should be added in this degree - |
---|
2133 | ideal P,Pplus,sPplus,CI,B; // P: will contain primary invariants, |
---|
2134 | // Pplus: P+B, CI: a complete |
---|
2135 | // intersection with the same Hilbert |
---|
2136 | // function as P |
---|
2137 | ideal sP=groebner(P); |
---|
2138 | dB=1; // used as degree bound |
---|
2139 | int i=0; |
---|
2140 | //-------------- loop that searches for primary invariants ------------------ |
---|
2141 | while(1) // repeat until n primary invariants are |
---|
2142 | { // found - |
---|
2143 | p(1..2)=partial_molien(M,1,p(2)); // next term of the partial expansion - |
---|
2144 | d=deg(p(1)); // degree where we'll search - |
---|
2145 | cd=int(coef(p(1),v1)[2,1]); // dimension of the homogeneous space of |
---|
2146 | // inviarants of degree d |
---|
2147 | if (v) |
---|
2148 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
2149 | } |
---|
2150 | B=invariant_basis_reynolds(REY,d,intvec(cd,6)); // basis of invariants of |
---|
2151 | // degree d |
---|
2152 | if (B[1]<>0) |
---|
2153 | { Pplus=P+B; |
---|
2154 | sPplus=groebner(Pplus); |
---|
2155 | newdim=dim(sPplus); |
---|
2156 | dif=n-i-newdim; |
---|
2157 | } |
---|
2158 | else |
---|
2159 | { dif=0; |
---|
2160 | } |
---|
2161 | if (dif<>0) // we have to find dif new primary |
---|
2162 | { // invariants |
---|
2163 | if (cd<>dif) |
---|
2164 | { P,sP,CI,dB=search(n,d,B,cd,P,sP,i,dif,dB,CI); // searching for dif invariants |
---|
2165 | } // i.e. we can take all of B |
---|
2166 | else |
---|
2167 | { for(j=i+1;j<=i+dif;j++) |
---|
2168 | { CI=CI+ideal(var(j)^d); |
---|
2169 | } |
---|
2170 | dB=dB+dif*(d-1); |
---|
2171 | P=Pplus; |
---|
2172 | sP=sPplus; |
---|
2173 | } |
---|
2174 | if (v) |
---|
2175 | { for (j=1;j<=dif;j++) |
---|
2176 | { " We find: "+string(P[i+j]); |
---|
2177 | } |
---|
2178 | } |
---|
2179 | i=i+dif; |
---|
2180 | if (i==n) // found all primary invariants |
---|
2181 | { if (v) |
---|
2182 | { ""; |
---|
2183 | " We found all primary invariants."; |
---|
2184 | ""; |
---|
2185 | } |
---|
2186 | return(matrix(P)); |
---|
2187 | } |
---|
2188 | } // done with degree d |
---|
2189 | } |
---|
2190 | } |
---|
2191 | example |
---|
2192 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
2193 | ring R=0,(x,y,z),dp; |
---|
2194 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
2195 | matrix REY,M=reynolds_molien(A); |
---|
2196 | matrix P=primary_char0(REY,M); |
---|
2197 | print(P); |
---|
2198 | } |
---|
2199 | /////////////////////////////////////////////////////////////////////////////// |
---|
2200 | |
---|
2201 | proc primary_charp (matrix REY,string ring_name,list #) |
---|
2202 | "USAGE: primary_charp(REY,ringname[,v]); |
---|
2203 | REY: a <matrix> representing the Reynolds operator, ringname: a |
---|
2204 | <string> giving the name of a ring where the Molien series is stored, |
---|
2205 | v: an optional <int> |
---|
2206 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien and |
---|
2207 | ringname gives the name of a ring of characteristic 0 that has been |
---|
2208 | created by molien or reynolds_molien |
---|
2209 | DISPLAY: information about the various stages of the programme if v does not |
---|
2210 | equal 0 |
---|
2211 | RETURN: primary invariants (type <matrix>) of the invariant ring |
---|
2212 | THEORY: Bases of homogeneous invariants are generated successively and those |
---|
2213 | are chosen as primary invariants that lower the dimension of the ideal |
---|
2214 | generated by the previously found invariants (see paper \"Generating a |
---|
2215 | Noetherian Normalization of the Invariant Ring of a Finite Group\" by |
---|
2216 | Decker, Heydtmann, Schreyer (1998)). |
---|
2217 | EXAMPLE: example primary_charp; shows an example |
---|
2218 | " |
---|
2219 | { degBound=0; |
---|
2220 | // ---------------- checking input and setting verbose mode ------------------- |
---|
2221 | if (char(basering)==0) |
---|
2222 | { "ERROR: primary_charp should only be used with rings of characteristic p>0."; |
---|
2223 | return(); |
---|
2224 | } |
---|
2225 | if (size(#)>1) |
---|
2226 | { "ERROR: primary_charp can only have three parameters."; |
---|
2227 | return(); |
---|
2228 | } |
---|
2229 | if (size(#)==1) |
---|
2230 | { if (typeof(#[1])<>"int") |
---|
2231 | { "ERROR: The third parameter should be of type <int>."; |
---|
2232 | return(); |
---|
2233 | } |
---|
2234 | else |
---|
2235 | { int v=#[1]; |
---|
2236 | } |
---|
2237 | } |
---|
2238 | else |
---|
2239 | { int v=0; |
---|
2240 | } |
---|
2241 | def br=basering; |
---|
2242 | int n=nvars(br); // n is the number of variables, as well |
---|
2243 | // as the size of the matrices, as well |
---|
2244 | // as the number of primary invariants, |
---|
2245 | // we should get |
---|
2246 | if (ncols(REY)<>n) |
---|
2247 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
2248 | return(); |
---|
2249 | } |
---|
2250 | if (typeof(`ring_name`)<>"ring") |
---|
2251 | { "ERROR: Second parameter ought to the name of a ring where the Molien"; |
---|
2252 | " is stored."; |
---|
2253 | return(); |
---|
2254 | } |
---|
2255 | //---------------------------------------------------------------------------- |
---|
2256 | if (v && voice<>2) |
---|
2257 | { " We can start looking for primary invariants..."; |
---|
2258 | ""; |
---|
2259 | } |
---|
2260 | if (v && voice==2) |
---|
2261 | { ""; |
---|
2262 | } |
---|
2263 | //----------------------- initializing variables ----------------------------- |
---|
2264 | int dB; |
---|
2265 | setring `ring_name`; // the Molien series is stores here - |
---|
2266 | poly p(1..2); // p(1) will be used for single terms of |
---|
2267 | // the partial expansion, p(2) to store |
---|
2268 | p(1..2)=partial_molien(M,1); // the intermediate result - |
---|
2269 | poly v1=var(1); // we need v1 to split off coefficients |
---|
2270 | // in the partial expansion of M (which |
---|
2271 | // is in terms of the first variable) |
---|
2272 | setring br; |
---|
2273 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
2274 | // space of invariants of degree d, |
---|
2275 | // newdim: dimension the ideal generated |
---|
2276 | // the primary invariants plus basis |
---|
2277 | // elements, dif=n-i-newdim, i.e. the |
---|
2278 | // number of new primary invairants that |
---|
2279 | // should be added in this degree - |
---|
2280 | ideal P,Pplus,sPplus,CI,B; // P: will contain primary invariants, |
---|
2281 | // Pplus: P+B, CI: a complete |
---|
2282 | // intersection with the same Hilbert |
---|
2283 | // function as P |
---|
2284 | ideal sP=groebner(P); |
---|
2285 | dB=1; // used as degree bound |
---|
2286 | int i=0; |
---|
2287 | //---------------- loop that searches for primary invariants ----------------- |
---|
2288 | while(1) // repeat until n primary invariants are |
---|
2289 | { // found |
---|
2290 | setring `ring_name`; |
---|
2291 | p(1..2)=partial_molien(M,1,p(2)); // next term of the partial expansion - |
---|
2292 | d=deg(p(1)); // degree where we'll search - |
---|
2293 | cd=int(coef(p(1),v1)[2,1]); // dimension of the homogeneous space of |
---|
2294 | // inviarants of degree d |
---|
2295 | setring br; |
---|
2296 | if (v) |
---|
2297 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
2298 | } |
---|
2299 | B=invariant_basis_reynolds(REY,d,intvec(cd,6)); // basis of invariants of |
---|
2300 | // degree d |
---|
2301 | if (ncols(B)<cd) |
---|
2302 | { |
---|
2303 | " warning: expected ",cd," invars, found ",ncols(B); |
---|
2304 | } |
---|
2305 | if (B[1]<>0) |
---|
2306 | { Pplus=P+B; |
---|
2307 | sPplus=groebner(Pplus); |
---|
2308 | newdim=dim(sPplus); |
---|
2309 | dif=n-i-newdim; |
---|
2310 | } |
---|
2311 | else |
---|
2312 | { dif=0; |
---|
2313 | } |
---|
2314 | if (dif<>0) // we have to find dif new primary |
---|
2315 | { // invariants |
---|
2316 | if (cd<>dif) |
---|
2317 | { P,sP,CI,dB=p_search(n,d,B,cd,P,sP,i,dif,dB,CI); |
---|
2318 | } |
---|
2319 | else // i.e. we can take all of B |
---|
2320 | { for(j=i+1;j>i+dif;j++) |
---|
2321 | { CI=CI+ideal(var(j)^d); |
---|
2322 | } |
---|
2323 | dB=dB+dif*(d-1); |
---|
2324 | P=Pplus; |
---|
2325 | sP=sPplus; |
---|
2326 | } |
---|
2327 | if (v) |
---|
2328 | { for (j=1;j<=size(P)-i;j++) |
---|
2329 | { " We find: "+string(P[i+j]); |
---|
2330 | } |
---|
2331 | } |
---|
2332 | i=size(P); |
---|
2333 | if (i==n) // found all primary invariants |
---|
2334 | { if (v) |
---|
2335 | { ""; |
---|
2336 | " We found all primary invariants."; |
---|
2337 | ""; |
---|
2338 | } |
---|
2339 | return(matrix(P)); |
---|
2340 | } |
---|
2341 | } // done with degree d |
---|
2342 | } |
---|
2343 | } |
---|
2344 | example |
---|
2345 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7 (changed into char 3)"; echo=2; |
---|
2346 | ring R=3,(x,y,z),dp; |
---|
2347 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
2348 | list L=group_reynolds(A); |
---|
2349 | string newring="alskdfj"; |
---|
2350 | molien(L[2..size(L)],newring); |
---|
2351 | matrix P=primary_charp(L[1],newring); |
---|
2352 | kill `newring`; |
---|
2353 | print(P); |
---|
2354 | } |
---|
2355 | /////////////////////////////////////////////////////////////////////////////// |
---|
2356 | |
---|
2357 | proc primary_char0_no_molien (matrix REY, list #) |
---|
2358 | "USAGE: primary_char0_no_molien(REY[,v]); |
---|
2359 | REY: a <matrix> representing the Reynolds operator, v: an optional |
---|
2360 | <int> |
---|
2361 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien |
---|
2362 | DISPLAY: information about the various stages of the programme if v does not |
---|
2363 | equal 0 |
---|
2364 | RETURN: primary invariants (type <matrix>) of the invariant ring and an |
---|
2365 | <intvec> listing some of the degrees where no non-trivial homogeneous |
---|
2366 | invariants are to be found |
---|
2367 | THEORY: Bases of homogeneous invariants are generated successively and those |
---|
2368 | are chosen as primary invariants that lower the dimension of the ideal |
---|
2369 | generated by the previously found invariants (see paper \"Generating a |
---|
2370 | Noetherian Normalization of the Invariant Ring of a Finite Group\" by |
---|
2371 | Decker, Heydtmann, Schreyer (1998)). |
---|
2372 | EXAMPLE: example primary_char0_no_molien; shows an example |
---|
2373 | " |
---|
2374 | { degBound=0; |
---|
2375 | //-------------- checking input and setting verbose mode --------------------- |
---|
2376 | if (char(basering)<>0) |
---|
2377 | { "ERROR: primary_char0_no_molien should only be used with rings of"; |
---|
2378 | " characteristic 0."; |
---|
2379 | return(); |
---|
2380 | } |
---|
2381 | if (size(#)>1) |
---|
2382 | { "ERROR: primary_char0_no_molien can only have two parameters."; |
---|
2383 | return(); |
---|
2384 | } |
---|
2385 | if (size(#)==1) |
---|
2386 | { if (typeof(#[1])<>"int") |
---|
2387 | { "ERROR: The second parameter should be of type <int>."; |
---|
2388 | return(); |
---|
2389 | } |
---|
2390 | else |
---|
2391 | { int v=#[1]; |
---|
2392 | } |
---|
2393 | } |
---|
2394 | else |
---|
2395 | { int v=0; |
---|
2396 | } |
---|
2397 | int n=nvars(basering); // n is the number of variables, as well |
---|
2398 | // as the size of the matrices, as well |
---|
2399 | // as the number of primary invariants, |
---|
2400 | // we should get |
---|
2401 | if (ncols(REY)<>n) |
---|
2402 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
2403 | return(); |
---|
2404 | } |
---|
2405 | //---------------------------------------------------------------------------- |
---|
2406 | if (v && voice<>2) |
---|
2407 | { " We can start looking for primary invariants..."; |
---|
2408 | ""; |
---|
2409 | } |
---|
2410 | if (v && voice==2) |
---|
2411 | { ""; |
---|
2412 | } |
---|
2413 | //----------------------- initializing variables ----------------------------- |
---|
2414 | int dB; |
---|
2415 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
2416 | // space of invariants of degree d, |
---|
2417 | // newdim: dimension the ideal generated |
---|
2418 | // the primary invariants plus basis |
---|
2419 | // elements, dif=n-i-newdim, i.e. the |
---|
2420 | // number of new primary invairants that |
---|
2421 | // should be added in this degree - |
---|
2422 | ideal P,Pplus,CI,B; // P: will contain primary invariants, |
---|
2423 | // Pplus: P+B, CI: a complete |
---|
2424 | // intersection with the same Hilbert |
---|
2425 | // function as P |
---|
2426 | ideal sP=groebner(P); |
---|
2427 | dB=1; // used as degree bound - |
---|
2428 | d=0; // initializing |
---|
2429 | int i=0; |
---|
2430 | intvec deg_vector; |
---|
2431 | //------------------ loop that searches for primary invariants --------------- |
---|
2432 | while(1) // repeat until n primary invariants are |
---|
2433 | { // found - |
---|
2434 | d++; // degree where we'll search |
---|
2435 | if (v) |
---|
2436 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
2437 | } |
---|
2438 | B=invariant_basis_reynolds(REY,d,intvec(-1,6)); // basis of invariants of |
---|
2439 | // degree d |
---|
2440 | if (B[1]<>0) |
---|
2441 | { Pplus=P+B; |
---|
2442 | newdim=dim(groebner(Pplus)); |
---|
2443 | dif=n-i-newdim; |
---|
2444 | } |
---|
2445 | else |
---|
2446 | { dif=0; |
---|
2447 | deg_vector=deg_vector,d; |
---|
2448 | } |
---|
2449 | if (dif<>0) // we have to find dif new primary |
---|
2450 | { // invariants |
---|
2451 | cd=size(B); |
---|
2452 | if (cd<>dif) |
---|
2453 | { P,sP,CI,dB=search(n,d,B,cd,P,sP,i,dif,dB,CI); |
---|
2454 | } |
---|
2455 | else // i.e. we can take all of B |
---|
2456 | { for(j=i+1;j<=i+dif;j++) |
---|
2457 | { CI=CI+ideal(var(j)^d); |
---|
2458 | } |
---|
2459 | dB=dB+dif*(d-1); |
---|
2460 | P=Pplus; |
---|
2461 | sP=groebner(P); |
---|
2462 | } |
---|
2463 | if (v) |
---|
2464 | { for (j=1;j<=dif;j++) |
---|
2465 | { " We find: "+string(P[i+j]); |
---|
2466 | } |
---|
2467 | } |
---|
2468 | i=i+dif; |
---|
2469 | if (i==n) // found all primary invariants |
---|
2470 | { if (v) |
---|
2471 | { ""; |
---|
2472 | " We found all primary invariants."; |
---|
2473 | ""; |
---|
2474 | } |
---|
2475 | if (deg_vector==0) |
---|
2476 | { return(matrix(P)); |
---|
2477 | } |
---|
2478 | else |
---|
2479 | { return(matrix(P),compress(deg_vector)); |
---|
2480 | } |
---|
2481 | } |
---|
2482 | } // done with degree d |
---|
2483 | else |
---|
2484 | { if (v) |
---|
2485 | { " None here..."; |
---|
2486 | } |
---|
2487 | } |
---|
2488 | } |
---|
2489 | } |
---|
2490 | example |
---|
2491 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
2492 | ring R=0,(x,y,z),dp; |
---|
2493 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
2494 | list L=group_reynolds(A); |
---|
2495 | list l=primary_char0_no_molien(L[1]); |
---|
2496 | print(l[1]); |
---|
2497 | } |
---|
2498 | /////////////////////////////////////////////////////////////////////////////// |
---|
2499 | |
---|
2500 | proc primary_charp_no_molien (matrix REY, list #) |
---|
2501 | "USAGE: primary_charp_no_molien(REY[,v]); |
---|
2502 | REY: a <matrix> representing the Reynolds operator, v: an optional |
---|
2503 | <int> |
---|
2504 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien |
---|
2505 | DISPLAY: information about the various stages of the programme if v does not |
---|
2506 | equal 0 |
---|
2507 | RETURN: primary invariants (type <matrix>) of the invariant ring and an |
---|
2508 | <intvec> listing some of the degrees where no non-trivial homogeneous |
---|
2509 | invariants are to be found |
---|
2510 | THEORY: Bases of homogeneous invariants are generated successively and those |
---|
2511 | are chosen as primary invariants that lower the dimension of the ideal |
---|
2512 | generated by the previously found invariants (see paper \"Generating a |
---|
2513 | Noetherian Normalization of the Invariant Ring of a Finite Group\" by |
---|
2514 | Decker, Heydtmann, Schreyer (1998)). |
---|
2515 | EXAMPLE: example primary_charp_no_molien; shows an example |
---|
2516 | " |
---|
2517 | { degBound=0; |
---|
2518 | //----------------- checking input and setting verbose mode ------------------ |
---|
2519 | if (char(basering)==0) |
---|
2520 | { "ERROR: primary_charp_no_molien should only be used with rings of"; |
---|
2521 | " characteristic p>0."; |
---|
2522 | return(); |
---|
2523 | } |
---|
2524 | if (size(#)>1) |
---|
2525 | { "ERROR: primary_charp_no_molien can only have two parameters."; |
---|
2526 | return(); |
---|
2527 | } |
---|
2528 | if (size(#)==1) |
---|
2529 | { if (typeof(#[1])<>"int") |
---|
2530 | { "ERROR: The second parameter should be of type <int>."; |
---|
2531 | return(); |
---|
2532 | } |
---|
2533 | else |
---|
2534 | { int v=#[1]; } |
---|
2535 | } |
---|
2536 | else |
---|
2537 | { int v=0; |
---|
2538 | } |
---|
2539 | int n=nvars(basering); // n is the number of variables, as well |
---|
2540 | // as the size of the matrices, as well |
---|
2541 | // as the number of primary invariants, |
---|
2542 | // we should get |
---|
2543 | if (ncols(REY)<>n) |
---|
2544 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
2545 | return(); |
---|
2546 | } |
---|
2547 | //---------------------------------------------------------------------------- |
---|
2548 | if (v && voice<>2) |
---|
2549 | { " We can start looking for primary invariants..."; |
---|
2550 | ""; |
---|
2551 | } |
---|
2552 | if (v && voice==2) |
---|
2553 | { ""; } |
---|
2554 | //-------------------- initializing variables -------------------------------- |
---|
2555 | int dB; |
---|
2556 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
2557 | // space of invariants of degree d, |
---|
2558 | // newdim: dimension the ideal generated |
---|
2559 | // the primary invariants plus basis |
---|
2560 | // elements, dif=n-i-newdim, i.e. the |
---|
2561 | // number of new primary invairants that |
---|
2562 | // should be added in this degree - |
---|
2563 | ideal P,Pplus,sPplus,CI,B; // P: will contain primary invariants, |
---|
2564 | // Pplus: P+B, CI: a complete |
---|
2565 | // intersection with the same Hilbert |
---|
2566 | // function as P |
---|
2567 | ideal sP=groebner(P); |
---|
2568 | dB=1; // used as degree bound - |
---|
2569 | d=0; // initializing |
---|
2570 | int i=0; |
---|
2571 | intvec deg_vector; |
---|
2572 | //------------------ loop that searches for primary invariants --------------- |
---|
2573 | while(1) // repeat until n primary invariants are |
---|
2574 | { // found - |
---|
2575 | d++; // degree where we'll search |
---|
2576 | if (v) |
---|
2577 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
2578 | } |
---|
2579 | B=invariant_basis_reynolds(REY,d,intvec(-1,6)); // basis of invariants of |
---|
2580 | // degree d |
---|
2581 | if (B[1]<>0) |
---|
2582 | { Pplus=P+B; |
---|
2583 | sPplus=groebner(Pplus); |
---|
2584 | newdim=dim(sPplus); |
---|
2585 | dif=n-i-newdim; |
---|
2586 | } |
---|
2587 | else |
---|
2588 | { dif=0; |
---|
2589 | deg_vector=deg_vector,d; |
---|
2590 | } |
---|
2591 | if (dif<>0) // we have to find dif new primary |
---|
2592 | { // invariants |
---|
2593 | cd=size(B); |
---|
2594 | if (cd<>dif) |
---|
2595 | { P,sP,CI,dB=p_search(n,d,B,cd,P,sP,i,dif,dB,CI); |
---|
2596 | } |
---|
2597 | else // i.e. we can take all of B |
---|
2598 | { for(j=i+1;j<=i+dif;j++) |
---|
2599 | { CI=CI+ideal(var(j)^d); |
---|
2600 | } |
---|
2601 | dB=dB+dif*(d-1); |
---|
2602 | P=Pplus; |
---|
2603 | sP=sPplus; |
---|
2604 | } |
---|
2605 | if (v) |
---|
2606 | { for (j=1;j<=size(P)-i;j++) |
---|
2607 | { " We find: "+string(P[i+j]); |
---|
2608 | } |
---|
2609 | } |
---|
2610 | i=size(P); |
---|
2611 | if (i==n) // found all primary invariants |
---|
2612 | { if (v) |
---|
2613 | { ""; |
---|
2614 | " We found all primary invariants."; |
---|
2615 | ""; |
---|
2616 | } |
---|
2617 | if (deg_vector==0) |
---|
2618 | { return(matrix(P)); |
---|
2619 | } |
---|
2620 | else |
---|
2621 | { return(matrix(P),compress(deg_vector)); |
---|
2622 | } |
---|
2623 | } |
---|
2624 | } // done with degree d |
---|
2625 | else |
---|
2626 | { if (v) |
---|
2627 | { " None here..."; |
---|
2628 | } |
---|
2629 | } |
---|
2630 | } |
---|
2631 | } |
---|
2632 | example |
---|
2633 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7 (changed into char 3)"; echo=2; |
---|
2634 | ring R=3,(x,y,z),dp; |
---|
2635 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
2636 | list L=group_reynolds(A); |
---|
2637 | list l=primary_charp_no_molien(L[1]); |
---|
2638 | print(l[1]); |
---|
2639 | } |
---|
2640 | /////////////////////////////////////////////////////////////////////////////// |
---|
2641 | |
---|
2642 | proc primary_charp_without (list #) |
---|
2643 | "USAGE: primary_charp_without(G1,G2,...[,v]); |
---|
2644 | G1,G2,...: <matrices> generating a finite matrix group, v: an optional |
---|
2645 | <int> |
---|
2646 | DISPLAY: information about the various stages of the programme if v does not |
---|
2647 | equal 0 |
---|
2648 | RETURN: primary invariants (type <matrix>) of the invariant ring |
---|
2649 | THEORY: Bases of homogeneous invariants are generated successively and those |
---|
2650 | are chosen as primary invariants that lower the dimension of the ideal |
---|
2651 | generated by the previously found invariants (see paper \"Generating a |
---|
2652 | Noetherian Normalization of the Invariant Ring of a Finite Group\" by |
---|
2653 | Decker, Heydtmann, Schreyer (1998)). No Reynolds |
---|
2654 | operator or Molien series is used. |
---|
2655 | EXAMPLE: example primary_charp_without; shows an example |
---|
2656 | " |
---|
2657 | { degBound=0; |
---|
2658 | //--------------------- checking input and setting verbose mode -------------- |
---|
2659 | if (char(basering)==0) |
---|
2660 | { "ERROR: primary_charp_without should only be used with rings of"; |
---|
2661 | " characteristic 0."; |
---|
2662 | return(); |
---|
2663 | } |
---|
2664 | if (size(#)==0) |
---|
2665 | { "ERROR: There are no parameters."; |
---|
2666 | return(); |
---|
2667 | } |
---|
2668 | if (typeof(#[size(#)])=="int") |
---|
2669 | { int v=#[size(#)]; |
---|
2670 | int gen_num=size(#)-1; |
---|
2671 | if (gen_num==0) |
---|
2672 | { "ERROR: There are no generators of a finite matrix group given."; |
---|
2673 | return(); |
---|
2674 | } |
---|
2675 | } |
---|
2676 | else |
---|
2677 | { int v=0; |
---|
2678 | int gen_num=size(#); |
---|
2679 | } |
---|
2680 | int n=nvars(basering); // n is the number of variables, as well |
---|
2681 | // as the size of the matrices, as well |
---|
2682 | // as the number of primary invariants, |
---|
2683 | // we should get |
---|
2684 | for (int i=1;i<=gen_num;i++) |
---|
2685 | { if (typeof(#[i])=="matrix") |
---|
2686 | { if (nrows(#[i])<>n or ncols(#[i])<>n) |
---|
2687 | { "ERROR: The number of variables of the base ring needs to be the same"; |
---|
2688 | " as the dimension of the square matrices"; |
---|
2689 | return(); |
---|
2690 | } |
---|
2691 | } |
---|
2692 | else |
---|
2693 | { "ERROR: The first parameters should be a list of matrices"; |
---|
2694 | return(); |
---|
2695 | } |
---|
2696 | } |
---|
2697 | //---------------------------------------------------------------------------- |
---|
2698 | if (v && voice==2) |
---|
2699 | { ""; |
---|
2700 | } |
---|
2701 | //---------------------------- initializing variables ------------------------ |
---|
2702 | int dB; |
---|
2703 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
2704 | // space of invariants of degree d, |
---|
2705 | // newdim: dimension the ideal generated |
---|
2706 | // the primary invariants plus basis |
---|
2707 | // elements, dif=n-i-newdim, i.e. the |
---|
2708 | // number of new primary invairants that |
---|
2709 | // should be added in this degree - |
---|
2710 | ideal P,Pplus,sPplus,CI,B; // P: will contain primary invariants, |
---|
2711 | // Pplus: P+B, CI: a complete |
---|
2712 | // intersection with the same Hilbert |
---|
2713 | // function as P |
---|
2714 | ideal sP=groebner(P); |
---|
2715 | dB=1; // used as degree bound - |
---|
2716 | d=0; // initializing |
---|
2717 | i=0; |
---|
2718 | intvec deg_vector; |
---|
2719 | //-------------------- loop that searches for primary invariants ------------- |
---|
2720 | while(1) // repeat until n primary invariants are |
---|
2721 | { // found - |
---|
2722 | d++; // degree where we'll search |
---|
2723 | if (v) |
---|
2724 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
2725 | } |
---|
2726 | B=invariant_basis(d,#[1..gen_num]); // basis of invariants of degree d |
---|
2727 | if (B[1]<>0) |
---|
2728 | { Pplus=P+B; |
---|
2729 | sPplus=groebner(Pplus); |
---|
2730 | newdim=dim(sPplus); |
---|
2731 | dif=n-i-newdim; |
---|
2732 | } |
---|
2733 | else |
---|
2734 | { dif=0; |
---|
2735 | deg_vector=deg_vector,d; |
---|
2736 | } |
---|
2737 | if (dif<>0) // we have to find dif new primary |
---|
2738 | { // invariants |
---|
2739 | cd=size(B); |
---|
2740 | if (cd<>dif) |
---|
2741 | { P,sP,CI,dB=p_search(n,d,B,cd,P,sP,i,dif,dB,CI); |
---|
2742 | } |
---|
2743 | else // i.e. we can take all of B |
---|
2744 | { for(j=i+1;j<=i+dif;j++) |
---|
2745 | { CI=CI+ideal(var(j)^d); |
---|
2746 | } |
---|
2747 | dB=dB+dif*(d-1); |
---|
2748 | P=Pplus; |
---|
2749 | sP=sPplus; |
---|
2750 | } |
---|
2751 | if (v) |
---|
2752 | { for (j=1;j<=size(P)-i;j++) |
---|
2753 | { " We find: "+string(P[i+j]); |
---|
2754 | } |
---|
2755 | } |
---|
2756 | i=size(P); |
---|
2757 | if (i==n) // found all primary invariants |
---|
2758 | { if (v) |
---|
2759 | { ""; |
---|
2760 | " We found all primary invariants."; |
---|
2761 | ""; |
---|
2762 | } |
---|
2763 | return(matrix(P)); |
---|
2764 | } |
---|
2765 | } // done with degree d |
---|
2766 | else |
---|
2767 | { if (v) |
---|
2768 | { " None here..."; |
---|
2769 | } |
---|
2770 | } |
---|
2771 | } |
---|
2772 | } |
---|
2773 | example |
---|
2774 | { "EXAMPLE:"; echo=2; |
---|
2775 | ring R=2,(x,y,z),dp; |
---|
2776 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
2777 | matrix P=primary_charp_without(A); |
---|
2778 | print(P); |
---|
2779 | } |
---|
2780 | /////////////////////////////////////////////////////////////////////////////// |
---|
2781 | |
---|
2782 | proc primary_invariants (list #) |
---|
2783 | "USAGE: primary_invariants(G1,G2,...[,flags]); |
---|
2784 | G1,G2,...: <matrices> generating a finite matrix group, flags: an |
---|
2785 | optional <intvec> with three entries, if the first one equals 0 (also |
---|
2786 | the default), the programme attempts to compute the Molien series and |
---|
2787 | Reynolds operator, if it equals 1, the programme is told that the |
---|
2788 | Molien series should not be computed, if it equals -1 characteristic 0 |
---|
2789 | is simulated, i.e. the Molien series is computed as if the base field |
---|
2790 | were characteristic 0 (the user must choose a field of large prime |
---|
2791 | characteristic, e.g. 32003) and if the first one is anything else, it |
---|
2792 | means that the characteristic of the base field divides the group |
---|
2793 | order, the second component should give the size of intervals between |
---|
2794 | canceling common factors in the expansion of the Molien series, 0 (the |
---|
2795 | default) means only once after generating all terms, in prime |
---|
2796 | characteristic also a negative number can be given to indicate that |
---|
2797 | common factors should always be canceled when the expansion is simple |
---|
2798 | (the root of the extension field occurs not among the coefficients) |
---|
2799 | DISPLAY: information about the various stages of the programme if the third |
---|
2800 | flag does not equal 0 |
---|
2801 | RETURN: primary invariants (type <matrix>) of the invariant ring and if |
---|
2802 | computable Reynolds operator (type <matrix>) and Molien series (type |
---|
2803 | <matrix>) or ring name (type string) where the Molien series |
---|
2804 | can be found in the char p case; if the first flag is 1 and we are in |
---|
2805 | the non-modular case then an <intvec> is returned giving some of the |
---|
2806 | degrees where no non-trivial homogeneous invariants can be found |
---|
2807 | THEORY: Bases of homogeneous invariants are generated successively and those |
---|
2808 | are chosen as primary invariants that lower the dimension of the ideal |
---|
2809 | generated by the previously found invariants (see paper \"Generating a |
---|
2810 | Noetherian Normalization of the Invariant Ring of a Finite Group\" by |
---|
2811 | Decker, Heydtmann, Schreyer (1998)). |
---|
2812 | EXAMPLE: example primary_invariants; shows an example |
---|
2813 | " |
---|
2814 | { |
---|
2815 | // ----------------- checking input and setting flags ------------------------ |
---|
2816 | if (size(#)==0) |
---|
2817 | { "ERROR: There are no parameters."; |
---|
2818 | return(); |
---|
2819 | } |
---|
2820 | int ch=char(basering); // the algorithms depend very much on the |
---|
2821 | // characteristic of the ground field |
---|
2822 | int n=nvars(basering); // n is the number of variables, as well |
---|
2823 | // as the size of the matrices, as well |
---|
2824 | // as the number of primary invariants, |
---|
2825 | // we should get |
---|
2826 | int gen_num; |
---|
2827 | int mol_flag,v; |
---|
2828 | if (typeof(#[size(#)])=="intvec") |
---|
2829 | { if (size(#[size(#)])<>3) |
---|
2830 | { "ERROR: <intvec> should have three entries."; |
---|
2831 | return(); |
---|
2832 | } |
---|
2833 | gen_num=size(#)-1; |
---|
2834 | mol_flag=#[size(#)][1]; |
---|
2835 | if (#[size(#)][2]<0 && (ch==0 or (ch<>0 && mol_flag==-1))) |
---|
2836 | { "ERROR: the second component of <intvec> should be >=0"; |
---|
2837 | return(); |
---|
2838 | } |
---|
2839 | int interval=#[size(#)][2]; |
---|
2840 | v=#[size(#)][3]; |
---|
2841 | if (gen_num==0) |
---|
2842 | { "ERROR: There are no generators of a finite matrix group given."; |
---|
2843 | return(); |
---|
2844 | } |
---|
2845 | } |
---|
2846 | else |
---|
2847 | { gen_num=size(#); |
---|
2848 | mol_flag=0; |
---|
2849 | int interval=0; |
---|
2850 | v=0; |
---|
2851 | } |
---|
2852 | for (int i=1;i<=gen_num;i++) |
---|
2853 | { if (typeof(#[i])=="matrix") |
---|
2854 | { if (nrows(#[i])<>n or ncols(#[i])<>n) |
---|
2855 | { "ERROR: The number of variables of the base ring needs to be the same"; |
---|
2856 | " as the dimension of the square matrices"; |
---|
2857 | return(); |
---|
2858 | } |
---|
2859 | } |
---|
2860 | else |
---|
2861 | { "ERROR: The first parameters should be a list of matrices"; |
---|
2862 | return(); |
---|
2863 | } |
---|
2864 | } |
---|
2865 | //---------------------------------------------------------------------------- |
---|
2866 | if (mol_flag==0) |
---|
2867 | { if (ch==0) |
---|
2868 | { matrix REY,M=reynolds_molien(#[1..gen_num],intvec(mol_flag,interval,v)); |
---|
2869 | // one will contain Reynolds operator and |
---|
2870 | // the other enumerator and denominator |
---|
2871 | // of Molien series |
---|
2872 | matrix P=primary_char0(REY,M,v); |
---|
2873 | return(P,REY,M); |
---|
2874 | } |
---|
2875 | else |
---|
2876 | { list L=group_reynolds(#[1..gen_num],v); |
---|
2877 | if (L[1]<>0) // testing whether we are in the modular |
---|
2878 | { string newring="aksldfalkdsflkj"; // case |
---|
2879 | if (minpoly==0) |
---|
2880 | { if (v) |
---|
2881 | { " We are dealing with the non-modular case."; |
---|
2882 | } |
---|
2883 | if (typeof(L[2])=="int") |
---|
2884 | { molien(L[3..size(L)],newring,L[2],intvec(mol_flag,interval,v)); |
---|
2885 | } |
---|
2886 | else |
---|
2887 | { molien(L[2..size(L)],newring,intvec(mol_flag,interval,v)); |
---|
2888 | } |
---|
2889 | matrix P=primary_charp(L[1],newring,v); |
---|
2890 | return(P,L[1],newring); |
---|
2891 | } |
---|
2892 | else |
---|
2893 | { if (v) |
---|
2894 | { " Since it is impossible for this programme to calculate the Molien series for"; |
---|
2895 | " invariant rings over extension fields of prime characteristic, we have to"; |
---|
2896 | " continue without it."; |
---|
2897 | ""; |
---|
2898 | |
---|
2899 | } |
---|
2900 | list l=primary_charp_no_molien(L[1],v); |
---|
2901 | if (size(l)==2) |
---|
2902 | { return(l[1],L[1],l[2]); |
---|
2903 | } |
---|
2904 | else |
---|
2905 | { return(l[1],L[1]); |
---|
2906 | } |
---|
2907 | } |
---|
2908 | } |
---|
2909 | else // the modular case |
---|
2910 | { if (v) |
---|
2911 | { " There is also no Molien series, we can make use of..."; |
---|
2912 | ""; |
---|
2913 | " We can start looking for primary invariants..."; |
---|
2914 | ""; |
---|
2915 | } |
---|
2916 | return(primary_charp_without(#[1..gen_num],v)); |
---|
2917 | } |
---|
2918 | } |
---|
2919 | } |
---|
2920 | if (mol_flag==1) // the user wants no calculation of the |
---|
2921 | { list L=group_reynolds(#[1..gen_num],v); // Molien series |
---|
2922 | if (ch==0) |
---|
2923 | { list l=primary_char0_no_molien(L[1],v); |
---|
2924 | if (size(l)==2) |
---|
2925 | { return(l[1],L[1],l[2]); |
---|
2926 | } |
---|
2927 | else |
---|
2928 | { return(l[1],L[1]); |
---|
2929 | } |
---|
2930 | } |
---|
2931 | else |
---|
2932 | { if (L[1]<>0) // testing whether we are in the modular |
---|
2933 | { list l=primary_charp_no_molien(L[1],v); // case |
---|
2934 | if (size(l)==2) |
---|
2935 | { return(l[1],L[1],l[2]); |
---|
2936 | } |
---|
2937 | else |
---|
2938 | { return(l[1],L[1]); |
---|
2939 | } |
---|
2940 | } |
---|
2941 | else // the modular case |
---|
2942 | { if (v) |
---|
2943 | { " We can start looking for primary invariants..."; |
---|
2944 | ""; |
---|
2945 | } |
---|
2946 | return(primary_charp_without(#[1..gen_num],v)); |
---|
2947 | } |
---|
2948 | } |
---|
2949 | } |
---|
2950 | if (mol_flag==-1) |
---|
2951 | { if (ch==0) |
---|
2952 | { "ERROR: Characteristic 0 can only be simulated in characteristic p>>0."; |
---|
2953 | return(); |
---|
2954 | } |
---|
2955 | list L=group_reynolds(#[1..gen_num],v); |
---|
2956 | string newring="aksldfalkdsflkj"; |
---|
2957 | molien(L[2..size(L)],newring,intvec(1,interval,v)); |
---|
2958 | matrix P=primary_charp(L[1],newring,v); |
---|
2959 | return(P,L[1],newring); |
---|
2960 | } |
---|
2961 | else // the user specified that the |
---|
2962 | { if (ch==0) // characteristic divides the group order |
---|
2963 | { "ERROR: The characteristic cannot divide the group order when it is 0."; |
---|
2964 | return(); |
---|
2965 | } |
---|
2966 | if (v) |
---|
2967 | { ""; |
---|
2968 | } |
---|
2969 | return(primary_charp_without(#[1..gen_num],v)); |
---|
2970 | } |
---|
2971 | } |
---|
2972 | example |
---|
2973 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
2974 | echo=2; |
---|
2975 | ring R=0,(x,y,z),dp; |
---|
2976 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
2977 | list L=primary_invariants(A); |
---|
2978 | print(L[1]); |
---|
2979 | } |
---|
2980 | |
---|
2981 | /////////////////////////////////////////////////////////////////////////////// |
---|
2982 | // This procedure finds dif primary invariants in degree d. It returns all |
---|
2983 | // primary invariants found so far. The coefficients lie in a field of |
---|
2984 | // characteristic 0. |
---|
2985 | /////////////////////////////////////////////////////////////////////////////// |
---|
2986 | proc search_random (int n,int d,ideal B,int cd,ideal P,int i,int dif,int dB,ideal CI,int max) |
---|
2987 | { string answer; |
---|
2988 | degBound=0; |
---|
2989 | int j,k,test_dim,flag; |
---|
2990 | matrix test_matrix[1][dif]; // the linear combination to test |
---|
2991 | intvec h; // Hilbert series |
---|
2992 | for (j=i+1;j<=i+dif;j++) |
---|
2993 | { CI=CI+ideal(var(j)^d); // homogeneous polynomial of the same |
---|
2994 | // degree as the one we're looking for |
---|
2995 | // is added |
---|
2996 | } |
---|
2997 | ideal TEST; |
---|
2998 | // h=hilb(std(CI),1); |
---|
2999 | dB=dB+dif*(d-1); // used as degBound |
---|
3000 | while (1) |
---|
3001 | { test_matrix=matrix(B)*random(max,cd,dif); |
---|
3002 | // degBound=dB; |
---|
3003 | TEST=P+ideal(test_matrix); |
---|
3004 | attrib(TEST,"isSB",1); |
---|
3005 | test_dim=dim(TEST); |
---|
3006 | // degBound=0; |
---|
3007 | if (n-test_dim==i+dif) |
---|
3008 | { break; |
---|
3009 | } |
---|
3010 | // degBound=dB; |
---|
3011 | test_dim=dim(groebner(TEST)); |
---|
3012 | // test_dim=dim(std(TEST,h)); // Hilbert driven std-calculation |
---|
3013 | // degBound=0; |
---|
3014 | if (n-test_dim==i+dif) |
---|
3015 | { break; |
---|
3016 | } |
---|
3017 | else |
---|
3018 | { "HELP: The "+string(dif)+" random combination(s) of the "+string(cd)+" basis elements with"; |
---|
3019 | " coefficients in the range from -"+string(max)+" to "+string(max)+" did not lower the"; |
---|
3020 | " dimension by "+string(dif)+". You can abort, try again or give a new range:"; |
---|
3021 | answer=""; |
---|
3022 | while (answer<>"n |
---|
3023 | " && answer<>"y |
---|
3024 | ") |
---|
3025 | { " Do you want to abort (y/n)?"; |
---|
3026 | answer=read(""); |
---|
3027 | } |
---|
3028 | if (answer=="y |
---|
3029 | ") |
---|
3030 | { flag=1; |
---|
3031 | break; |
---|
3032 | } |
---|
3033 | answer=""; |
---|
3034 | while (answer<>"n |
---|
3035 | " && answer<>"y |
---|
3036 | ") |
---|
3037 | { " Do you want to try again (y/n)?"; |
---|
3038 | answer=read(""); |
---|
3039 | } |
---|
3040 | if (answer=="n |
---|
3041 | ") |
---|
3042 | { flag=1; |
---|
3043 | while (flag) |
---|
3044 | { " Give a new <int> > "+string(max)+" that bounds the range of coefficients:"; |
---|
3045 | answer=read(""); |
---|
3046 | for (j=1;j<=size(answer)-1;j++) |
---|
3047 | { for (k=0;k<=9;k++) |
---|
3048 | { if (answer[j]==string(k)) |
---|
3049 | { break; |
---|
3050 | } |
---|
3051 | } |
---|
3052 | if (k>9) |
---|
3053 | { flag=1; |
---|
3054 | break; |
---|
3055 | } |
---|
3056 | flag=0; |
---|
3057 | } |
---|
3058 | if (not(flag)) |
---|
3059 | { execute("test_dim="+string(answer[1..size(answer)])); |
---|
3060 | if (test_dim<=max) |
---|
3061 | { flag=1; |
---|
3062 | } |
---|
3063 | else |
---|
3064 | { max=test_dim; |
---|
3065 | } |
---|
3066 | } |
---|
3067 | } |
---|
3068 | } |
---|
3069 | } |
---|
3070 | } |
---|
3071 | if (not(flag)) |
---|
3072 | { P[(i+1)..(i+dif)]=test_matrix[1,1..dif]; |
---|
3073 | } |
---|
3074 | return(P,CI,dB); |
---|
3075 | } |
---|
3076 | |
---|
3077 | /////////////////////////////////////////////////////////////////////////////// |
---|
3078 | // This procedure finds at most dif primary invariants in degree d. It returns |
---|
3079 | // all primary invariants found so far. The coefficients lie in the field of |
---|
3080 | // characteristic p>0. |
---|
3081 | /////////////////////////////////////////////////////////////////////////////// |
---|
3082 | proc p_search_random (int n,int d,ideal B,int cd,ideal P,int i,int dif,int dB,ideal CI,int max) |
---|
3083 | { string answer; |
---|
3084 | degBound=0; |
---|
3085 | int j,k,test_dim,flag; |
---|
3086 | matrix test_matrix[1][dif]; // the linear combination to test |
---|
3087 | intvec h; // Hilbert series |
---|
3088 | ideal TEST; |
---|
3089 | while (dif>0) |
---|
3090 | { for (j=i+1;j<=i+dif;j++) |
---|
3091 | { CI=CI+ideal(var(j)^d); // homogeneous polynomial of the same |
---|
3092 | // degree as the one we're looking for |
---|
3093 | // is added |
---|
3094 | } |
---|
3095 | // h=hilb(std(CI),1); |
---|
3096 | dB=dB+dif*(d-1); // used as degBound |
---|
3097 | test_matrix=matrix(B)*random(max,cd,dif); |
---|
3098 | // degBound=dB; |
---|
3099 | TEST=P+ideal(test_matrix); |
---|
3100 | attrib(TEST,"isSB",1); |
---|
3101 | test_dim=dim(TEST); |
---|
3102 | // degBound=0; |
---|
3103 | if (n-test_dim==i+dif) |
---|
3104 | { break; |
---|
3105 | } |
---|
3106 | // degBound=dB; |
---|
3107 | test_dim=dim(groebner(TEST)); |
---|
3108 | // test_dim=dim(std(TEST,h)); // Hilbert driven std-calculation |
---|
3109 | // degBound=0; |
---|
3110 | if (n-test_dim==i+dif) |
---|
3111 | { break; |
---|
3112 | } |
---|
3113 | else |
---|
3114 | { "HELP: The "+string(dif)+" random combination(s) of the "+string(cd)+" basis elements with"; |
---|
3115 | " coefficients in the range from -"+string(max)+" to "+string(max)+" did not lower the"; |
---|
3116 | " dimension by "+string(dif)+". You can abort, try again, lower the number of"; |
---|
3117 | " combinations searched for by 1 or give a larger coefficient range:"; |
---|
3118 | answer=""; |
---|
3119 | while (answer<>"n |
---|
3120 | " && answer<>"y |
---|
3121 | ") |
---|
3122 | { " Do you want to abort (y/n)?"; |
---|
3123 | answer=read(""); |
---|
3124 | } |
---|
3125 | if (answer=="y |
---|
3126 | ") |
---|
3127 | { flag=1; |
---|
3128 | break; |
---|
3129 | } |
---|
3130 | answer=""; |
---|
3131 | while (answer<>"n |
---|
3132 | " && answer<>"y |
---|
3133 | ") |
---|
3134 | { " Do you want to try again (y/n)?"; |
---|
3135 | answer=read(""); |
---|
3136 | } |
---|
3137 | if (answer=="n |
---|
3138 | ") |
---|
3139 | { answer=""; |
---|
3140 | while (answer<>"n |
---|
3141 | " && answer<>"y |
---|
3142 | ") |
---|
3143 | { " Do you want to lower the number of combinations by 1 (y/n)?"; |
---|
3144 | answer=read(""); |
---|
3145 | } |
---|
3146 | if (answer=="y |
---|
3147 | ") |
---|
3148 | { dif=dif-1; |
---|
3149 | } |
---|
3150 | else |
---|
3151 | { flag=1; |
---|
3152 | while (flag) |
---|
3153 | { " Give a new <int> > "+string(max)+" that bounds the range of coefficients:"; |
---|
3154 | answer=read(""); |
---|
3155 | for (j=1;j<=size(answer)-1;j++) |
---|
3156 | { for (k=0;k<=9;k++) |
---|
3157 | { if (answer[j]==string(k)) |
---|
3158 | { break; |
---|
3159 | } |
---|
3160 | } |
---|
3161 | if (k>9) |
---|
3162 | { flag=1; |
---|
3163 | break; |
---|
3164 | } |
---|
3165 | flag=0; |
---|
3166 | } |
---|
3167 | if (not(flag)) |
---|
3168 | { execute("test_dim="+string(answer[1..size(answer)])); |
---|
3169 | if (test_dim<=max) |
---|
3170 | { flag=1; |
---|
3171 | } |
---|
3172 | else |
---|
3173 | { max=test_dim; |
---|
3174 | } |
---|
3175 | } |
---|
3176 | } |
---|
3177 | } |
---|
3178 | } |
---|
3179 | } |
---|
3180 | CI=CI[1..i]; |
---|
3181 | dB=dB-dif*(d-1); |
---|
3182 | } |
---|
3183 | if (dif && not(flag)) |
---|
3184 | { P[(i+1)..(i+dif)]=test_matrix[1,1..dif]; |
---|
3185 | } |
---|
3186 | if (dif && flag) |
---|
3187 | { P[n+1]=0; |
---|
3188 | } |
---|
3189 | return(P,CI,dB); |
---|
3190 | } |
---|
3191 | /////////////////////////////////////////////////////////////////////////////// |
---|
3192 | |
---|
3193 | proc primary_char0_random (matrix REY,matrix M,int max,list #) |
---|
3194 | "USAGE: primary_char0_random(REY,M,r[,v]); |
---|
3195 | REY: a <matrix> representing the Reynolds operator, M: a 1x2 <matrix> |
---|
3196 | representing the Molien series, r: an <int> where -|r| to |r| is the |
---|
3197 | range of coefficients of the random combinations of bases elements, |
---|
3198 | v: an optional <int> |
---|
3199 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien and |
---|
3200 | M the one of molien or the second one of reynolds_molien |
---|
3201 | DISPLAY: information about the various stages of the programme if v does not |
---|
3202 | equal 0 |
---|
3203 | RETURN: primary invariants (type <matrix>) of the invariant ring |
---|
3204 | THEORY: Bases of homogeneous invariants are generated successively and random |
---|
3205 | linear combinations are chosen as primary invariants that lower the |
---|
3206 | dimension of the ideal generated by the previously found invariants |
---|
3207 | (see \"Generating a Noetherian Normalization of the Invariant Ring of |
---|
3208 | a Finite Group\" by Decker, Heydtmann, Schreyer (1998)). |
---|
3209 | EXAMPLE: example primary_char0_random; shows an example |
---|
3210 | " |
---|
3211 | { degBound=0; |
---|
3212 | if (char(basering)<>0) |
---|
3213 | { "ERROR: primary_char0_random should only be used with rings of"; |
---|
3214 | " characteristic 0."; |
---|
3215 | return(); |
---|
3216 | } |
---|
3217 | //----------------- checking input and setting verbose mode ------------------ |
---|
3218 | if (size(#)>1) |
---|
3219 | { "ERROR: primary_char0_random can only have four parameters."; |
---|
3220 | return(); |
---|
3221 | } |
---|
3222 | if (size(#)==1) |
---|
3223 | { if (typeof(#[1])<>"int") |
---|
3224 | { "ERROR: The fourth parameter should be of type <int>."; |
---|
3225 | return(); |
---|
3226 | } |
---|
3227 | else |
---|
3228 | { int v=#[1]; |
---|
3229 | } |
---|
3230 | } |
---|
3231 | else |
---|
3232 | { int v=0; |
---|
3233 | } |
---|
3234 | int n=nvars(basering); // n is the number of variables, as well |
---|
3235 | // as the size of the matrices, as well |
---|
3236 | // as the number of primary invariants, |
---|
3237 | // we should get |
---|
3238 | if (ncols(REY)<>n) |
---|
3239 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
3240 | return(); |
---|
3241 | } |
---|
3242 | if (ncols(M)<>2 or nrows(M)<>1) |
---|
3243 | { "ERROR: Second parameter ought to be the Molien series." |
---|
3244 | return(); |
---|
3245 | } |
---|
3246 | //---------------------------------------------------------------------------- |
---|
3247 | if (v && voice<>2) |
---|
3248 | { " We can start looking for primary invariants..."; |
---|
3249 | ""; |
---|
3250 | } |
---|
3251 | if (v && voice==2) |
---|
3252 | { ""; |
---|
3253 | } |
---|
3254 | //------------------------- initializing variables --------------------------- |
---|
3255 | int dB; |
---|
3256 | poly p(1..2); // p(1) will be used for single terms of |
---|
3257 | // the partial expansion, p(2) to store |
---|
3258 | p(1..2)=partial_molien(M,1); // the intermediate result - |
---|
3259 | poly v1=var(1); // we need v1 to split off coefficients |
---|
3260 | // in the partial expansion of M (which |
---|
3261 | // is in terms of the first variable) - |
---|
3262 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
3263 | // space of invariants of degree d, |
---|
3264 | // newdim: dimension the ideal generated |
---|
3265 | // the primary invariants plus basis |
---|
3266 | // elements, dif=n-i-newdim, i.e. the |
---|
3267 | // number of new primary invairants that |
---|
3268 | // should be added in this degree - |
---|
3269 | ideal P,Pplus,CI,B; // P: will contain primary invariants, |
---|
3270 | // Pplus: P+B,CI: a complete |
---|
3271 | // intersection with the same Hilbert |
---|
3272 | // function as P - |
---|
3273 | dB=1; // used as degree bound |
---|
3274 | int i=0; |
---|
3275 | //-------------- loop that searches for primary invariants ------------------ |
---|
3276 | while(1) // repeat until n primary invariants are |
---|
3277 | { // found - |
---|
3278 | p(1..2)=partial_molien(M,1,p(2)); // next term of the partial expansion - |
---|
3279 | d=deg(p(1)); // degree where we'll search - |
---|
3280 | cd=int(coef(p(1),v1)[2,1]); // dimension of the homogeneous space of |
---|
3281 | // inviarants of degree d |
---|
3282 | if (v) |
---|
3283 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
3284 | } |
---|
3285 | B=invariant_basis_reynolds(REY,d,intvec(cd,6)); // basis of invariants of |
---|
3286 | // degree d |
---|
3287 | if (B[1]<>0) |
---|
3288 | { Pplus=P+B; |
---|
3289 | newdim=dim(groebner(Pplus)); |
---|
3290 | dif=n-i-newdim; |
---|
3291 | } |
---|
3292 | else |
---|
3293 | { dif=0; |
---|
3294 | } |
---|
3295 | if (dif<>0) // we have to find dif new primary |
---|
3296 | { // invariants |
---|
3297 | if (cd<>dif) |
---|
3298 | { P,CI,dB=search_random(n,d,B,cd,P,i,dif,dB,CI,max); // searching for |
---|
3299 | } // dif invariants - |
---|
3300 | else // i.e. we can take all of B |
---|
3301 | { for(j=i+1;j>i+dif;j++) |
---|
3302 | { CI=CI+ideal(var(j)^d); |
---|
3303 | } |
---|
3304 | dB=dB+dif*(d-1); |
---|
3305 | P=Pplus; |
---|
3306 | } |
---|
3307 | if (ncols(P)==i) |
---|
3308 | { "WARNING: The return value is not a set of primary invariants, but"; |
---|
3309 | " polynomials qualifying as the first "+string(i)+" primary invariants."; |
---|
3310 | return(matrix(P)); |
---|
3311 | } |
---|
3312 | if (v) |
---|
3313 | { for (j=1;j<=dif;j++) |
---|
3314 | { " We find: "+string(P[i+j]); |
---|
3315 | } |
---|
3316 | } |
---|
3317 | i=i+dif; |
---|
3318 | if (i==n) // found all primary invariants |
---|
3319 | { if (v) |
---|
3320 | { ""; |
---|
3321 | " We found all primary invariants."; |
---|
3322 | ""; |
---|
3323 | } |
---|
3324 | return(matrix(P)); |
---|
3325 | } |
---|
3326 | } // done with degree d |
---|
3327 | } |
---|
3328 | } |
---|
3329 | example |
---|
3330 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
3331 | ring R=0,(x,y,z),dp; |
---|
3332 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
3333 | matrix REY,M=reynolds_molien(A); |
---|
3334 | matrix P=primary_char0_random(REY,M,1); |
---|
3335 | print(P); |
---|
3336 | } |
---|
3337 | /////////////////////////////////////////////////////////////////////////////// |
---|
3338 | |
---|
3339 | proc primary_charp_random (matrix REY,string ring_name,int max,list #) |
---|
3340 | "USAGE: primary_charp_random(REY,ringname,r[,v]); |
---|
3341 | REY: a <matrix> representing the Reynolds operator, ringname: a |
---|
3342 | <string> giving the name of a ring where the Molien series is stored, |
---|
3343 | r: an <int> where -|r| to |r| is the range of coefficients of the |
---|
3344 | random combinations of bases elements, v: an optional <int> |
---|
3345 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien and |
---|
3346 | ringname gives the name of a ring of characteristic 0 that has been |
---|
3347 | created by molien or reynolds_molien |
---|
3348 | DISPLAY: information about the various stages of the programme if v does not |
---|
3349 | equal 0 |
---|
3350 | RETURN: primary invariants (type <matrix>) of the invariant ring |
---|
3351 | THEORY: Bases of homogeneous invariants are generated successively and random |
---|
3352 | linear combinations are chosen as primary invariants that lower the |
---|
3353 | dimension of the ideal generated by the previously found invariants |
---|
3354 | (see \"Generating a Noetherian Normalization of the Invariant Ring of |
---|
3355 | a Finite Group\" by Decker, Heydtmann, Schreyer (1998)). |
---|
3356 | EXAMPLE: example primary_charp_random; shows an example |
---|
3357 | " |
---|
3358 | { degBound=0; |
---|
3359 | // ---------------- checking input and setting verbose mode ------------------ |
---|
3360 | if (char(basering)==0) |
---|
3361 | { "ERROR: primary_charp_random should only be used with rings of"; |
---|
3362 | " characteristic p>0."; |
---|
3363 | return(); |
---|
3364 | } |
---|
3365 | if (size(#)>1) |
---|
3366 | { "ERROR: primary_charp_random can only have four parameters."; |
---|
3367 | return(); |
---|
3368 | } |
---|
3369 | if (size(#)==1) |
---|
3370 | { if (typeof(#[1])<>"int") |
---|
3371 | { "ERROR: The fourth parameter should be of type <int>."; |
---|
3372 | return(); |
---|
3373 | } |
---|
3374 | else |
---|
3375 | { int v=#[1]; |
---|
3376 | } |
---|
3377 | } |
---|
3378 | else |
---|
3379 | { int v=0; |
---|
3380 | } |
---|
3381 | def br=basering; |
---|
3382 | int n=nvars(br); // n is the number of variables, as well |
---|
3383 | // as the size of the matrices, as well |
---|
3384 | // as the number of primary invariants, |
---|
3385 | // we should get |
---|
3386 | if (ncols(REY)<>n) |
---|
3387 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
3388 | return(); |
---|
3389 | } |
---|
3390 | if (typeof(`ring_name`)<>"ring") |
---|
3391 | { "ERROR: Second parameter ought to the name of a ring where the Molien"; |
---|
3392 | " is stored."; |
---|
3393 | return(); |
---|
3394 | } |
---|
3395 | //---------------------------------------------------------------------------- |
---|
3396 | if (v && voice<>2) |
---|
3397 | { " We can start looking for primary invariants..."; |
---|
3398 | ""; |
---|
3399 | } |
---|
3400 | if (v && voice==2) |
---|
3401 | { ""; |
---|
3402 | } |
---|
3403 | //----------------------- initializing variables ----------------------------- |
---|
3404 | int dB; |
---|
3405 | setring `ring_name`; // the Molien series is stores here - |
---|
3406 | poly p(1..2); // p(1) will be used for single terms of |
---|
3407 | // the partial expansion, p(2) to store |
---|
3408 | p(1..2)=partial_molien(M,1); // the intermediate result - |
---|
3409 | poly v1=var(1); // we need v1 to split off coefficients |
---|
3410 | // in the partial expansion of M (which |
---|
3411 | // is in terms of the first variable) |
---|
3412 | setring br; |
---|
3413 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
3414 | // space of invariants of degree d, |
---|
3415 | // newdim: dimension the ideal generated |
---|
3416 | // the primary invariants plus basis |
---|
3417 | // elements, dif=n-i-newdim, i.e. the |
---|
3418 | // number of new primary invairants that |
---|
3419 | // should be added in this degree - |
---|
3420 | ideal P,Pplus,CI,B; // P: will contain primary invariants, |
---|
3421 | // Pplus: P+B, CI: a complete |
---|
3422 | // intersection with the same Hilbert |
---|
3423 | // function as P - |
---|
3424 | dB=1; // used as degree bound |
---|
3425 | int i=0; |
---|
3426 | //---------------- loop that searches for primary invariants ----------------- |
---|
3427 | while(1) // repeat until n primary invariants are |
---|
3428 | { // found |
---|
3429 | setring `ring_name`; |
---|
3430 | p(1..2)=partial_molien(M,1,p(2)); // next term of the partial expansion - |
---|
3431 | d=deg(p(1)); // degree where we'll search - |
---|
3432 | cd=int(coef(p(1),v1)[2,1]); // dimension of the homogeneous space of |
---|
3433 | // inviarants of degree d |
---|
3434 | setring br; |
---|
3435 | if (v) |
---|
3436 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
3437 | } |
---|
3438 | B=invariant_basis_reynolds(REY,d,intvec(cd,6)); // basis of invariants of |
---|
3439 | // degree d |
---|
3440 | if (B[1]<>0) |
---|
3441 | { Pplus=P+B; |
---|
3442 | newdim=dim(groebner(Pplus)); |
---|
3443 | dif=n-i-newdim; |
---|
3444 | } |
---|
3445 | else |
---|
3446 | { dif=0; |
---|
3447 | } |
---|
3448 | if (dif<>0) // we have to find dif new primary |
---|
3449 | { // invariants |
---|
3450 | if (cd<>dif) |
---|
3451 | { P,CI,dB=p_search_random(n,d,B,cd,P,i,dif,dB,CI,max); |
---|
3452 | } |
---|
3453 | else // i.e. we can take all of B |
---|
3454 | { for(j=i+1;j>i+dif;j++) |
---|
3455 | { CI=CI+ideal(var(j)^d); |
---|
3456 | } |
---|
3457 | dB=dB+dif*(d-1); |
---|
3458 | P=Pplus; |
---|
3459 | } |
---|
3460 | if (ncols(P)==n+1) |
---|
3461 | { "WARNING: The first return value is not a set of primary invariants,"; |
---|
3462 | " but polynomials qualifying as the first "+string(i)+" primary invariants."; |
---|
3463 | return(matrix(P)); |
---|
3464 | } |
---|
3465 | if (v) |
---|
3466 | { for (j=1;j<=size(P)-i;j++) |
---|
3467 | { " We find: "+string(P[i+j]); |
---|
3468 | } |
---|
3469 | } |
---|
3470 | i=size(P); |
---|
3471 | if (i==n) // found all primary invariants |
---|
3472 | { if (v) |
---|
3473 | { ""; |
---|
3474 | " We found all primary invariants."; |
---|
3475 | ""; |
---|
3476 | } |
---|
3477 | return(matrix(P)); |
---|
3478 | } |
---|
3479 | } // done with degree d |
---|
3480 | } |
---|
3481 | } |
---|
3482 | example |
---|
3483 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7 (changed into char 3)"; echo=2; |
---|
3484 | ring R=3,(x,y,z),dp; |
---|
3485 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
3486 | list L=group_reynolds(A); |
---|
3487 | string newring="alskdfj"; |
---|
3488 | molien(L[2..size(L)],newring); |
---|
3489 | matrix P=primary_charp_random(L[1],newring,1); |
---|
3490 | kill `newring`; |
---|
3491 | print(P); |
---|
3492 | } |
---|
3493 | /////////////////////////////////////////////////////////////////////////////// |
---|
3494 | |
---|
3495 | proc primary_char0_no_molien_random (matrix REY, int max, list #) |
---|
3496 | "USAGE: primary_char0_no_molien_random(REY,r[,v]); |
---|
3497 | REY: a <matrix> representing the Reynolds operator, r: an <int> where |
---|
3498 | -|r| to |r| is the range of coefficients of the random combinations of |
---|
3499 | bases elements, v: an optional <int> |
---|
3500 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien |
---|
3501 | DISPLAY: information about the various stages of the programme if v does not |
---|
3502 | equal 0 |
---|
3503 | RETURN: primary invariants (type <matrix>) of the invariant ring and an |
---|
3504 | <intvec> listing some of the degrees where no non-trivial homogeneous |
---|
3505 | invariants are to be found |
---|
3506 | THEORY: Bases of homogeneous invariants are generated successively and random |
---|
3507 | linear combinations are chosen as primary invariants that lower the |
---|
3508 | dimension of the ideal generated by the previously found invariants |
---|
3509 | (see \"Generating a Noetherian Normalization of the Invariant Ring of |
---|
3510 | a Finite Group\" by Decker, Heydtmann, Schreyer (1998)). |
---|
3511 | EXAMPLE: example primary_char0_no_molien_random; shows an example |
---|
3512 | " |
---|
3513 | { degBound=0; |
---|
3514 | //-------------- checking input and setting verbose mode --------------------- |
---|
3515 | if (char(basering)<>0) |
---|
3516 | { "ERROR: primary_char0_no_molien_random should only be used with rings of"; |
---|
3517 | " characteristic 0."; |
---|
3518 | return(); |
---|
3519 | } |
---|
3520 | if (size(#)>1) |
---|
3521 | { "ERROR: primary_char0_no_molien_random can only have three parameters."; |
---|
3522 | return(); |
---|
3523 | } |
---|
3524 | if (size(#)==1) |
---|
3525 | { if (typeof(#[1])<>"int") |
---|
3526 | { "ERROR: The third parameter should be of type <int>."; |
---|
3527 | return(); |
---|
3528 | } |
---|
3529 | else |
---|
3530 | { int v=#[1]; |
---|
3531 | } |
---|
3532 | } |
---|
3533 | else |
---|
3534 | { int v=0; |
---|
3535 | } |
---|
3536 | int n=nvars(basering); // n is the number of variables, as well |
---|
3537 | // as the size of the matrices, as well |
---|
3538 | // as the number of primary invariants, |
---|
3539 | // we should get |
---|
3540 | if (ncols(REY)<>n) |
---|
3541 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
3542 | return(); |
---|
3543 | } |
---|
3544 | //---------------------------------------------------------------------------- |
---|
3545 | if (v && voice<>2) |
---|
3546 | { " We can start looking for primary invariants..."; |
---|
3547 | ""; |
---|
3548 | } |
---|
3549 | if (v && voice==2) |
---|
3550 | { ""; |
---|
3551 | } |
---|
3552 | //----------------------- initializing variables ----------------------------- |
---|
3553 | int dB; |
---|
3554 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
3555 | // space of invariants of degree d, |
---|
3556 | // newdim: dimension the ideal generated |
---|
3557 | // the primary invariants plus basis |
---|
3558 | // elements, dif=n-i-newdim, i.e. the |
---|
3559 | // number of new primary invairants that |
---|
3560 | // should be added in this degree - |
---|
3561 | ideal P,Pplus,CI,B; // P: will contain primary invariants, |
---|
3562 | // Pplus: P+B, CI: a complete |
---|
3563 | // intersection with the same Hilbert |
---|
3564 | // function as P - |
---|
3565 | dB=1; // used as degree bound - |
---|
3566 | d=0; // initializing |
---|
3567 | int i=0; |
---|
3568 | intvec deg_vector; |
---|
3569 | //------------------ loop that searches for primary invariants --------------- |
---|
3570 | while(1) // repeat until n primary invariants are |
---|
3571 | { // found - |
---|
3572 | d++; // degree where we'll search |
---|
3573 | if (v) |
---|
3574 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
3575 | } |
---|
3576 | B=invariant_basis_reynolds(REY,d,intvec(-1,6)); // basis of invariants of |
---|
3577 | // degree d |
---|
3578 | if (B[1]<>0) |
---|
3579 | { Pplus=P+B; |
---|
3580 | newdim=dim(groebner(Pplus)); |
---|
3581 | dif=n-i-newdim; |
---|
3582 | } |
---|
3583 | else |
---|
3584 | { dif=0; |
---|
3585 | deg_vector=deg_vector,d; |
---|
3586 | } |
---|
3587 | if (dif<>0) // we have to find dif new primary |
---|
3588 | { // invariants |
---|
3589 | cd=size(B); |
---|
3590 | if (cd<>dif) |
---|
3591 | { P,CI,dB=search_random(n,d,B,cd,P,i,dif,dB,CI,max); |
---|
3592 | } |
---|
3593 | else // i.e. we can take all of B |
---|
3594 | { for(j=i+1;j<=i+dif;j++) |
---|
3595 | { CI=CI+ideal(var(j)^d); |
---|
3596 | } |
---|
3597 | dB=dB+dif*(d-1); |
---|
3598 | P=Pplus; |
---|
3599 | } |
---|
3600 | if (ncols(P)==i) |
---|
3601 | { "WARNING: The first return value is not a set of primary invariants,"; |
---|
3602 | " but polynomials qualifying as the first "+string(i)+" primary invariants."; |
---|
3603 | return(matrix(P)); |
---|
3604 | } |
---|
3605 | if (v) |
---|
3606 | { for (j=1;j<=dif;j++) |
---|
3607 | { " We find: "+string(P[i+j]); |
---|
3608 | } |
---|
3609 | } |
---|
3610 | i=i+dif; |
---|
3611 | if (i==n) // found all primary invariants |
---|
3612 | { if (v) |
---|
3613 | { ""; |
---|
3614 | " We found all primary invariants."; |
---|
3615 | ""; |
---|
3616 | } |
---|
3617 | if (deg_vector==0) |
---|
3618 | { return(matrix(P)); |
---|
3619 | } |
---|
3620 | else |
---|
3621 | { return(matrix(P),compress(deg_vector)); |
---|
3622 | } |
---|
3623 | } |
---|
3624 | } // done with degree d |
---|
3625 | else |
---|
3626 | { if (v) |
---|
3627 | { " None here..."; |
---|
3628 | } |
---|
3629 | } |
---|
3630 | } |
---|
3631 | } |
---|
3632 | example |
---|
3633 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
3634 | ring R=0,(x,y,z),dp; |
---|
3635 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
3636 | list L=group_reynolds(A); |
---|
3637 | list l=primary_char0_no_molien_random(L[1],1); |
---|
3638 | print(l[1]); |
---|
3639 | } |
---|
3640 | /////////////////////////////////////////////////////////////////////////////// |
---|
3641 | |
---|
3642 | proc primary_charp_no_molien_random (matrix REY, int max, list #) |
---|
3643 | "USAGE: primary_charp_no_molien_random(REY,r[,v]); |
---|
3644 | REY: a <matrix> representing the Reynolds operator, r: an <int> where |
---|
3645 | -|r| to |r| is the range of coefficients of the random combinations of |
---|
3646 | bases elements, v: an optional <int> |
---|
3647 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien |
---|
3648 | DISPLAY: information about the various stages of the programme if v does not |
---|
3649 | equal 0 |
---|
3650 | RETURN: primary invariants (type <matrix>) of the invariant ring and an |
---|
3651 | <intvec> listing some of the degrees where no non-trivial homogeneous |
---|
3652 | invariants are to be found |
---|
3653 | THEORY: Bases of homogeneous invariants are generated successively and random |
---|
3654 | linear combinations are chosen as primary invariants that lower the |
---|
3655 | dimension of the ideal generated by the previously found invariants |
---|
3656 | (see \"Generating a Noetherian Normalization of the Invariant Ring of |
---|
3657 | a Finite Group\" by Decker, Heydtmann, Schreyer (1998)). |
---|
3658 | EXAMPLE: example primary_charp_no_molien_random; shows an example |
---|
3659 | " |
---|
3660 | { degBound=0; |
---|
3661 | //----------------- checking input and setting verbose mode ------------------ |
---|
3662 | if (char(basering)==0) |
---|
3663 | { "ERROR: primary_charp_no_molien_random should only be used with rings of"; |
---|
3664 | " characteristic p>0."; |
---|
3665 | return(); |
---|
3666 | } |
---|
3667 | if (size(#)>1) |
---|
3668 | { "ERROR: primary_charp_no_molien_random can only have three parameters."; |
---|
3669 | return(); |
---|
3670 | } |
---|
3671 | if (size(#)==1) |
---|
3672 | { if (typeof(#[1])<>"int") |
---|
3673 | { "ERROR: The third parameter should be of type <int>."; |
---|
3674 | return(); |
---|
3675 | } |
---|
3676 | else |
---|
3677 | { int v=#[1]; |
---|
3678 | } |
---|
3679 | } |
---|
3680 | else |
---|
3681 | { int v=0; |
---|
3682 | } |
---|
3683 | int n=nvars(basering); // n is the number of variables, as well |
---|
3684 | // as the size of the matrices, as well |
---|
3685 | // as the number of primary invariants, |
---|
3686 | // we should get |
---|
3687 | if (ncols(REY)<>n) |
---|
3688 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
3689 | return(); |
---|
3690 | } |
---|
3691 | //---------------------------------------------------------------------------- |
---|
3692 | if (v && voice<>2) |
---|
3693 | { " We can start looking for primary invariants..."; |
---|
3694 | ""; |
---|
3695 | } |
---|
3696 | if (v && voice==2) |
---|
3697 | { ""; |
---|
3698 | } |
---|
3699 | //-------------------- initializing variables -------------------------------- |
---|
3700 | int dB; |
---|
3701 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
3702 | // space of invariants of degree d, |
---|
3703 | // newdim: dimension the ideal generated |
---|
3704 | // the primary invariants plus basis |
---|
3705 | // elements, dif=n-i-newdim, i.e. the |
---|
3706 | // number of new primary invairants that |
---|
3707 | // should be added in this degree - |
---|
3708 | ideal P,Pplus,CI,B; // P: will contain primary invariants, |
---|
3709 | // Pplus: P+B, CI: a complete |
---|
3710 | // intersection with the same Hilbert |
---|
3711 | // function as P - |
---|
3712 | dB=1; // used as degree bound - |
---|
3713 | d=0; // initializing |
---|
3714 | int i=0; |
---|
3715 | intvec deg_vector; |
---|
3716 | //------------------ loop that searches for primary invariants --------------- |
---|
3717 | while(1) // repeat until n primary invariants are |
---|
3718 | { // found - |
---|
3719 | d++; // degree where we'll search |
---|
3720 | if (v) |
---|
3721 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
3722 | } |
---|
3723 | B=invariant_basis_reynolds(REY,d,intvec(-1,6)); // basis of invariants of |
---|
3724 | // degree d |
---|
3725 | if (B[1]<>0) |
---|
3726 | { Pplus=P+B; |
---|
3727 | newdim=dim(groebner(Pplus)); |
---|
3728 | dif=n-i-newdim; |
---|
3729 | } |
---|
3730 | else |
---|
3731 | { dif=0; |
---|
3732 | deg_vector=deg_vector,d; |
---|
3733 | } |
---|
3734 | if (dif<>0) // we have to find dif new primary |
---|
3735 | { // invariants |
---|
3736 | cd=size(B); |
---|
3737 | if (cd<>dif) |
---|
3738 | { P,CI,dB=p_search_random(n,d,B,cd,P,i,dif,dB,CI,max); |
---|
3739 | } |
---|
3740 | else // i.e. we can take all of B |
---|
3741 | { for(j=i+1;j<=i+dif;j++) |
---|
3742 | { CI=CI+ideal(var(j)^d); |
---|
3743 | } |
---|
3744 | dB=dB+dif*(d-1); |
---|
3745 | P=Pplus; |
---|
3746 | } |
---|
3747 | if (ncols(P)==n+1) |
---|
3748 | { "WARNING: The first return value is not a set of primary invariants,"; |
---|
3749 | " but polynomials qualifying as the first "+string(i)+" primary invariants."; |
---|
3750 | return(matrix(P)); |
---|
3751 | } |
---|
3752 | if (v) |
---|
3753 | { for (j=1;j<=size(P)-i;j++) |
---|
3754 | { " We find: "+string(P[i+j]); |
---|
3755 | } |
---|
3756 | } |
---|
3757 | i=size(P); |
---|
3758 | if (i==n) // found all primary invariants |
---|
3759 | { if (v) |
---|
3760 | { ""; |
---|
3761 | " We found all primary invariants."; |
---|
3762 | ""; |
---|
3763 | } |
---|
3764 | if (deg_vector==0) |
---|
3765 | { return(matrix(P)); |
---|
3766 | } |
---|
3767 | else |
---|
3768 | { return(matrix(P),compress(deg_vector)); |
---|
3769 | } |
---|
3770 | } |
---|
3771 | } // done with degree d |
---|
3772 | else |
---|
3773 | { if (v) |
---|
3774 | { " None here..."; |
---|
3775 | } |
---|
3776 | } |
---|
3777 | } |
---|
3778 | } |
---|
3779 | example |
---|
3780 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7 (changed into char 3)"; echo=2; |
---|
3781 | ring R=3,(x,y,z),dp; |
---|
3782 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
3783 | list L=group_reynolds(A); |
---|
3784 | list l=primary_charp_no_molien_random(L[1],1); |
---|
3785 | print(l[1]); |
---|
3786 | } |
---|
3787 | /////////////////////////////////////////////////////////////////////////////// |
---|
3788 | |
---|
3789 | proc primary_charp_without_random (list #) |
---|
3790 | "USAGE: primary_charp_without_random(G1,G2,...,r[,v]); |
---|
3791 | G1,G2,...: <matrices> generating a finite matrix group, r: an <int> |
---|
3792 | where -|r| to |r| is the range of coefficients of the random |
---|
3793 | combinations of bases elements, v: an optional <int> |
---|
3794 | DISPLAY: information about the various stages of the programme if v does not |
---|
3795 | equal 0 |
---|
3796 | RETURN: primary invariants (type <matrix>) of the invariant ring |
---|
3797 | THEORY: Bases of homogeneous invariants are generated successively and random |
---|
3798 | linear combinations are chosen as primary invariants that lower the |
---|
3799 | dimension of the ideal generated by the previously found invariants |
---|
3800 | (see \"Generating a Noetherian Normalization of the Invariant Ring of |
---|
3801 | a Finite Group\" by Decker, Heydtmann, Schreyer (1998)). No Reynolds |
---|
3802 | operator or Molien series is used. |
---|
3803 | EXAMPLE: example primary_charp_without_random; shows an example |
---|
3804 | " |
---|
3805 | { degBound=0; |
---|
3806 | //--------------------- checking input and setting verbose mode -------------- |
---|
3807 | if (char(basering)==0) |
---|
3808 | { "ERROR: primary_charp_without_random should only be used with rings of"; |
---|
3809 | " characteristic 0."; |
---|
3810 | return(); |
---|
3811 | } |
---|
3812 | if (size(#)<2) |
---|
3813 | { "ERROR: There are too few parameters."; |
---|
3814 | return(); |
---|
3815 | } |
---|
3816 | if (typeof(#[size(#)])=="int" && typeof(#[size(#)-1])=="int") |
---|
3817 | { int v=#[size(#)]; |
---|
3818 | int max=#[size(#)-1]; |
---|
3819 | int gen_num=size(#)-2; |
---|
3820 | if (gen_num==0) |
---|
3821 | { "ERROR: There are no generators of a finite matrix group given."; |
---|
3822 | return(); |
---|
3823 | } |
---|
3824 | } |
---|
3825 | else |
---|
3826 | { if (typeof(#[size(#)])=="int") |
---|
3827 | { int max=#[size(#)]; |
---|
3828 | int v=0; |
---|
3829 | int gen_num=size(#)-1; |
---|
3830 | } |
---|
3831 | else |
---|
3832 | { "ERROR: The last parameter should be an <int>."; |
---|
3833 | return(); |
---|
3834 | } |
---|
3835 | } |
---|
3836 | int n=nvars(basering); // n is the number of variables, as well |
---|
3837 | // as the size of the matrices, as well |
---|
3838 | // as the number of primary invariants, |
---|
3839 | // we should get |
---|
3840 | for (int i=1;i<=gen_num;i++) |
---|
3841 | { if (typeof(#[i])=="matrix") |
---|
3842 | { if (nrows(#[i])<>n or ncols(#[i])<>n) |
---|
3843 | { "ERROR: The number of variables of the base ring needs to be the same"; |
---|
3844 | " as the dimension of the square matrices"; |
---|
3845 | return(); |
---|
3846 | } |
---|
3847 | } |
---|
3848 | else |
---|
3849 | { "ERROR: The first parameters should be a list of matrices"; |
---|
3850 | return(); |
---|
3851 | } |
---|
3852 | } |
---|
3853 | //---------------------------------------------------------------------------- |
---|
3854 | if (v && voice==2) |
---|
3855 | { ""; |
---|
3856 | } |
---|
3857 | //---------------------------- initializing variables ------------------------ |
---|
3858 | int dB; |
---|
3859 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
3860 | // space of invariants of degree d, |
---|
3861 | // newdim: dimension the ideal generated |
---|
3862 | // the primary invariants plus basis |
---|
3863 | // elements, dif=n-i-newdim, i.e. the |
---|
3864 | // number of new primary invairants that |
---|
3865 | // should be added in this degree - |
---|
3866 | ideal P,Pplus,CI,B; // P: will contain primary invariants, |
---|
3867 | // Pplus: P+B, CI: a complete |
---|
3868 | // intersection with the same Hilbert |
---|
3869 | // function as P - |
---|
3870 | dB=1; // used as degree bound - |
---|
3871 | d=0; // initializing |
---|
3872 | i=0; |
---|
3873 | intvec deg_vector; |
---|
3874 | //-------------------- loop that searches for primary invariants ------------- |
---|
3875 | while(1) // repeat until n primary invariants are |
---|
3876 | { // found - |
---|
3877 | d++; // degree where we'll search |
---|
3878 | if (v) |
---|
3879 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
3880 | } |
---|
3881 | B=invariant_basis(d,#[1..gen_num]); // basis of invariants of degree d |
---|
3882 | if (B[1]<>0) |
---|
3883 | { Pplus=P+B; |
---|
3884 | newdim=dim(groebner(Pplus)); |
---|
3885 | dif=n-i-newdim; |
---|
3886 | } |
---|
3887 | else |
---|
3888 | { dif=0; |
---|
3889 | deg_vector=deg_vector,d; |
---|
3890 | } |
---|
3891 | if (dif<>0) // we have to find dif new primary |
---|
3892 | { // invariants |
---|
3893 | cd=size(B); |
---|
3894 | if (cd<>dif) |
---|
3895 | { P,CI,dB=p_search_random(n,d,B,cd,P,i,dif,dB,CI,max); |
---|
3896 | } |
---|
3897 | else // i.e. we can take all of B |
---|
3898 | { for(j=i+1;j<=i+dif;j++) |
---|
3899 | { CI=CI+ideal(var(j)^d); |
---|
3900 | } |
---|
3901 | dB=dB+dif*(d-1); |
---|
3902 | P=Pplus; |
---|
3903 | } |
---|
3904 | if (ncols(P)==n+1) |
---|
3905 | { "WARNING: The first return value is not a set of primary invariants,"; |
---|
3906 | " but polynomials qualifying as the first "+string(i)+" primary invariants."; |
---|
3907 | return(matrix(P)); |
---|
3908 | } |
---|
3909 | if (v) |
---|
3910 | { for (j=1;j<=size(P)-i;j++) |
---|
3911 | { " We find: "+string(P[i+j]); |
---|
3912 | } |
---|
3913 | } |
---|
3914 | i=size(P); |
---|
3915 | if (i==n) // found all primary invariants |
---|
3916 | { if (v) |
---|
3917 | { ""; |
---|
3918 | " We found all primary invariants."; |
---|
3919 | ""; |
---|
3920 | } |
---|
3921 | return(matrix(P)); |
---|
3922 | } |
---|
3923 | } // done with degree d |
---|
3924 | else |
---|
3925 | { if (v) |
---|
3926 | { " None here..."; |
---|
3927 | } |
---|
3928 | } |
---|
3929 | } |
---|
3930 | } |
---|
3931 | example |
---|
3932 | { "EXAMPLE:"; echo=2; |
---|
3933 | ring R=2,(x,y,z),dp; |
---|
3934 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
3935 | matrix P=primary_charp_without_random(A,1); |
---|
3936 | print(P); |
---|
3937 | } |
---|
3938 | /////////////////////////////////////////////////////////////////////////////// |
---|
3939 | |
---|
3940 | proc primary_invariants_random (list #) |
---|
3941 | "USAGE: primary_invariants_random(G1,G2,...,r[,flags]); |
---|
3942 | G1,G2,...: <matrices> generating a finite matrix group, r: an <int> |
---|
3943 | where -|r| to |r| is the range of coefficients of the random |
---|
3944 | combinations of bases elements, flags: an optional <intvec> with three |
---|
3945 | entries, if the first one equals 0 (also the default), the programme |
---|
3946 | attempts to compute the Molien series and Reynolds operator, if it |
---|
3947 | equals 1, the programme is told that the Molien series should not be |
---|
3948 | computed, if it equals -1 characteristic 0 is simulated, i.e. the |
---|
3949 | Molien series is computed as if the base field were characteristic 0 |
---|
3950 | (the user must choose a field of large prime characteristic, e.g. |
---|
3951 | 32003) and if the first one is anything else, it means that the |
---|
3952 | characteristic of the base field divides the group order, the second |
---|
3953 | component should give the size of intervals between canceling common |
---|
3954 | factors in the expansion of the Molien series, 0 (the default) means |
---|
3955 | only once after generating all terms, in prime characteristic also a |
---|
3956 | negative number can be given to indicate that common factors should |
---|
3957 | always be canceled when the expansion is simple (the root of the |
---|
3958 | extension field does not occur among the coefficients) |
---|
3959 | DISPLAY: information about the various stages of the programme if the third |
---|
3960 | flag does not equal 0 |
---|
3961 | RETURN: primary invariants (type <matrix>) of the invariant ring and if |
---|
3962 | computable Reynolds operator (type <matrix>) and Molien series (type |
---|
3963 | <matrix>), if the first flag is 1 and we are in the non-modular case |
---|
3964 | then an <intvec> is returned giving some of the degrees where no |
---|
3965 | non-trivial homogeneous invariants can be found |
---|
3966 | THEORY: Bases of homogeneous invariants are generated successively and random |
---|
3967 | linear combinations are chosen as primary invariants that lower the |
---|
3968 | dimension of the ideal generated by the previously found invariants |
---|
3969 | (see \"Generating a Noetherian Normalization of the Invariant Ring of |
---|
3970 | a Finite Group\" by Decker, Heydtmann, Schreyer (1998)). |
---|
3971 | EXAMPLE: example primary_invariants_random; shows an example |
---|
3972 | " |
---|
3973 | { |
---|
3974 | // ----------------- checking input and setting flags ------------------------ |
---|
3975 | if (size(#)<2) |
---|
3976 | { "ERROR: There are too few parameters."; |
---|
3977 | return(); |
---|
3978 | } |
---|
3979 | int ch=char(basering); // the algorithms depend very much on the |
---|
3980 | // characteristic of the ground field |
---|
3981 | int n=nvars(basering); // n is the number of variables, as well |
---|
3982 | // as the size of the matrices, as well |
---|
3983 | // as the number of primary invariants, |
---|
3984 | // we should get |
---|
3985 | int gen_num; |
---|
3986 | int mol_flag,v; |
---|
3987 | if (typeof(#[size(#)])=="intvec" && typeof(#[size(#)-1])=="int") |
---|
3988 | { if (size(#[size(#)])<>3) |
---|
3989 | { "ERROR: <intvec> should have three entries."; |
---|
3990 | return(); |
---|
3991 | } |
---|
3992 | gen_num=size(#)-2; |
---|
3993 | mol_flag=#[size(#)][1]; |
---|
3994 | if (#[size(#)][2]<0 && (ch==0 or (ch<>0 && mol_flag<>0))) |
---|
3995 | { "ERROR: the second component of <intvec> should be >=0"; |
---|
3996 | return(); |
---|
3997 | } |
---|
3998 | int interval=#[size(#)][2]; |
---|
3999 | v=#[size(#)][3]; |
---|
4000 | int max=#[size(#)-1]; |
---|
4001 | if (gen_num==0) |
---|
4002 | { "ERROR: There are no generators of a finite matrix group given."; |
---|
4003 | return(); |
---|
4004 | } |
---|
4005 | } |
---|
4006 | else |
---|
4007 | { if (typeof(#[size(#)])=="int") |
---|
4008 | { gen_num=size(#)-1; |
---|
4009 | mol_flag=0; |
---|
4010 | int interval=0; |
---|
4011 | v=0; |
---|
4012 | int max=#[size(#)]; |
---|
4013 | } |
---|
4014 | else |
---|
4015 | { "ERROR: If the two last parameters are not <int> and <intvec>, the last"; |
---|
4016 | " parameter should be an <int>."; |
---|
4017 | return(); |
---|
4018 | } |
---|
4019 | } |
---|
4020 | for (int i=1;i<=gen_num;i++) |
---|
4021 | { if (typeof(#[i])=="matrix") |
---|
4022 | { if (nrows(#[i])<>n or ncols(#[i])<>n) |
---|
4023 | { "ERROR: The number of variables of the base ring needs to be the same"; |
---|
4024 | " as the dimension of the square matrices"; |
---|
4025 | return(); |
---|
4026 | } |
---|
4027 | } |
---|
4028 | else |
---|
4029 | { "ERROR: The first parameters should be a list of matrices"; |
---|
4030 | return(); |
---|
4031 | } |
---|
4032 | } |
---|
4033 | //---------------------------------------------------------------------------- |
---|
4034 | if (mol_flag==0) |
---|
4035 | { if (ch==0) |
---|
4036 | { matrix REY,M=reynolds_molien(#[1..gen_num],intvec(0,interval,v)); |
---|
4037 | // one will contain Reynolds operator and |
---|
4038 | // the other enumerator and denominator |
---|
4039 | // of Molien series |
---|
4040 | matrix P=primary_char0_random(REY,M,max,v); |
---|
4041 | return(P,REY,M); |
---|
4042 | } |
---|
4043 | else |
---|
4044 | { list L=group_reynolds(#[1..gen_num],v); |
---|
4045 | if (L[1]<>0) // testing whether we are in the modular |
---|
4046 | { string newring="aksldfalkdsflkj"; // case |
---|
4047 | if (minpoly==0) |
---|
4048 | { if (v) |
---|
4049 | { " We are dealing with the non-modular case."; |
---|
4050 | } |
---|
4051 | if (typeof(L[2])=="int") |
---|
4052 | { molien(L[3..size(L)],newring,L[2],intvec(0,interval,v)); |
---|
4053 | } |
---|
4054 | else |
---|
4055 | { molien(L[2..size(L)],newring,intvec(0,interval,v)); |
---|
4056 | } |
---|
4057 | matrix P=primary_charp_random(L[1],newring,max,v); |
---|
4058 | return(P,L[1],newring); |
---|
4059 | } |
---|
4060 | else |
---|
4061 | { if (v) |
---|
4062 | { " Since it is impossible for this programme to calculate the Molien series for"; |
---|
4063 | " invariant rings over extension fields of prime characteristic, we have to"; |
---|
4064 | " continue without it."; |
---|
4065 | ""; |
---|
4066 | |
---|
4067 | } |
---|
4068 | list l=primary_charp_no_molien_random(L[1],max,v); |
---|
4069 | if (size(l)==2) |
---|
4070 | { return(l[1],L[1],l[2]); |
---|
4071 | } |
---|
4072 | else |
---|
4073 | { return(l[1],L[1]); |
---|
4074 | } |
---|
4075 | } |
---|
4076 | } |
---|
4077 | else // the modular case |
---|
4078 | { if (v) |
---|
4079 | { " There is also no Molien series, we can make use of..."; |
---|
4080 | ""; |
---|
4081 | " We can start looking for primary invariants..."; |
---|
4082 | ""; |
---|
4083 | } |
---|
4084 | return(primary_charp_without_random(#[1..gen_num],max,v)); |
---|
4085 | } |
---|
4086 | } |
---|
4087 | } |
---|
4088 | if (mol_flag==1) // the user wants no calculation of the |
---|
4089 | { list L=group_reynolds(#[1..gen_num],v); // Molien series |
---|
4090 | if (ch==0) |
---|
4091 | { list l=primary_char0_no_molien_random(L[1],max,v); |
---|
4092 | if (size(l)==2) |
---|
4093 | { return(l[1],L[1],l[2]); |
---|
4094 | } |
---|
4095 | else |
---|
4096 | { return(l[1],L[1]); |
---|
4097 | } |
---|
4098 | } |
---|
4099 | else |
---|
4100 | { if (L[1]<>0) // testing whether we are in the modular |
---|
4101 | { list l=primary_charp_no_molien_random(L[1],max,v); // case |
---|
4102 | if (size(l)==2) |
---|
4103 | { return(l[1],L[1],l[2]); |
---|
4104 | } |
---|
4105 | else |
---|
4106 | { return(l[1],L[1]); |
---|
4107 | } |
---|
4108 | } |
---|
4109 | else // the modular case |
---|
4110 | { if (v) |
---|
4111 | { " We can start looking for primary invariants..."; |
---|
4112 | ""; |
---|
4113 | } |
---|
4114 | return(primary_charp_without_random(#[1..gen_num],max,v)); |
---|
4115 | } |
---|
4116 | } |
---|
4117 | } |
---|
4118 | if (mol_flag==-1) |
---|
4119 | { if (ch==0) |
---|
4120 | { "ERROR: Characteristic 0 can only be simulated in characteristic p>>0."; |
---|
4121 | return(); |
---|
4122 | } |
---|
4123 | list L=group_reynolds(#[1..gen_num],v); |
---|
4124 | string newring="aksldfalkdsflkj"; |
---|
4125 | if (typeof(L[2])=="int") |
---|
4126 | { molien(L[3..size(L)],newring,L[2],intvec(0,interval,v)); |
---|
4127 | } |
---|
4128 | else |
---|
4129 | { molien(L[2..size(L)],newring,intvec(0,interval,v)); |
---|
4130 | } |
---|
4131 | matrix P=primary_charp_random(L[1],newring,max,v); |
---|
4132 | return(P,L[1],newring); |
---|
4133 | } |
---|
4134 | else // the user specified that the |
---|
4135 | { if (ch==0) // characteristic divides the group order |
---|
4136 | { "ERROR: The characteristic cannot divide the group order when it is 0."; |
---|
4137 | return(); |
---|
4138 | } |
---|
4139 | if (v) |
---|
4140 | { ""; |
---|
4141 | } |
---|
4142 | return(primary_charp_without_random(#[1..gen_num],max,v)); |
---|
4143 | } |
---|
4144 | } |
---|
4145 | example |
---|
4146 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
4147 | ring R=0,(x,y,z),dp; |
---|
4148 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
4149 | list L=primary_invariants_random(A,1); |
---|
4150 | print(L[1]); |
---|
4151 | } |
---|
4152 | /////////////////////////////////////////////////////////////////////////////// |
---|
4153 | |
---|
4154 | proc concat_intmat(intmat A,intmat B) |
---|
4155 | { int n=nrows(A); |
---|
4156 | int m1=ncols(A); |
---|
4157 | int m2=ncols(B); |
---|
4158 | intmat C[n][m1+m2]; |
---|
4159 | C[1..n,1..m1]=A[1..n,1..m1]; |
---|
4160 | C[1..n,m1+1..m1+m2]=B[1..n,1..m2]; |
---|
4161 | return(C); |
---|
4162 | } |
---|
4163 | /////////////////////////////////////////////////////////////////////////////// |
---|
4164 | |
---|
4165 | proc power_products(intvec deg_vec,int d) |
---|
4166 | "USAGE: power_products(dv,d); |
---|
4167 | dv: an <intvec> giving the degrees of homogeneous polynomials, d: the |
---|
4168 | degree of the desired power products |
---|
4169 | RETURN: a size(dv)*m <intmat> where each column ought to be interpreted as |
---|
4170 | containing the exponents of the corresponding polynomials. The product |
---|
4171 | of the powers is then homogeneous of degree d. |
---|
4172 | EXAMPLE: example power_products; shows an example |
---|
4173 | " |
---|
4174 | { ring R=0,x,dp; |
---|
4175 | if (d<=0) |
---|
4176 | { "ERROR: The <int> may not be <= 0"; |
---|
4177 | return(); |
---|
4178 | } |
---|
4179 | int d_neu,j,nc; |
---|
4180 | int s=size(deg_vec); |
---|
4181 | intmat PP[s][1]; |
---|
4182 | intmat TEST[s][1]; |
---|
4183 | for (int i=1;i<=s;i++) |
---|
4184 | { if (i<0) |
---|
4185 | { "ERROR: The entries of <intvec> may not be <= 0"; |
---|
4186 | return(); |
---|
4187 | } |
---|
4188 | d_neu=d-deg_vec[i]; |
---|
4189 | if (d_neu>0) |
---|
4190 | { intmat PPd_neu=power_products(intvec(deg_vec[i..s]),d_neu); |
---|
4191 | if (size(ideal(PPd_neu))<>0) |
---|
4192 | { nc=ncols(PPd_neu); |
---|
4193 | intmat PPd_neu_gross[s][nc]; |
---|
4194 | PPd_neu_gross[i..s,1..nc]=PPd_neu[1..s-i+1,1..nc]; |
---|
4195 | for (j=1;j<=nc;j++) |
---|
4196 | { PPd_neu_gross[i,j]=PPd_neu_gross[i,j]+1; |
---|
4197 | } |
---|
4198 | PP=concat_intmat(PP,PPd_neu_gross); |
---|
4199 | kill PPd_neu_gross; |
---|
4200 | } |
---|
4201 | kill PPd_neu; |
---|
4202 | } |
---|
4203 | if (d_neu==0) |
---|
4204 | { intmat PPd_neu[s][1]; |
---|
4205 | PPd_neu[i,1]=1; |
---|
4206 | PP=concat_intmat(PP,PPd_neu); |
---|
4207 | kill PPd_neu; |
---|
4208 | } |
---|
4209 | } |
---|
4210 | if (matrix(PP)<>matrix(TEST)) |
---|
4211 | { PP=compress(PP); |
---|
4212 | } |
---|
4213 | return(PP); |
---|
4214 | } |
---|
4215 | example |
---|
4216 | { "EXAMPLE:"; echo=2; |
---|
4217 | intvec dv=5,5,5,10,10; |
---|
4218 | print(power_products(dv,10)); |
---|
4219 | print(power_products(dv,7)); |
---|
4220 | } |
---|
4221 | /////////////////////////////////////////////////////////////////////////////// |
---|
4222 | |
---|
4223 | static proc old_secondary_char0 (matrix P, matrix REY, matrix M, list #) |
---|
4224 | "USAGE: secondary_char0(P,REY,M[,v]); |
---|
4225 | P: a 1xn <matrix> with primary invariants, |
---|
4226 | REY: a gxn <matrix> representing the Reynolds operator, |
---|
4227 | M: a 1x2 <matrix> giving numerator and denominator of the Molien series, |
---|
4228 | v: an optional <int> |
---|
4229 | ASSUME: n is the number of variables of the basering, g the size of the group, |
---|
4230 | REY is the 1st return value of group_reynolds(), reynolds_molien() or |
---|
4231 | the second one of primary_invariants(), M the return value of molien() |
---|
4232 | or the second one of reynolds_molien() or the third one of |
---|
4233 | primary_invariants() |
---|
4234 | RETURN: secondary invariants of the invariant ring (type <matrix>) and |
---|
4235 | irreducible secondary invariants (type <matrix>) |
---|
4236 | DISPLAY: information if v does not equal 0 |
---|
4237 | THEORY: The secondary invariants are calculated by finding a basis (in terms |
---|
4238 | of monomials) of the basering modulo the primary invariants, mapping |
---|
4239 | those to invariants with the Reynolds operator and using these images |
---|
4240 | or their power products such that they are linearly independent modulo |
---|
4241 | the primary invariants (see paper \"Some Algorithms in Invariant |
---|
4242 | Theory of Finite Groups\" by Kemper and Steel (1997)). |
---|
4243 | " |
---|
4244 | { def br=basering; |
---|
4245 | degBound=0; |
---|
4246 | //----------------- checking input and setting verbose mode ------------------ |
---|
4247 | if (char(br)<>0) |
---|
4248 | { "ERROR: secondary_char0 should only be used with rings of characteristic 0."; |
---|
4249 | return(); |
---|
4250 | } |
---|
4251 | int i; |
---|
4252 | if (size(#)>0) |
---|
4253 | { if (typeof(#[size(#)])=="int") |
---|
4254 | { int v=#[size(#)]; |
---|
4255 | } |
---|
4256 | else |
---|
4257 | { int v=0; |
---|
4258 | } |
---|
4259 | } |
---|
4260 | else |
---|
4261 | { int v=0; |
---|
4262 | } |
---|
4263 | int n=nvars(br); // n is the number of variables, as well |
---|
4264 | // as the size of the matrices, as well |
---|
4265 | // as the number of primary invariants, |
---|
4266 | // we should get |
---|
4267 | if (ncols(P)<>n) |
---|
4268 | { "ERROR: The first parameter ought to be the matrix of the primary"; |
---|
4269 | " invariants." |
---|
4270 | return(); |
---|
4271 | } |
---|
4272 | if (ncols(REY)<>n) |
---|
4273 | { "ERROR: The second parameter ought to be the Reynolds operator." |
---|
4274 | return(); |
---|
4275 | } |
---|
4276 | if (ncols(M)<>2 or nrows(M)<>1) |
---|
4277 | { "ERROR: The third parameter ought to be the Molien series." |
---|
4278 | return(); |
---|
4279 | } |
---|
4280 | if (v && voice==2) |
---|
4281 | { ""; |
---|
4282 | } |
---|
4283 | int j, m, counter; |
---|
4284 | //- finding the polynomial giving number and degrees of secondary invariants - |
---|
4285 | poly p=1; |
---|
4286 | for (j=1;j<=n;j++) // calculating the denominator of the |
---|
4287 | { p=p*(1-var(1)^deg(P[j])); // Hilbert series of the ring generated |
---|
4288 | } // by the primary invariants - |
---|
4289 | matrix s[1][2]=M[1,1]*p,M[1,2]; // s is used for canceling |
---|
4290 | s=matrix(syz(ideal(s))); |
---|
4291 | p=s[2,1]; // the polynomial telling us where to |
---|
4292 | // search for secondary invariants |
---|
4293 | map slead=br,ideal(0); |
---|
4294 | p=1/slead(p)*p; // smallest term of p needs to be 1 |
---|
4295 | if (v) |
---|
4296 | { " Polynomial telling us where to look for secondary invariants:"; |
---|
4297 | " "+string(p); |
---|
4298 | ""; |
---|
4299 | } |
---|
4300 | matrix dimmat=coeffs(p,var(1)); // dimmat will contain the number of |
---|
4301 | // secondary invariants, we need to find |
---|
4302 | // of a certain degree - |
---|
4303 | m=nrows(dimmat); // m-1 is the highest degree |
---|
4304 | if (v) |
---|
4305 | { " In degree 0 we have: 1"; |
---|
4306 | ""; |
---|
4307 | } |
---|
4308 | //-------------------------- initializing variables -------------------------- |
---|
4309 | intmat PP; |
---|
4310 | poly pp; |
---|
4311 | int k; |
---|
4312 | intvec deg_vec; |
---|
4313 | ideal sP=groebner(ideal(P)); |
---|
4314 | ideal TEST,B,IS; |
---|
4315 | ideal S=1; // 1 is the first secondary invariant - |
---|
4316 | //--------------------- generating secondary invariants ---------------------- |
---|
4317 | for (i=2;i<=m;i++) // going through dimmat - |
---|
4318 | { if (int(dimmat[i,1])<>0) // when it is == 0 we need to find 0 |
---|
4319 | { // elements in the current degree (i-1) |
---|
4320 | if (v) |
---|
4321 | { " Searching in degree "+string(i-1)+", we need to find "+string(int(dimmat[i,1]))+" invariant(s)..."; |
---|
4322 | } |
---|
4323 | TEST=sP; |
---|
4324 | counter=0; // we'll count up to degvec[i] |
---|
4325 | if (IS[1]<>0) |
---|
4326 | { PP=power_products(deg_vec,i-1); // finding power products of irreducible |
---|
4327 | } // secondary invariants |
---|
4328 | if (size(ideal(PP))<>0) |
---|
4329 | { for (j=1;j<=ncols(PP);j++) // going through all the power products |
---|
4330 | { pp=1; |
---|
4331 | for (k=1;k<=nrows(PP);k++) |
---|
4332 | { pp=pp*IS[1,k]^PP[k,j]; |
---|
4333 | } |
---|
4334 | if (reduce(pp,TEST)<>0) |
---|
4335 | { S=S,pp; |
---|
4336 | counter++; |
---|
4337 | if (v) |
---|
4338 | { " We find: "+string(pp); |
---|
4339 | } |
---|
4340 | if (int(dimmat[i,1])<>counter) |
---|
4341 | { // TEST=std(TEST+ideal(NF(pp,TEST))); // should be replaced by next |
---|
4342 | // line soon |
---|
4343 | TEST=std(TEST,pp); |
---|
4344 | } |
---|
4345 | else |
---|
4346 | { break; |
---|
4347 | } |
---|
4348 | } |
---|
4349 | } |
---|
4350 | } |
---|
4351 | if (int(dimmat[i,1])<>counter) |
---|
4352 | { B=sort_of_invariant_basis(sP,REY,i-1,int(dimmat[i,1])*6); // B contains |
---|
4353 | // images of kbase(sP,i-1) under the |
---|
4354 | // Reynolds operator that are linearly |
---|
4355 | // independent and that don't reduce to |
---|
4356 | // 0 modulo sP - |
---|
4357 | if (counter==0 && ncols(B)==int(dimmat[i,1])) // then we can take all of |
---|
4358 | { S=S,B; // B |
---|
4359 | IS=IS+B; |
---|
4360 | if (deg_vec[1]==0) |
---|
4361 | { deg_vec=i-1; |
---|
4362 | if (v) |
---|
4363 | { " We find: "+string(B[1]); |
---|
4364 | } |
---|
4365 | for (j=2;j<=int(dimmat[i,1]);j++) |
---|
4366 | { deg_vec=deg_vec,i-1; |
---|
4367 | if (v) |
---|
4368 | { " We find: "+string(B[j]); |
---|
4369 | } |
---|
4370 | } |
---|
4371 | } |
---|
4372 | else |
---|
4373 | { for (j=1;j<=int(dimmat[i,1]);j++) |
---|
4374 | { deg_vec=deg_vec,i-1; |
---|
4375 | if (v) |
---|
4376 | { " We find: "+string(B[j]); |
---|
4377 | } |
---|
4378 | } |
---|
4379 | } |
---|
4380 | } |
---|
4381 | else |
---|
4382 | { j=0; // j goes through all of B - |
---|
4383 | while (int(dimmat[i,1])<>counter) // need to find dimmat[i,1] |
---|
4384 | { // invariants that are linearly |
---|
4385 | // independent modulo TEST |
---|
4386 | j++; |
---|
4387 | if (reduce(B[j],TEST)<>0) // B[j] should be added |
---|
4388 | { S=S,B[j]; |
---|
4389 | IS=IS+ideal(B[j]); |
---|
4390 | if (deg_vec[1]==0) |
---|
4391 | { deg_vec[1]=i-1; |
---|
4392 | } |
---|
4393 | else |
---|
4394 | { deg_vec=deg_vec,i-1; |
---|
4395 | } |
---|
4396 | counter++; |
---|
4397 | if (v) |
---|
4398 | { " We find: "+string(B[j]); |
---|
4399 | } |
---|
4400 | if (int(dimmat[i,1])<>counter) |
---|
4401 | { //TEST=std(TEST+ideal(NF(B[j],TEST))); // should be replaced by |
---|
4402 | // next line |
---|
4403 | TEST=std(TEST,B[j]); |
---|
4404 | } |
---|
4405 | } |
---|
4406 | } |
---|
4407 | } |
---|
4408 | } |
---|
4409 | if (v) |
---|
4410 | { ""; |
---|
4411 | } |
---|
4412 | } |
---|
4413 | } |
---|
4414 | if (v) |
---|
4415 | { " We're done!"; |
---|
4416 | ""; |
---|
4417 | } |
---|
4418 | return(matrix(S),matrix(IS)); |
---|
4419 | } |
---|
4420 | |
---|
4421 | /////////////////////////////////////////////////////////////////////////////// |
---|
4422 | |
---|
4423 | |
---|
4424 | proc secondary_char0 (matrix P, matrix REY, matrix M, list #) |
---|
4425 | " |
---|
4426 | USAGE: secondary_char0(P,REY,M[,v][,\"AH\"]); |
---|
4427 | @* P: a 1xn <matrix> with homogeneous primary invariants, where |
---|
4428 | n is the number of variables of the basering; |
---|
4429 | @* REY: a gxn <matrix> representing the Reynolds operator, where |
---|
4430 | g the size of the corresponding group; |
---|
4431 | @* M: a 1x2 <matrix> giving numerator and denominator of the Molien |
---|
4432 | series; |
---|
4433 | @* v: an optional <int>; |
---|
4434 | @* \"AH\": if this string occurs as (optional) parameter, then an |
---|
4435 | old version of secondary_char0 is used (for downward |
---|
4436 | compatibility) |
---|
4437 | ASSUME: REY is the 1st return value of group_reynolds(), reynolds_molien() or |
---|
4438 | the second one of primary_invariants(); |
---|
4439 | @* M is the return value of molien() |
---|
4440 | or the second one of reynolds_molien() or the third one of |
---|
4441 | primary_invariants() |
---|
4442 | RETURN: Homogeneous secondary invariants and irreducible secondary |
---|
4443 | invariants of the invariant ring (both type <matrix>) |
---|
4444 | DISPLAY: Information on the progress of the computations if v is an integer |
---|
4445 | different from 0. |
---|
4446 | THEORY: The secondary invariants are calculated by finding a basis (in terms |
---|
4447 | of monomials) of the basering modulo the primary invariants, mapping |
---|
4448 | those to invariants with the Reynolds operator. Among these images |
---|
4449 | or their power products we pick a maximal subset that is linearly |
---|
4450 | independent modulo the primary invariants (see paper \"Some |
---|
4451 | Algorithms in Invariant Theory of Finite Groups\" by Kemper and |
---|
4452 | Steel (1997)). The size of this set can be read off from the |
---|
4453 | Molien series. |
---|
4454 | NOTE: Secondary invariants are not uniquely determined by the given data. |
---|
4455 | Specifically, the output of secondary_char0(P,REY,M,"AH") will |
---|
4456 | differ from the output of secondary_char0(P,REY,M). However, the |
---|
4457 | ideal generated by the irreducible homogeneous |
---|
4458 | secondary invariants will be the same in both cases. |
---|
4459 | @* There is an internal parameter \"pieces\" that, by default, equals 15. |
---|
4460 | Setting \"pieces\" to a smaller value might decrease the memory consumption, |
---|
4461 | but increase the running time. |
---|
4462 | SEE ALSO: irred_secondary_char0; |
---|
4463 | EXAMPLE: example secondary_char0; shows an example |
---|
4464 | " |
---|
4465 | { def br=basering; |
---|
4466 | |
---|
4467 | //----------------- checking input and setting verbose mode ------------------ |
---|
4468 | if (char(br)<>0) |
---|
4469 | { "ERROR: secondary_char0 should only be used with rings of characteristic 0."; |
---|
4470 | return(); |
---|
4471 | } |
---|
4472 | int i; |
---|
4473 | if (size(#)>0) |
---|
4474 | { if (typeof(#[1])=="int") |
---|
4475 | { int v=#[1]; |
---|
4476 | } |
---|
4477 | else |
---|
4478 | { int v=0; |
---|
4479 | } |
---|
4480 | } |
---|
4481 | else |
---|
4482 | { int v=0; |
---|
4483 | } |
---|
4484 | if (size(#)>0) |
---|
4485 | { if (typeof(#[size(#)])=="string") |
---|
4486 | { if (#[size(#)]=="AH") |
---|
4487 | { if (typeof(#[1])=="int") |
---|
4488 | { matrix S,IS = old_secondary_char0(P,REY,M,#[1]); |
---|
4489 | return(S,IS); |
---|
4490 | } |
---|
4491 | else |
---|
4492 | { matrix S,IS = old_secondary_char0(P,REY,M); |
---|
4493 | return(S,IS); |
---|
4494 | } |
---|
4495 | } |
---|
4496 | else |
---|
4497 | { "ERROR: If the last optional parameter is a string, it should be \"AH\"."; |
---|
4498 | return(matrix(ideal()),matrix(ideal())); |
---|
4499 | } |
---|
4500 | } |
---|
4501 | } |
---|
4502 | int n=nvars(br); // n is the number of variables, as well |
---|
4503 | // as the size of the matrices, as well |
---|
4504 | // as the number of primary invariants, |
---|
4505 | // we should get |
---|
4506 | if (ncols(P)<>n) |
---|
4507 | { "ERROR: The first parameter ought to be the matrix of the primary"; |
---|
4508 | " invariants."; |
---|
4509 | return(); |
---|
4510 | } |
---|
4511 | if (ncols(REY)<>n) |
---|
4512 | { "ERROR: The second parameter ought to be the Reynolds operator."; |
---|
4513 | return(); |
---|
4514 | } |
---|
4515 | if (ncols(M)<>2 or nrows(M)<>1) |
---|
4516 | { "ERROR: The third parameter ought to be the Molien series."; |
---|
4517 | return(); |
---|
4518 | } |
---|
4519 | if (v && voice==2) |
---|
4520 | { ""; |
---|
4521 | } |
---|
4522 | int j, m, counter, irrcounter; |
---|
4523 | //- finding the polynomial giving number and degrees of secondary invariants - |
---|
4524 | poly p=1; |
---|
4525 | for (j=1;j<=n;j++) // calculating the denominator of the |
---|
4526 | { p=p*(1-var(1)^deg(P[j])); // Hilbert series of the ring generated |
---|
4527 | } // by the primary invariants - |
---|
4528 | matrix s[1][2]=M[1,1]*p,M[1,2]; // s is used for canceling |
---|
4529 | s=matrix(syz(ideal(s))); |
---|
4530 | p=s[2,1]; // the polynomial telling us where to |
---|
4531 | // search for secondary invariants |
---|
4532 | map slead=br,ideal(0); |
---|
4533 | p=1/slead(p)*p; // smallest [constant] term of |
---|
4534 | // p needs to be 1 |
---|
4535 | matrix dimmat=coeffs(p,var(1)); // dimmat will contain the number of |
---|
4536 | // secondary invariants, we need to find |
---|
4537 | // of a certain degree - |
---|
4538 | m=nrows(dimmat); // m-1 is the highest degree |
---|
4539 | if (v) |
---|
4540 | { "We need to find"; |
---|
4541 | for (j=1;j<=m;j++) |
---|
4542 | { if (int(dimmat[j,1])<>1) |
---|
4543 | { int(dimmat[j,1]), "secondary invariants in degree",j-1; |
---|
4544 | } |
---|
4545 | else |
---|
4546 | { "1 secondary invariant in degree",j-1; |
---|
4547 | } |
---|
4548 | } |
---|
4549 | } |
---|
4550 | if (v) |
---|
4551 | { " In degree 0 we have: 1"; |
---|
4552 | ""; |
---|
4553 | } |
---|
4554 | //-------------------------- initializing variables -------------------------- |
---|
4555 | ideal ProdCand; // contains products of secondary invariants, |
---|
4556 | // i.e., candidates for reducible sec. inv. |
---|
4557 | ideal Mul1,Mul2; |
---|
4558 | |
---|
4559 | int dgb=degBound; |
---|
4560 | degBound = 0; |
---|
4561 | option(redSB); |
---|
4562 | ideal sP = groebner(ideal(P)); |
---|
4563 | ideal Reductor,SaveRed; // sP union Reductor is a Groebner basis up to degree i-1 |
---|
4564 | int SizeSave; |
---|
4565 | |
---|
4566 | list SSort; // sec. inv. first sorted by degree and then sorted by the |
---|
4567 | // minimal degree of a non-constant invariant factor. |
---|
4568 | list ISSort; // irr. sec. inv. sorted by degree |
---|
4569 | |
---|
4570 | poly helpP; |
---|
4571 | ideal helpI; |
---|
4572 | ideal Indicator; // will tell us which candidates for sec. inv. we can choose |
---|
4573 | ideal ReducedCandidates; |
---|
4574 | int helpint; |
---|
4575 | int k,k2,k3,minD; |
---|
4576 | int ii; |
---|
4577 | int saveAttr; |
---|
4578 | ideal B,IS; // IS will contain all irr. sec. inv. |
---|
4579 | |
---|
4580 | int pieces = 15; // If this parameter is small, the memory consumption will decrease. |
---|
4581 | // In some cases, a careful choice will speed up computations |
---|
4582 | |
---|
4583 | for (i=1;i<m;i++) |
---|
4584 | { SSort[i] = list(); |
---|
4585 | for (k=1;k<=i;k++) |
---|
4586 | { SSort[i][k]=ideal(); |
---|
4587 | attrib(SSort[i][k],"size",0); |
---|
4588 | } |
---|
4589 | } |
---|
4590 | //--------------------- generating secondary invariants ---------------------- |
---|
4591 | for (i=2;i<=m;i++) // going through dimmat - |
---|
4592 | { if (int(dimmat[i,1])<>0) // when it is == 0 we need to find no |
---|
4593 | { // elements in the current degree (i-1) |
---|
4594 | if (v) |
---|
4595 | { " Searching in degree "+string(i-1)+", we need to find "+string(int(dimmat[i,1]))+ |
---|
4596 | " invariant(s)..."; |
---|
4597 | " Looking for Power Products..."; |
---|
4598 | } |
---|
4599 | counter = 0; // we'll count up to dimmat[i,1] |
---|
4600 | Reductor = ideal(0); |
---|
4601 | helpint = 0; |
---|
4602 | SaveRed = Reductor; |
---|
4603 | SizeSave = 0; |
---|
4604 | attrib(Reductor,"isSB",1); |
---|
4605 | attrib(SaveRed,"isSB",1); |
---|
4606 | |
---|
4607 | // We start searching for reducible secondary invariants in degree i-1, i.e., those |
---|
4608 | // that are power products of irreducible secondary invariants. |
---|
4609 | // It suffices to restrict the search at products of one _irreducible_ sec. inv. (Mul1) |
---|
4610 | // with some sec. inv. (Mul2). |
---|
4611 | // Moreover, we avoid to consider power products twice since we take a product |
---|
4612 | // into account only if the minimal degree of a non-constant invariant factor in "Mul2" is not |
---|
4613 | // smaller than the degree of "Mul1". |
---|
4614 | for (k=1;k<i-1;k++) |
---|
4615 | { if (int(dimmat[i,1])==counter) |
---|
4616 | { break; |
---|
4617 | } |
---|
4618 | if (typeof(ISSort[k])<>"none") |
---|
4619 | { Mul1 = ISSort[k]; |
---|
4620 | } |
---|
4621 | else |
---|
4622 | { Mul1 = ideal(0); |
---|
4623 | } |
---|
4624 | if ((int(dimmat[i-k,1])>0) && (Mul1[1]<>0)) |
---|
4625 | { for (minD=k;minD<i-k;minD++) |
---|
4626 | { if (int(dimmat[i,1])==counter) |
---|
4627 | { break; |
---|
4628 | } |
---|
4629 | for (k2=1;k2 <= ((attrib(SSort[i-k-1][minD],"size")-1) div pieces)+1; k2++) |
---|
4630 | { if (int(dimmat[i,1])==counter) |
---|
4631 | { break; |
---|
4632 | } |
---|
4633 | Mul2=ideal(0); |
---|
4634 | if (attrib(SSort[i-k-1][minD],"size")>=k2*pieces) |
---|
4635 | { for (k3=1;k3<=pieces;k3++) |
---|
4636 | { Mul2[k3] = SSort[i-k-1][minD][((k2-1)*pieces)+k3]; |
---|
4637 | } |
---|
4638 | } |
---|
4639 | else |
---|
4640 | { for (k3=1;k3<=(attrib(SSort[i-k-1][minD],"size") mod pieces);k3++) |
---|
4641 | { Mul2[k3] = SSort[i-k-1][minD][((k2-1)*pieces)+k3]; |
---|
4642 | } |
---|
4643 | } |
---|
4644 | ProdCand = Mul1*Mul2; |
---|
4645 | ReducedCandidates = reduce(ProdCand,sP); |
---|
4646 | // sP union SaveRed union Reductor is a homogeneous Groebner basis |
---|
4647 | // up to degree i-1. |
---|
4648 | // We first reduce by sP (which is fixed, so we can do it once for all), |
---|
4649 | // then by SaveRed resp. by Reductor (which is modified during |
---|
4650 | // the computations). |
---|
4651 | Indicator = reduce(ReducedCandidates,SaveRed); |
---|
4652 | // If Indicator[ii]==0 then ReducedCandidates it the reduction |
---|
4653 | // of an invariant that is in the algebra generated by primary |
---|
4654 | // invariants and previously computed secondary invariants. |
---|
4655 | // Otherwise ProdCand[ii] can be taken as secondary invariant. |
---|
4656 | if (size(Indicator)<>0) |
---|
4657 | { for (ii=1;ii<=ncols(ProdCand);ii++) // going through all the power products |
---|
4658 | { helpP = reduce(Indicator[ii],Reductor); |
---|
4659 | if (helpP <> 0) |
---|
4660 | { counter++; |
---|
4661 | saveAttr = attrib(SSort[i-1][k],"size")+1; |
---|
4662 | SSort[i-1][k][saveAttr] = ProdCand[ii]; |
---|
4663 | // By construction, this is a _reducible_ s.i. |
---|
4664 | attrib(SSort[i-1][k],"size",saveAttr); |
---|
4665 | if (v) |
---|
4666 | { "We found",counter, " of", int(dimmat[i,1])," secondaries in degree",i-1; |
---|
4667 | } |
---|
4668 | if (int(dimmat[i,1])<>counter) |
---|
4669 | { helpint++; |
---|
4670 | Reductor[helpint] = helpP; |
---|
4671 | // Lemma: If G is a homogeneous Groebner basis up to degree i-1 and p is a |
---|
4672 | // homogeneous polynomial of degree i-1 then G union NF(p,G) is |
---|
4673 | // a homogeneous Groebner basis up to degree i-1. |
---|
4674 | attrib(Reductor, "isSB",1); |
---|
4675 | // if Reductor becomes too large, we reduce the whole of Indicator by |
---|
4676 | // it, save it in SaveRed, and work with a smaller Reductor. This turns |
---|
4677 | // out to save a little time. |
---|
4678 | if (ncols(Reductor)>100) |
---|
4679 | { Indicator=reduce(Indicator,Reductor); |
---|
4680 | for (k3=1;k3<=ncols(Reductor);k3++) |
---|
4681 | { SaveRed[SizeSave+k3] = Reductor[k3]; |
---|
4682 | } |
---|
4683 | SizeSave=SizeSave+size(Reductor); |
---|
4684 | Reductor = ideal(0); |
---|
4685 | helpint = 0; |
---|
4686 | attrib(SaveRed, "isSB",1); |
---|
4687 | attrib(Reductor, "isSB",1); |
---|
4688 | } |
---|
4689 | } |
---|
4690 | else |
---|
4691 | { break; |
---|
4692 | } |
---|
4693 | } |
---|
4694 | } |
---|
4695 | } |
---|
4696 | for (k3=1;k3<=size(Reductor);k3++) |
---|
4697 | { SaveRed[SizeSave+k3] = Reductor[k3]; |
---|
4698 | } |
---|
4699 | SizeSave=SizeSave+size(Reductor); |
---|
4700 | Reductor = ideal(0); |
---|
4701 | helpint = 0; |
---|
4702 | attrib(SaveRed, "isSB",1); |
---|
4703 | attrib(Reductor, "isSB",1); |
---|
4704 | } |
---|
4705 | } |
---|
4706 | } |
---|
4707 | } |
---|
4708 | |
---|
4709 | // The remaining sec. inv. are irreducible! |
---|
4710 | if (int(dimmat[i,1])<>counter) // need more than all the power products |
---|
4711 | { if (v) |
---|
4712 | { "There are irreducible secondary invariants in degree ", i-1," !!"; |
---|
4713 | } |
---|
4714 | B=sort_of_invariant_basis(sP,REY,i-1,int(dimmat[i,1])*6); |
---|
4715 | // B contains |
---|
4716 | // images of kbase(sP,i-1) under the |
---|
4717 | // Reynolds operator that are linearly |
---|
4718 | // independent and that don't reduce to |
---|
4719 | // 0 modulo sP - |
---|
4720 | if (counter==0 && ncols(B)==int(dimmat[i,1])) // then we can take all of B |
---|
4721 | { IS=IS+B; |
---|
4722 | saveAttr = attrib(SSort[i-1][i-1],"size")+int(dimmat[i,1]); |
---|
4723 | SSort[i-1][i-1] = SSort[i-1][i-1] + B; |
---|
4724 | attrib(SSort[i-1][i-1],"size", saveAttr); |
---|
4725 | if (typeof(ISSort[i-1]) <> "none") |
---|
4726 | { ISSort[i-1] = ISSort[i-1] + B; |
---|
4727 | } |
---|
4728 | else |
---|
4729 | { ISSort[i-1] = B; |
---|
4730 | } |
---|
4731 | if (v) {" We found: "; print(B);} |
---|
4732 | } |
---|
4733 | else |
---|
4734 | { irrcounter=0; |
---|
4735 | j=0; // j goes through all of B - |
---|
4736 | // Compare the comments on the computation of reducible sec. inv.! |
---|
4737 | ReducedCandidates = reduce(B,sP); |
---|
4738 | Indicator = reduce(ReducedCandidates,SaveRed); |
---|
4739 | while (int(dimmat[i,1])<>counter) |
---|
4740 | { j++; |
---|
4741 | helpP = reduce(Indicator[j],Reductor); |
---|
4742 | if (helpP <>0) // B[j] should be added |
---|
4743 | { counter++; irrcounter++; |
---|
4744 | IS=IS,B[j]; |
---|
4745 | saveAttr = attrib(SSort[i-1][i-1],"size")+1; |
---|
4746 | SSort[i-1][i-1][saveAttr] = B[j]; |
---|
4747 | attrib(SSort[i-1][i-1],"size",saveAttr); |
---|
4748 | if (typeof(ISSort[i-1]) <> "none") |
---|
4749 | { ISSort[i-1][irrcounter] = B[j]; |
---|
4750 | } |
---|
4751 | else |
---|
4752 | { ISSort[i-1] = ideal(B[j]); |
---|
4753 | } |
---|
4754 | if (v) |
---|
4755 | { " We found the irreducible sec. inv. "+string(B[j]); |
---|
4756 | } |
---|
4757 | helpint++; |
---|
4758 | Reductor[helpint] = helpP; |
---|
4759 | attrib(Reductor, "isSB",1); |
---|
4760 | if (ncols(Reductor)>100) |
---|
4761 | { Indicator=reduce(Indicator,Reductor); |
---|
4762 | for (k3=1;k3<=ncols(Reductor);k3++) |
---|
4763 | { SaveRed[SizeSave+k3] = Reductor[k3]; |
---|
4764 | } |
---|
4765 | SizeSave=SizeSave+size(Reductor); |
---|
4766 | Reductor = ideal(0); |
---|
4767 | helpint = 0; |
---|
4768 | attrib(SaveRed, "isSB",1); |
---|
4769 | attrib(Reductor, "isSB",1); |
---|
4770 | } |
---|
4771 | } |
---|
4772 | } |
---|
4773 | } |
---|
4774 | } |
---|
4775 | if (v) |
---|
4776 | { ""; |
---|
4777 | } |
---|
4778 | } |
---|
4779 | } |
---|
4780 | if (v) |
---|
4781 | { " We're done!"; |
---|
4782 | ""; |
---|
4783 | } |
---|
4784 | degBound = dgb; |
---|
4785 | |
---|
4786 | // Prepare return: |
---|
4787 | int TotalNumber; |
---|
4788 | for (k=1;k<=m;k++) |
---|
4789 | { TotalNumber = TotalNumber + int(dimmat[k,1]); |
---|
4790 | } |
---|
4791 | matrix S[1][TotalNumber]; |
---|
4792 | S[1,1]=1; |
---|
4793 | j=1; |
---|
4794 | for (k=1;k<m;k++) |
---|
4795 | { for (k2=1;k2<=k;k2++) |
---|
4796 | { if (typeof(attrib(SSort[k][k2],"size"))=="int") |
---|
4797 | {for (i=1;i<=attrib(SSort[k][k2],"size");i++) |
---|
4798 | { j++; |
---|
4799 | S[1,j] = SSort[k][k2][i]; |
---|
4800 | } |
---|
4801 | SSort[k][k2]=ideal(); |
---|
4802 | } |
---|
4803 | } |
---|
4804 | } |
---|
4805 | return(S,matrix(compress(IS))); |
---|
4806 | } |
---|
4807 | |
---|
4808 | |
---|
4809 | example |
---|
4810 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
4811 | ring R=0,(x,y,z),dp; |
---|
4812 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
4813 | list L=primary_invariants(A); |
---|
4814 | matrix S,IS=secondary_char0(L[1..3],1); |
---|
4815 | print(S); |
---|
4816 | print(IS); |
---|
4817 | } |
---|
4818 | //example |
---|
4819 | //{ "EXAMPLE: S. King"; echo=2; |
---|
4820 | //ring r= 0, (V1,V2,V3,V4,V5,V6),dp; |
---|
4821 | //matrix A1[6][6] = 0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0; |
---|
4822 | //matrix A2[6][6] = 0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0; |
---|
4823 | //list L = primary_invariants(A1,A2); // sorry, this takes a while... |
---|
4824 | //matrix S, IS = secondary_char0(L[1],L[2],L[3],1); |
---|
4825 | //S; |
---|
4826 | //IS; |
---|
4827 | //} |
---|
4828 | /////////////////////////////////////////////////////////////////////////////// |
---|
4829 | |
---|
4830 | |
---|
4831 | proc irred_secondary_char0 (matrix P, matrix REY, matrix M, list #) |
---|
4832 | " |
---|
4833 | USAGE: irred_secondary_char0(P,REY,M[,v]); |
---|
4834 | @* P: a 1xn <matrix> with homogeneous primary invariants, where |
---|
4835 | n is the number of variables of the basering; |
---|
4836 | @* REY: a gxn <matrix> representing the Reynolds operator, where |
---|
4837 | g the size of the corresponding group; |
---|
4838 | @* M: a 1x2 <matrix> giving numerator and denominator of the |
---|
4839 | Molien series; |
---|
4840 | @* v: an optional <int>; |
---|
4841 | RETURN: Irreducible homogeneous secondary invariants of the invariant ring |
---|
4842 | (type <matrix>) |
---|
4843 | ASSUME: REY is the 1st return value of group_reynolds(), reynolds_molien() or |
---|
4844 | the second one of primary_invariants(); |
---|
4845 | M is the return value of molien() or the second one of |
---|
4846 | reynolds_molien() or the third one of primary_invariants() |
---|
4847 | DISPLAY: Information on the progress of computations if v does not equal 0 |
---|
4848 | THEORY: The secondary invariants are calculated by finding a basis (in terms |
---|
4849 | of monomials) of the basering modulo the primary invariants, mapping |
---|
4850 | those to invariants with the Reynolds operator. Among these images |
---|
4851 | or their power products we pick a maximal subset that is linearly |
---|
4852 | independent modulo the primary invariants (see paper \"Some |
---|
4853 | Algorithms in Invariant Theory of Finite Groups\" by Kemper and |
---|
4854 | Steel (1997)). The size of this set can be read off from the |
---|
4855 | Molien series. Here, only irreducible secondary |
---|
4856 | invariants are explicitly computed, which saves time and memory. |
---|
4857 | NOTE: There is an internal parameter \"pieces\" that, by default, equals 15. |
---|
4858 | Setting \"pieces\" to a smaller value might decrease the memory consumption, |
---|
4859 | but increase the running time. |
---|
4860 | SEE ALSO: secondary_char0 |
---|
4861 | KEYWORDS: irreducible secondary invariant |
---|
4862 | EXAMPLE: example irred_secondary_char0; shows an example |
---|
4863 | " |
---|
4864 | { def br=basering; |
---|
4865 | |
---|
4866 | //----------------- checking input and setting verbose mode ------------------ |
---|
4867 | if (char(br)<>0) |
---|
4868 | { "ERROR: irred_secondary_char0 should only be used with rings of characteristic 0."; |
---|
4869 | return(); |
---|
4870 | } |
---|
4871 | int i; |
---|
4872 | if (size(#)>0) |
---|
4873 | { if (typeof(#[size(#)])=="int") |
---|
4874 | { int v=#[size(#)]; |
---|
4875 | } |
---|
4876 | else |
---|
4877 | { int v=0; |
---|
4878 | } |
---|
4879 | } |
---|
4880 | else |
---|
4881 | { int v=0; |
---|
4882 | } |
---|
4883 | int n=nvars(br); // n is the number of variables, as well |
---|
4884 | // as the size of the matrices, as well |
---|
4885 | // as the number of primary invariants, |
---|
4886 | // we should get |
---|
4887 | if (ncols(P)<>n) |
---|
4888 | { "ERROR: The first parameter ought to be the matrix of the primary"; |
---|
4889 | " invariants." |
---|
4890 | return(); |
---|
4891 | } |
---|
4892 | if (ncols(REY)<>n) |
---|
4893 | { "ERROR: The second parameter ought to be the Reynolds operator." |
---|
4894 | return(); |
---|
4895 | } |
---|
4896 | if (ncols(M)<>2 or nrows(M)<>1) |
---|
4897 | { "ERROR: The third parameter ought to be the Molien series." |
---|
4898 | return(); |
---|
4899 | } |
---|
4900 | if (v && voice==2) |
---|
4901 | { ""; |
---|
4902 | } |
---|
4903 | int j, m, counter, irrcounter; |
---|
4904 | //- finding the polynomial giving number and degrees of secondary invariants - |
---|
4905 | poly p=1; |
---|
4906 | for (j=1;j<=n;j++) // calculating the denominator of the |
---|
4907 | { p=p*(1-var(1)^deg(P[j])); // Hilbert series of the ring generated |
---|
4908 | } // by the primary invariants - |
---|
4909 | matrix s[1][2]=M[1,1]*p,M[1,2]; // s is used for canceling |
---|
4910 | s=matrix(syz(ideal(s))); |
---|
4911 | p=s[2,1]; // the polynomial telling us where to |
---|
4912 | // search for secondary invariants |
---|
4913 | map slead=br,ideal(0); |
---|
4914 | p=1/slead(p)*p; // smallest [constant] term of |
---|
4915 | // p needs to be 1 |
---|
4916 | matrix dimmat=coeffs(p,var(1)); // dimmat will contain the number of |
---|
4917 | // secondary invariants, we need to find |
---|
4918 | // of a certain degree - |
---|
4919 | m=nrows(dimmat); // m-1 is the highest degree |
---|
4920 | if (v) |
---|
4921 | { "We need to find"; |
---|
4922 | for (j=1;j<=m;j++) |
---|
4923 | { if (int(dimmat[j,1])<>1) |
---|
4924 | { int(dimmat[j,1]), "secondary invariants in degree",j-1; |
---|
4925 | } |
---|
4926 | else |
---|
4927 | { "1 secondary invariant in degree",j-1; |
---|
4928 | } |
---|
4929 | } |
---|
4930 | } |
---|
4931 | if (v) |
---|
4932 | { " In degree 0 we have: 1"; |
---|
4933 | ""; |
---|
4934 | } |
---|
4935 | //-------------------------- initializing variables -------------------------- |
---|
4936 | ideal ProdCand; // contains products of secondary invariants, |
---|
4937 | // i.e., candidates for reducible sec. inv. |
---|
4938 | |
---|
4939 | ideal Mul1,Mul2; |
---|
4940 | |
---|
4941 | int dgb=degBound; |
---|
4942 | degBound = 0; |
---|
4943 | option(redSB); |
---|
4944 | ideal sP = groebner(ideal(P)); |
---|
4945 | ideal Reductor,SaveRed; // sP union Reductor is a Groebner basis up to degree i-1 |
---|
4946 | int SizeSave; |
---|
4947 | |
---|
4948 | list RedSSort; // sec. Inv. reduced by sP, sorted first by degree and then |
---|
4949 | // by the minimal degree of a non-constant invariant factor |
---|
4950 | list RedISSort; // irr. sec. Inv. reduced by sP, sorted by degree |
---|
4951 | |
---|
4952 | poly helpP; |
---|
4953 | ideal helpI; |
---|
4954 | ideal Indicator; // will tell us which candidates for sec. inv. we can choose |
---|
4955 | ideal ReducedCandidates; |
---|
4956 | int helpint; |
---|
4957 | int k,k2,k3,minD; |
---|
4958 | int ii; |
---|
4959 | int saveAttr; |
---|
4960 | ideal B,IS; // IS will contain all irr. sec. inv. |
---|
4961 | |
---|
4962 | int pieces = 15; // If this parameter is small, the memory consumption will decrease. |
---|
4963 | // In some cases, a careful choice will speed up computations |
---|
4964 | |
---|
4965 | for (i=1;i<m;i++) |
---|
4966 | { RedSSort[i] = list(); |
---|
4967 | for (k=1;k<=i;k++) |
---|
4968 | { RedSSort[i][k]=ideal(); |
---|
4969 | attrib(RedSSort[i][k],"size",0); |
---|
4970 | } |
---|
4971 | } |
---|
4972 | //--------------------- generating secondary invariants ---------------------- |
---|
4973 | for (i=2;i<=m;i++) // going through dimmat - |
---|
4974 | { if (int(dimmat[i,1])<>0) // when it is == 0 we need to find no |
---|
4975 | { // elements in the current degree (i-1) |
---|
4976 | if (v) |
---|
4977 | { " Searching in degree "+string(i-1)+", we need to find "+ |
---|
4978 | string(int(dimmat[i,1]))+" invariant(s)..."; |
---|
4979 | " Looking for Power Products..."; |
---|
4980 | } |
---|
4981 | counter = 0; // we'll count up to dimmat[i,1] |
---|
4982 | Reductor = ideal(0); |
---|
4983 | helpint = 0; |
---|
4984 | SaveRed = Reductor; |
---|
4985 | SizeSave = 0; |
---|
4986 | attrib(Reductor,"isSB",1); |
---|
4987 | attrib(SaveRed,"isSB",1); |
---|
4988 | |
---|
4989 | // We start searching for reducible secondary invariants in degree i-1, i.e., those |
---|
4990 | // that are power products of irreducible secondary invariants. |
---|
4991 | // It suffices to restrict the search at products of one _irreducible_ sec. inv. (Mul1) |
---|
4992 | // with some sec. inv. (Mul2). |
---|
4993 | // Moreover, we avoid to consider power products twice since we take a product |
---|
4994 | // into account only if the minimal degree of a non-constant invariant factor in "Mul2" is not |
---|
4995 | // smaller than the degree of "Mul1". |
---|
4996 | // Finally, as we are not interested in the reducible sec. inv., we will only |
---|
4997 | // work with their reduction modulo sP --- this allows to detect a secondary invariant |
---|
4998 | // without to actually compute it! |
---|
4999 | for (k=1;k<i-1;k++) |
---|
5000 | { if (int(dimmat[i,1])==counter) |
---|
5001 | { break; |
---|
5002 | } |
---|
5003 | if (typeof(RedISSort[k])<>"none") |
---|
5004 | { Mul1 = RedISSort[k]; |
---|
5005 | } |
---|
5006 | else |
---|
5007 | { Mul1 = ideal(0); |
---|
5008 | } |
---|
5009 | if ((int(dimmat[i-k,1])>0) && (Mul1[1]<>0)) |
---|
5010 | { for (minD=k;minD<i-k;minD++) |
---|
5011 | { if (int(dimmat[i,1])==counter) |
---|
5012 | { break; |
---|
5013 | } |
---|
5014 | for (k2=1;k2 <= ((attrib(RedSSort[i-k-1][minD],"size")-1) div pieces)+1; k2++) |
---|
5015 | { if (int(dimmat[i,1])==counter) |
---|
5016 | { break; |
---|
5017 | } |
---|
5018 | Mul2=ideal(0); |
---|
5019 | if (attrib(RedSSort[i-k-1][minD],"size")>=k2*pieces) |
---|
5020 | { for (k3=1;k3<=pieces;k3++) |
---|
5021 | { Mul2[k3] = RedSSort[i-k-1][minD][((k2-1)*pieces)+k3]; |
---|
5022 | } |
---|
5023 | } |
---|
5024 | else |
---|
5025 | { for (k3=1;k3<=(attrib(RedSSort[i-k-1][minD],"size") mod pieces);k3++) |
---|
5026 | { Mul2[k3] = RedSSort[i-k-1][minD][((k2-1)*pieces)+k3]; |
---|
5027 | } |
---|
5028 | } |
---|
5029 | ProdCand = Mul1*Mul2; |
---|
5030 | ReducedCandidates = reduce(ProdCand,sP); |
---|
5031 | // sP union SaveRed union Reductor is a homogeneous Groebner basis |
---|
5032 | // up to degree i-1. |
---|
5033 | // We first reduce by sP (which is fixed, so we can do it once for all), |
---|
5034 | // then by SaveRed resp. by Reductor (which is modified during |
---|
5035 | // the computations). |
---|
5036 | Indicator = reduce(ReducedCandidates,SaveRed); |
---|
5037 | // If Indicator[ii]==0 then ReducedCandidates it the reduction |
---|
5038 | // of an invariant that is in the algebra generated by primary |
---|
5039 | // invariants and previously computed secondary invariants. |
---|
5040 | // Otherwise ReducedCandidates[ii] is the reduction of an invariant |
---|
5041 | // that we can take as secondary invariant. |
---|
5042 | if (size(Indicator)<>0) |
---|
5043 | { for (ii=1;ii<=ncols(ProdCand);ii++) // going through all the power products |
---|
5044 | { helpP = reduce(Indicator[ii],Reductor); |
---|
5045 | if (helpP <> 0) |
---|
5046 | { counter++; |
---|
5047 | saveAttr = attrib(RedSSort[i-1][k],"size")+1; |
---|
5048 | RedSSort[i-1][k][saveAttr] = ReducedCandidates[ii]; |
---|
5049 | // By construction, this is the reduction of a reducible s.i. |
---|
5050 | // of degree i-1. |
---|
5051 | attrib(RedSSort[i-1][k],"size",saveAttr); |
---|
5052 | if (v) |
---|
5053 | { "We found",counter, " of", int(dimmat[i,1])," secondaries in degree",i-1; |
---|
5054 | } |
---|
5055 | if (int(dimmat[i,1])<>counter) |
---|
5056 | { helpint++; |
---|
5057 | Reductor[helpint] = helpP; |
---|
5058 | // Lemma: If G is a homogeneous Groebner basis up to degree i-1 and p is a |
---|
5059 | // homogeneous polynomial of degree i-1 then G union NF(p,G) is |
---|
5060 | // a homogeneous Groebner basis up to degree i-1. |
---|
5061 | attrib(Reductor, "isSB",1); |
---|
5062 | // if Reductor becomes too large, we reduce the whole of Indicator by |
---|
5063 | // it, save it in SaveRed, and work with a smaller Reductor. This turns |
---|
5064 | // out to save a little time. |
---|
5065 | if (ncols(Reductor)>100) |
---|
5066 | { Indicator=reduce(Indicator,Reductor); |
---|
5067 | for (k3=1;k3<=ncols(Reductor);k3++) |
---|
5068 | { SaveRed[SizeSave+k3] = Reductor[k3]; |
---|
5069 | } |
---|
5070 | SizeSave=SizeSave+size(Reductor); |
---|
5071 | Reductor = ideal(0); |
---|
5072 | helpint = 0; |
---|
5073 | attrib(SaveRed, "isSB",1); |
---|
5074 | attrib(Reductor, "isSB",1); |
---|
5075 | } |
---|
5076 | } |
---|
5077 | else |
---|
5078 | { break; |
---|
5079 | } |
---|
5080 | } |
---|
5081 | } |
---|
5082 | } |
---|
5083 | for (k3=1;k3<=size(Reductor);k3++) |
---|
5084 | { SaveRed[SizeSave+k3] = Reductor[k3]; |
---|
5085 | } |
---|
5086 | SizeSave=SizeSave+size(Reductor); |
---|
5087 | Reductor = ideal(0); |
---|
5088 | helpint = 0; |
---|
5089 | attrib(SaveRed, "isSB",1); |
---|
5090 | attrib(Reductor, "isSB",1); |
---|
5091 | } |
---|
5092 | } |
---|
5093 | } |
---|
5094 | } |
---|
5095 | |
---|
5096 | // The remaining sec. inv. are irreducible! |
---|
5097 | if (int(dimmat[i,1])<>counter) // need more than all the power products |
---|
5098 | { if (v) |
---|
5099 | { "There are irreducible secondary invariants in degree ", i-1," !!"; |
---|
5100 | } |
---|
5101 | B=sort_of_invariant_basis(sP,REY,i-1,int(dimmat[i,1])*6); |
---|
5102 | // B is a set of |
---|
5103 | // images of kbase(sP,i-1) under the |
---|
5104 | // Reynolds operator that are linearly |
---|
5105 | // independent and that do not reduce to |
---|
5106 | // 0 modulo sP - |
---|
5107 | if (counter==0 && ncols(B)==int(dimmat[i,1])) // then we can take all of B |
---|
5108 | { IS=IS+B; |
---|
5109 | saveAttr = attrib(RedSSort[i-1][i-1],"size")+int(dimmat[i,1]); |
---|
5110 | RedSSort[i-1][i-1] = RedSSort[i-1][i-1] + B; |
---|
5111 | attrib(RedSSort[i-1][i-1],"size", saveAttr); |
---|
5112 | if (typeof(RedISSort[i-1]) <> "none") |
---|
5113 | { RedISSort[i-1] = RedISSort[i-1] + B; |
---|
5114 | } |
---|
5115 | else |
---|
5116 | { RedISSort[i-1] = B; |
---|
5117 | } |
---|
5118 | if (v) {" We found: ";print(B);} |
---|
5119 | } |
---|
5120 | else |
---|
5121 | { irrcounter=0; |
---|
5122 | j=0; // j goes through all of B - |
---|
5123 | // Compare the comments on the computation of reducible sec. inv.! |
---|
5124 | ReducedCandidates = reduce(B,sP); |
---|
5125 | Indicator = reduce(ReducedCandidates,SaveRed); |
---|
5126 | while (int(dimmat[i,1])<>counter) // need to find dimmat[i,1] |
---|
5127 | { // invariants that are linearly independent |
---|
5128 | j++; |
---|
5129 | helpP = reduce(Indicator[j],Reductor); |
---|
5130 | if (helpP <>0) // B[j] should be added |
---|
5131 | { counter++; irrcounter++; |
---|
5132 | IS=IS,B[j]; |
---|
5133 | saveAttr = attrib(RedSSort[i-1][i-1],"size")+1; |
---|
5134 | RedSSort[i-1][i-1][saveAttr] = ReducedCandidates[j]; |
---|
5135 | attrib(RedSSort[i-1][i-1],"size",saveAttr); |
---|
5136 | if (typeof(RedISSort[i-1]) <> "none") |
---|
5137 | { RedISSort[i-1][irrcounter] = ReducedCandidates[j]; |
---|
5138 | } |
---|
5139 | else |
---|
5140 | { RedISSort[i-1] = ideal(ReducedCandidates[j]); |
---|
5141 | } |
---|
5142 | if (v) |
---|
5143 | { " We found the irreducible sec. inv. "+string(B[j]); |
---|
5144 | } |
---|
5145 | helpint++; |
---|
5146 | Reductor[helpint] = helpP; |
---|
5147 | attrib(Reductor, "isSB",1); |
---|
5148 | if (ncols(Reductor)>100) |
---|
5149 | { Indicator=reduce(Indicator,Reductor); |
---|
5150 | for (k3=1;k3<=ncols(Reductor);k3++) |
---|
5151 | { SaveRed[SizeSave+k3] = Reductor[k3]; |
---|
5152 | } |
---|
5153 | SizeSave=SizeSave+size(Reductor); |
---|
5154 | Reductor = ideal(0); |
---|
5155 | helpint = 0; |
---|
5156 | attrib(SaveRed, "isSB",1); |
---|
5157 | attrib(Reductor, "isSB",1); |
---|
5158 | } |
---|
5159 | } |
---|
5160 | } |
---|
5161 | } |
---|
5162 | } |
---|
5163 | if (v) |
---|
5164 | { ""; |
---|
5165 | } |
---|
5166 | } |
---|
5167 | } |
---|
5168 | if (v) |
---|
5169 | { " We're done!"; |
---|
5170 | ""; |
---|
5171 | } |
---|
5172 | degBound = dgb; |
---|
5173 | |
---|
5174 | return(matrix(compress(IS))); |
---|
5175 | } |
---|
5176 | |
---|
5177 | |
---|
5178 | //example |
---|
5179 | //{ "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
5180 | // ring R=0,(x,y,z),dp; |
---|
5181 | // matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
5182 | // list L=primary_invariants(A); |
---|
5183 | // matrix IS=irred_secondary_char0(L[1..3],1); |
---|
5184 | // print(IS); |
---|
5185 | //} |
---|
5186 | example |
---|
5187 | { "EXAMPLE: S. King"; echo=2; |
---|
5188 | ring r= 0, (a,b,c,d,e,f),dp; |
---|
5189 | matrix A1[6][6] = 0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0; |
---|
5190 | matrix A2[6][6] = 0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0; |
---|
5191 | list L = primary_invariants(A1,A2); // sorry, this takes a while... |
---|
5192 | matrix IS = irred_secondary_char0(L[1],L[2],L[3],0); |
---|
5193 | IS; |
---|
5194 | } |
---|
5195 | |
---|
5196 | |
---|
5197 | /////////////////////////////////////////////////////////////////////////////// |
---|
5198 | |
---|
5199 | |
---|
5200 | proc secondary_charp (matrix P, matrix REY, string ring_name, list #) |
---|
5201 | "USAGE: secondary_charp(P,REY,ringname[,v]); |
---|
5202 | P: a 1xn <matrix> with primary invariants, REY: a gxn <matrix> |
---|
5203 | representing the Reynolds operator, ringname: a <string> giving the |
---|
5204 | name of a ring of characteristic 0 where the Molien series is stored, |
---|
5205 | v: an optional <int> |
---|
5206 | ASSUME: n is the number of variables of the basering, g the size of the group, |
---|
5207 | REY is the 1st return value of group_reynolds(), reynolds_molien() or |
---|
5208 | the second one of primary_invariants(), `ringname` is a ring of |
---|
5209 | char 0 that has been created by molien() or reynolds_molien() or |
---|
5210 | primary_invariants() |
---|
5211 | RETURN: secondary invariants of the invariant ring (type <matrix>) and |
---|
5212 | irreducible secondary invariants (type <matrix>) |
---|
5213 | DISPLAY: information if v does not equal 0 |
---|
5214 | THEORY: Secondary invariants are calculated by finding a basis (in terms of |
---|
5215 | monomials) of the basering modulo primary invariants, mapping those |
---|
5216 | to invariants with the Reynolds operator and using these images or |
---|
5217 | their power products such that they are linearly independent modulo |
---|
5218 | the primary invariants (see paper \"Some Algorithms in Invariant |
---|
5219 | Theory of Finite Groups\" by Kemper and Steel (1997)). |
---|
5220 | EXAMPLE: example secondary_charp; shows an example |
---|
5221 | " |
---|
5222 | { def br=basering; |
---|
5223 | degBound=0; |
---|
5224 | //---------------- checking input and setting verbose mode ------------------- |
---|
5225 | if (char(br)==0) |
---|
5226 | { "ERROR: secondary_charp should only be used with rings of characteristic p>0."; |
---|
5227 | return(); |
---|
5228 | } |
---|
5229 | int i; |
---|
5230 | if (size(#)>0) |
---|
5231 | { if (typeof(#[size(#)])=="int") |
---|
5232 | { int v=#[size(#)]; |
---|
5233 | } |
---|
5234 | else |
---|
5235 | { int v=0; |
---|
5236 | } |
---|
5237 | } |
---|
5238 | else |
---|
5239 | { int v=0; |
---|
5240 | } |
---|
5241 | int n=nvars(br); // n is the number of variables, as well |
---|
5242 | // as the size of the matrices, as well |
---|
5243 | // as the number of primary invariants, |
---|
5244 | // we should get |
---|
5245 | if (ncols(P)<>n) |
---|
5246 | { "ERROR: The first parameter ought to be the matrix of the primary"; |
---|
5247 | " invariants." |
---|
5248 | return(); |
---|
5249 | } |
---|
5250 | if (ncols(REY)<>n) |
---|
5251 | { "ERROR: The second parameter ought to be the Reynolds operator." |
---|
5252 | return(); |
---|
5253 | } |
---|
5254 | if (typeof(`ring_name`)<>"ring") |
---|
5255 | { "ERROR: The <string> should give the name of the ring where the Molien." |
---|
5256 | " series is stored."; |
---|
5257 | return(); |
---|
5258 | } |
---|
5259 | if (v && voice==2) |
---|
5260 | { ""; |
---|
5261 | } |
---|
5262 | int j, m, counter, d; |
---|
5263 | intvec deg_dim_vec; |
---|
5264 | //- finding the polynomial giving number and degrees of secondary invariants - |
---|
5265 | for (j=1;j<=n;j++) |
---|
5266 | { deg_dim_vec[j]=deg(P[j]); |
---|
5267 | } |
---|
5268 | setring `ring_name`; |
---|
5269 | poly p=1; |
---|
5270 | for (j=1;j<=n;j++) // calculating the denominator of the |
---|
5271 | { p=p*(1-var(1)^deg_dim_vec[j]); // Hilbert series of the ring generated |
---|
5272 | } // by the primary invariants - |
---|
5273 | matrix s[1][2]=M[1,1]*p,M[1,2]; // s is used for canceling |
---|
5274 | s=matrix(syz(ideal(s))); |
---|
5275 | p=s[2,1]; // the polynomial telling us where to |
---|
5276 | // search for secondary invariants |
---|
5277 | map slead=basering,ideal(0); |
---|
5278 | p=1/slead(p)*p; // smallest term of p needs to be 1 |
---|
5279 | if (v) |
---|
5280 | { " Polynomial telling us where to look for secondary invariants:"; |
---|
5281 | " "+string(p); |
---|
5282 | ""; |
---|
5283 | } |
---|
5284 | matrix dimmat=coeffs(p,var(1)); // dimmat will contain the number of |
---|
5285 | // secondary invariants, we need to find |
---|
5286 | // of a certain degree - |
---|
5287 | m=nrows(dimmat); // m-1 is the highest degree |
---|
5288 | deg_dim_vec=1; |
---|
5289 | for (j=2;j<=m;j++) |
---|
5290 | { deg_dim_vec=deg_dim_vec,int(dimmat[j,1]); |
---|
5291 | } |
---|
5292 | if (v) |
---|
5293 | { " In degree 0 we have: 1"; |
---|
5294 | ""; |
---|
5295 | } |
---|
5296 | //------------------------ initializing variables ---------------------------- |
---|
5297 | setring br; |
---|
5298 | intmat PP; |
---|
5299 | poly pp; |
---|
5300 | int k; |
---|
5301 | intvec deg_vec; |
---|
5302 | ideal sP=groebner(ideal(P)); |
---|
5303 | ideal TEST,B,IS; |
---|
5304 | ideal S=1; // 1 is the first secondary invariant |
---|
5305 | //------------------- generating secondary invariants ------------------------ |
---|
5306 | for (i=2;i<=m;i++) // going through deg_dim_vec - |
---|
5307 | { if (deg_dim_vec[i]<>0) // when it is == 0 we need to find 0 |
---|
5308 | { // elements in the current degree (i-1) |
---|
5309 | if (v) |
---|
5310 | { " Searching in degree "+string(i-1)+", we need to find "+string(deg_dim_vec[i])+" invariant(s)..."; |
---|
5311 | } |
---|
5312 | TEST=sP; |
---|
5313 | counter=0; // we'll count up to degvec[i] |
---|
5314 | if (IS[1]<>0) |
---|
5315 | { PP=power_products(deg_vec,i-1); // generating power products of |
---|
5316 | } // irreducible secondary invariants |
---|
5317 | if (size(ideal(PP))<>0) |
---|
5318 | { for (j=1;j<=ncols(PP);j++) // going through all of those |
---|
5319 | { pp=1; |
---|
5320 | for (k=1;k<=nrows(PP);k++) |
---|
5321 | { pp=pp*IS[1,k]^PP[k,j]; |
---|
5322 | } |
---|
5323 | if (reduce(pp,TEST)<>0) |
---|
5324 | { S=S,pp; |
---|
5325 | counter++; |
---|
5326 | if (v) |
---|
5327 | { " We find: "+string(pp); |
---|
5328 | } |
---|
5329 | if (deg_dim_vec[i]<>counter) |
---|
5330 | { //TEST=std(TEST+ideal(NF(pp,TEST))); // should be soon replaced by |
---|
5331 | // next line |
---|
5332 | TEST=std(TEST,pp); |
---|
5333 | } |
---|
5334 | else |
---|
5335 | { break; |
---|
5336 | } |
---|
5337 | } |
---|
5338 | } |
---|
5339 | } |
---|
5340 | if (deg_dim_vec[i]<>counter) |
---|
5341 | { B=sort_of_invariant_basis(sP,REY,i-1,deg_dim_vec[i]*6); // B contains |
---|
5342 | // images of kbase(sP,i-1) under the |
---|
5343 | // Reynolds operator that are linearly |
---|
5344 | // independent and that don't reduce to |
---|
5345 | // 0 modulo sP - |
---|
5346 | if (counter==0 && ncols(B)==deg_dim_vec[i]) // then we can add all of B |
---|
5347 | { S=S,B; |
---|
5348 | IS=IS+B; |
---|
5349 | if (deg_vec[1]==0) |
---|
5350 | { deg_vec=i-1; |
---|
5351 | if (v) |
---|
5352 | { " We find: "+string(B[1]); |
---|
5353 | } |
---|
5354 | for (j=2;j<=deg_dim_vec[i];j++) |
---|
5355 | { deg_vec=deg_vec,i-1; |
---|
5356 | if (v) |
---|
5357 | { " We find: "+string(B[j]); |
---|
5358 | } |
---|
5359 | } |
---|
5360 | } |
---|
5361 | else |
---|
5362 | { for (j=1;j<=deg_dim_vec[i];j++) |
---|
5363 | { deg_vec=deg_vec,i-1; |
---|
5364 | if (v) |
---|
5365 | { " We find: "+string(B[j]); |
---|
5366 | } |
---|
5367 | } |
---|
5368 | } |
---|
5369 | } |
---|
5370 | else |
---|
5371 | { j=0; // j goes through all of B - |
---|
5372 | while (deg_dim_vec[i]<>counter) // need to find deg_dim_vec[i] |
---|
5373 | { // invariants that are linearly |
---|
5374 | // independent modulo TEST |
---|
5375 | j++; |
---|
5376 | if (reduce(B[j],TEST)<>0) // B[j] should be added |
---|
5377 | { S=S,B[j]; |
---|
5378 | IS=IS+ideal(B[j]); |
---|
5379 | if (deg_vec[1]==0) |
---|
5380 | { deg_vec[1]=i-1; |
---|
5381 | } |
---|
5382 | else |
---|
5383 | { deg_vec=deg_vec,i-1; |
---|
5384 | } |
---|
5385 | counter++; |
---|
5386 | if (v) |
---|
5387 | { " We find: "+string(B[j]); |
---|
5388 | } |
---|
5389 | if (deg_dim_vec[i]<>counter) |
---|
5390 | { //TEST=std(TEST+ideal(NF(B[j],TEST))); // should be soon replaced |
---|
5391 | // by next line |
---|
5392 | TEST=std(TEST,B[j]); |
---|
5393 | } |
---|
5394 | } |
---|
5395 | } |
---|
5396 | } |
---|
5397 | } |
---|
5398 | if (v) |
---|
5399 | { ""; |
---|
5400 | } |
---|
5401 | } |
---|
5402 | } |
---|
5403 | if (v) |
---|
5404 | { " We're done!"; |
---|
5405 | ""; |
---|
5406 | } |
---|
5407 | if (ring_name=="aksldfalkdsflkj") |
---|
5408 | { kill `ring_name`; |
---|
5409 | } |
---|
5410 | return(matrix(S),matrix(IS)); |
---|
5411 | } |
---|
5412 | example |
---|
5413 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7 (changed into char 3)"; echo=2; |
---|
5414 | ring R=3,(x,y,z),dp; |
---|
5415 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
5416 | list L=primary_invariants(A); |
---|
5417 | matrix S,IS=secondary_charp(L[1..size(L)]); |
---|
5418 | print(S); |
---|
5419 | print(IS); |
---|
5420 | } |
---|
5421 | /////////////////////////////////////////////////////////////////////////////// |
---|
5422 | |
---|
5423 | proc secondary_no_molien (matrix P, matrix REY, list #) |
---|
5424 | "USAGE: secondary_no_molien(P,REY[,deg_vec,v]); |
---|
5425 | P: a 1xn <matrix> with primary invariants, REY: a gxn <matrix> |
---|
5426 | representing the Reynolds operator, deg_vec: an optional <intvec> |
---|
5427 | listing some degrees where no non-trivial homogeneous invariants can |
---|
5428 | be found, v: an optional <int> |
---|
5429 | ASSUME: n is the number of variables of the basering, g the size of the group, |
---|
5430 | REY is the 1st return value of group_reynolds(), reynolds_molien() or |
---|
5431 | the second one of primary_invariants(), deg_vec is the second return |
---|
5432 | value of primary_char0_no_molien(), primary_charp_no_molien(), |
---|
5433 | primary_char0_no_molien_random() or primary_charp_no_molien_random() |
---|
5434 | RETURN: secondary invariants of the invariant ring (type <matrix>) |
---|
5435 | DISPLAY: information if v does not equal 0 |
---|
5436 | THEORY: Secondary invariants are calculated by finding a basis (in terms of |
---|
5437 | monomials) of the basering modulo primary invariants, mapping those to |
---|
5438 | invariants with the Reynolds operator and using these images as |
---|
5439 | candidates for secondary invariants. |
---|
5440 | EXAMPLE: example secondary_no_molien; shows an example |
---|
5441 | " |
---|
5442 | { int i; |
---|
5443 | degBound=0; |
---|
5444 | //------------------ checking input and setting verbose ---------------------- |
---|
5445 | if (size(#)==1 or size(#)==2) |
---|
5446 | { if (typeof(#[size(#)])=="int") |
---|
5447 | { if (size(#)==2) |
---|
5448 | { if (typeof(#[size(#)-1])=="intvec") |
---|
5449 | { intvec deg_vec=#[size(#)-1]; |
---|
5450 | } |
---|
5451 | else |
---|
5452 | { "ERROR: the third parameter should be an <intvec>"; |
---|
5453 | return(); |
---|
5454 | } |
---|
5455 | } |
---|
5456 | int v=#[size(#)]; |
---|
5457 | } |
---|
5458 | else |
---|
5459 | { if (size(#)==1) |
---|
5460 | { if (typeof(#[size(#)])=="intvec") |
---|
5461 | { intvec deg_vec=#[size(#)]; |
---|
5462 | int v=0; |
---|
5463 | } |
---|
5464 | else |
---|
5465 | { "ERROR: the third parameter should be an <intvec>"; |
---|
5466 | return(); |
---|
5467 | } |
---|
5468 | } |
---|
5469 | else |
---|
5470 | { "ERROR: wrong list of parameters"; |
---|
5471 | return(); |
---|
5472 | } |
---|
5473 | } |
---|
5474 | } |
---|
5475 | else |
---|
5476 | { if (size(#)>2) |
---|
5477 | { "ERROR: there are too many parameters"; |
---|
5478 | return(); |
---|
5479 | } |
---|
5480 | int v=0; |
---|
5481 | } |
---|
5482 | int n=nvars(basering); // n is the number of variables, as well |
---|
5483 | // as the size of the matrices, as well |
---|
5484 | // as the number of primary invariants, |
---|
5485 | // we should get |
---|
5486 | if (ncols(P)<>n) |
---|
5487 | { "ERROR: The first parameter ought to be the matrix of the primary"; |
---|
5488 | " invariants." |
---|
5489 | return(); |
---|
5490 | } |
---|
5491 | if (ncols(REY)<>n) |
---|
5492 | { "ERROR: The second parameter ought to be the Reynolds operator." |
---|
5493 | return(); |
---|
5494 | } |
---|
5495 | if (v && voice==2) |
---|
5496 | { ""; |
---|
5497 | } |
---|
5498 | int j, m, d; |
---|
5499 | int max=1; |
---|
5500 | for (j=1;j<=n;j++) |
---|
5501 | { max=max*deg(P[j]); |
---|
5502 | } |
---|
5503 | max=max/nrows(REY); |
---|
5504 | if (v) |
---|
5505 | { " We need to find "+string(max)+" secondary invariants."; |
---|
5506 | ""; |
---|
5507 | " In degree 0 we have: 1"; |
---|
5508 | ""; |
---|
5509 | } |
---|
5510 | //------------------------- initializing variables --------------------------- |
---|
5511 | ideal sP=groebner(ideal(P)); |
---|
5512 | ideal B, TEST; |
---|
5513 | ideal S=1; // 1 is the first secondary invariant |
---|
5514 | int counter=1; |
---|
5515 | i=0; |
---|
5516 | if (defined(deg_vec)<>voice) |
---|
5517 | { intvec deg_vec; |
---|
5518 | } |
---|
5519 | int k=1; |
---|
5520 | //--------------------- generating secondary invariants ---------------------- |
---|
5521 | while (counter<>max) |
---|
5522 | { i++; |
---|
5523 | if (deg_vec[k]<>i) |
---|
5524 | { if (v) |
---|
5525 | { " Searching in degree "+string(i)+"..."; |
---|
5526 | } |
---|
5527 | B=sort_of_invariant_basis(sP,REY,i,max); // B contains images of |
---|
5528 | // kbase(sP,i) under the Reynolds |
---|
5529 | // operator that are linearly independent |
---|
5530 | // and that don't reduce to 0 modulo sP |
---|
5531 | TEST=sP; |
---|
5532 | for (j=1;j<=ncols(B);j++) |
---|
5533 | { // that are linearly independent modulo |
---|
5534 | // TEST |
---|
5535 | if (reduce(B[j],TEST)<>0) // B[j] should be added |
---|
5536 | { S=S,B[j]; |
---|
5537 | counter++; |
---|
5538 | if (v) |
---|
5539 | { " We find: "+string(B[j]); |
---|
5540 | } |
---|
5541 | if (counter==max) |
---|
5542 | { break; |
---|
5543 | } |
---|
5544 | else |
---|
5545 | { if (j<>ncols(B)) |
---|
5546 | { //TEST=std(TEST+ideal(NF(B[j],TEST))); // should soon be replaced by |
---|
5547 | // next line |
---|
5548 | TEST=std(TEST,B[j]); |
---|
5549 | } |
---|
5550 | } |
---|
5551 | } |
---|
5552 | } |
---|
5553 | } |
---|
5554 | else |
---|
5555 | { if (size(deg_vec)==k) |
---|
5556 | { k=1; } |
---|
5557 | else |
---|
5558 | { k++; } |
---|
5559 | } |
---|
5560 | } |
---|
5561 | if (v) |
---|
5562 | { ""; } |
---|
5563 | if (v) |
---|
5564 | { " We're done!"; ""; } |
---|
5565 | return(matrix(S)); |
---|
5566 | } |
---|
5567 | example |
---|
5568 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
5569 | ring R=0,(x,y,z),dp; |
---|
5570 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
5571 | list L=primary_invariants(A,intvec(1,1,0)); |
---|
5572 | matrix S=secondary_no_molien(L[1..3]); |
---|
5573 | print(S); |
---|
5574 | } |
---|
5575 | /////////////////////////////////////////////////////////////////////////////// |
---|
5576 | |
---|
5577 | proc secondary_and_irreducibles_no_molien (matrix P, matrix REY, list #) |
---|
5578 | "USAGE: secondary_and_irreducibles_no_molien(P,REY[,v]); |
---|
5579 | P: a 1xn <matrix> with primary invariants, REY: a gxn <matrix> |
---|
5580 | representing the Reynolds operator, v: an optional <int> |
---|
5581 | ASSUME: n is the number of variables of the basering, g the size of the group, |
---|
5582 | REY is the 1st return value of group_reynolds(), reynolds_molien() or |
---|
5583 | the second one of primary_invariants() |
---|
5584 | RETURN: secondary invariants of the invariant ring (type <matrix>) and |
---|
5585 | irreducible secondary invariants (type <matrix>) |
---|
5586 | DISPLAY: information if v does not equal 0 |
---|
5587 | THEORY: Secondary invariants are calculated by finding a basis (in terms of |
---|
5588 | monomials) of the basering modulo primary invariants, mapping those to |
---|
5589 | invariants with the Reynolds operator and using these images or their |
---|
5590 | power products such that they are linearly independent modulo the |
---|
5591 | primary invariants (see paper \"Some Algorithms in Invariant Theory of |
---|
5592 | Finite Groups\" by Kemper and Steel (1997)). |
---|
5593 | EXAMPLE: example secondary_and_irreducibles_no_molien; shows an example |
---|
5594 | " |
---|
5595 | { int i; |
---|
5596 | degBound=0; |
---|
5597 | //--------------------- checking input and setting verbose mode -------------- |
---|
5598 | if (size(#)==1 or size(#)==2) |
---|
5599 | { if (typeof(#[size(#)])=="int") |
---|
5600 | { if (size(#)==2) |
---|
5601 | { if (typeof(#[size(#)-1])=="intvec") |
---|
5602 | { intvec deg_vec=#[size(#)-1]; |
---|
5603 | } |
---|
5604 | else |
---|
5605 | { "ERROR: the third parameter should be an <intvec>"; |
---|
5606 | return(); |
---|
5607 | } |
---|
5608 | } |
---|
5609 | int v=#[size(#)]; |
---|
5610 | } |
---|
5611 | else |
---|
5612 | { if (size(#)==1) |
---|
5613 | { if (typeof(#[size(#)])=="intvec") |
---|
5614 | { intvec deg_vec=#[size(#)]; |
---|
5615 | int v=0; |
---|
5616 | } |
---|
5617 | else |
---|
5618 | { "ERROR: the third parameter should be an <intvec>"; |
---|
5619 | return(); |
---|
5620 | } |
---|
5621 | } |
---|
5622 | else |
---|
5623 | { "ERROR: wrong list of parameters"; |
---|
5624 | return(); |
---|
5625 | } |
---|
5626 | } |
---|
5627 | } |
---|
5628 | else |
---|
5629 | { if (size(#)>2) |
---|
5630 | { "ERROR: there are too many parameters"; |
---|
5631 | return(); |
---|
5632 | } |
---|
5633 | int v=0; |
---|
5634 | } |
---|
5635 | int n=nvars(basering); // n is the number of variables, as well |
---|
5636 | // as the size of the matrices, as well |
---|
5637 | // as the number of primary invariants, |
---|
5638 | // we should get |
---|
5639 | if (ncols(P)<>n) |
---|
5640 | { "ERROR: The first parameter ought to be the matrix of the primary"; |
---|
5641 | " invariants." |
---|
5642 | return(); |
---|
5643 | } |
---|
5644 | if (ncols(REY)<>n) |
---|
5645 | { "ERROR: The second parameter ought to be the Reynolds operator." |
---|
5646 | return(); |
---|
5647 | } |
---|
5648 | if (v && voice==2) |
---|
5649 | { ""; |
---|
5650 | } |
---|
5651 | int j, m, d; |
---|
5652 | int max=1; |
---|
5653 | for (j=1;j<=n;j++) |
---|
5654 | { max=max*deg(P[j]); |
---|
5655 | } |
---|
5656 | max=max/nrows(REY); |
---|
5657 | if (v) |
---|
5658 | { " We need to find "+string(max)+" secondary invariants."; |
---|
5659 | ""; |
---|
5660 | " In degree 0 we have: 1"; |
---|
5661 | ""; |
---|
5662 | } |
---|
5663 | //------------------------ initializing variables ---------------------------- |
---|
5664 | intmat PP; |
---|
5665 | poly pp; |
---|
5666 | int k; |
---|
5667 | intvec irreducible_deg_vec; |
---|
5668 | ideal sP=groebner(ideal(P)); |
---|
5669 | ideal B,TEST,IS; |
---|
5670 | ideal S=1; // 1 is the first secondary invariant |
---|
5671 | int counter=1; |
---|
5672 | i=0; |
---|
5673 | if (defined(deg_vec)<>voice) |
---|
5674 | { intvec deg_vec; |
---|
5675 | } |
---|
5676 | int l=1; |
---|
5677 | //------------------- generating secondary invariants ------------------------ |
---|
5678 | while (counter<>max) |
---|
5679 | { i++; |
---|
5680 | if (deg_vec[l]<>i) |
---|
5681 | { if (v) |
---|
5682 | { " Searching in degree "+string(i)+"..."; |
---|
5683 | } |
---|
5684 | TEST=sP; |
---|
5685 | if (IS[1]<>0) |
---|
5686 | { PP=power_products(irreducible_deg_vec,i); // generating all power |
---|
5687 | } // products of irreducible secondary |
---|
5688 | // invariants |
---|
5689 | if (size(ideal(PP))<>0) |
---|
5690 | { for (j=1;j<=ncols(PP);j++) // going through all those power products |
---|
5691 | { pp=1; |
---|
5692 | for (k=1;k<=nrows(PP);k++) |
---|
5693 | { pp=pp*IS[1,k]^PP[k,j]; |
---|
5694 | } |
---|
5695 | if (reduce(pp,TEST)<>0) |
---|
5696 | { S=S,pp; |
---|
5697 | counter++; |
---|
5698 | if (v) |
---|
5699 | { " We find: "+string(pp); |
---|
5700 | } |
---|
5701 | if (counter<>max) |
---|
5702 | { //TEST=std(TEST+ideal(NF(pp,TEST))); // should soon be replaced by |
---|
5703 | // next line |
---|
5704 | TEST=std(TEST,pp); |
---|
5705 | } |
---|
5706 | else |
---|
5707 | { break; |
---|
5708 | } |
---|
5709 | } |
---|
5710 | } |
---|
5711 | } |
---|
5712 | if (max<>counter) |
---|
5713 | { B=sort_of_invariant_basis(sP,REY,i,max); // B contains images of |
---|
5714 | // kbase(sP,i) under the Reynolds |
---|
5715 | // operator that are linearly independent |
---|
5716 | // and that don't reduce to 0 modulo sP |
---|
5717 | for (j=1;j<=ncols(B);j++) |
---|
5718 | { if (reduce(B[j],TEST)<>0) // B[j] should be added |
---|
5719 | { S=S,B[j]; |
---|
5720 | IS=IS+ideal(B[j]); |
---|
5721 | if (irreducible_deg_vec[1]==0) |
---|
5722 | { irreducible_deg_vec[1]=i; |
---|
5723 | } |
---|
5724 | else |
---|
5725 | { irreducible_deg_vec=irreducible_deg_vec,i; |
---|
5726 | } |
---|
5727 | counter++; |
---|
5728 | if (v) |
---|
5729 | { " We find: "+string(B[j]); |
---|
5730 | } |
---|
5731 | if (counter==max) |
---|
5732 | { break; |
---|
5733 | } |
---|
5734 | else |
---|
5735 | { if (j<>ncols(B)) |
---|
5736 | { //TEST=std(TEST+ideal(NF(B[j],TEST))); // should soon be replaced |
---|
5737 | // by next line |
---|
5738 | TEST=std(TEST,B[j]); |
---|
5739 | } |
---|
5740 | } |
---|
5741 | } |
---|
5742 | } |
---|
5743 | } |
---|
5744 | } |
---|
5745 | else |
---|
5746 | { if (size(deg_vec)==l) |
---|
5747 | { l=1; |
---|
5748 | } |
---|
5749 | else |
---|
5750 | { l++; |
---|
5751 | } |
---|
5752 | } |
---|
5753 | } |
---|
5754 | if (v) |
---|
5755 | { ""; |
---|
5756 | } |
---|
5757 | if (v) |
---|
5758 | { " We're done!"; |
---|
5759 | ""; |
---|
5760 | } |
---|
5761 | return(matrix(S),matrix(IS)); |
---|
5762 | } |
---|
5763 | example |
---|
5764 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
5765 | ring R=0,(x,y,z),dp; |
---|
5766 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
5767 | list L=primary_invariants(A,intvec(1,1,0)); |
---|
5768 | matrix S,IS=secondary_and_irreducibles_no_molien(L[1..2]); |
---|
5769 | print(S); |
---|
5770 | print(IS); |
---|
5771 | } |
---|
5772 | /////////////////////////////////////////////////////////////////////////////// |
---|
5773 | |
---|
5774 | proc secondary_not_cohen_macaulay (matrix P, list #) |
---|
5775 | "USAGE: secondary_not_cohen_macaulay(P,G1,G2,...[,v]); |
---|
5776 | P: a 1xn <matrix> with primary invariants, G1,G2,...: nxn <matrices> |
---|
5777 | generating a finite matrix group, v: an optional <int> |
---|
5778 | ASSUME: n is the number of variables of the basering |
---|
5779 | RETURN: secondary invariants of the invariant ring (type <matrix>) |
---|
5780 | DISPLAY: information if v does not equal 0 |
---|
5781 | THEORY: Secondary invariants are generated following \"Generating Invariant |
---|
5782 | Rings of Finite Groups over Arbitrary Fields\" by Kemper (1996). |
---|
5783 | EXAMPLE: example secondary_not_cohen_macaulay; shows an example |
---|
5784 | " |
---|
5785 | { int i, j; |
---|
5786 | degBound=0; |
---|
5787 | def br=basering; |
---|
5788 | int n=nvars(br); // n is the number of variables, as well |
---|
5789 | // as the size of the matrices, as well |
---|
5790 | // as the number of primary invariants, |
---|
5791 | // we should get - |
---|
5792 | if (size(#)>0) // checking input and setting verbose |
---|
5793 | { if (typeof(#[size(#)])=="int") |
---|
5794 | { int gen_num=size(#)-1; |
---|
5795 | if (gen_num==0) |
---|
5796 | { "ERROR: There are no generators of the finite matrix group given."; |
---|
5797 | return(); |
---|
5798 | } |
---|
5799 | int v=#[size(#)]; |
---|
5800 | for (i=1;i<=gen_num;i++) |
---|
5801 | { if (typeof(#[i])<>"matrix") |
---|
5802 | { "ERROR: These parameters should be generators of the finite matrix group."; |
---|
5803 | return(); |
---|
5804 | } |
---|
5805 | if ((n<>nrows(#[i])) or (n<>ncols(#[i]))) |
---|
5806 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
5807 | return(); |
---|
5808 | } |
---|
5809 | } |
---|
5810 | } |
---|
5811 | else |
---|
5812 | { int v=0; |
---|
5813 | int gen_num=size(#); |
---|
5814 | for (i=1;i<=gen_num;i++) |
---|
5815 | { if (typeof(#[i])<>"matrix") |
---|
5816 | { "ERROR: These parameters should be generators of the finite matrix group."; |
---|
5817 | return(); |
---|
5818 | } |
---|
5819 | if ((n<>nrows(#[i])) or (n<>ncols(#[i]))) |
---|
5820 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
5821 | return(); |
---|
5822 | } |
---|
5823 | } |
---|
5824 | } |
---|
5825 | } |
---|
5826 | else |
---|
5827 | { "ERROR: There are no generators of the finite matrix group given."; |
---|
5828 | return(); |
---|
5829 | } |
---|
5830 | if (ncols(P)<>n) |
---|
5831 | { "ERROR: The first parameter ought to be the matrix of the primary"; |
---|
5832 | " invariants." |
---|
5833 | return(); |
---|
5834 | } |
---|
5835 | if (v && voice==2) |
---|
5836 | { ""; |
---|
5837 | } |
---|
5838 | ring alskdfalkdsj=0,x,dp; |
---|
5839 | matrix M[1][2]=1,(1-x)^n; // we look at our primary invariants as |
---|
5840 | export alskdfalkdsj; |
---|
5841 | export M; |
---|
5842 | setring br; // such of the subgroup that only |
---|
5843 | matrix REY=matrix(maxideal(1)); // contains the identity, this means that |
---|
5844 | // ch does not divide the order anymore, |
---|
5845 | // this means that we can make use of the |
---|
5846 | // Molien series again - M[1,1]/M[1,2] is |
---|
5847 | // the Molien series of that group, we |
---|
5848 | // now calculate the secondary invariants |
---|
5849 | // of this subgroup in the usual fashion |
---|
5850 | // where the primary invariants are the |
---|
5851 | // ones from the bigger group |
---|
5852 | if (v) |
---|
5853 | { " The procedure secondary_charp() is called to calculate secondary invariants"; |
---|
5854 | " of the invariant ring of the trivial group with respect to the primary"; |
---|
5855 | " invariants found previously."; |
---|
5856 | ""; |
---|
5857 | } |
---|
5858 | matrix trivialS, trivialSI=secondary_charp(P,REY,"alskdfalkdsj",v); |
---|
5859 | kill trivialSI; |
---|
5860 | kill alskdfalkdsj; |
---|
5861 | // now we have those secondary invariants |
---|
5862 | int k=ncols(trivialS); // k is the number of the secondary |
---|
5863 | // invariants, we just calculated |
---|
5864 | if (v) |
---|
5865 | { " We calculate secondary invariants from the ones found for the trivial"; |
---|
5866 | " subgroup."; |
---|
5867 | ""; |
---|
5868 | } |
---|
5869 | map f; // used to let generators act on |
---|
5870 | // secondary invariants with respect to |
---|
5871 | // the trivial group - |
---|
5872 | matrix M(1)[gen_num][k]; // M(1) will contain a module |
---|
5873 | ideal B; |
---|
5874 | for (i=1;i<=gen_num;i++) |
---|
5875 | { B=ideal(matrix(maxideal(1))*transpose(#[i])); // image of the various |
---|
5876 | // variables under the i-th generator - |
---|
5877 | f=br,B; // the corresponding mapping - |
---|
5878 | B=f(trivialS)-trivialS; // these relations should be 0 - |
---|
5879 | M(1)[i,1..k]=B[1..k]; // we will look for the syzygies of M(1) |
---|
5880 | } |
---|
5881 | //intvec save_opts=option(get); |
---|
5882 | //option(returnSB,redSB); |
---|
5883 | //module M(2)=syz(M(1)); // nres(M(1),2)[2]; |
---|
5884 | //option(set,save_opts); |
---|
5885 | module M(2)=nres(M(1),2)[2]; |
---|
5886 | int m=ncols(M(2)); // number of generators of the module |
---|
5887 | // M(2) - |
---|
5888 | // the following steps calculates the intersection of the module M(2) with |
---|
5889 | // the algebra A^k where A denote the subalgebra of the usual polynomial |
---|
5890 | // ring, generated by the primary invariants |
---|
5891 | string mp=string(minpoly); // generating a ring where we can do |
---|
5892 | // elimination |
---|
5893 | execute("ring R=("+charstr(br)+"),(x(1..n),y(1..n),h),dp;"); |
---|
5894 | execute("minpoly=number("+mp+");"); |
---|
5895 | map f=br,maxideal(1); // canonical mapping |
---|
5896 | matrix M[k][m+k*n]; |
---|
5897 | M[1..k,1..m]=matrix(f(M(2))); // will contain a module - |
---|
5898 | matrix P=f(P); // primary invariants in the new ring |
---|
5899 | for (i=1;i<=n;i++) |
---|
5900 | { for (j=1;j<=k;j++) |
---|
5901 | { M[j,m+(i-1)*k+j]=y(i)-P[1,i]; |
---|
5902 | } |
---|
5903 | } |
---|
5904 | M=elim(module(M),1,n); // eliminating x(1..n), std-calculation |
---|
5905 | // is done internally - |
---|
5906 | M=homog(module(M),h); // homogenize for 'minbase' |
---|
5907 | M=minbase(module(M)); |
---|
5908 | setring br; |
---|
5909 | ideal substitute=maxideal(1),ideal(P),1; |
---|
5910 | f=R,substitute; // replacing y(1..n) by primary |
---|
5911 | // invariants - |
---|
5912 | M(2)=f(M); // M(2) is the new module |
---|
5913 | m=ncols(M(2)); |
---|
5914 | matrix S[1][m]; |
---|
5915 | S=matrix(trivialS)*matrix(M(2)); // S now contains the secondary |
---|
5916 | // invariants |
---|
5917 | for (i=1; i<=m;i++) |
---|
5918 | { S[1,i]=S[1,i]/leadcoef(S[1,i]); // making elements nice |
---|
5919 | } |
---|
5920 | S=sort(ideal(S))[1]; |
---|
5921 | if (v) |
---|
5922 | { " These are the secondary invariants: "; |
---|
5923 | for (i=1;i<=m;i++) |
---|
5924 | { " "+string(S[1,i]); |
---|
5925 | } |
---|
5926 | ""; |
---|
5927 | " We're done!"; |
---|
5928 | ""; |
---|
5929 | } |
---|
5930 | if ((v or (voice==2)) && (m>1)) |
---|
5931 | { " WARNING: The invariant ring might not have a Hironaka decomposition"; |
---|
5932 | " if the characteristic of the coefficient field divides the"; |
---|
5933 | " group order."; |
---|
5934 | } |
---|
5935 | return(S); |
---|
5936 | } |
---|
5937 | example |
---|
5938 | { "EXAMPLE:"; echo=2; |
---|
5939 | ring R=2,(x,y,z),dp; |
---|
5940 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
5941 | list L=primary_invariants(A); |
---|
5942 | matrix S=secondary_not_cohen_macaulay(L[1],A); |
---|
5943 | print(S); |
---|
5944 | } |
---|
5945 | /////////////////////////////////////////////////////////////////////////////// |
---|
5946 | |
---|
5947 | proc invariant_ring (list #) |
---|
5948 | "USAGE: invariant_ring(G1,G2,...[,flags]); |
---|
5949 | G1,G2,...: <matrices> generating a finite matrix group, flags: an |
---|
5950 | optional <intvec> with three entries: if the first one equals 0, the |
---|
5951 | program attempts to compute the Molien series and Reynolds operator, |
---|
5952 | if it equals 1, the program is told that the Molien series should not |
---|
5953 | be computed, if it equals -1 characteristic 0 is simulated, i.e. the |
---|
5954 | Molien series is computed as if the base field were characteristic 0 |
---|
5955 | (the user must choose a field of large prime characteristic, e.g. |
---|
5956 | 32003) and if the first one is anything else, it means that the |
---|
5957 | characteristic of the base field divides the group order (i.e. it will |
---|
5958 | not even be attempted to compute the Reynolds operator or Molien |
---|
5959 | series), the second component should give the size of intervals |
---|
5960 | between canceling common factors in the expansion of Molien series, 0 |
---|
5961 | (the default) means only once after generating all terms, in prime |
---|
5962 | characteristic also a negative number can be given to indicate that |
---|
5963 | common factors should always be canceled when the expansion is simple |
---|
5964 | (the root of the extension field occurs not among the coefficients) |
---|
5965 | RETURN: primary and secondary invariants (both of type <matrix>) generating |
---|
5966 | the invariant ring with respect to the matrix group generated by the |
---|
5967 | matrices in the input and irreducible secondary invariants (type |
---|
5968 | <matrix>) if the Molien series was available |
---|
5969 | DISPLAY: information about the various stages of the program if the third flag |
---|
5970 | does not equal 0 |
---|
5971 | THEORY: Bases of homogeneous invariants are generated successively and those |
---|
5972 | are chosen as primary invariants that lower the dimension of the ideal |
---|
5973 | generated by the previously found invariants (see \"Generating a |
---|
5974 | Noetherian Normalization of the Invariant Ring of a Finite Group\" by |
---|
5975 | Decker, Heydtmann, Schreyer (1998)). In the |
---|
5976 | non-modular case secondary invariants are calculated by finding a |
---|
5977 | basis (in terms of monomials) of the basering modulo the primary |
---|
5978 | invariants, mapping to invariants with the Reynolds operator and using |
---|
5979 | those or their power products such that they are linearly independent |
---|
5980 | modulo the primary invariants (see \"Some Algorithms in Invariant |
---|
5981 | Theory of Finite Groups\" by Kemper and Steel (1997)). In the modular |
---|
5982 | case they are generated according to \"Generating Invariant Rings of |
---|
5983 | Finite Groups over Arbitrary Fields\" by Kemper (1996). |
---|
5984 | EXAMPLE: example invariant_ring; shows an example |
---|
5985 | " |
---|
5986 | { if (size(#)==0) |
---|
5987 | { "ERROR: There are no generators given."; |
---|
5988 | return(); |
---|
5989 | } |
---|
5990 | int ch=char(basering); // the algorithms depend very much on the |
---|
5991 | // characteristic of the ground field - |
---|
5992 | int n=nvars(basering); // n is the number of variables, as well |
---|
5993 | // as the size of the matrices, as well |
---|
5994 | // as the number of primary invariants, |
---|
5995 | // we should get |
---|
5996 | int gen_num; |
---|
5997 | int mol_flag, v; |
---|
5998 | //------------------- checking input and setting flags ----------------------- |
---|
5999 | if (typeof(#[size(#)])=="intvec") |
---|
6000 | { if (size(#[size(#)])<>3) |
---|
6001 | { "ERROR: The <intvec> should have three entries."; |
---|
6002 | return(); |
---|
6003 | } |
---|
6004 | gen_num=size(#)-1; |
---|
6005 | mol_flag=#[size(#)][1]; |
---|
6006 | if (#[size(#)][2]<0 && (ch==0 or (ch<>0 && mol_flag<>0))) |
---|
6007 | { "ERROR: the second component of <intvec> should be >=0"; |
---|
6008 | return(); |
---|
6009 | } |
---|
6010 | int interval=#[size(#)][2]; |
---|
6011 | v=#[size(#)][3]; |
---|
6012 | } |
---|
6013 | else |
---|
6014 | { gen_num=size(#); |
---|
6015 | mol_flag=0; |
---|
6016 | int interval=0; |
---|
6017 | v=0; |
---|
6018 | } |
---|
6019 | //---------------------------------------------------------------------------- |
---|
6020 | if (mol_flag==0) // calculation Molien series will be |
---|
6021 | { if (ch==0) // attempted - |
---|
6022 | { matrix REY,M=reynolds_molien(#[1..gen_num],intvec(0,interval,v)); // one |
---|
6023 | // will contain Reynolds operator and the |
---|
6024 | // other enumerator and denominator of |
---|
6025 | // Molien series |
---|
6026 | matrix P=primary_char0(REY,M,v); |
---|
6027 | matrix S,IS=secondary_char0(P,REY,M,v); |
---|
6028 | return(P,S,IS); |
---|
6029 | } |
---|
6030 | else |
---|
6031 | { list L=group_reynolds(#[1..gen_num],v); |
---|
6032 | if (L[1]<>0) // testing whether we are in the modular |
---|
6033 | { string newring="aksldfalkdsflkj"; // case |
---|
6034 | if (minpoly==0) |
---|
6035 | { if (v) |
---|
6036 | { " We are dealing with the non-modular case."; |
---|
6037 | } |
---|
6038 | if (typeof(L[2])=="int") |
---|
6039 | { molien(L[3..size(L)],newring,L[2],intvec(0,interval,v)); |
---|
6040 | } |
---|
6041 | else |
---|
6042 | { molien(L[2..size(L)],newring,intvec(0,interval,v)); |
---|
6043 | } |
---|
6044 | matrix P=primary_charp(L[1],newring,v); |
---|
6045 | matrix S,IS=secondary_charp(P,L[1],newring,v); |
---|
6046 | if (defined(aksldfalkdsflkj)==2) |
---|
6047 | { kill aksldfalkdsflkj; |
---|
6048 | } |
---|
6049 | return(P,S,IS); |
---|
6050 | } |
---|
6051 | else |
---|
6052 | { if (v) |
---|
6053 | { " Since it is impossible for this programme to calculate the Molien |
---|
6054 | series for"; |
---|
6055 | " invariant rings over extension fields of prime characteristic, we |
---|
6056 | have to"; |
---|
6057 | " continue without it."; |
---|
6058 | ""; |
---|
6059 | |
---|
6060 | } |
---|
6061 | list l=primary_charp_no_molien(L[1],v); |
---|
6062 | if (size(l)==2) |
---|
6063 | { matrix S=secondary_no_molien(l[1],L[1],l[2],v); |
---|
6064 | } |
---|
6065 | else |
---|
6066 | { matrix S=secondary_no_molien(l[1],L[1],v); |
---|
6067 | } |
---|
6068 | return(l[1],S); |
---|
6069 | } |
---|
6070 | } |
---|
6071 | else // the modular case |
---|
6072 | { if (v) |
---|
6073 | { " There is also no Molien series or Reynolds operator, we can make use of..."; |
---|
6074 | ""; |
---|
6075 | " We can start looking for primary invariants..."; |
---|
6076 | ""; |
---|
6077 | } |
---|
6078 | matrix P=primary_charp_without(#[1..gen_num],v); |
---|
6079 | matrix S=secondary_not_cohen_macaulay(P,#[1..gen_num],v); |
---|
6080 | return(P,S); |
---|
6081 | } |
---|
6082 | } |
---|
6083 | } |
---|
6084 | if (mol_flag==1) // the user wants no calculation of the |
---|
6085 | { list L=group_reynolds(#[1..gen_num],v); // Molien series |
---|
6086 | if (ch==0) |
---|
6087 | { list l=primary_char0_no_molien(L[1],v); |
---|
6088 | if (size(l)==2) |
---|
6089 | { matrix S=secondary_no_molien(l[1],L[1],l[2],v); |
---|
6090 | } |
---|
6091 | else |
---|
6092 | { matrix S=secondary_no_molien(l[1],L[1],v); |
---|
6093 | } |
---|
6094 | return(l[1],S); |
---|
6095 | } |
---|
6096 | else |
---|
6097 | { if (L[1]<>0) // testing whether we are in the modular |
---|
6098 | { list l=primary_charp_no_molien(L[1],v); // case |
---|
6099 | if (size(l)==2) |
---|
6100 | { matrix S=secondary_no_molien(l[1],L[1],l[2],v); |
---|
6101 | } |
---|
6102 | else |
---|
6103 | { matrix S=secondary_no_molien(l[1],L[1],v); |
---|
6104 | } |
---|
6105 | return(l[1],S); |
---|
6106 | } |
---|
6107 | else // the modular case |
---|
6108 | { if (v) |
---|
6109 | { " We can start looking for primary invariants..."; |
---|
6110 | ""; |
---|
6111 | } |
---|
6112 | matrix P=primary_charp_without(#[1..gen_num],v); |
---|
6113 | matrix S=secondary_not_cohen_macaulay(P,#[1..gen_num],v); |
---|
6114 | return(L[1],S); |
---|
6115 | } |
---|
6116 | } |
---|
6117 | } |
---|
6118 | if (mol_flag==-1) |
---|
6119 | { if (ch==0) |
---|
6120 | { "ERROR: Characteristic 0 can only be simulated in characteristic p>>0. |
---|
6121 | "; |
---|
6122 | return(); |
---|
6123 | } |
---|
6124 | list L=group_reynolds(#[1..gen_num],v); |
---|
6125 | string newring="aksldfalkdsflkj"; |
---|
6126 | if (typeof(L[2])=="int") |
---|
6127 | { molien(L[3..size(L)],newring,L[2],intvec(1,interval,v)); |
---|
6128 | } |
---|
6129 | else |
---|
6130 | { molien(L[2..size(L)],newring,intvec(1,interval,v)); |
---|
6131 | } |
---|
6132 | matrix P=primary_charp(L[1],newring,v); |
---|
6133 | matrix S,IS=secondary_charp(P,L[1],newring,v); |
---|
6134 | kill aksldfalkdsflkj; |
---|
6135 | return(P,S,IS); |
---|
6136 | } |
---|
6137 | else // the user specified that the |
---|
6138 | { if (ch==0) // characteristic divides the group order |
---|
6139 | { "ERROR: The characteristic cannot divide the group order when it is 0. |
---|
6140 | "; |
---|
6141 | return(); |
---|
6142 | } |
---|
6143 | if (v) |
---|
6144 | { ""; |
---|
6145 | } |
---|
6146 | matrix P=primary_charp_without(#[1..gen_num],v); |
---|
6147 | matrix S=secondary_not_cohen_macaulay(P,#[1..gen_num],v); |
---|
6148 | return(L[1],S); |
---|
6149 | } |
---|
6150 | } |
---|
6151 | example |
---|
6152 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
6153 | ring R=0,(x,y,z),dp; |
---|
6154 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
6155 | matrix P,S,IS=invariant_ring(A); |
---|
6156 | print(P); |
---|
6157 | print(S); |
---|
6158 | print(IS); |
---|
6159 | } |
---|
6160 | /////////////////////////////////////////////////////////////////////////////// |
---|
6161 | |
---|
6162 | proc invariant_ring_random (list #) |
---|
6163 | "USAGE: invariant_ring_random(G1,G2,...,r[,flags]); |
---|
6164 | G1,G2,...: <matrices> generating a finite matrix group, r: an <int> |
---|
6165 | where -|r| to |r| is the range of coefficients of random |
---|
6166 | combinations of bases elements that serve as primary invariants, |
---|
6167 | flags: an optional <intvec> with three entries: if the first equals 0, |
---|
6168 | the program attempts to compute the Molien series and Reynolds |
---|
6169 | operator, if it equals 1, the program is told that the Molien series |
---|
6170 | should not be computed, if it equals -1 characteristic 0 is simulated, |
---|
6171 | i.e. the Molien series is computed as if the base field were |
---|
6172 | characteristic 0 (the user must choose a field of large prime |
---|
6173 | characteristic, e.g. 32003) and if the first one is anything else, |
---|
6174 | then the characteristic of the base field divides the group order |
---|
6175 | (i.e. we will not even attempt to compute the Reynolds operator or |
---|
6176 | Molien series), the second component should give the size of intervals |
---|
6177 | between canceling common factors in the expansion of the Molien |
---|
6178 | series, 0 (the default) means only once after generating all terms, |
---|
6179 | in prime characteristic also a negative number can be given to |
---|
6180 | indicate that common factors should always be canceled when the |
---|
6181 | expansion is simple (the root of the extension field does not occur |
---|
6182 | among the coefficients) |
---|
6183 | RETURN: primary and secondary invariants (both of type <matrix>) generating |
---|
6184 | invariant ring with respect to the matrix group generated by the |
---|
6185 | matrices in the input and irreducible secondary invariants (type |
---|
6186 | <matrix>) if the Molien series was available |
---|
6187 | DISPLAY: information about the various stages of the program if the third flag |
---|
6188 | does not equal 0 |
---|
6189 | THEORY: is the same as for invariant_ring except that random combinations of |
---|
6190 | basis elements are chosen as candidates for primary invariants and |
---|
6191 | hopefully they lower the dimension of the previously found primary |
---|
6192 | invariants by the right amount. |
---|
6193 | EXAMPLE: example invariant_ring_random; shows an example |
---|
6194 | " |
---|
6195 | { if (size(#)<2) |
---|
6196 | { "ERROR: There are too few parameters."; |
---|
6197 | return(); |
---|
6198 | } |
---|
6199 | int ch=char(basering); // the algorithms depend very much on the |
---|
6200 | // characteristic of the ground field |
---|
6201 | int n=nvars(basering); // n is the number of variables, as well |
---|
6202 | // as the size of the matrices, as well |
---|
6203 | // as the number of primary invariants, |
---|
6204 | // we should get |
---|
6205 | int gen_num; |
---|
6206 | int mol_flag, v; |
---|
6207 | //------------------- checking input and setting flags ----------------------- |
---|
6208 | if (typeof(#[size(#)])=="intvec" && typeof(#[size(#)-1])=="int") |
---|
6209 | { if (size(#[size(#)])<>3) |
---|
6210 | { "ERROR: <intvec> should have three entries."; |
---|
6211 | return(); |
---|
6212 | } |
---|
6213 | gen_num=size(#)-2; |
---|
6214 | mol_flag=#[size(#)][1]; |
---|
6215 | if (#[size(#)][2]<0 && (ch==0 or (ch<>0 && mol_flag<>0))) |
---|
6216 | { "ERROR: the second component of <intvec> should be >=0"; |
---|
6217 | return(); |
---|
6218 | } |
---|
6219 | int interval=#[size(#)][2]; |
---|
6220 | v=#[size(#)][3]; |
---|
6221 | int max=#[size(#)-1]; |
---|
6222 | if (gen_num==0) |
---|
6223 | { "ERROR: There are no generators of a finite matrix group given."; |
---|
6224 | return(); |
---|
6225 | } |
---|
6226 | } |
---|
6227 | else |
---|
6228 | { if (typeof(#[size(#)])=="int") |
---|
6229 | { gen_num=size(#)-1; |
---|
6230 | mol_flag=0; |
---|
6231 | int interval=0; |
---|
6232 | v=0; |
---|
6233 | int max=#[size(#)]; |
---|
6234 | } |
---|
6235 | else |
---|
6236 | { "ERROR: If the two last parameters are not <int> and <intvec>, the last"; |
---|
6237 | " parameter should be an <int>."; |
---|
6238 | return(); |
---|
6239 | } |
---|
6240 | } |
---|
6241 | for (int i=1;i<=gen_num;i++) |
---|
6242 | { if (typeof(#[i])=="matrix") |
---|
6243 | { if (nrows(#[i])<>n or ncols(#[i])<>n) |
---|
6244 | { "ERROR: The number of variables of the base ring needs to be the same"; |
---|
6245 | " as the dimension of the square matrices"; |
---|
6246 | return(); |
---|
6247 | } |
---|
6248 | } |
---|
6249 | else |
---|
6250 | { "ERROR: The first parameters should be a list of matrices"; |
---|
6251 | return(); |
---|
6252 | } |
---|
6253 | } |
---|
6254 | //---------------------------------------------------------------------------- |
---|
6255 | if (mol_flag==0) |
---|
6256 | { if (ch==0) |
---|
6257 | { matrix REY,M=reynolds_molien(#[1..gen_num],intvec(0,interval,v)); // one |
---|
6258 | // will contain Reynolds operator and the |
---|
6259 | // other enumerator and denominator of |
---|
6260 | // Molien series |
---|
6261 | matrix P=primary_char0_random(REY,M,max,v); |
---|
6262 | matrix S,IS=secondary_char0(P,REY,M,v); |
---|
6263 | return(P,S,IS); |
---|
6264 | } |
---|
6265 | else |
---|
6266 | { list L=group_reynolds(#[1..gen_num],v); |
---|
6267 | if (L[1]<>0) // testing whether we are in the modular |
---|
6268 | { string newring="aksldfalkdsflkj"; // case |
---|
6269 | if (minpoly==0) |
---|
6270 | { if (v) |
---|
6271 | { " We are dealing with the non-modular case."; |
---|
6272 | } |
---|
6273 | if (typeof(L[2])=="int") |
---|
6274 | { molien(L[3..size(L)],newring,L[2],intvec(0,interval,v)); |
---|
6275 | } |
---|
6276 | else |
---|
6277 | { molien(L[2..size(L)],newring,intvec(0,interval,v)); |
---|
6278 | } |
---|
6279 | matrix P=primary_charp_random(L[1],newring,max,v); |
---|
6280 | matrix S,IS=secondary_charp(P,L[1],newring,v); |
---|
6281 | if (voice==2) |
---|
6282 | { kill aksldfalkdsflkj; |
---|
6283 | } |
---|
6284 | return(P,S,IS); |
---|
6285 | } |
---|
6286 | else |
---|
6287 | { if (v) |
---|
6288 | { " Since it is impossible for this programme to calculate the Molien |
---|
6289 | series for"; |
---|
6290 | " invariant rings over extension fields of prime characteristic, we |
---|
6291 | have to"; |
---|
6292 | " continue without it."; |
---|
6293 | ""; |
---|
6294 | |
---|
6295 | } |
---|
6296 | list l=primary_charp_no_molien_random(L[1],max,v); |
---|
6297 | if (size(l)==2) |
---|
6298 | { matrix S=secondary_no_molien(l[1],L[1],l[2],v); |
---|
6299 | } |
---|
6300 | else |
---|
6301 | { matrix S=secondary_no_molien(l[1],L[1],v); |
---|
6302 | } |
---|
6303 | return(l[1],S); |
---|
6304 | } |
---|
6305 | } |
---|
6306 | else // the modular case |
---|
6307 | { if (v) |
---|
6308 | { " There is also no Molien series, we can make use of..."; |
---|
6309 | ""; |
---|
6310 | " We can start looking for primary invariants..."; |
---|
6311 | ""; |
---|
6312 | } |
---|
6313 | matrix P=primary_charp_without_random(#[1..gen_num],max,v); |
---|
6314 | matrix S=secondary_not_cohen_macaulay(P,#[1..gen_num],v); |
---|
6315 | return(P,S); |
---|
6316 | } |
---|
6317 | } |
---|
6318 | } |
---|
6319 | if (mol_flag==1) // the user wants no calculation of the |
---|
6320 | { list L=group_reynolds(#[1..gen_num],v); // Molien series |
---|
6321 | if (ch==0) |
---|
6322 | { list l=primary_char0_no_molien_random(L[1],max,v); |
---|
6323 | if (size(l)==2) |
---|
6324 | { matrix S=secondary_no_molien(l[1],L[1],l[2],v); |
---|
6325 | } |
---|
6326 | else |
---|
6327 | { matrix S=secondary_no_molien(l[1],L[1],v); |
---|
6328 | } |
---|
6329 | return(l[1],S); |
---|
6330 | } |
---|
6331 | else |
---|
6332 | { if (L[1]<>0) // testing whether we are in the modular |
---|
6333 | { list l=primary_charp_no_molien_random(L[1],max,v); // case |
---|
6334 | if (size(l)==2) |
---|
6335 | { matrix S=secondary_no_molien(l[1],L[1],l[2],v); |
---|
6336 | } |
---|
6337 | else |
---|
6338 | { matrix S=secondary_no_molien(l[1],L[1],v); |
---|
6339 | } |
---|
6340 | return(l[1],S); |
---|
6341 | } |
---|
6342 | else // the modular case |
---|
6343 | { if (v) |
---|
6344 | { " We can start looking for primary invariants..."; |
---|
6345 | ""; |
---|
6346 | } |
---|
6347 | matrix P=primary_charp_without_random(#[1..gen_num],max,v); |
---|
6348 | matrix S=secondary_not_cohen_macaulay(P,#[1..gen_num],v); |
---|
6349 | return(L[1],S); |
---|
6350 | } |
---|
6351 | } |
---|
6352 | } |
---|
6353 | if (mol_flag==-1) |
---|
6354 | { if (ch==0) |
---|
6355 | { "ERROR: Characteristic 0 can only be simulated in characteristic p>>0. |
---|
6356 | "; |
---|
6357 | return(); |
---|
6358 | } |
---|
6359 | list L=group_reynolds(#[1..gen_num],v); |
---|
6360 | string newring="aksldfalkdsflkj"; |
---|
6361 | if (typeof(L[2])=="int") |
---|
6362 | { molien(L[3..size(L)],newring,L[2],intvec(mol_flag,interval,v)); |
---|
6363 | } |
---|
6364 | else |
---|
6365 | { molien(L[2..size(L)],newring,intvec(mol_flag,interval,v)); |
---|
6366 | } |
---|
6367 | matrix P=primary_charp_random(L[1],newring,max,v); |
---|
6368 | matrix S,IS=secondary_charp(P,L[1],newring,v); |
---|
6369 | kill aksldfalkdsflkj; |
---|
6370 | return(P,S,IS); |
---|
6371 | } |
---|
6372 | else // the user specified that the |
---|
6373 | { if (ch==0) // characteristic divides the group order |
---|
6374 | { "ERROR: The characteristic cannot divide the group order when it is 0. |
---|
6375 | "; |
---|
6376 | return(); |
---|
6377 | } |
---|
6378 | if (v) |
---|
6379 | { ""; |
---|
6380 | } |
---|
6381 | matrix P=primary_charp_without_random(#[1..gen_num],max,v); |
---|
6382 | matrix S=secondary_not_cohen_macaulay(P,#[1..gen_num],v); |
---|
6383 | return(L[1],S); |
---|
6384 | } |
---|
6385 | } |
---|
6386 | example |
---|
6387 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
6388 | ring R=0,(x,y,z),dp; |
---|
6389 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
6390 | matrix P,S,IS=invariant_ring_random(A,1); |
---|
6391 | print(P); |
---|
6392 | print(S); |
---|
6393 | print(IS); |
---|
6394 | } |
---|
6395 | /////////////////////////////////////////////////////////////////////////////// |
---|
6396 | |
---|
6397 | proc orbit_variety (matrix F,string newring) |
---|
6398 | "USAGE: orbit_variety(F,s); |
---|
6399 | F: a 1xm <matrix> defing an invariant ring, s: a <string> giving the |
---|
6400 | name for a new ring |
---|
6401 | RETURN: a Groebner basis (type <ideal>, named G) for the ideal defining the |
---|
6402 | orbit variety (i.e. the syzygy ideal) in the new ring (named `s`) |
---|
6403 | THEORY: The ideal of algebraic relations of the invariant ring generators is |
---|
6404 | calculated, then the variables of the original ring are eliminated and |
---|
6405 | the polynomials that are left over define the orbit variety |
---|
6406 | EXAMPLE: example orbit_variety; shows an example |
---|
6407 | " |
---|
6408 | { if (newring=="") |
---|
6409 | { "ERROR: the second parameter may not be an empty <string>"; |
---|
6410 | return(); |
---|
6411 | } |
---|
6412 | if (nrows(F)==1) |
---|
6413 | { def br=basering; |
---|
6414 | int n=nvars(br); |
---|
6415 | int m=ncols(F); |
---|
6416 | string mp=string(minpoly); |
---|
6417 | execute("ring R=("+charstr(br)+"),("+varstr(br)+",y(1..m)),dp;"); |
---|
6418 | execute("minpoly=number("+mp+");"); |
---|
6419 | ideal I=ideal(imap(br,F)); |
---|
6420 | for (int i=1;i<=m;i++) |
---|
6421 | { I[i]=I[i]-y(i); |
---|
6422 | } |
---|
6423 | I=elim(I,1,n); |
---|
6424 | execute("ring "+newring+"=("+charstr(br)+"),(y(1..m)),dp(m);"); |
---|
6425 | execute("minpoly=number("+mp+");"); |
---|
6426 | ideal vars; |
---|
6427 | for (i=2;i<=n;i++) |
---|
6428 | { vars[i]=0; |
---|
6429 | } |
---|
6430 | vars=vars,y(1..m); |
---|
6431 | map emb=R,vars; |
---|
6432 | ideal G=emb(I); |
---|
6433 | kill emb, vars, R; |
---|
6434 | keepring `newring`; |
---|
6435 | return(); |
---|
6436 | } |
---|
6437 | else |
---|
6438 | { "ERROR: the <matrix> may only have one row"; |
---|
6439 | return(); |
---|
6440 | } |
---|
6441 | } |
---|
6442 | example |
---|
6443 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
6444 | ring R=0,(x,y,z),dp; |
---|
6445 | matrix F[1][7]=x2+y2,z2,x4+y4,1,x2z-1y2z,xyz,x3y-1xy3; |
---|
6446 | string newring="E"; |
---|
6447 | orbit_variety(F,newring); |
---|
6448 | print(G); |
---|
6449 | basering; |
---|
6450 | } |
---|
6451 | /////////////////////////////////////////////////////////////////////////////// |
---|
6452 | |
---|
6453 | static proc degvec(ideal I) |
---|
6454 | "USAGE: degvec(I); |
---|
6455 | I an <ideal>. |
---|
6456 | RETURN: the <intvec> of degrees of the generators of I. |
---|
6457 | " |
---|
6458 | { intvec v; |
---|
6459 | for (int j = 1;j<=ncols(I);j++) |
---|
6460 | { v[j]=deg(I[j]); |
---|
6461 | } |
---|
6462 | return(v); |
---|
6463 | } |
---|
6464 | |
---|
6465 | /////////////////////////////////////////////////////////////////////////////// |
---|
6466 | |
---|
6467 | proc rel_orbit_variety(ideal I,matrix F,list #) |
---|
6468 | "USAGE: rel_orbit_variety(I,F[,s]); |
---|
6469 | @* I: an <ideal> invariant under the action of a group, |
---|
6470 | @* F: a 1xm <matrix> defining the invariant ring of this group. |
---|
6471 | @* s: optional <string>; if s is present then (for downward |
---|
6472 | compatibility) the old procedure <relative_orbit_variety> |
---|
6473 | is called, and in this case s gives the name of a new <ring>. |
---|
6474 | RETURN: Without optional s, a list L of two rings is returned. |
---|
6475 | @* The ring L[1] carries a weighted degree order with variables |
---|
6476 | y(1..m), the weight of y(k) equal to the degree of the |
---|
6477 | k-th generators F[1,k] of the invariant ring. |
---|
6478 | L[1] contains a Groebner basis (type <ideal>, named G) of the |
---|
6479 | ideal defining the relative orbit variety with respect to I. |
---|
6480 | @* The ring L[2] has the variables of the basering together with y(1..m) |
---|
6481 | and carries a block order: The first block is the order of the |
---|
6482 | basering, the second is the weighted degree order occuring in L[1]. |
---|
6483 | L[2] contains G and a Groebner basis (type <ideal>, named Conv) |
---|
6484 | such that if p is any invariant polynomial expressed in the |
---|
6485 | variables of the basering then reduce(p,Conv) is a polynomial in |
---|
6486 | the new variables y(1..m) such that evaluation at the generators |
---|
6487 | of the invariant ring yields p. This can be used to avoid the |
---|
6488 | application of <algebra_containment> |
---|
6489 | (see @ref{algebra_containment}). |
---|
6490 | @* For the case of optional s, see @ref{relative_orbit_variety}. |
---|
6491 | THEORY: A Groebner basis of the ideal of algebraic relations of the invariant |
---|
6492 | ring generators is calculated, then one of the basis elements plus |
---|
6493 | the ideal generators. The variables of the original ring are |
---|
6494 | eliminated and the polynomials that are left define the relative |
---|
6495 | orbit variety with respect to I. The elimination is done by a |
---|
6496 | weighted blockorder that has the advantage of dealing with |
---|
6497 | quasi-homogeneous ideals. |
---|
6498 | NOTE: We provide the ring L[1] for the sake of downward compatibility, |
---|
6499 | since it is closer to the ring returned by relative_orbit_variety |
---|
6500 | than L[2]. However, L[1] carries a weighted degree order, whereas |
---|
6501 | the ring returned by relative_orbit_variety is lexicographically |
---|
6502 | ordered. |
---|
6503 | SEE ALSO: relative_orbit_variety |
---|
6504 | EXAMPLE: example rel_orbit_variety; shows an example. |
---|
6505 | " |
---|
6506 | { if (size(#)>0) |
---|
6507 | { if (typeof(#[1])=="string") |
---|
6508 | { relative_orbit_variety(I,F,#[1]); |
---|
6509 | keepring basering; |
---|
6510 | return(); |
---|
6511 | } |
---|
6512 | else |
---|
6513 | { "ERROR: the third parameter may either be empty or a <string>."; |
---|
6514 | return(); |
---|
6515 | } |
---|
6516 | } |
---|
6517 | degBound=0; |
---|
6518 | if (nrows(F)==1) |
---|
6519 | { option(redSB,noredTail); |
---|
6520 | def br=basering; |
---|
6521 | int n=nvars(br); |
---|
6522 | int m=ncols(F); |
---|
6523 | // In the following ring definition, any elimination order would work. |
---|
6524 | list rlist = ringlist(br); |
---|
6525 | int i; |
---|
6526 | for (i=1;i<=m;i++) |
---|
6527 | { rlist[2][n+i]="y("+string(i)+")"; |
---|
6528 | } |
---|
6529 | rlist[3][size(rlist[3])+1] = rlist[3][size(rlist[3])]; |
---|
6530 | rlist[3][size(rlist[3])-1][1] = "wp"; |
---|
6531 | rlist[3][size(rlist[3])-1][2] = degvec(ideal(F)); |
---|
6532 | def newring = ring(rlist); |
---|
6533 | |
---|
6534 | list smallrlist; |
---|
6535 | smallrlist[1]=rlist[1]; |
---|
6536 | smallrlist[2]=list(); |
---|
6537 | for (i=1;i<=m;i++) |
---|
6538 | { smallrlist[2][i]="y("+string(i)+")"; |
---|
6539 | } |
---|
6540 | smallrlist[3]=list(); |
---|
6541 | smallrlist[3][1]=list(); |
---|
6542 | smallrlist[3][1][1] = "wp"; |
---|
6543 | smallrlist[3][1][2] = degvec(ideal(F)); |
---|
6544 | smallrlist[3][2] = rlist[3][size(rlist[3])]; |
---|
6545 | smallrlist[4] = rlist[4]; |
---|
6546 | def smallring = ring(smallrlist); |
---|
6547 | |
---|
6548 | setring(newring); |
---|
6549 | ideal J=ideal(imap(br,F)); |
---|
6550 | ideal I=imap(br,I); |
---|
6551 | for (i=1;i<=m;i++) |
---|
6552 | { J[i]=J[i]-y(i); |
---|
6553 | } |
---|
6554 | // We chose the weighted block order since this makes J quasi-homogeneous! |
---|
6555 | ideal Conv=groebner(J); |
---|
6556 | J=Conv,I; |
---|
6557 | J=groebner(J); |
---|
6558 | ideal vars; |
---|
6559 | vars[n]=0; |
---|
6560 | vars=vars,y(1..m); |
---|
6561 | map emb=newring,vars; |
---|
6562 | ideal G=emb(J); |
---|
6563 | J=J-G; |
---|
6564 | for (i=1;i<=ncols(G);i++) |
---|
6565 | { if (J[i]<>0) |
---|
6566 | { G[i]=0; |
---|
6567 | } |
---|
6568 | } |
---|
6569 | G=compress(G); |
---|
6570 | vars=ideal(); |
---|
6571 | for (i=2;i<=n;i++) |
---|
6572 | { vars[i]=0; |
---|
6573 | } |
---|
6574 | vars=vars,y(1..m); |
---|
6575 | emb=newring,vars; |
---|
6576 | G=compress(emb(G)); |
---|
6577 | export(G); |
---|
6578 | export(Conv); |
---|
6579 | |
---|
6580 | setring(smallring); |
---|
6581 | ideal G = compress(imap(newring,G)); |
---|
6582 | export(G); |
---|
6583 | |
---|
6584 | list L; |
---|
6585 | L[1]=smallring; |
---|
6586 | L[2]=newring; |
---|
6587 | |
---|
6588 | dbprint( printlevel-voice+3," |
---|
6589 | // 'rel_orbit_variety' created a list of two rings. |
---|
6590 | // If L is the name of that list, you can access |
---|
6591 | // the first ring by |
---|
6592 | // def R = L[1]; setring R; |
---|
6593 | // (similarly for the second ring)"); |
---|
6594 | return(L); |
---|
6595 | } |
---|
6596 | else |
---|
6597 | { "ERROR: the <matrix> may only have one row"; |
---|
6598 | return(); |
---|
6599 | } |
---|
6600 | } |
---|
6601 | example |
---|
6602 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.6.3:"; echo=2; |
---|
6603 | ring R=0,(x,y,z),dp; |
---|
6604 | matrix F[1][3]=x+y+z,xy+xz+yz,xyz; |
---|
6605 | ideal I=x2+y2+z2-1,x2y+y2z+z2x-2x-2y-2z,xy2+yz2+zx2-2x-2y-2z; |
---|
6606 | list L = rel_orbit_variety(I,F); |
---|
6607 | def AllR = L[2]; |
---|
6608 | setring(AllR); |
---|
6609 | print(G); |
---|
6610 | print(Conv); |
---|
6611 | basering; |
---|
6612 | } |
---|
6613 | /////////////////////////////////////////////////////////////////////////////// |
---|
6614 | |
---|
6615 | proc relative_orbit_variety(ideal I,matrix F,string newring) |
---|
6616 | " |
---|
6617 | USAGE: relative_orbit_variety(I,F,s); |
---|
6618 | I: an <ideal> invariant under the action of a group, |
---|
6619 | @* F: a 1xm <matrix> defining the invariant ring of this group, |
---|
6620 | @* s: a <string> giving a name for a new ring |
---|
6621 | RETURN: The procedure ends with a new ring named s. |
---|
6622 | It contains a Groebner basis |
---|
6623 | (type <ideal>, named G) for the ideal defining the |
---|
6624 | relative orbit variety with respect to I in the new ring. |
---|
6625 | THEORY: A Groebner basis of the ideal of algebraic relations of the invariant |
---|
6626 | ring generators is calculated, then one of the basis elements plus the |
---|
6627 | ideal generators. The variables of the original ring are eliminated |
---|
6628 | and the polynomials that are left define the relative orbit variety |
---|
6629 | with respect to I. |
---|
6630 | NOTE: This procedure is now replaced by rel_orbit_variety |
---|
6631 | (see @ref{rel_orbit_variety}), which uses a different elemination |
---|
6632 | order that should usually allow faster computations. |
---|
6633 | SEE ALSO: rel_orbit_variety |
---|
6634 | EXAMPLE: example relative_orbit_variety; shows an example |
---|
6635 | " |
---|
6636 | { if (newring=="") |
---|
6637 | { "ERROR: the third parameter may not be empty a <string>"; |
---|
6638 | return(); |
---|
6639 | } |
---|
6640 | degBound=0; |
---|
6641 | if (nrows(F)==1) |
---|
6642 | { def br=basering; |
---|
6643 | int n=nvars(br); |
---|
6644 | int m=ncols(F); |
---|
6645 | string mp=string(minpoly); |
---|
6646 | execute("ring R=("+charstr(br)+"),("+varstr(br)+",y(1..m)),lp;"); |
---|
6647 | execute("minpoly=number("+mp+");"); |
---|
6648 | ideal J=ideal(imap(br,F)); |
---|
6649 | ideal I=imap(br,I); |
---|
6650 | for (int i=1;i<=m;i++) |
---|
6651 | { J[i]=J[i]-y(i); |
---|
6652 | } |
---|
6653 | J=std(J); |
---|
6654 | J=J,I; |
---|
6655 | option(redSB); |
---|
6656 | J=std(J); |
---|
6657 | ideal vars; |
---|
6658 | //for (i=1;i<=n;i=i+1) |
---|
6659 | //{ vars[i]=0; |
---|
6660 | //} |
---|
6661 | vars[n]=0; |
---|
6662 | vars=vars,y(1..m); |
---|
6663 | map emb=R,vars; |
---|
6664 | ideal G=emb(J); |
---|
6665 | J=J-G; |
---|
6666 | for (i=1;i<=ncols(G);i++) |
---|
6667 | { if (J[i]<>0) |
---|
6668 | { G[i]=0; |
---|
6669 | } |
---|
6670 | } |
---|
6671 | G=compress(G); |
---|
6672 | execute("ring "+newring+"=("+charstr(br)+"),(y(1..m)),lp;"); |
---|
6673 | execute("minpoly=number("+mp+");"); |
---|
6674 | ideal vars; |
---|
6675 | for (i=2;i<=n;i++) |
---|
6676 | { vars[i]=0; |
---|
6677 | } |
---|
6678 | vars=vars,y(1..m); |
---|
6679 | map emb=R,vars; |
---|
6680 | ideal G=emb(G); |
---|
6681 | kill vars, emb; |
---|
6682 | keepring `newring`; |
---|
6683 | return(); |
---|
6684 | } |
---|
6685 | else |
---|
6686 | { "ERROR: the <matrix> may only have one row"; |
---|
6687 | return(); |
---|
6688 | } |
---|
6689 | } |
---|
6690 | |
---|
6691 | example |
---|
6692 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.6.3:"; echo=2; |
---|
6693 | ring R=0,(x,y,z),dp; |
---|
6694 | matrix F[1][3]=x+y+z,xy+xz+yz,xyz; |
---|
6695 | ideal I=x2+y2+z2-1,x2y+y2z+z2x-2x-2y-2z,xy2+yz2+zx2-2x-2y-2z; |
---|
6696 | string newring="E"; |
---|
6697 | relative_orbit_variety(I,F,newring); |
---|
6698 | print(G); |
---|
6699 | basering; |
---|
6700 | } |
---|
6701 | |
---|
6702 | /////////////////////////////////////////////////////////////////////////////// |
---|
6703 | |
---|
6704 | proc image_of_variety(ideal I,matrix F) |
---|
6705 | "USAGE: image_of_variety(I,F); |
---|
6706 | @* I: an arbitray <ideal>, |
---|
6707 | @* F: a 1xm <matrix> defining an invariant ring of some matrix group |
---|
6708 | RETURN: The <ideal> defining the image under that group of the variety defined |
---|
6709 | by I |
---|
6710 | THEORY: rel_orbit_variety(I,F) is called and the newly introduced |
---|
6711 | @* variables in the output are replaced by the generators of the |
---|
6712 | @* invariant ring. This ideal in the original variables defines the image |
---|
6713 | @* of the variety defined by I |
---|
6714 | EXAMPLE: example image_of_variety; shows an example |
---|
6715 | " |
---|
6716 | { if (nrows(F)==1) |
---|
6717 | { def br=basering; |
---|
6718 | int n=nvars(br); |
---|
6719 | list L = rel_orbit_variety(I,F); |
---|
6720 | def newring=L[2]; |
---|
6721 | setring(newring); |
---|
6722 | ideal F=imap(br,F); |
---|
6723 | for (int i=1;i<=n;i++) |
---|
6724 | { F=0,F; |
---|
6725 | } |
---|
6726 | map emb2=newring,F; |
---|
6727 | ideal trafo = compress(emb2(G)); |
---|
6728 | setring br; |
---|
6729 | return(imap(newring,trafo)); |
---|
6730 | } |
---|
6731 | else |
---|
6732 | { "ERROR: the <matrix> may only have one row"; |
---|
6733 | return(); |
---|
6734 | } |
---|
6735 | } |
---|
6736 | example |
---|
6737 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.6.8:"; echo=2; |
---|
6738 | ring R=0,(x,y,z),dp; |
---|
6739 | matrix F[1][3]=x+y+z,xy+xz+yz,xyz; |
---|
6740 | ideal I=xy; |
---|
6741 | print(image_of_variety(I,F)); |
---|
6742 | } |
---|
6743 | /////////////////////////////////////////////////////////////////////////////// |
---|
6744 | |
---|
6745 | |
---|
6746 | |
---|
6747 | |
---|