[5811fb] | 1 | // $Id: finvar.lib,v 1.15 1998-12-02 10:52:08 Singular Exp $ |
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[29aa4bf] | 2 | // author: Agnes Eileen Heydtmann, email:agnes@math.uni-sb.de |
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[5811fb] | 3 | // last change: 98/11/05 |
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[29aa4bf] | 4 | ////////////////////////////////////////////////////////////////////////////// |
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[5811fb] | 5 | version="$Id: finvar.lib,v 1.15 1998-12-02 10:52:08 Singular Exp $" |
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[29aa4bf] | 6 | info=" |
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| 7 | LIBRARY: finvar.lib LIBRARY TO CALCULATE INVARIANT RINGS & MORE |
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| 8 | |
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[5811fb] | 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 Decker et al.. |
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| 12 | Author: Agnes E. Heydtmann, agnes@math.uni-sb.de |
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| 13 | |
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| 14 | MAIN PROCEDURES: |
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| 15 | invariant_ring() generators of the invariant ring (i.r.) |
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| 16 | invariant_ring_random() generators of the i.r., randomized alg. |
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| 17 | primary_invariants() primary invariants (p.i.) |
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| 18 | primary_invariants_random() primary invariants, randomized alg. |
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[29aa4bf] | 19 | |
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[5811fb] | 20 | SUB-PROCEDURES: |
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| 21 | cyclotomic() cyclotomic polynomial |
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| 22 | group_reynolds() finite group and Reynolds operator (R.o.) |
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| 23 | molien() Molien series (M.s.) |
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| 24 | reynolds_molien() Reynolds operator and Molien series |
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| 25 | partial_molien() partial expansion of Molien series |
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| 26 | evaluate_reynolds() image under the Reynolds operator |
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| 27 | invariant_basis() basis of homogeneous invariants of a degree |
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| 28 | invariant_basis_reynolds() as invariant_basis(), with R.o. |
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| 29 | primary_char0() primary invariants in char 0 |
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| 30 | primary_charp() primary invariant in char p |
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| 31 | primary_char0_no_molien() p.i., char 0, without Molien series |
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| 32 | primary_charp_no_molien() p.i., char p, without Molien series |
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| 33 | primary_charp_without() p.i., char p, without R.o. or Molien series |
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| 34 | primary_char0_random() primary invariants in char 0, randomized |
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| 35 | primary_charp_random() primary invariants in char p, randomized |
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| 36 | primary_char0_no_molien_random() p.i., char 0, without M.s., randomized |
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| 37 | primary_charp_no_molien_random() p.i., char p, without M.s., randomized |
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| 38 | primary_charp_without_random() p.i., char p, without R.o. or M.s., random. |
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| 39 | power_products() exponents for power products |
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| 40 | secondary_char0() secondary (s.i.) 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 | algebra_containment() query of algebra containment |
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| 46 | module_containment() query of module containment |
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| 47 | orbit_variety() ideal of the orbit variety |
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| 48 | relative_orbit_variety() ideal of a relative orbit variety |
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| 49 | image_of_variety() ideal of the image of a variety |
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| 50 | "; |
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| 51 | //////////////////////////////////////////////////////////////////////////////// |
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| 52 | // perhaps useful procedures (no help provided): |
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| 53 | // unique() is a matrix among other matrices? |
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| 54 | // exponent() gives the exponent of a number |
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| 55 | // sort_of_invariant_basis() lin. ind. invariants of a degree mod p.i. |
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| 56 | // next_vector lists all of Z^n with first nonzero entry 1 |
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| 57 | // int_number_map integers 1..q are maped to q field elements |
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| 58 | // search searches a number of p.i., char 0 |
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| 59 | // p_search searches a number of p.i., char p |
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| 60 | // search_random searches a # of p.i., char 0, randomized |
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| 61 | // p_search_random searches a # of p.i., char p, randomized |
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| 62 | // concat_intmat concatenates two integer matrices |
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[29aa4bf] | 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 | //////////////////////////////////////////////////////////////////////////////// |
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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 | { for (int i=1;i<size(#);i=i+1) |
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| 75 | { if (#[i]==#[size(#)]) |
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| 76 | { return(0); |
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| 77 | } |
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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 | proc cyclotomic (int i) |
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| 83 | "USAGE: cyclotomic(i); |
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| 84 | i: an <int> > 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 | EXAMPLE: example cyclotomic; shows an example |
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| 88 | THEORY: x^i-1 is divided by the j-th cyclotomic polynomial where j takes on the |
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| 89 | value of proper divisors of i |
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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=j+1; |
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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 | { 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 (used in molien); in general all other list |
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| 143 | elements are nxn <matrices> listing all elements of the finite group |
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| 144 | DISPLAY: information if v does not equal 0 |
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| 145 | EXAMPLE: example group_reynolds; shows an example |
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| 146 | THEORY: The entire matrix group is generated by getting all left products of |
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| 147 | the 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 there are no new elements found in one run. Additionally each time |
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| 151 | a new group element is found the corresponding ring mapping of which |
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| 152 | the Reynolds operator is made up is generated. They are stored in the |
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| 153 | rows of the first return value. |
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| 154 | " |
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| 155 | { int ch=char(basering); // the existance of the Reynolds operator |
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| 156 | // is dependent on the characteristic of |
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| 157 | // the base field |
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| 158 | int gen_num; // number of generators |
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| 159 | //------------------------ making sure the input is okay --------------------- |
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| 160 | if (typeof(#[size(#)])=="int") |
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| 161 | { if (size(#)==1) |
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| 162 | { "ERROR: there are no matrices given among the parameters"; |
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| 163 | return(); |
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| 164 | } |
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| 165 | int v=#[size(#)]; |
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| 166 | gen_num=size(#)-1; |
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| 167 | } |
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| 168 | else // last parameter is not <int> |
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| 169 | { int v=0; // no information is default |
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| 170 | gen_num=size(#); |
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| 171 | } |
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| 172 | if (typeof(#[1])<>"matrix") |
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| 173 | { "ERROR: the parameters must be a list of matrices and maybe an <int>"; |
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| 174 | return(); |
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| 175 | } |
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| 176 | int n=nrows(#[1]); |
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| 177 | if (n<>nvars(basering)) |
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| 178 | { "ERROR: the number of variables of the basering needs to be the same"; |
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| 179 | " as the dimension of the matrices"; |
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| 180 | return(); |
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| 181 | } |
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| 182 | if (n<>ncols(#[1])) |
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| 183 | { "ERROR: matrices need to be square and of the same dimensions"; |
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| 184 | return(); |
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| 185 | } |
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| 186 | matrix vars=matrix(maxideal(1)); // creating an nx1-matrix containing the |
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| 187 | vars=transpose(vars); // variables of the ring - |
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| 188 | matrix REY=#[1]*vars; // calculating the first ring mapping - |
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| 189 | // REY will contain the Reynolds |
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| 190 | // operator - |
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| 191 | matrix G(1)=#[1]; // G(k) are elements of the group - |
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| 192 | if (ch<>0 && minpoly==0 && gen_num<>1) // finding out of which order the group |
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| 193 | { matrix I=diag(1,n); // element is |
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| 194 | matrix TEST=G(1); |
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| 195 | int o1=1; |
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| 196 | int o2; |
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| 197 | while (TEST<>I) |
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| 198 | { TEST=TEST*G(1); |
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| 199 | o1=o1+1; |
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| 200 | } |
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| 201 | } |
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| 202 | int i=1; |
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| 203 | // -------------- doubles among the generators should be avoided ------------- |
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| 204 | for (int j=2;j<=gen_num;j=j+1) // this loop adds the parameters to the |
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| 205 | { // group, leaving out doubles and |
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| 206 | // checking whether the parameters are |
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| 207 | // compatible with the task of the |
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| 208 | // procedure |
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| 209 | if (not(typeof(#[j])=="matrix")) |
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| 210 | { "ERROR: the parameters must be a list of matrices and maybe an <int>"; |
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| 211 | return(); |
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| 212 | } |
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| 213 | if ((n!=nrows(#[j])) or (n!=ncols(#[j]))) |
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| 214 | { "ERROR: matrices need to be square and of the same dimensions"; |
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| 215 | return(); |
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| 216 | } |
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| 217 | if (unique(G(1..i),#[j])) |
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| 218 | { i=i+1; |
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| 219 | matrix G(i)=#[j]; |
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| 220 | if (ch<>0 && minpoly==0) // finding out of which order the group |
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| 221 | { TEST=G(i); // element is |
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| 222 | o2=1; |
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| 223 | while (TEST<>I) |
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| 224 | { TEST=TEST*G(i); |
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| 225 | o2=o2+1; |
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| 226 | } |
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| 227 | o1=o1*o2/gcd(o1,o2); // lowest common multiple of the element |
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| 228 | } // orders - |
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| 229 | REY=concat(REY,#[j]*vars); // adding ring homomorphisms to REY |
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| 230 | } |
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| 231 | } |
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| 232 | int g=i; // G(1)..G(i) are generators without |
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| 233 | // doubles - g generally is the number |
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| 234 | // of elements in the group so far - |
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| 235 | j=i; // j is the number of new elements that |
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| 236 | // we use as factors |
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| 237 | int k, m, l; |
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| 238 | if (v) |
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| 239 | { ""; |
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| 240 | " Generating the entire matrix group and the Reynolds operator..."; |
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| 241 | ""; |
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| 242 | } |
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| 243 | // -------------- main loop that finds all the group elements ---------------- |
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| 244 | while (1) |
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| 245 | { l=0; // l is the number of products we get in |
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| 246 | // one going |
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| 247 | for (m=g-j+1;m<=g;m=m+1) |
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| 248 | { for (k=1;k<=i;k=k+1) |
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| 249 | { l=l+1; |
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| 250 | matrix P(l)=G(k)*G(m); // possible new element |
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| 251 | } |
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| 252 | } |
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| 253 | j=0; |
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| 254 | for (k=1;k<=l;k=k+1) |
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| 255 | { if (unique(G(1..g),P(k))) |
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| 256 | { j=j+1; // a new factor for next run |
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| 257 | g=g+1; |
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| 258 | matrix G(g)=P(k); // a new group element - |
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| 259 | if (ch<>0 && minpoly==0 && i<>1) // finding out of which order the group |
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| 260 | { TEST=G(g); // element is |
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| 261 | o2=1; |
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| 262 | while (TEST<>I) |
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| 263 | { TEST=TEST*G(g); |
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| 264 | o2=o2+1; |
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| 265 | } |
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| 266 | o1=o1*o2/gcd(o1,o2); // lowest common multiple of the element |
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| 267 | } // orders - |
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| 268 | REY=concat(REY,P(k)*vars); // adding new mapping to REY |
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| 269 | if (v) |
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| 270 | { " Group element "+string(g)+" has been found."; |
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| 271 | } |
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| 272 | } |
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| 273 | kill P(k); |
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| 274 | } |
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| 275 | if (j==0) // when we didn't add any new elements |
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| 276 | { break; // in one run through the while loop |
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| 277 | } // we are done |
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| 278 | } |
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| 279 | if (v) |
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| 280 | { if (g<=i) |
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| 281 | { " There are only "+string(g)+" group elements."; |
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| 282 | } |
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| 283 | ""; |
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| 284 | } |
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| 285 | REY=transpose(REY); // when we evaluate the Reynolds operator |
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| 286 | // later on, we actually want 1xn |
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| 287 | // matrices |
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| 288 | if (ch<>0) |
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| 289 | { if ((g%ch)==0) |
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| 290 | { if (voice==2) |
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| 291 | { "WARNING: The characteristic of the coefficient field divides the group order."; |
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| 292 | " Proceed without the Reynolds operator!"; |
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| 293 | } |
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| 294 | else |
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| 295 | { if (v) |
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| 296 | { " The characteristic of the base field divides the group order."; |
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| 297 | " We have to continue without Reynolds operator..."; |
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| 298 | ""; |
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| 299 | } |
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| 300 | } |
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| 301 | kill REY; |
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| 302 | matrix REY[1][1]=0; |
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| 303 | return(REY,G(1..g)); |
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| 304 | } |
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| 305 | if (minpoly==0) |
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| 306 | { if (i>1) |
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| 307 | { return(REY,o1,G(1..g)); |
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| 308 | } |
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| 309 | return(REY,G(1..g)); |
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| 310 | } |
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| 311 | } |
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| 312 | if (v) |
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| 313 | { " Done generating the group and the Reynolds operator."; |
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| 314 | ""; |
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| 315 | } |
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| 316 | return(REY,G(1..g)); |
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| 317 | } |
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| 318 | example |
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| 319 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
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| 320 | 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=i+1; |
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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 | proc molien (list #) |
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| 340 | "USAGE: molien(G1,G2,...[,ringname,lcm,flags]); |
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| 341 | G1,G2,...: nxn <matrices> generating a finite matrix group, ringname: |
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| 342 | a <string> giving a name for a new ring of characteristic 0 for the |
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| 343 | Molien series in case of prime characteristic, lcm: an <int> giving the |
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| 344 | lowest common multiple of the elements' orders in case of prime |
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| 345 | characteristic, minpoly==0 and a non-cyclic group, flags: an optional |
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| 346 | <intvec> with three components: if the first element is not equal to 0 |
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| 347 | characteristic 0 is simulated, i.e. the Molien series is computed as |
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| 348 | if the base field were characteristic 0 (the user must choose a field |
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| 349 | of large prime characteristic, e.g. 32003), the second component should |
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| 350 | give the size of intervals between canceling common factors in the |
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| 351 | expansion of the Molien series, 0 (the default) means only once after |
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| 352 | generating all terms, in prime characteristic also a negative number |
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| 353 | can be given to indicate that common factors should always be canceled |
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| 354 | when the expansion is simple (the root of the extension field does not |
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| 355 | occur among the coefficients) |
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| 356 | ASSUME: n is the number of variables of the basering, G1,G2... are the group |
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| 357 | elements generated by group_reynolds(), lcm is the second return value |
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| 358 | of group_reynolds() |
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| 359 | RETURN: in case of characteristic 0 a 1x2 <matrix> giving enumerator and |
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| 360 | denominator of Molien series; in case of prime characteristic a ring |
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| 361 | with the name `ringname` of characteristic 0 is created where the same |
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| 362 | Molien series (named M) is stored |
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| 363 | DISPLAY: information if the third component of flags does not equal 0 |
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| 364 | EXAMPLE: example molien; shows an example |
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| 365 | THEORY: In characteristic 0 the terms 1/det(1-xE) for all group elements of the |
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| 366 | 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 factors |
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| 369 | have been canceled. |
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| 370 | " |
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| 371 | { def br=basering; // the Molien series depends on the |
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| 372 | int ch=char(br); // characteristic of the coefficient |
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| 373 | // field - |
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| 374 | int g; // size of the group |
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| 375 | //---------------------- making sure the input is okay ----------------------- |
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| 376 | if (typeof(#[size(#)])=="intvec") |
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| 377 | { if (size(#[size(#)])==3) |
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| 378 | { int mol_flag=#[size(#)][1]; |
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| 379 | if (#[size(#)][2]<0 && (ch==0 or (ch<>0 && mol_flag<>0))) |
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| 380 | { "ERROR: the second component of <intvec> should be >=0" |
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| 381 | return(); |
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| 382 | } |
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| 383 | int interval=#[size(#)][2]; |
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| 384 | int v=#[size(#)][3]; |
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| 385 | } |
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| 386 | else |
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| 387 | { "ERROR: <intvec> should have three components"; |
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| 388 | return(); |
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| 389 | } |
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| 390 | if (ch<>0) |
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| 391 | { if (typeof(#[size(#)-1])=="int") |
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| 392 | { int r=#[size(#)-1]; |
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| 393 | if (typeof(#[size(#)-2])<>"string") |
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| 394 | { "ERROR: in characteristic p>0 a <string> must be given for the name of a new"; |
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| 395 | " ring where the Molien series can be stored"; |
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| 396 | return(); |
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| 397 | } |
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| 398 | else |
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| 399 | { if (#[size(#)-2]=="") |
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| 400 | { "ERROR: <string> may not be empty"; |
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| 401 | return(); |
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| 402 | } |
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| 403 | string newring=#[size(#)-2]; |
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| 404 | g=size(#)-3; |
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| 405 | } |
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| 406 | } |
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| 407 | else |
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| 408 | { if (typeof(#[size(#)-1])<>"string") |
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| 409 | { "ERROR: in characteristic p>0 a <string> must be given for the name of a new"; |
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| 410 | " ring where the Molien series can be stored"; |
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| 411 | return(); |
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| 412 | } |
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| 413 | else |
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| 414 | { if (#[size(#)-1]=="") |
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| 415 | { "ERROR: <string> may not be empty"; |
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| 416 | return(); |
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| 417 | } |
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| 418 | string newring=#[size(#)-1]; |
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| 419 | g=size(#)-2; |
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| 420 | int r=g; |
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| 421 | } |
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| 422 | } |
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| 423 | } |
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| 424 | else // then <string> ist not needed |
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| 425 | { g=size(#)-1; |
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| 426 | } |
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| 427 | } |
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| 428 | else // last parameter is not <intvec> |
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| 429 | { int v=0; // no information is default |
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| 430 | int mol_flag=0; // computing of Molien series is default |
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| 431 | int interval=0; |
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| 432 | if (ch<>0) |
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| 433 | { if (typeof(#[size(#)])=="int") |
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| 434 | { int r=#[size(#)]; |
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| 435 | if (typeof(#[size(#)-1])<>"string") |
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| 436 | { "ERROR: in characteristic p>0 a <string> must be given for the name of a new"; |
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| 437 | " ring where the Molien series can be stored"; |
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| 438 | return(); |
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| 439 | } |
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| 440 | else |
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| 441 | { if (#[size(#)-1]=="") |
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| 442 | { "ERROR: <string> may not be empty"; |
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| 443 | return(); |
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| 444 | } |
---|
| 445 | string newring=#[size(#)-1]; |
---|
| 446 | g=size(#)-2; |
---|
| 447 | } |
---|
| 448 | } |
---|
| 449 | else |
---|
| 450 | { if (typeof(#[size(#)])<>"string") |
---|
| 451 | { "ERROR: in characteristic p>0 a <string> must be given for the name of a new"; |
---|
| 452 | " ring where the Molien series can be stored"; |
---|
| 453 | return(); |
---|
| 454 | } |
---|
| 455 | else |
---|
| 456 | { if (#[size(#)]=="") |
---|
| 457 | { "ERROR: <string> may not be empty"; |
---|
| 458 | return(); |
---|
| 459 | } |
---|
| 460 | string newring=#[size(#)]; |
---|
| 461 | g=size(#)-1; |
---|
| 462 | int r=g; |
---|
| 463 | } |
---|
| 464 | } |
---|
| 465 | } |
---|
| 466 | else |
---|
| 467 | { g=size(#); |
---|
| 468 | } |
---|
| 469 | } |
---|
| 470 | if (ch<>0) |
---|
| 471 | { if ((g/r)*r<>g) |
---|
| 472 | { "ERROR: <int> should divide the group order." |
---|
| 473 | return(); |
---|
| 474 | } |
---|
| 475 | } |
---|
| 476 | if (ch<>0) |
---|
| 477 | { if ((g%ch)==0) |
---|
| 478 | { if (voice==2) |
---|
| 479 | { "WARNING: The characteristic of the coefficient field divides the group"; |
---|
| 480 | " order. Proceed without the Molien series!"; |
---|
| 481 | } |
---|
| 482 | else |
---|
| 483 | { if (v) |
---|
| 484 | { " The characteristic of the base field divides the group order."; |
---|
| 485 | " We have to continue without Molien series..."; |
---|
| 486 | ""; |
---|
| 487 | } |
---|
| 488 | } |
---|
| 489 | } |
---|
| 490 | if (minpoly<>0 && mol_flag==0) |
---|
| 491 | { if (voice==2) |
---|
| 492 | { "WARNING: It is impossible for this program to calculate the Molien series"; |
---|
| 493 | " for finite groups over extension fields of prime characteristic."; |
---|
| 494 | } |
---|
| 495 | else |
---|
| 496 | { if (v) |
---|
| 497 | { " Since it is impossible for this program to calculate the Molien series for"; |
---|
| 498 | " invariant rings over extension fields of prime characteristic, we have to"; |
---|
| 499 | " continue without it."; |
---|
| 500 | ""; |
---|
| 501 | } |
---|
| 502 | } |
---|
| 503 | return(); |
---|
| 504 | } |
---|
| 505 | } |
---|
| 506 | //---------------------------------------------------------------------------- |
---|
| 507 | if (not(typeof(#[1])=="matrix")) |
---|
| 508 | { "ERROR: the parameters must be a list of matrices and maybe an <intvec>"; |
---|
| 509 | return(); |
---|
| 510 | } |
---|
| 511 | int n=nrows(#[1]); |
---|
| 512 | if (n<>nvars(br)) |
---|
| 513 | { "ERROR: the number of variables of the basering needs to be the same"; |
---|
| 514 | " as the dimension of the square matrices"; |
---|
| 515 | return(); |
---|
| 516 | } |
---|
| 517 | if (v && voice<>2) |
---|
| 518 | { ""; |
---|
| 519 | " Generating the Molien series..."; |
---|
| 520 | ""; |
---|
| 521 | } |
---|
| 522 | if (v && voice==2) |
---|
| 523 | { ""; |
---|
| 524 | } |
---|
| 525 | //------------- calculating Molien series in characteristic 0 ---------------- |
---|
| 526 | if (ch==0) // when ch==0 we can calculate the Molien |
---|
| 527 | { matrix I=diag(1,n); // series in any case - |
---|
| 528 | poly v1=maxideal(1)[1]; // the Molien series will be in terms of |
---|
| 529 | // the first variable of the current |
---|
| 530 | // ring - |
---|
| 531 | matrix M[1][2]; // M will contain the Molien series - |
---|
| 532 | M[1,1]=0; // M[1,1] will be the numerator - |
---|
| 533 | M[1,2]=1; // M[1,2] will be the denominator - |
---|
| 534 | matrix s; // will help us canceling in the |
---|
| 535 | // fraction |
---|
| 536 | poly p; // will contain the denominator of the |
---|
| 537 | // new term of the Molien series |
---|
| 538 | //------------ computing 1/det(1+xE) for all E in the group ------------------ |
---|
| 539 | for (int j=1;j<=g;j=j+1) |
---|
| 540 | { if (not(typeof(#[j])=="matrix")) |
---|
| 541 | { "ERROR: the parameters must be a list of matrices and maybe an <intvec>"; |
---|
| 542 | return(); |
---|
| 543 | } |
---|
| 544 | if ((n<>nrows(#[j])) or (n<>ncols(#[j]))) |
---|
| 545 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
| 546 | return(); |
---|
| 547 | } |
---|
| 548 | p=det(I-v1*#[j]); // denominator of new term - |
---|
| 549 | M[1,1]=M[1,1]*p+M[1,2]; // expanding M[1,1]/M[1,2] + 1/p |
---|
| 550 | M[1,2]=M[1,2]*p; |
---|
| 551 | if (interval<>0) // canceling common terms of denominator |
---|
| 552 | { if ((j/interval)*interval==j or j==g) // and enumerator - |
---|
| 553 | { s=matrix(syz(ideal(M))); // once gcd() is faster than syz() these |
---|
| 554 | M[1,1]=-s[2,1]; // three lines should be replaced by the |
---|
| 555 | M[1,2]=s[1,1]; // following three |
---|
| 556 | // p=gcd(M[1,1],M[1,2]); |
---|
| 557 | // M[1,1]=M[1,1]/p; |
---|
| 558 | // M[1,2]=M[1,2]/p; |
---|
| 559 | } |
---|
| 560 | } |
---|
| 561 | if (v) |
---|
| 562 | { " Term "+string(j)+" of the Molien series has been computed."; |
---|
| 563 | } |
---|
| 564 | } |
---|
| 565 | if (interval==0) // canceling common terms of denominator |
---|
| 566 | { // and enumerator - |
---|
| 567 | s=matrix(syz(ideal(M))); // once gcd() is faster than syz() these |
---|
| 568 | M[1,1]=-s[2,1]; // three lines should be replaced by the |
---|
| 569 | M[1,2]=s[1,1]; // following three |
---|
| 570 | // p=gcd(M[1,1],M[1,2]); |
---|
| 571 | // M[1,1]=M[1,1]/p; |
---|
| 572 | // M[1,2]=M[1,2]/p; |
---|
| 573 | } |
---|
| 574 | map slead=br,ideal(0); |
---|
| 575 | s=slead(M); |
---|
| 576 | M[1,1]=1/s[1,1]*M[1,1]; // numerator and denominator have to have |
---|
| 577 | M[1,2]=1/s[1,2]*M[1,2]; // a constant term of 1 |
---|
| 578 | if (v) |
---|
| 579 | { ""; |
---|
| 580 | " We are done calculating the Molien series."; |
---|
| 581 | ""; |
---|
| 582 | } |
---|
| 583 | return(M); |
---|
| 584 | } |
---|
| 585 | //---- calculating Molien series in prime characteristic with Brauer lift ---- |
---|
| 586 | if (ch<>0 && mol_flag==0) |
---|
| 587 | { if (g<>1) |
---|
| 588 | { matrix G(1..g)=#[1..g]; |
---|
| 589 | if (interval<0) |
---|
| 590 | { string Mstring; |
---|
| 591 | } |
---|
| 592 | //------ preparing everything for Brauer lifts into characteristic 0 --------- |
---|
| 593 | ring Q=0,x,dp; // we want to extend our ring as well as |
---|
| 594 | // the ring of rational numbers Q to |
---|
| 595 | // contain r-th primitive roots of unity |
---|
| 596 | // in order to factor characteristic |
---|
| 597 | // polynomials of group elements into |
---|
| 598 | // linear factors and lift eigenvalues to |
---|
| 599 | // characteristic 0 - |
---|
| 600 | poly minq=cyclotomic(r); // minq now contains the size-of-group-th |
---|
| 601 | // cyclotomic polynomial of Q, it is |
---|
| 602 | // irreducible there |
---|
| 603 | ring `newring`=(0,e),x,dp; |
---|
| 604 | map f=Q,ideal(e); |
---|
| 605 | minpoly=number(f(minq)); // e is now a r-th primitive root of |
---|
| 606 | // unity - |
---|
| 607 | kill Q, f; // no longer needed - |
---|
| 608 | poly p=1; // used to build the denominator of the |
---|
| 609 | // new term in the Molien series |
---|
| 610 | matrix s[1][2]; // used for canceling - |
---|
| 611 | matrix M[1][2]=0,1; // will contain Molien series - |
---|
| 612 | ring v1br=char(br),x,dp; // we calculate the r-th cyclotomic |
---|
| 613 | poly minp=cyclotomic(r); // polynomial of the base field and pick |
---|
| 614 | minp=factorize(minp)[1][2]; // an irreducible factor of it - |
---|
| 615 | if (deg(minp)==1) // in this case the base field contains |
---|
| 616 | { ring bre=char(br),x,dp; // r-th roots of unity already |
---|
| 617 | map f1=v1br,ideal(0); |
---|
| 618 | number e=-number((f1(minp))); // e is a r-th primitive root of unity |
---|
| 619 | } |
---|
| 620 | else |
---|
| 621 | { ring bre=(char(br),e),x,dp; |
---|
| 622 | map f1=v1br,ideal(e); |
---|
| 623 | minpoly=number(f1(minp)); // e is a r-th primitive root of unity |
---|
| 624 | } |
---|
| 625 | map f2=br,ideal(0); // we need f2 to map our group elements |
---|
| 626 | // to this new extension field bre |
---|
| 627 | matrix xI=diag(x,n); |
---|
| 628 | poly p; // used for the characteristic polynomial |
---|
| 629 | // to factor - |
---|
| 630 | list L; // will contain the linear factors of the |
---|
| 631 | ideal F; // characteristic polynomial of the group |
---|
| 632 | intvec C; // elements and their powers |
---|
| 633 | int i, j, k; |
---|
| 634 | // -------------- finding all the terms of the Molien series ----------------- |
---|
| 635 | for (i=1;i<=g;i=i+1) |
---|
| 636 | { setring br; |
---|
| 637 | if (not(typeof(#[i])=="matrix")) |
---|
| 638 | { "ERROR: the parameters must be a list of matrices and maybe an <intvec>"; |
---|
| 639 | return(); |
---|
| 640 | } |
---|
| 641 | if ((n<>nrows(#[i])) or (n<>ncols(#[i]))) |
---|
| 642 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
| 643 | return(); |
---|
| 644 | } |
---|
| 645 | setring bre; |
---|
| 646 | p=det(xI-f2(G(i))); // characteristic polynomial of G(i) |
---|
| 647 | L=factorize(p); |
---|
| 648 | F=L[1]; |
---|
| 649 | C=L[2]; |
---|
| 650 | for (j=2;j<=ncols(F);j=j+1) |
---|
| 651 | { F[j]=-1*(F[j]-x); // F[j] is now an eigenvalue of G(i), |
---|
| 652 | // it is a power of a primitive r-th root |
---|
| 653 | // of unity - |
---|
| 654 | k=exponent(number(F[j]),e); // F[j]==e^k |
---|
| 655 | setring `newring`; |
---|
| 656 | p=p*(1-x*(e^k))^C[j]; // building the denominator of the new |
---|
| 657 | setring bre; // term |
---|
| 658 | } |
---|
| 659 | // ----------- |
---|
| 660 | // k=0; |
---|
| 661 | // while(k<r) |
---|
| 662 | // { map f3=basering,ideal(e^k); |
---|
| 663 | // while (f3(p)==0) |
---|
| 664 | // { p=p/(x-e^k); |
---|
| 665 | // setring `newring`; |
---|
| 666 | // p=p*(1-x*(e^k)); // building the denominator of the new |
---|
| 667 | // setring bre; |
---|
| 668 | // } |
---|
| 669 | // kill f3; |
---|
| 670 | // if (p==1) |
---|
| 671 | // { break; |
---|
| 672 | // } |
---|
| 673 | // k=k+1; |
---|
| 674 | // } |
---|
| 675 | setring `newring`; |
---|
| 676 | M[1,1]=M[1,1]*p+M[1,2]; // expanding M[1,1]/M[1,2] + 1/p |
---|
| 677 | M[1,2]=M[1,2]*p; |
---|
| 678 | if (interval<0) |
---|
| 679 | { if (i<>g) |
---|
| 680 | { Mstring=string(M); |
---|
| 681 | for (j=1;j<=size(Mstring);j=j+1) |
---|
| 682 | { if (Mstring[j]=="e") |
---|
| 683 | { interval=0; |
---|
| 684 | break; |
---|
| 685 | } |
---|
| 686 | } |
---|
| 687 | } |
---|
| 688 | if (interval<>0) |
---|
| 689 | { s=matrix(syz(ideal(M))); // once gcd() is faster than syz() |
---|
| 690 | M[1,1]=-s[2,1]; // these three lines should be |
---|
| 691 | M[1,2]=s[1,1]; // replaced by the following three |
---|
| 692 | // p=gcd(M[1,1],M[1,2]); |
---|
| 693 | // M[1,1]=M[1,1]/p; |
---|
| 694 | // M[1,2]=M[1,2]/p; |
---|
| 695 | } |
---|
| 696 | else |
---|
| 697 | { interval=-1; |
---|
| 698 | } |
---|
| 699 | } |
---|
| 700 | else |
---|
| 701 | { if (interval<>0) // canceling common terms of denominator |
---|
| 702 | { if ((i/interval)*interval==i or i==g) // and enumerator |
---|
| 703 | { s=matrix(syz(ideal(M))); // once gcd() is faster than syz() |
---|
| 704 | M[1,1]=-s[2,1]; // these three lines should be |
---|
| 705 | M[1,2]=s[1,1]; // replaced by the following three |
---|
| 706 | // p=gcd(M[1,1],M[1,2]); |
---|
| 707 | // M[1,1]=M[1,1]/p; |
---|
| 708 | // M[1,2]=M[1,2]/p; |
---|
| 709 | } |
---|
| 710 | } |
---|
| 711 | } |
---|
| 712 | p=1; |
---|
| 713 | setring bre; |
---|
| 714 | if (v) |
---|
| 715 | { " Term "+string(i)+" of the Molien series has been computed."; |
---|
| 716 | } |
---|
| 717 | } |
---|
| 718 | if (v) |
---|
| 719 | { ""; |
---|
| 720 | } |
---|
| 721 | setring `newring`; |
---|
| 722 | if (interval==0) // canceling common terms of denominator |
---|
| 723 | { // and enumerator - |
---|
| 724 | s=matrix(syz(ideal(M))); // once gcd() is faster than syz() these |
---|
| 725 | M[1,1]=-s[2,1]; // three lines should be replaced by the |
---|
| 726 | M[1,2]=s[1,1]; // following three |
---|
| 727 | // p=gcd(M[1,1],M[1,2]); |
---|
| 728 | // M[1,1]=M[1,1]/p; |
---|
| 729 | // M[1,2]=M[1,2]/p; |
---|
| 730 | } |
---|
| 731 | map slead=`newring`,ideal(0); |
---|
| 732 | s=slead(M); // forcing the constant term of numerator |
---|
| 733 | M[1,1]=1/s[1,1]*M[1,1]; // and denominator to be 1 |
---|
| 734 | M[1,2]=1/s[1,2]*M[1,2]; |
---|
| 735 | kill slead; |
---|
| 736 | kill s; |
---|
| 737 | kill p; |
---|
| 738 | } |
---|
| 739 | else // if the group only contains an identity |
---|
| 740 | { ring `newring`=0,x,dp; // element, it is very easy to calculate |
---|
| 741 | matrix M[1][2]=1,(1-x)^n; // the Molien series |
---|
| 742 | } |
---|
| 743 | export `newring`; // we keep the ring where we computed the |
---|
| 744 | export M; // Molien series in such that we can |
---|
| 745 | setring br; // keep it |
---|
| 746 | if (v) |
---|
| 747 | { " We are done calculating the Molien series."; |
---|
| 748 | ""; |
---|
| 749 | } |
---|
| 750 | } |
---|
| 751 | else // i.e. char<>0 and mol_flag<>0, the user |
---|
| 752 | { // has specified that we are dealing with |
---|
| 753 | // a ring of large characteristic which |
---|
| 754 | // can be treated like a ring of |
---|
| 755 | // characteristic 0; we'll avoid the |
---|
| 756 | // Brauer lifts |
---|
| 757 | //----------------------- simulating characteristic 0 ------------------------ |
---|
| 758 | string chst=charstr(br); |
---|
| 759 | for (int i=1;i<=size(chst);i=i+1) |
---|
| 760 | { if (chst[i]==",") |
---|
| 761 | { break; |
---|
| 762 | } |
---|
| 763 | } |
---|
| 764 | //----------------- generating ring of characteristic 0 ---------------------- |
---|
| 765 | if (minpoly==0) |
---|
| 766 | { if (i>size(chst)) |
---|
| 767 | { execute "ring "+newring+"=0,("+varstr(br)+"),("+ordstr(br)+")"; |
---|
| 768 | } |
---|
| 769 | else |
---|
| 770 | { chst=chst[i..size(chst)]; |
---|
| 771 | execute "ring "+newring+"=(0"+chst+"),("+varstr(br)+"),("+ordstr(br)+")"; |
---|
| 772 | } |
---|
| 773 | } |
---|
| 774 | else |
---|
| 775 | { string minp=string(minpoly); |
---|
| 776 | minp=minp[2..size(minp)-1]; |
---|
| 777 | chst=chst[i..size(chst)]; |
---|
| 778 | execute "ring "+newring+"=(0"+chst+"),("+varstr(br)+"),("+ordstr(br)+")"; |
---|
| 779 | execute "minpoly="+minp; |
---|
| 780 | } |
---|
| 781 | matrix I=diag(1,n); |
---|
| 782 | poly v1=maxideal(1)[1]; // the Molien series will be in terms of |
---|
| 783 | // the first variable of the current |
---|
| 784 | // ring - |
---|
| 785 | matrix M[1][2]; // M will contain the Molien series - |
---|
| 786 | M[1,1]=0; // M[1,1] will be the numerator - |
---|
| 787 | M[1,2]=1; // M[1,2] will be the denominator - |
---|
| 788 | matrix s; // will help us canceling in the |
---|
| 789 | // fraction |
---|
| 790 | poly p; // will contain the denominator of the |
---|
| 791 | // new term of the Molien series |
---|
| 792 | int j; |
---|
| 793 | string links, rechts; |
---|
| 794 | //----------------- finding all terms of the Molien series ------------------- |
---|
| 795 | for (i=1;i<=g;i=i+1) |
---|
| 796 | { setring br; |
---|
| 797 | if (not(typeof(#[i])=="matrix")) |
---|
| 798 | { "ERROR: the parameters must be a list of matrices and maybe an <intvec>"; |
---|
| 799 | return(); |
---|
| 800 | } |
---|
| 801 | if ((n<>nrows(#[i])) or (n<>ncols(#[i]))) |
---|
| 802 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
| 803 | return(); |
---|
| 804 | } |
---|
| 805 | string stM(i)=string(#[i]); |
---|
| 806 | for (j=1;j<=size(stM(i));j=j+1) |
---|
| 807 | { if (stM(i)[j]==" |
---|
| 808 | ") |
---|
| 809 | { links=stM(i)[1..j-1]; |
---|
| 810 | rechts=stM(i)[j+1..size(stM(i))]; |
---|
| 811 | stM(i)=links+rechts; |
---|
| 812 | } |
---|
| 813 | } |
---|
| 814 | setring `newring`; |
---|
| 815 | execute "matrix G(i)["+string(n)+"]["+string(n)+"]="+stM(i); |
---|
| 816 | p=det(I-v1*G(i)); // denominator of new term - |
---|
| 817 | M[1,1]=M[1,1]*p+M[1,2]; // expanding M[1,1]/M[1,2] + 1/p |
---|
| 818 | M[1,2]=M[1,2]*p; |
---|
| 819 | if (interval<>0) // canceling common terms of denominator |
---|
| 820 | { if ((i/interval)*interval==i or i==g) // and enumerator |
---|
| 821 | { |
---|
| 822 | s=matrix(syz(ideal(M))); // once gcd() is faster than syz() these |
---|
| 823 | M[1,1]=-s[2,1]; // three lines should be replaced by the |
---|
| 824 | M[1,2]=s[1,1]; // following three |
---|
| 825 | // p=gcd(M[1,1],M[1,2]); |
---|
| 826 | // M[1,1]=M[1,1]/p; |
---|
| 827 | // M[1,2]=M[1,2]/p; |
---|
| 828 | } |
---|
| 829 | } |
---|
| 830 | if (v) |
---|
| 831 | { " Term "+string(i)+" of the Molien series has been computed."; |
---|
| 832 | } |
---|
| 833 | } |
---|
| 834 | if (interval==0) // canceling common terms of denominator |
---|
| 835 | { // and enumerator - |
---|
| 836 | s=matrix(syz(ideal(M))); // once gcd() is faster than syz() these |
---|
| 837 | M[1,1]=-s[2,1]; // three lines should be replaced by the |
---|
| 838 | M[1,2]=s[1,1]; // following three |
---|
| 839 | // p=gcd(M[1,1],M[1,2]); |
---|
| 840 | // M[1,1]=M[1,1]/p; |
---|
| 841 | // M[1,2]=M[1,2]/p; |
---|
| 842 | } |
---|
| 843 | map slead=`newring`,ideal(0); |
---|
| 844 | s=slead(M); |
---|
| 845 | M[1,1]=1/s[1,1]*M[1,1]; // numerator and denominator have to have |
---|
| 846 | M[1,2]=1/s[1,2]*M[1,2]; // a constant term of 1 |
---|
| 847 | if (v) |
---|
| 848 | { ""; |
---|
| 849 | " We are done calculating the Molien series."; |
---|
| 850 | ""; |
---|
| 851 | } |
---|
| 852 | kill G(1..g), s, slead, p, v1, I; |
---|
| 853 | export `newring`; // we keep the ring where we computed the |
---|
| 854 | export M; // the Molien series such that we can |
---|
| 855 | setring br; // keep it |
---|
| 856 | } |
---|
| 857 | } |
---|
| 858 | example |
---|
| 859 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
| 860 | " note the case of prime characteristic"; |
---|
| 861 | echo=2; |
---|
| 862 | ring R=0,(x,y,z),dp; |
---|
| 863 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 864 | list L=group_reynolds(A); |
---|
| 865 | matrix M=molien(L[2..size(L)]); |
---|
| 866 | print(M); |
---|
| 867 | ring S=3,(x,y,z),dp; |
---|
| 868 | string newring="alksdfjlaskdjf"; |
---|
| 869 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 870 | list L=group_reynolds(A); |
---|
| 871 | molien(L[2..size(L)],newring); |
---|
| 872 | setring alksdfjlaskdjf; |
---|
| 873 | print(M); |
---|
| 874 | setring S; |
---|
| 875 | kill alksdfjlaskdjf; |
---|
| 876 | } |
---|
| 877 | |
---|
| 878 | proc reynolds_molien (list #) |
---|
| 879 | "USAGE: reynolds_molien(G1,G2,...[,ringname,flags]); |
---|
| 880 | G1,G2,...: nxn <matrices> generating a finite matrix group, ringname: |
---|
| 881 | a <string> giving a name for a new ring of characteristic 0 for the |
---|
| 882 | Molien series in case of prime characteristic, flags: an optional |
---|
| 883 | <intvec> with three components: if the first element is not equal to 0 |
---|
| 884 | characteristic 0 is simulated, i.e. the Molien series is computed as |
---|
| 885 | if the base field were characteristic 0 (the user must choose a field |
---|
| 886 | of large prime characteristic, e.g. 32003) the second component should |
---|
| 887 | give the size of intervals between canceling common factors in the |
---|
| 888 | expansion of the Molien series, 0 (the default) means only once after |
---|
| 889 | generating all terms, in prime characteristic also a negative number |
---|
| 890 | can be given to indicate that common factors should always be canceled |
---|
| 891 | when the expansion is simple (the root of the extension field does not |
---|
| 892 | occur among the coefficients) |
---|
| 893 | ASSUME: n is the number of variables of the basering, G1,G2... are the group |
---|
| 894 | elements generated by group_reynolds(), g is the size of the group |
---|
| 895 | RETURN: a gxn <matrix> representing the Reynolds operator is the first return |
---|
| 896 | value and in case of characteristic 0 a 1x2 <matrix> giving enumerator |
---|
| 897 | and denominator of Molien series is the second one; in case of prime |
---|
| 898 | characteristic a ring with the name `ringname` of characteristic 0 is |
---|
| 899 | created where the same Molien series (named M) is stored |
---|
| 900 | DISPLAY: information if the third component of flags does not equal 0 |
---|
| 901 | EXAMPLE: example reynolds_molien; shows an example |
---|
| 902 | THEORY: The entire matrix group is generated by getting all left products of |
---|
| 903 | the generators with the new elements from the last run through the loop |
---|
| 904 | (or the generators themselves during the first run). All the ones that |
---|
| 905 | have been generated before are thrown out and the program terminates |
---|
| 906 | when there are no new elements found in one run. Additionally each time |
---|
| 907 | a new group element is found the corresponding ring mapping of which |
---|
| 908 | the Reynolds operator is made up is generated. They are stored in the |
---|
| 909 | rows of the first return value. In characteristic 0 the terms |
---|
| 910 | 1/det(1-xE) is computed whenever a new element E is found. In prime |
---|
| 911 | characteristic a Brauer lift is involved and the terms are only |
---|
| 912 | computed after the entire matrix group is generated (to avoid the |
---|
| 913 | modular case). The returned matrix gives enumerator and denominator of |
---|
| 914 | the expanded version where common factors have been canceled. |
---|
| 915 | " |
---|
| 916 | { def br=basering; // the Molien series depends on the |
---|
| 917 | int ch=char(br); // characteristic of the coefficient |
---|
| 918 | // field |
---|
| 919 | int gen_num; |
---|
| 920 | //------------------- making sure the input is okay -------------------------- |
---|
| 921 | if (typeof(#[size(#)])=="intvec") |
---|
| 922 | { if (size(#[size(#)])==3) |
---|
| 923 | { int mol_flag=#[size(#)][1]; |
---|
| 924 | if (#[size(#)][2]<0 && (ch==0 or (ch<>0 && mol_flag<>0))) |
---|
| 925 | { "ERROR: the second component of the <intvec> should be >=0"; |
---|
| 926 | return(); |
---|
| 927 | } |
---|
| 928 | int interval=#[size(#)][2]; |
---|
| 929 | int v=#[size(#)][3]; |
---|
| 930 | } |
---|
| 931 | else |
---|
| 932 | { "ERROR: <intvec> should have three components"; |
---|
| 933 | return(); |
---|
| 934 | } |
---|
| 935 | if (ch<>0) |
---|
| 936 | { if (typeof(#[size(#)-1])<>"string") |
---|
| 937 | { "ERROR: in characteristic p a <string> must be given for the name"; |
---|
| 938 | " of a new ring where the Molien series can be stored"; |
---|
| 939 | return(); |
---|
| 940 | } |
---|
| 941 | else |
---|
| 942 | { if (#[size(#)-1]=="") |
---|
| 943 | { "ERROR: <string> may not be empty"; |
---|
| 944 | return(); |
---|
| 945 | } |
---|
| 946 | string newring=#[size(#)-1]; |
---|
| 947 | gen_num=size(#)-2; |
---|
| 948 | } |
---|
| 949 | } |
---|
| 950 | else // then <string> ist not needed |
---|
| 951 | { gen_num=size(#)-1; |
---|
| 952 | } |
---|
| 953 | } |
---|
| 954 | else // last parameter is not <intvec> |
---|
| 955 | { int v=0; // no information is default |
---|
| 956 | int interval; |
---|
| 957 | int mol_flag=0; // computing of Molien series is default |
---|
| 958 | if (ch<>0) |
---|
| 959 | { if (typeof(#[size(#)])<>"string") |
---|
| 960 | { "ERROR: in characteristic p a <string> must be given for the name"; |
---|
| 961 | " of a new ring where the Molien series can be stored"; |
---|
| 962 | return(); |
---|
| 963 | } |
---|
| 964 | else |
---|
| 965 | { if (#[size(#)]=="") |
---|
| 966 | { "ERROR: <string> may not be empty"; |
---|
| 967 | return(); |
---|
| 968 | } |
---|
| 969 | string newring=#[size(#)]; |
---|
| 970 | gen_num=size(#)-1; |
---|
| 971 | } |
---|
| 972 | } |
---|
| 973 | else |
---|
| 974 | { gen_num=size(#); |
---|
| 975 | } |
---|
| 976 | } |
---|
| 977 | // ----------------- computing the terms with Brauer lift -------------------- |
---|
| 978 | if (ch<>0 && mol_flag==0) |
---|
| 979 | { list L=group_reynolds(#[1..gen_num],v); |
---|
| 980 | if (L[1]==0) |
---|
| 981 | { if (voice==2) |
---|
| 982 | { "WARNING: The characteristic of the coefficient field divides the group order."; |
---|
| 983 | " Proceed without the Reynolds operator or the Molien series!"; |
---|
| 984 | return(); |
---|
| 985 | } |
---|
| 986 | if (v) |
---|
| 987 | { " The characteristic of the base field divides the group order."; |
---|
| 988 | " We have to continue without Reynolds operator or the Molien series..."; |
---|
| 989 | return(); |
---|
| 990 | } |
---|
| 991 | } |
---|
| 992 | if (minpoly<>0) |
---|
| 993 | { if (voice==2) |
---|
| 994 | { "WARNING: It is impossible for this program to calculate the Molien series"; |
---|
| 995 | " for finite groups over extension fields of prime characteristic."; |
---|
| 996 | return(L[1]); |
---|
| 997 | } |
---|
| 998 | else |
---|
| 999 | { if (v) |
---|
| 1000 | { " Since it is impossible for this program to calculate the Molien series for"; |
---|
| 1001 | " invariant rings over extension fields of prime characteristic, we have to"; |
---|
| 1002 | " continue without it."; |
---|
| 1003 | return(L[1]); |
---|
| 1004 | } |
---|
| 1005 | } |
---|
| 1006 | } |
---|
| 1007 | if (typeof(L[2])=="int") |
---|
| 1008 | { molien(L[3..size(L)],newring,L[2],intvec(mol_flag,interval,v)); |
---|
| 1009 | } |
---|
| 1010 | else |
---|
| 1011 | { molien(L[2..size(L)],newring,intvec(mol_flag,interval,v)); |
---|
| 1012 | } |
---|
| 1013 | return(L[1]); |
---|
| 1014 | } |
---|
| 1015 | //----------- computing Molien series in the straight forward way ------------ |
---|
| 1016 | if (ch==0) |
---|
| 1017 | { if (typeof(#[1])<>"matrix") |
---|
| 1018 | { "ERROR: the parameters must be a list of matrices and maybe an <intvec>"; |
---|
| 1019 | return(); |
---|
| 1020 | } |
---|
| 1021 | int n=nrows(#[1]); |
---|
| 1022 | if (n<>nvars(br)) |
---|
| 1023 | { "ERROR: the number of variables of the basering needs to be the same"; |
---|
| 1024 | " as the dimension of the matrices"; |
---|
| 1025 | return(); |
---|
| 1026 | } |
---|
| 1027 | if (n<>ncols(#[1])) |
---|
| 1028 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
| 1029 | return(); |
---|
| 1030 | } |
---|
| 1031 | matrix vars=matrix(maxideal(1)); // creating an nx1-matrix containing the |
---|
| 1032 | vars=transpose(vars); // variables of the ring - |
---|
| 1033 | matrix A(1)=#[1]*vars; // calculating the first ring mapping - |
---|
| 1034 | // A(1) will contain the Reynolds |
---|
| 1035 | // operator - |
---|
| 1036 | poly v1=vars[1,1]; // the Molien series will be in terms of |
---|
| 1037 | // the first variable of the current |
---|
| 1038 | // ring |
---|
| 1039 | matrix I=diag(1,n); |
---|
| 1040 | matrix A(2)[1][2]; // A(2) will contain the Molien series - |
---|
| 1041 | A(2)[1,1]=1; // A(2)[1,1] will be the numerator |
---|
| 1042 | matrix G(1)=#[1]; // G(k) are elements of the group - |
---|
| 1043 | A(2)[1,2]=det(I-v1*(G(1))); // A(2)[1,2] will be the denominator - |
---|
| 1044 | matrix s; // will help us canceling in the |
---|
| 1045 | // fraction |
---|
| 1046 | poly p; // will contain the denominator of the |
---|
| 1047 | // new term of the Molien series |
---|
| 1048 | int i=1; |
---|
| 1049 | for (int j=2;j<=gen_num;j=j+1) // this loop adds the parameters to the |
---|
| 1050 | { // group, leaving out doubles and |
---|
| 1051 | // checking whether the parameters are |
---|
| 1052 | // compatible with the task of the |
---|
| 1053 | // procedure |
---|
| 1054 | if (not(typeof(#[j])=="matrix")) |
---|
| 1055 | { "ERROR: the parameters must be a list of matrices and maybe an <intvec>"; |
---|
| 1056 | return(); |
---|
| 1057 | } |
---|
| 1058 | if ((n!=nrows(#[j])) or (n!=ncols(#[j]))) |
---|
| 1059 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
| 1060 | return(); |
---|
| 1061 | } |
---|
| 1062 | if (unique(G(1..i),#[j])) |
---|
| 1063 | { i=i+1; |
---|
| 1064 | matrix G(i)=#[j]; |
---|
| 1065 | A(1)=concat(A(1),#[j]*vars); // adding ring homomorphisms to A(1) - |
---|
| 1066 | p=det(I-v1*#[j]); // denominator of new term - |
---|
| 1067 | A(2)[1,1]=A(2)[1,1]*p+A(2)[1,2]; // expanding A(2)[1,1]/A(2)[1,2] +1/p |
---|
| 1068 | A(2)[1,2]=A(2)[1,2]*p; |
---|
| 1069 | if (interval<>0) // canceling common terms of denominator |
---|
| 1070 | { if ((i/interval)*interval==i) // and enumerator |
---|
| 1071 | { |
---|
| 1072 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() these |
---|
| 1073 | A(2)[1,1]=-s[2,1]; // three lines should be replaced by the |
---|
| 1074 | A(2)[1,2]=s[1,1]; // following three |
---|
| 1075 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
| 1076 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
| 1077 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
| 1078 | } |
---|
| 1079 | } |
---|
| 1080 | } |
---|
| 1081 | } |
---|
| 1082 | int g=i; // G(1)..G(i) are generators without |
---|
| 1083 | // doubles - g generally is the number |
---|
| 1084 | // of elements in the group so far - |
---|
| 1085 | j=i; // j is the number of new elements that |
---|
| 1086 | // we use as factors |
---|
| 1087 | int k, m, l; |
---|
| 1088 | if (v) |
---|
| 1089 | { ""; |
---|
| 1090 | " Generating the entire matrix group. Whenever a new group element is found,"; |
---|
| 1091 | " the coressponding ring homomorphism of the Reynolds operator and the"; |
---|
| 1092 | " corresponding term of the Molien series is generated."; |
---|
| 1093 | ""; |
---|
| 1094 | } |
---|
| 1095 | //----------- computing 1/det(I-xE) whenever a new element E is found -------- |
---|
| 1096 | while (1) |
---|
| 1097 | { l=0; // l is the number of products we get in |
---|
| 1098 | // one going |
---|
| 1099 | for (m=g-j+1;m<=g;m=m+1) |
---|
| 1100 | { for (k=1;k<=i;k=k+1) |
---|
| 1101 | { l=l+1; |
---|
| 1102 | matrix P(l)=G(k)*G(m); // possible new element |
---|
| 1103 | } |
---|
| 1104 | } |
---|
| 1105 | j=0; |
---|
| 1106 | for (k=1;k<=l;k=k+1) |
---|
| 1107 | { if (unique(G(1..g),P(k))) |
---|
| 1108 | { j=j+1; // a new factor for next run |
---|
| 1109 | g=g+1; |
---|
| 1110 | matrix G(g)=P(k); // a new group element - |
---|
| 1111 | A(1)=concat(A(1),P(k)*vars); // adding new mapping to A(1) |
---|
| 1112 | p=det(I-v1*P(k)); // denominator of new term |
---|
| 1113 | A(2)[1,1]=A(2)[1,1]*p+A(2)[1,2]; |
---|
| 1114 | A(2)[1,2]=A(2)[1,2]*p; // expanding A(2)[1,1]/A(2)[1,2] + 1/p - |
---|
| 1115 | if (interval<>0) // canceling common terms of denominator |
---|
| 1116 | { if ((g/interval)*interval==g) // and enumerator |
---|
| 1117 | { |
---|
| 1118 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() |
---|
| 1119 | A(2)[1,1]=-s[2,1]; // these three lines should be replaced |
---|
| 1120 | A(2)[1,2]=s[1,1]; // by the following three |
---|
| 1121 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
| 1122 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
| 1123 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
| 1124 | } |
---|
| 1125 | } |
---|
| 1126 | if (v) |
---|
| 1127 | { " Group element "+string(g)+" has been found."; |
---|
| 1128 | } |
---|
| 1129 | } |
---|
| 1130 | kill P(k); |
---|
| 1131 | } |
---|
| 1132 | if (j==0) // when we didn't add any new elements |
---|
| 1133 | { break; // in one run through the while loop |
---|
| 1134 | } // we are done |
---|
| 1135 | } |
---|
| 1136 | if (v) |
---|
| 1137 | { if (g<=i) |
---|
| 1138 | { " There are only "+string(g)+" group elements."; |
---|
| 1139 | } |
---|
| 1140 | ""; |
---|
| 1141 | } |
---|
| 1142 | A(1)=transpose(A(1)); // when we evaluate the Reynolds operator |
---|
| 1143 | // later on, we actually want 1xn |
---|
| 1144 | // matrices |
---|
| 1145 | if (interval==0) // canceling common terms of denominator |
---|
| 1146 | { // and enumerator - |
---|
| 1147 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() |
---|
| 1148 | A(2)[1,1]=-s[2,1]; // these three lines should be replaced |
---|
| 1149 | A(2)[1,2]=s[1,1]; // by the following three |
---|
| 1150 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
| 1151 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
| 1152 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
| 1153 | } |
---|
| 1154 | if (interval<>0) // canceling common terms of denominator |
---|
| 1155 | { if ((g/interval)*interval<>g) // and enumerator |
---|
| 1156 | { |
---|
| 1157 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() |
---|
| 1158 | A(2)[1,1]=-s[2,1]; // these three lines should be replaced |
---|
| 1159 | A(2)[1,2]=s[1,1]; // by the following three |
---|
| 1160 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
| 1161 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
| 1162 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
| 1163 | } |
---|
| 1164 | } |
---|
| 1165 | map slead=br,ideal(0); |
---|
| 1166 | s=slead(A(2)); |
---|
| 1167 | A(2)[1,1]=1/s[1,1]*A(2)[1,1]; // numerator and denominator have to have |
---|
| 1168 | A(2)[1,2]=1/s[1,2]*A(2)[1,2]; // a constant term of 1 |
---|
| 1169 | if (v) |
---|
| 1170 | { " Now we are done calculating Molien series and Reynolds operator."; |
---|
| 1171 | ""; |
---|
| 1172 | } |
---|
| 1173 | return(A(1..2)); |
---|
| 1174 | } |
---|
| 1175 | //------------------------ simulating characteristic 0 ----------------------- |
---|
| 1176 | else // if ch<>0 and mol_flag<>0 |
---|
| 1177 | { if (typeof(#[1])<>"matrix") |
---|
| 1178 | { "ERROR: the parameters must be a list of matrices and maybe an <intvec>"; |
---|
| 1179 | return(); |
---|
| 1180 | } |
---|
| 1181 | int n=nrows(#[1]); |
---|
| 1182 | if (n<>nvars(br)) |
---|
| 1183 | { "ERROR: the number of variables of the basering needs to be the same"; |
---|
| 1184 | " as the dimension of the matrices"; |
---|
| 1185 | return(); |
---|
| 1186 | } |
---|
| 1187 | if (n<>ncols(#[1])) |
---|
| 1188 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
| 1189 | return(); |
---|
| 1190 | } |
---|
| 1191 | matrix vars=matrix(maxideal(1)); // creating an nx1-matrix containing the |
---|
| 1192 | vars=transpose(vars); // variables of the ring - |
---|
| 1193 | matrix A(1)=#[1]*vars; // calculating the first ring mapping - |
---|
| 1194 | // A(1) will contain the Reynolds |
---|
| 1195 | // operator |
---|
| 1196 | string chst=charstr(br); |
---|
| 1197 | for (int i=1;i<=size(chst);i=i+1) |
---|
| 1198 | { if (chst[i]==",") |
---|
| 1199 | { break; |
---|
| 1200 | } |
---|
| 1201 | } |
---|
| 1202 | if (minpoly==0) |
---|
| 1203 | { if (i>size(chst)) |
---|
| 1204 | { execute "ring "+newring+"=0,("+varstr(br)+"),("+ordstr(br)+")"; |
---|
| 1205 | } |
---|
| 1206 | else |
---|
| 1207 | { chst=chst[i..size(chst)]; |
---|
| 1208 | execute "ring "+newring+"=(0"+chst+"),("+varstr(br)+"),("+ordstr(br)+")"; |
---|
| 1209 | } |
---|
| 1210 | } |
---|
| 1211 | else |
---|
| 1212 | { string minp=string(minpoly); |
---|
| 1213 | minp=minp[2..size(minp)-1]; |
---|
| 1214 | chst=chst[i..size(chst)]; |
---|
| 1215 | execute "ring "+newring+"=(0"+chst+"),("+varstr(br)+"),("+ordstr(br)+")"; |
---|
| 1216 | execute "minpoly="+minp; |
---|
| 1217 | } |
---|
| 1218 | poly v1=var(1); // the Molien series will be in terms of |
---|
| 1219 | // the first variable of the current |
---|
| 1220 | // ring |
---|
| 1221 | matrix I=diag(1,n); |
---|
| 1222 | int o; |
---|
| 1223 | setring br; |
---|
| 1224 | matrix G(1)=#[1]; |
---|
| 1225 | string links, rechts; |
---|
| 1226 | string stM(1)=string(#[1]); |
---|
| 1227 | for (o=1;o<=size(stM(1));o=o+1) |
---|
| 1228 | { if (stM(1)[o]==" |
---|
| 1229 | ") |
---|
| 1230 | { links=stM(1)[1..o-1]; |
---|
| 1231 | rechts=stM(1)[o+1..size(stM(1))]; |
---|
| 1232 | stM(1)=links+rechts; |
---|
| 1233 | } |
---|
| 1234 | } |
---|
| 1235 | setring `newring`; |
---|
| 1236 | execute "matrix G(1)["+string(n)+"]["+string(n)+"]="+stM(1); |
---|
| 1237 | matrix A(2)[1][2]; // A(2) will contain the Molien series - |
---|
| 1238 | A(2)[1,1]=1; // A(2)[1,1] will be the numerator |
---|
| 1239 | A(2)[1,2]=det(I-v1*(G(1))); // A(2)[1,2] will be the denominator - |
---|
| 1240 | matrix s; // will help us canceling in the |
---|
| 1241 | // fraction |
---|
| 1242 | poly p; // will contain the denominator of the |
---|
| 1243 | // new term of the Molien series |
---|
| 1244 | i=1; |
---|
| 1245 | for (int j=2;j<=gen_num;j=j+1) // this loop adds the parameters to the |
---|
| 1246 | { // group, leaving out doubles and |
---|
| 1247 | // checking whether the parameters are |
---|
| 1248 | // compatible with the task of the |
---|
| 1249 | // procedure |
---|
| 1250 | setring br; |
---|
| 1251 | if (not(typeof(#[j])=="matrix")) |
---|
| 1252 | { "ERROR: the parameters must be a list of matrices and maybe an <intvec>"; |
---|
| 1253 | return(); |
---|
| 1254 | } |
---|
| 1255 | if ((n!=nrows(#[j])) or (n!=ncols(#[j]))) |
---|
| 1256 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
| 1257 | return(); |
---|
| 1258 | } |
---|
| 1259 | if (unique(G(1..i),#[j])) |
---|
| 1260 | { i=i+1; |
---|
| 1261 | matrix G(i)=#[j]; |
---|
| 1262 | A(1)=concat(A(1),G(i)*vars); // adding ring homomorphisms to A(1) |
---|
| 1263 | string stM(i)=string(G(i)); |
---|
| 1264 | for (o=1;o<=size(stM(i));o=o+1) |
---|
| 1265 | { if (stM(i)[o]==" |
---|
| 1266 | ") |
---|
| 1267 | { links=stM(i)[1..o-1]; |
---|
| 1268 | rechts=stM(i)[o+1..size(stM(i))]; |
---|
| 1269 | stM(i)=links+rechts; |
---|
| 1270 | } |
---|
| 1271 | } |
---|
| 1272 | setring `newring`; |
---|
| 1273 | execute "matrix G(i)["+string(n)+"]["+string(n)+"]="+stM(i); |
---|
| 1274 | p=det(I-v1*G(i)); // denominator of new term - |
---|
| 1275 | A(2)[1,1]=A(2)[1,1]*p+A(2)[1,2]; // expanding A(2)[1,1]/A(2)[1,2] +1/p |
---|
| 1276 | A(2)[1,2]=A(2)[1,2]*p; |
---|
| 1277 | if (interval<>0) // canceling common terms of denominator |
---|
| 1278 | { if ((i/interval)*interval==i) // and enumerator |
---|
| 1279 | { |
---|
| 1280 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() these |
---|
| 1281 | A(2)[1,1]=-s[2,1]; // three lines should be replaced by the |
---|
| 1282 | A(2)[1,2]=s[1,1]; // following three |
---|
| 1283 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
| 1284 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
| 1285 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
| 1286 | } |
---|
| 1287 | } |
---|
| 1288 | setring br; |
---|
| 1289 | } |
---|
| 1290 | } |
---|
| 1291 | int g=i; // G(1)..G(i) are generators without |
---|
| 1292 | // doubles - g generally is the number |
---|
| 1293 | // of elements in the group so far - |
---|
| 1294 | j=i; // j is the number of new elements that |
---|
| 1295 | // we use as factors |
---|
| 1296 | int k, m, l; |
---|
| 1297 | if (v) |
---|
| 1298 | { ""; |
---|
| 1299 | " Generating the entire matrix group. Whenever a new group element is found,"; |
---|
| 1300 | " the coressponding ring homomorphism of the Reynolds operator and the"; |
---|
| 1301 | " corresponding term of the Molien series is generated."; |
---|
| 1302 | ""; |
---|
| 1303 | } |
---|
| 1304 | // taking all elements in a ring of characteristic 0 and computing the terms |
---|
| 1305 | // of the Molien series there |
---|
| 1306 | while (1) |
---|
| 1307 | { l=0; // l is the number of products we get in |
---|
| 1308 | // one going |
---|
| 1309 | for (m=g-j+1;m<=g;m=m+1) |
---|
| 1310 | { for (k=1;k<=i;k=k+1) |
---|
| 1311 | { l=l+1; |
---|
| 1312 | matrix P(l)=G(k)*G(m); // possible new element |
---|
| 1313 | } |
---|
| 1314 | } |
---|
| 1315 | j=0; |
---|
| 1316 | for (k=1;k<=l;k=k+1) |
---|
| 1317 | { if (unique(G(1..g),P(k))) |
---|
| 1318 | { j=j+1; // a new factor for next run |
---|
| 1319 | g=g+1; |
---|
| 1320 | matrix G(g)=P(k); // a new group element - |
---|
| 1321 | A(1)=concat(A(1),G(g)*vars); // adding new mapping to A(1) |
---|
| 1322 | string stM(g)=string(G(g)); |
---|
| 1323 | for (o=1;o<=size(stM(g));o=o+1) |
---|
| 1324 | { if (stM(g)[o]==" |
---|
| 1325 | ") |
---|
| 1326 | { links=stM(g)[1..o-1]; |
---|
| 1327 | rechts=stM(g)[o+1..size(stM(g))]; |
---|
| 1328 | stM(g)=links+rechts; |
---|
| 1329 | } |
---|
| 1330 | } |
---|
| 1331 | setring `newring`; |
---|
| 1332 | execute "matrix G(g)["+string(n)+"]["+string(n)+"]="+stM(g); |
---|
| 1333 | p=det(I-v1*G(g)); // denominator of new term |
---|
| 1334 | A(2)[1,1]=A(2)[1,1]*p+A(2)[1,2]; |
---|
| 1335 | A(2)[1,2]=A(2)[1,2]*p; // expanding A(2)[1,1]/A(2)[1,2] + 1/p - |
---|
| 1336 | if (interval<>0) // canceling common terms of denominator |
---|
| 1337 | { if ((g/interval)*interval==g) // and enumerator |
---|
| 1338 | { |
---|
| 1339 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() |
---|
| 1340 | A(2)[1,1]=-s[2,1]; // these three lines should be replaced |
---|
| 1341 | A(2)[1,2]=s[1,1]; // by the following three |
---|
| 1342 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
| 1343 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
| 1344 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
| 1345 | } |
---|
| 1346 | } |
---|
| 1347 | if (v) |
---|
| 1348 | { " Group element "+string(g)+" has been found."; |
---|
| 1349 | } |
---|
| 1350 | setring br; |
---|
| 1351 | } |
---|
| 1352 | kill P(k); |
---|
| 1353 | } |
---|
| 1354 | if (j==0) // when we didn't add any new elements |
---|
| 1355 | { break; // in one run through the while loop |
---|
| 1356 | } // we are done |
---|
| 1357 | } |
---|
| 1358 | if (v) |
---|
| 1359 | { if (g<=i) |
---|
| 1360 | { " There are only "+string(g)+" group elements."; |
---|
| 1361 | } |
---|
| 1362 | ""; |
---|
| 1363 | } |
---|
| 1364 | A(1)=transpose(A(1)); // when we evaluate the Reynolds operator |
---|
| 1365 | // later on, we actually want 1xn |
---|
| 1366 | // matrices |
---|
| 1367 | setring `newring`; |
---|
| 1368 | if (interval==0) // canceling common terms of denominator |
---|
| 1369 | { // and enumerator - |
---|
| 1370 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() |
---|
| 1371 | A(2)[1,1]=-s[2,1]; // these three lines should be replaced |
---|
| 1372 | A(2)[1,2]=s[1,1]; // by the following three |
---|
| 1373 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
| 1374 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
| 1375 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
| 1376 | } |
---|
| 1377 | if (interval<>0) // canceling common terms of denominator |
---|
| 1378 | { if ((g/interval)*interval<>g) // and enumerator |
---|
| 1379 | { |
---|
| 1380 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() |
---|
| 1381 | A(2)[1,1]=-s[2,1]; // these three lines should be replaced |
---|
| 1382 | A(2)[1,2]=s[1,1]; // by the following three |
---|
| 1383 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
| 1384 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
| 1385 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
| 1386 | } |
---|
| 1387 | } |
---|
| 1388 | map slead=`newring`,ideal(0); |
---|
| 1389 | s=slead(A(2)); |
---|
| 1390 | A(2)[1,1]=1/s[1,1]*A(2)[1,1]; // numerator and denominator have to have |
---|
| 1391 | A(2)[1,2]=1/s[1,2]*A(2)[1,2]; // a constant term of 1 |
---|
| 1392 | if (v) |
---|
| 1393 | { " Now we are done calculating Molien series and Reynolds operator."; |
---|
| 1394 | ""; |
---|
| 1395 | } |
---|
| 1396 | matrix M=A(2); |
---|
| 1397 | kill G(1..g), s, slead, p, v1, I, A(2); |
---|
| 1398 | export `newring`; // we keep the ring where we computed the |
---|
| 1399 | export M; // the Molien series such that we can |
---|
| 1400 | setring br; // keep it |
---|
| 1401 | return(A(1)); |
---|
| 1402 | } |
---|
| 1403 | } |
---|
| 1404 | example |
---|
| 1405 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
| 1406 | " note the case of prime characteristic"; |
---|
| 1407 | echo=2; |
---|
| 1408 | ring R=0,(x,y,z),dp; |
---|
| 1409 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 1410 | matrix REY,M=reynolds_molien(A); |
---|
| 1411 | print(REY); |
---|
| 1412 | print(M); |
---|
| 1413 | ring S=3,(x,y,z),dp; |
---|
| 1414 | string newring="Qadjoint"; |
---|
| 1415 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 1416 | matrix REY=reynolds_molien(A,newring); |
---|
| 1417 | print(REY); |
---|
| 1418 | setring Qadjoint; |
---|
| 1419 | print(M); |
---|
| 1420 | setring S; |
---|
| 1421 | kill Qadjoint; |
---|
| 1422 | } |
---|
| 1423 | |
---|
| 1424 | proc partial_molien (matrix M, int n, list #) |
---|
| 1425 | "USAGE: partial_molien(M,n[,p]); |
---|
| 1426 | M: a 1x2 <matrix>, n: an <int> indicating number of terms in the |
---|
| 1427 | expansion, p: an optional <poly> |
---|
| 1428 | ASSUME: M is the return value of molien or the second return value of |
---|
| 1429 | reynolds_molien, p ought to be the second return value of a previous |
---|
| 1430 | run of partial_molien and avoids recalculating known terms |
---|
| 1431 | RETURN: n terms (type <poly>) of the partial expansion of the Molien series |
---|
| 1432 | (first n if there is no third parameter given, otherwise the next n |
---|
| 1433 | terms depending on a previous calculation) and an intermediate result |
---|
| 1434 | (type <poly>) of the calculation to be used as third parameter in a next |
---|
| 1435 | run of partial_molien |
---|
| 1436 | THEORY: The following calculation is implemented: |
---|
| 1437 | (1+a1x+a2x^2+...+anx^n)/(1+b1x+b2x^2+...+bmx^m)=(1+(a1-b1)x+... |
---|
| 1438 | (1+b1x+b2x^2+...+bmx^m) |
---|
| 1439 | ----------------------- |
---|
| 1440 | (a1-b1)x+(a2-b2)x^2+... |
---|
| 1441 | (a1-b1)x+b1(a1-b1)x^2+... |
---|
| 1442 | EXAMPLE: example partial_molien; shows an example |
---|
| 1443 | " |
---|
| 1444 | { poly A(2); // A(2) will contain the return value of |
---|
| 1445 | // the intermediate result |
---|
| 1446 | if (char(basering)<>0) |
---|
| 1447 | { "ERROR: you have to change to a basering of characteristic 0, one in"; |
---|
| 1448 | " which the Molien series is defined"; |
---|
| 1449 | } |
---|
| 1450 | if (ncols(M)==2 && nrows(M)==1 && n>0 && size(#)<2) |
---|
| 1451 | { def br=basering; // keeping track of the old ring |
---|
| 1452 | map slead=br,ideal(0); |
---|
| 1453 | matrix s=slead(M); |
---|
| 1454 | if (s[1,1]<>1 || s[1,2]<>1) |
---|
| 1455 | { "ERROR: the constant terms of enumerator and denominator are not 1"; |
---|
| 1456 | return(); |
---|
| 1457 | } |
---|
| 1458 | |
---|
| 1459 | if (size(#)==0) |
---|
| 1460 | { A(2)=M[1,1]; // if a third parameter is not given, the |
---|
| 1461 | // intermediate result from the last run |
---|
| 1462 | // corresponds to the numerator - we need |
---|
| 1463 | } // its smallest term |
---|
| 1464 | else |
---|
| 1465 | { if (typeof(#[1])=="poly") |
---|
| 1466 | { A(2)=#[1]; // if a third term is given we 'start' |
---|
| 1467 | } // with its smallest term |
---|
| 1468 | else |
---|
| 1469 | { "ERROR: <poly> as third parameter expected"; |
---|
| 1470 | return(); |
---|
| 1471 | } |
---|
| 1472 | } |
---|
| 1473 | poly A(1)=M[1,2]; // denominator of Molien series (for now) |
---|
| 1474 | string mp=string(minpoly); |
---|
| 1475 | execute "ring R=("+charstr(br)+"),("+varstr(br)+"),ds;"; |
---|
| 1476 | execute "minpoly=number("+mp+");"; |
---|
| 1477 | poly A(1)=0; // A(1) will contain the sum of n terms - |
---|
| 1478 | poly min; // min will be our smallest term - |
---|
| 1479 | poly A(2)=fetch(br,A(2)); // fetching A(2) and M[1,2] into R |
---|
| 1480 | poly den=fetch(br,A(1)); |
---|
| 1481 | for (int i=1; i<=n; i=i+1) // getting n terms and adding them up |
---|
| 1482 | { min=lead(A(2)); |
---|
| 1483 | A(1)=A(1)+min; |
---|
| 1484 | A(2)=A(2)-min*den; |
---|
| 1485 | } |
---|
| 1486 | setring br; // moving A(1) and A(2) back in the |
---|
| 1487 | A(1)=fetch(R,A(1)); // actual ring for output |
---|
| 1488 | A(2)=fetch(R,A(2)); |
---|
| 1489 | return(A(1..2)); |
---|
| 1490 | } |
---|
| 1491 | else |
---|
| 1492 | { "ERROR: the first parameter has to be a 1x2-matrix, i.e. the matrix"; |
---|
| 1493 | " returned by the procedure 'reynolds_molien', the second one"; |
---|
| 1494 | " should be > 0 and there should be no more than 3 parameters;" |
---|
| 1495 | return(); |
---|
| 1496 | } |
---|
| 1497 | } |
---|
| 1498 | example |
---|
| 1499 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
| 1500 | echo=2; |
---|
| 1501 | ring R=0,(x,y,z),dp; |
---|
| 1502 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 1503 | matrix REY,M=reynolds_molien(A); |
---|
| 1504 | poly p(1..2); |
---|
| 1505 | p(1..2)=partial_molien(M,5); |
---|
| 1506 | p(1); |
---|
| 1507 | p(1..2)=partial_molien(M,5,p(2)); |
---|
| 1508 | p(1); |
---|
| 1509 | } |
---|
| 1510 | |
---|
| 1511 | proc evaluate_reynolds (matrix REY, ideal I) |
---|
| 1512 | "USAGE: evaluate_reynolds(REY,I); |
---|
| 1513 | REY: a <matrix> representing the Reynolds operator, I: an arbitrary |
---|
| 1514 | <ideal> |
---|
| 1515 | ASSUME: REY is the first return value of group_reynolds() or reynolds_molien() |
---|
| 1516 | RETURNS: image of the polynomials defining I under the Reynolds operator |
---|
| 1517 | (type <ideal>) |
---|
| 1518 | NOTE: the characteristic of the coefficient field of the polynomial ring |
---|
| 1519 | should not divide the order of the finite matrix group |
---|
| 1520 | EXAMPLE: example evaluate_reynolds; shows an example |
---|
| 1521 | THEORY: REY has been constructed in such a way that each row serves as a ring |
---|
| 1522 | mapping of which the Reynolds operator is made up. |
---|
| 1523 | " |
---|
| 1524 | { def br=basering; |
---|
| 1525 | int n=nvars(br); |
---|
| 1526 | if (ncols(REY)==n) |
---|
| 1527 | { int m=nrows(REY); // we need m to 'cut' the ring |
---|
| 1528 | // homomorphisms 'out' of REY and to |
---|
| 1529 | // divide by the group order in the end |
---|
| 1530 | int num_poly=ncols(I); |
---|
| 1531 | matrix MI=matrix(I); |
---|
| 1532 | matrix MiI[1][num_poly]; |
---|
| 1533 | map pREY; |
---|
| 1534 | matrix rowREY[1][n]; |
---|
| 1535 | for (int i=1;i<=m;i=i+1) |
---|
| 1536 | { rowREY=REY[i,1..n]; |
---|
| 1537 | pREY=br,ideal(rowREY); // f is now the i-th ring homomorphism |
---|
| 1538 | MiI=pREY(MI)+MiI; |
---|
| 1539 | } |
---|
| 1540 | MiI=(1/number(m))*MiI; |
---|
| 1541 | return(ideal(MiI)); |
---|
| 1542 | } |
---|
| 1543 | else |
---|
| 1544 | { "ERROR: the number of columns in the <matrix> should be the same as the"; |
---|
| 1545 | " number of variables in the basering; in fact it should be first"; |
---|
| 1546 | " return value of group_reynolds() or reynolds_molien()."; |
---|
| 1547 | return(); |
---|
| 1548 | } |
---|
| 1549 | } |
---|
| 1550 | example |
---|
| 1551 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
| 1552 | echo=2; |
---|
| 1553 | ring R=0,(x,y,z),dp; |
---|
| 1554 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 1555 | list L=group_reynolds(A); |
---|
| 1556 | ideal I=x2,y2,z2; |
---|
| 1557 | print(evaluate_reynolds(L[1],I)); |
---|
| 1558 | } |
---|
| 1559 | |
---|
| 1560 | proc invariant_basis (int g,list #) |
---|
| 1561 | "USAGE: invariant_basis(g,G1,G2,...); |
---|
| 1562 | g: an <int> indicating of which degree (>0) the homogeneous basis |
---|
| 1563 | shoud be, G1,G2,...: <matrices> generating a finite matrix group |
---|
| 1564 | RETURNS: the basis (type <ideal>) of the space of invariants of degree g |
---|
| 1565 | EXAMPLE: example invariant_basis; shows an example |
---|
| 1566 | THEORY: A general polynomial of degree g is generated and the generators of the |
---|
| 1567 | matrix group applied. The difference ought to be 0 and this way a |
---|
| 1568 | system of linear equations is created. It is solved by computing |
---|
| 1569 | syzygies. |
---|
| 1570 | " |
---|
| 1571 | { if (g<=0) |
---|
| 1572 | { "ERROR: the first parameter should be > 0"; |
---|
| 1573 | return(); |
---|
| 1574 | } |
---|
| 1575 | def br=basering; |
---|
| 1576 | ideal mon=sort(maxideal(g))[1]; // needed for constructing a general |
---|
| 1577 | int m=ncols(mon); // homogeneous polynomial of degree g |
---|
| 1578 | mon=sort(mon,intvec(m..1))[1]; |
---|
| 1579 | int a=size(#); |
---|
| 1580 | int i; |
---|
| 1581 | int n=nvars(br); |
---|
| 1582 | //---------------------- checking that the input is ok ----------------------- |
---|
| 1583 | for (i=1;i<=a;i=i+1) |
---|
| 1584 | { if (typeof(#[i])=="matrix") |
---|
| 1585 | { if (nrows(#[i])==n && ncols(#[i])==n) |
---|
| 1586 | { matrix G(i)=#[i]; |
---|
| 1587 | } |
---|
| 1588 | else |
---|
| 1589 | { "ERROR: the number of variables of the base ring needs to be the same"; |
---|
| 1590 | " as the dimension of the square matrices"; |
---|
| 1591 | return(); |
---|
| 1592 | } |
---|
| 1593 | } |
---|
| 1594 | else |
---|
| 1595 | { "ERROR: the last parameters should be a list of matrices"; |
---|
| 1596 | return(); |
---|
| 1597 | } |
---|
| 1598 | } |
---|
| 1599 | //---------------------------------------------------------------------------- |
---|
| 1600 | execute "ring T=("+charstr(br)+"),("+varstr(br)+",p(1..m)),lp;"; |
---|
| 1601 | // p(1..m) are the general coefficients of the general polynomial of degree g |
---|
| 1602 | execute "ideal vars="+varstr(br)+";"; |
---|
| 1603 | map f; |
---|
| 1604 | ideal mon=imap(br,mon); |
---|
| 1605 | poly P=0; |
---|
| 1606 | for (i=m;i>=1;i=i-1) |
---|
| 1607 | { P=P+p(i)*mon[i]; // P is the general polynomial |
---|
| 1608 | } |
---|
| 1609 | ideal I; // will help substituting variables in P |
---|
| 1610 | // by linear combinations of variables - |
---|
| 1611 | poly Pnew,temp; // Pnew is P with substitutions - |
---|
| 1612 | matrix S[m*a][m]; // will contain system of linear |
---|
| 1613 | // equations |
---|
| 1614 | int j,k; |
---|
| 1615 | //------------------- building the system of linear equations ---------------- |
---|
| 1616 | for (i=1;i<=a;i=i+1) |
---|
| 1617 | { I=ideal(matrix(vars)*transpose(imap(br,G(i)))); |
---|
| 1618 | I=I,p(1..m); |
---|
| 1619 | f=T,I; |
---|
| 1620 | Pnew=f(P); |
---|
| 1621 | for (j=1;j<=m;j=j+1) |
---|
| 1622 | { temp=P/mon[j]-Pnew/mon[j]; |
---|
| 1623 | for (k=1;k<=m;k=k+1) |
---|
| 1624 | { S[m*(i-1)+j,k]=temp/p(k); |
---|
| 1625 | } |
---|
| 1626 | } |
---|
| 1627 | } |
---|
| 1628 | //---------------------------------------------------------------------------- |
---|
| 1629 | setring br; |
---|
| 1630 | map f=T,ideal(0); |
---|
| 1631 | matrix S=f(S); |
---|
| 1632 | matrix s=matrix(syz(S)); // s contains a basis of the space of |
---|
| 1633 | // solutions - |
---|
| 1634 | ideal I=ideal(matrix(mon)*s); // I contains a basis of homogeneous |
---|
| 1635 | if (I[1]<>0) // invariants of degree d |
---|
| 1636 | { for (i=1;i<=ncols(I);i=i+1) |
---|
| 1637 | { I[i]=I[i]/leadcoef(I[i]); // setting leading coefficients to 1 |
---|
| 1638 | } |
---|
| 1639 | } |
---|
| 1640 | return(I); |
---|
| 1641 | } |
---|
| 1642 | example |
---|
| 1643 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
| 1644 | echo=2; |
---|
| 1645 | ring R=0,(x,y,z),dp; |
---|
| 1646 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 1647 | print(invariant_basis(2,A)); |
---|
| 1648 | } |
---|
| 1649 | |
---|
| 1650 | proc invariant_basis_reynolds (matrix REY,int d,list #) |
---|
| 1651 | "USAGE: invariant_basis_reynolds(REY,d[,flags]); |
---|
| 1652 | REY: a <matrix> representing the Reynolds operator, d: an <int> |
---|
| 1653 | indicating of which degree (>0) the homogeneous basis shoud be, flags: |
---|
| 1654 | an optional <intvec> with two entries: its first component gives the |
---|
| 1655 | dimension of the space (default <0 meaning unknown) and its second |
---|
| 1656 | component is used as the number of polynomials that should be mapped |
---|
| 1657 | to invariants during one call of evaluate_reynolds if the dimension of |
---|
| 1658 | the space is unknown or the number such that number x dimension |
---|
| 1659 | polynomials are mapped to invariants during one call of |
---|
| 1660 | evaluate_reynolds |
---|
| 1661 | ASSUME: REY is the first return value of group_reynolds() or reynolds_molien() |
---|
| 1662 | and flags[1] given by partial_molien |
---|
| 1663 | RETURN: the basis (type <ideal>) of the space of invariants of degree d |
---|
| 1664 | EXAMPLE: example invariant_basis_reynolds; shows an example |
---|
| 1665 | THEORY: Monomials of degree d are mapped to invariants with the Reynolds |
---|
| 1666 | operator. A linearly independent set is generated with the help of |
---|
| 1667 | minbase. |
---|
| 1668 | " |
---|
| 1669 | { |
---|
| 1670 | //---------------------- checking that the input is ok ----------------------- |
---|
| 1671 | if (d<=0) |
---|
| 1672 | { " ERROR: the second parameter should be > 0"; |
---|
| 1673 | return(); |
---|
| 1674 | } |
---|
| 1675 | if (size(#)>1) |
---|
| 1676 | { " ERROR: there should be at most three parameters"; |
---|
| 1677 | return(); |
---|
| 1678 | } |
---|
| 1679 | if (size(#)==1) |
---|
| 1680 | { if (typeof(#[1])<>"intvec") |
---|
| 1681 | { " ERROR: the third parameter should be of type <intvec>"; |
---|
| 1682 | return(); |
---|
| 1683 | } |
---|
| 1684 | if (size(#[1])<>2) |
---|
| 1685 | { " ERROR: there should be two components in <intvec>"; |
---|
| 1686 | return(); |
---|
| 1687 | } |
---|
| 1688 | else |
---|
| 1689 | { int cd=#[1][1]; |
---|
| 1690 | int step_fac=#[1][2]; |
---|
| 1691 | } |
---|
| 1692 | if (step_fac<=0) |
---|
| 1693 | { " ERROR: the second component of <intvec> should be > 0"; |
---|
| 1694 | return(); |
---|
| 1695 | } |
---|
| 1696 | if (cd==0) |
---|
| 1697 | { return(ideal(0)); |
---|
| 1698 | } |
---|
| 1699 | } |
---|
| 1700 | else |
---|
| 1701 | { int step_fac=1; |
---|
| 1702 | int cd=-1; |
---|
| 1703 | } |
---|
| 1704 | if (ncols(REY)<>nvars(basering)) |
---|
| 1705 | { "ERROR: the number of columns in the <matrix> should be the same as the"; |
---|
| 1706 | " number of variables in the basering; in fact it should be first"; |
---|
| 1707 | " return value of group_reynolds() or reynolds_molien()."; |
---|
| 1708 | return(); |
---|
| 1709 | } |
---|
| 1710 | //---------------------------------------------------------------------------- |
---|
| 1711 | ideal mon=sort(maxideal(d))[1]; |
---|
| 1712 | degBound=d; |
---|
| 1713 | int j=ncols(mon); |
---|
| 1714 | mon=sort(mon,intvec(j..1))[1]; |
---|
| 1715 | ideal B; // will contain the basis |
---|
| 1716 | if (cd<0) |
---|
| 1717 | { if (step_fac>j) // all of mon will be mapped to |
---|
| 1718 | { B=evaluate_reynolds(REY,mon); // invariants at once |
---|
| 1719 | B=minbase(B); |
---|
| 1720 | degBound=0; |
---|
| 1721 | return(B); |
---|
| 1722 | } |
---|
| 1723 | } |
---|
| 1724 | else |
---|
| 1725 | { if (step_fac*cd>j) // all of mon will be mapped to |
---|
| 1726 | { B=evaluate_reynolds(REY,mon); // invariants at once |
---|
| 1727 | B=minbase(B); |
---|
| 1728 | degBound=0; |
---|
| 1729 | return(B); |
---|
| 1730 | } |
---|
| 1731 | } |
---|
| 1732 | int i,k; |
---|
| 1733 | int upper_bound=0; |
---|
| 1734 | int lower_bound=0; |
---|
| 1735 | ideal part_mon; // a part of mon of size step_fac*cd |
---|
| 1736 | while (1) |
---|
| 1737 | { lower_bound=upper_bound+1; |
---|
| 1738 | if (cd<0) |
---|
| 1739 | { upper_bound=upper_bound+step_fac; |
---|
| 1740 | } |
---|
| 1741 | else |
---|
| 1742 | { upper_bound=upper_bound+step_fac*cd; |
---|
| 1743 | } |
---|
| 1744 | if (upper_bound>j) |
---|
| 1745 | { upper_bound=j; |
---|
| 1746 | } |
---|
| 1747 | part_mon=mon[lower_bound..upper_bound]; |
---|
| 1748 | B=minbase(B+evaluate_reynolds(REY,part_mon)); |
---|
| 1749 | if ((ncols(B)==cd and B[1]<>0) or upper_bound==j) |
---|
| 1750 | { degBound=0; |
---|
| 1751 | return(B); |
---|
| 1752 | } |
---|
| 1753 | } |
---|
| 1754 | } |
---|
| 1755 | example |
---|
| 1756 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
| 1757 | echo=2; |
---|
| 1758 | ring R=0,(x,y,z),dp; |
---|
| 1759 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 1760 | intvec flags=0,1,0; |
---|
| 1761 | matrix REY,M=reynolds_molien(A,flags); |
---|
| 1762 | flags=8,6; |
---|
| 1763 | print(invariant_basis_reynolds(REY,6,flags)); |
---|
| 1764 | } |
---|
| 1765 | |
---|
| 1766 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 1767 | // This procedure generates linearly independent invariant polynomials of degree |
---|
| 1768 | // d that do not reduce to 0 modulo the primary invariants. It does this by |
---|
| 1769 | // applying the Reynolds operator to the monomials returned by kbase(sP,d). The |
---|
| 1770 | // result is used when computing secondary invariants. |
---|
| 1771 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 1772 | proc sort_of_invariant_basis (ideal sP,matrix REY,int d,int step_fac) |
---|
| 1773 | { ideal mon=kbase(sP,d); |
---|
| 1774 | degBound=d; |
---|
| 1775 | int j=ncols(mon); |
---|
| 1776 | int i; |
---|
| 1777 | mon=sort(mon,intvec(j..1))[1]; |
---|
| 1778 | ideal B; // will contain the "sort of basis" |
---|
| 1779 | if (step_fac>j) |
---|
| 1780 | { B=compress(evaluate_reynolds(REY,mon)); |
---|
| 1781 | for (i=1;i<=ncols(B);i=i+1) // those are taken our that are o mod sP |
---|
| 1782 | { if (reduce(B[i],sP)==0) |
---|
| 1783 | { B[i]=0; |
---|
| 1784 | } |
---|
| 1785 | } |
---|
| 1786 | B=minbase(B); // here are the linearly independent ones |
---|
| 1787 | degBound=0; |
---|
| 1788 | return(B); |
---|
| 1789 | } |
---|
| 1790 | int upper_bound=0; |
---|
| 1791 | int lower_bound=0; |
---|
| 1792 | ideal part_mon; // parts of mon |
---|
| 1793 | while (1) |
---|
| 1794 | { lower_bound=upper_bound+1; |
---|
| 1795 | upper_bound=upper_bound+step_fac; |
---|
| 1796 | if (upper_bound>j) |
---|
| 1797 | { upper_bound=j; |
---|
| 1798 | } |
---|
| 1799 | part_mon=mon[lower_bound..upper_bound]; |
---|
| 1800 | part_mon=compress(evaluate_reynolds(REY,part_mon)); |
---|
| 1801 | for (i=1;i<=ncols(part_mon);i=i+1) |
---|
| 1802 | { if (reduce(part_mon[i],sP)==0) |
---|
| 1803 | { part_mon[i]=0; |
---|
| 1804 | } |
---|
| 1805 | } |
---|
| 1806 | B=minbase(B+part_mon); // here are the linearly independent ones |
---|
| 1807 | if (upper_bound==j) |
---|
| 1808 | { degBound=0; |
---|
| 1809 | return(B); |
---|
| 1810 | } |
---|
| 1811 | } |
---|
| 1812 | } |
---|
| 1813 | |
---|
| 1814 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 1815 | // Procedure returning the succeeding vector after vec. It is used to list |
---|
| 1816 | // all the vectors of Z^n with first nonzero entry 1. They are listed by |
---|
| 1817 | // increasing sum of the absolute value of their entries. |
---|
| 1818 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 1819 | proc next_vector(intmat vec) |
---|
| 1820 | { int n=ncols(vec); // p: >0, n: <0, p0: >=0, n0: <=0 |
---|
| 1821 | for (int i=1;i<=n;i=i+1) // finding out which is the first |
---|
| 1822 | { if (vec[1,i]<>0) // component <>0 |
---|
| 1823 | { break; |
---|
| 1824 | } |
---|
| 1825 | } |
---|
| 1826 | intmat new[1][n]; |
---|
| 1827 | if (i>n) // 0,...,0 --> 1,0....,0 |
---|
| 1828 | { new[1,1]=1; |
---|
| 1829 | return(new); |
---|
| 1830 | } |
---|
| 1831 | if (i==n) // 0,...,1 --> 1,1,0,...,0 |
---|
| 1832 | { new[1,1..2]=1,1; |
---|
| 1833 | return(new); |
---|
| 1834 | } |
---|
| 1835 | if (i==n-1) |
---|
| 1836 | { if (vec[1,n]==0) // 0,...,0,1,0 --> 0,...,0,1 |
---|
| 1837 | { new[1,n]=1; |
---|
| 1838 | return(new); |
---|
| 1839 | } |
---|
| 1840 | if (vec[1,n]>0) // 0,..,0,1,p --> 0,...,0,1,-p |
---|
| 1841 | { new[1,1..n]=vec[1,1..n-1],-vec[1,n]; |
---|
| 1842 | return(new); |
---|
| 1843 | } |
---|
| 1844 | new[1,1..2]=1,1-vec[1,n]; // 0,..,0,1,n --> 1,1-n,0,..,0 |
---|
| 1845 | return(new); |
---|
| 1846 | } |
---|
| 1847 | if (i>1) |
---|
| 1848 | { intmat temp[1][n-i+1]=vec[1,i..n]; // 0,...,0,1,*,...,* --> 1,*,...,* |
---|
| 1849 | temp=next_vector(temp); |
---|
| 1850 | new[1,i..n]=temp[1,1..n-i+1]; |
---|
| 1851 | return(new); |
---|
| 1852 | } // case left: 1,*,...,* |
---|
| 1853 | for (i=2;i<=n;i=i+1) |
---|
| 1854 | { if (vec[1,i]>0) // make first positive negative and |
---|
| 1855 | { vec[1,i]=-vec[1,i]; // return |
---|
| 1856 | return(vec); |
---|
| 1857 | } |
---|
| 1858 | else |
---|
| 1859 | { vec[1,i]=-vec[1,i]; // make all negatives before positives |
---|
| 1860 | } // positive |
---|
| 1861 | } |
---|
| 1862 | for (i=2;i<=n-1;i=i+1) // case: 1,p,...,p after 1,n,...,n |
---|
| 1863 | { if (vec[1,i]>0) |
---|
| 1864 | { vec[1,2]=vec[1,i]-1; // shuffleing things around... |
---|
| 1865 | if (i>2) // same sum of absolute values of entries |
---|
| 1866 | { vec[1,i]=0; |
---|
| 1867 | } |
---|
| 1868 | vec[1,i+1]=vec[1,i+1]+1; |
---|
| 1869 | return(vec); |
---|
| 1870 | } |
---|
| 1871 | } // case left: 1,0,...,0 --> 1,1,0,...,0 |
---|
| 1872 | new[1,2..3]=1,vec[1,n]; // and: 1,0,...,0,1 --> 0,1,1,0,...,0 |
---|
| 1873 | return(new); |
---|
| 1874 | } |
---|
| 1875 | |
---|
| 1876 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 1877 | // Maps integers to elements of the base field. It is only called if the base |
---|
| 1878 | // field is of prime characteristic. If the base field has q elements (depending |
---|
| 1879 | // on minpoly) 1..q is mapped to those q elements. |
---|
| 1880 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 1881 | proc int_number_map (int i) |
---|
| 1882 | { int p=char(basering); |
---|
| 1883 | if (minpoly==0) // if no minpoly is given, we have p |
---|
| 1884 | { i=i%p; // elements in the field |
---|
| 1885 | return(number(i)); |
---|
| 1886 | } |
---|
| 1887 | int d=pardeg(minpoly); |
---|
| 1888 | if (i<0) |
---|
| 1889 | { int bool=1; |
---|
| 1890 | i=(-1)*i; |
---|
| 1891 | } |
---|
| 1892 | i=i%p^d; // base field has p^d elements - |
---|
| 1893 | number a=par(1); // a is the root of the minpoly - we have |
---|
| 1894 | number out=0; // to construct a linear combination of |
---|
| 1895 | int j=1; // a^k |
---|
| 1896 | int k; |
---|
| 1897 | while (1) |
---|
| 1898 | { if (i<p^j) // finding an upper bound on i |
---|
| 1899 | { for (k=0;k<j-1;k=k+1) |
---|
| 1900 | { out=out+((i/p^k)%p)*a^k; // finding how often p^k is contained in |
---|
| 1901 | } // i |
---|
| 1902 | out=out+(i/p^(j-1))*a^(j-1); |
---|
| 1903 | if (defined(bool)==voice) |
---|
| 1904 | { return((-1)*out); |
---|
| 1905 | } |
---|
| 1906 | return(out); |
---|
| 1907 | } |
---|
| 1908 | j=j+1; |
---|
| 1909 | } |
---|
| 1910 | } |
---|
| 1911 | |
---|
| 1912 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 1913 | // This procedure finds dif primary invariants in degree d. It returns all |
---|
| 1914 | // primary invariants found so far. The coefficients lie in a field of |
---|
| 1915 | // characteristic 0. |
---|
| 1916 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 1917 | proc search (int n,int d,ideal B,int cd,ideal P,ideal sP,int i,int dif,int dB,ideal CI) |
---|
| 1918 | { intmat vec[1][cd]; // the coefficients for the next |
---|
| 1919 | // combination - |
---|
| 1920 | degBound=0; |
---|
| 1921 | poly test_poly; // the linear combination to test |
---|
| 1922 | int test_dim; |
---|
| 1923 | intvec h; // Hilbert series |
---|
| 1924 | int j=i+1; |
---|
| 1925 | matrix tB=transpose(B); |
---|
| 1926 | ideal TEST; |
---|
| 1927 | while(j<=i+dif) |
---|
| 1928 | { CI=CI+ideal(var(j)^d); // homogeneous polynomial of the same |
---|
| 1929 | // degree as the one we're looking for is |
---|
| 1930 | // added |
---|
| 1931 | // h=hilb(std(CI),1); |
---|
| 1932 | dB=dB+d-1; // used as degBound |
---|
| 1933 | while(1) |
---|
| 1934 | { vec=next_vector(vec); // next vector |
---|
| 1935 | test_poly=(vec*tB)[1,1]; |
---|
| 1936 | // degBound=dB; |
---|
| 1937 | TEST=sP+ideal(test_poly); |
---|
| 1938 | attrib(TEST,"isSB",1); |
---|
| 1939 | test_dim=dim(TEST); |
---|
| 1940 | // degBound=0; |
---|
| 1941 | if (n-test_dim==j) // the dimension has been lowered by one |
---|
| 1942 | { sP=TEST; |
---|
| 1943 | break; |
---|
| 1944 | } |
---|
| 1945 | // degBound=dB; |
---|
| 1946 | TEST=std(sP+ideal(test_poly)); // should soon be replaced by next line |
---|
| 1947 | // TEST=std(sP,test_poly,h); // Hilbert driven std-calculation |
---|
| 1948 | test_dim=dim(TEST); |
---|
| 1949 | // degBound=0; |
---|
| 1950 | if (n-test_dim==j) // the dimension has been lowered by one |
---|
| 1951 | { sP=TEST; |
---|
| 1952 | break; |
---|
| 1953 | } |
---|
| 1954 | } |
---|
| 1955 | P[j]=test_poly; // test_poly ist added to primary |
---|
| 1956 | j=j+1; // invariants |
---|
| 1957 | } |
---|
| 1958 | return(P,sP,CI,dB); |
---|
| 1959 | } |
---|
| 1960 | |
---|
| 1961 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 1962 | // This procedure finds at most dif primary invariants in degree d. It returns |
---|
| 1963 | // all primary invariants found so far. The coefficients lie in the field of |
---|
| 1964 | // characteristic p>0. |
---|
| 1965 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 1966 | proc p_search (int n,int d,ideal B,int cd,ideal P,ideal sP,int i,int dif,int dB,ideal CI) |
---|
| 1967 | { def br=basering; |
---|
| 1968 | degBound=0; |
---|
| 1969 | matrix vec(1)[1][cd]; // starting with 0-vector - |
---|
| 1970 | intmat new[1][cd]; // the coefficients for the next |
---|
| 1971 | // combination - |
---|
| 1972 | matrix pnew[1][cd]; // new needs to be mapped into br - |
---|
| 1973 | int counter=1; // counts the vectors |
---|
| 1974 | int j; |
---|
| 1975 | int p=char(br); |
---|
| 1976 | if (minpoly<>0) |
---|
| 1977 | { int ext_deg=pardeg(minpoly); // field has p^d elements |
---|
| 1978 | } |
---|
| 1979 | else |
---|
| 1980 | { int ext_deg=1; // field has p^d elements |
---|
| 1981 | } |
---|
| 1982 | poly test_poly; // the linear combination to test |
---|
| 1983 | int test_dim; |
---|
| 1984 | ring R=0,x,dp; // just to calculate next variable |
---|
| 1985 | // bound - |
---|
| 1986 | number bound=(number(p)^(ext_deg*cd)-1)/(number(p)^ext_deg-1)+1; // this is |
---|
| 1987 | // how many linearly independent vectors |
---|
| 1988 | // of size cd exist having entries in the |
---|
| 1989 | // base field of br |
---|
| 1990 | setring br; |
---|
| 1991 | intvec h; // Hilbert series |
---|
| 1992 | int k=i+1; |
---|
| 1993 | matrix tB=transpose(B); |
---|
| 1994 | ideal TEST; |
---|
| 1995 | while (k<=i+dif) |
---|
| 1996 | { CI=CI+ideal(var(k)^d); // homogeneous polynomial of the same |
---|
| 1997 | // degree as the one we're looking for is |
---|
| 1998 | // added |
---|
| 1999 | // h=hilb(std(CI),1); |
---|
| 2000 | dB=dB+d-1; // used as degBound |
---|
| 2001 | setring R; |
---|
| 2002 | while (number(counter)<>bound) // otherwise, we are done |
---|
| 2003 | { setring br; |
---|
| 2004 | new=next_vector(new); |
---|
| 2005 | for (j=1;j<=cd;j=j+1) |
---|
| 2006 | { pnew[1,j]=int_number_map(new[1,j]); // mapping an integer into br |
---|
| 2007 | } |
---|
| 2008 | if (unique(vec(1..counter),pnew)) // checking whether we tried pnew before |
---|
| 2009 | { counter=counter+1; |
---|
| 2010 | matrix vec(counter)=pnew; // keeping track of the ones we tried - |
---|
| 2011 | test_poly=(vec(counter)*tB)[1,1]; // linear combination - |
---|
| 2012 | // degBound=dB; |
---|
| 2013 | TEST=sP+ideal(test_poly); |
---|
| 2014 | attrib(TEST,"isSB",1); |
---|
| 2015 | test_dim=dim(TEST); |
---|
| 2016 | // degBound=0; |
---|
| 2017 | if (n-test_dim==k) // the dimension has been lowered by one |
---|
| 2018 | { sP=TEST; |
---|
| 2019 | setring R; |
---|
| 2020 | break; |
---|
| 2021 | } |
---|
| 2022 | // degBound=dB; |
---|
| 2023 | TEST=std(sP+ideal(test_poly)); // should soon to be replaced by next |
---|
| 2024 | // line |
---|
| 2025 | // TEST=std(sP,test_poly,h); // Hilbert driven std-calculation |
---|
| 2026 | test_dim=dim(TEST); |
---|
| 2027 | // degBound=0; |
---|
| 2028 | if (n-test_dim==k) // the dimension has been lowered by one |
---|
| 2029 | { sP=TEST; |
---|
| 2030 | setring R; |
---|
| 2031 | break; |
---|
| 2032 | } |
---|
| 2033 | } |
---|
| 2034 | setring R; |
---|
| 2035 | } |
---|
| 2036 | if (number(counter)<=bound) |
---|
| 2037 | { setring br; |
---|
| 2038 | P[k]=test_poly; // test_poly ist added to primary |
---|
| 2039 | } // invariants |
---|
| 2040 | else |
---|
| 2041 | { setring br; |
---|
| 2042 | CI=CI[1..size(CI)-1]; |
---|
| 2043 | return(P,sP,CI,dB-d+1); |
---|
| 2044 | } |
---|
| 2045 | k=k+1; |
---|
| 2046 | } |
---|
| 2047 | return(P,sP,CI,dB); |
---|
| 2048 | } |
---|
| 2049 | |
---|
| 2050 | proc primary_char0 (matrix REY,matrix M,list #) |
---|
| 2051 | "USAGE: primary_char0(REY,M[,v]); |
---|
| 2052 | REY: a <matrix> representing the Reynolds operator, M: a 1x2 <matrix> |
---|
| 2053 | representing the Molien series, v: an optional <int> |
---|
| 2054 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien and |
---|
| 2055 | M the one of molien or the second one of reynolds_molien |
---|
| 2056 | DISPLAY: information about the various stages of the programme if v does not |
---|
| 2057 | equal 0 |
---|
| 2058 | RETURN: primary invariants (type <matrix>) of the invariant ring |
---|
| 2059 | EXAMPLE: example primary_char0; shows an example |
---|
| 2060 | THEORY: Bases of homogeneous invariants are generated successively and those |
---|
| 2061 | are chosen as primary invariants that lower the dimension of the ideal |
---|
| 2062 | generated by the previously found invariants (see paper \"Generating a |
---|
| 2063 | Noetherian Normalization of the Invariant Ring of a Finite Group\" by |
---|
| 2064 | Decker, Heydtmann, Schreyer (1997) to appear in JSC). |
---|
| 2065 | " |
---|
| 2066 | { degBound=0; |
---|
| 2067 | if (char(basering)<>0) |
---|
| 2068 | { "ERROR: primary_char0 should only be used with rings of characteristic 0."; |
---|
| 2069 | return(); |
---|
| 2070 | } |
---|
| 2071 | //----------------- checking input and setting verbose mode ------------------ |
---|
| 2072 | if (size(#)>1) |
---|
| 2073 | { "ERROR: primary_char0 can only have three parameters."; |
---|
| 2074 | return(); |
---|
| 2075 | } |
---|
| 2076 | if (size(#)==1) |
---|
| 2077 | { if (typeof(#[1])<>"int") |
---|
| 2078 | { "ERROR: The third parameter should be of type <int>."; |
---|
| 2079 | return(); |
---|
| 2080 | } |
---|
| 2081 | else |
---|
| 2082 | { int v=#[1]; |
---|
| 2083 | } |
---|
| 2084 | } |
---|
| 2085 | else |
---|
| 2086 | { int v=0; |
---|
| 2087 | } |
---|
| 2088 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 2089 | // as the size of the matrices, as well |
---|
| 2090 | // as the number of primary invariants, |
---|
| 2091 | // we should get |
---|
| 2092 | if (ncols(REY)<>n) |
---|
| 2093 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
| 2094 | return(); |
---|
| 2095 | } |
---|
| 2096 | if (ncols(M)<>2 or nrows(M)<>1) |
---|
| 2097 | { "ERROR: Second parameter ought to be the Molien series." |
---|
| 2098 | return(); |
---|
| 2099 | } |
---|
| 2100 | //---------------------------------------------------------------------------- |
---|
| 2101 | if (v && voice<>2) |
---|
| 2102 | { " We can start looking for primary invariants..."; |
---|
| 2103 | ""; |
---|
| 2104 | } |
---|
| 2105 | if (v && voice==2) |
---|
| 2106 | { ""; |
---|
| 2107 | } |
---|
| 2108 | //------------------------- initializing variables --------------------------- |
---|
| 2109 | int dB; |
---|
| 2110 | poly p(1..2); // p(1) will be used for single terms of |
---|
| 2111 | // the partial expansion, p(2) to store |
---|
| 2112 | p(1..2)=partial_molien(M,1); // the intermediate result - |
---|
| 2113 | poly v1=var(1); // we need v1 to split off coefficients |
---|
| 2114 | // in the partial expansion of M (which |
---|
| 2115 | // is in terms of the first variable) - |
---|
| 2116 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
| 2117 | // space of invariants of degree d, |
---|
| 2118 | // newdim: dimension the ideal generated |
---|
| 2119 | // the primary invariants plus basis |
---|
| 2120 | // elements, dif=n-i-newdim, i.e. the |
---|
| 2121 | // number of new primary invairants that |
---|
| 2122 | // should be added in this degree - |
---|
| 2123 | ideal P,Pplus,sPplus,CI,B; // P: will contain primary invariants, |
---|
| 2124 | // Pplus: P+B, CI: a complete |
---|
| 2125 | // intersection with the same Hilbert |
---|
| 2126 | // function as P |
---|
| 2127 | ideal sP=std(P); |
---|
| 2128 | dB=1; // used as degree bound |
---|
| 2129 | int i=0; |
---|
| 2130 | //-------------- loop that searches for primary invariants ------------------ |
---|
| 2131 | while(1) // repeat until n primary invariants are |
---|
| 2132 | { // found - |
---|
| 2133 | p(1..2)=partial_molien(M,1,p(2)); // next term of the partial expansion - |
---|
| 2134 | d=deg(p(1)); // degree where we'll search - |
---|
| 2135 | cd=int(coef(p(1),v1)[2,1]); // dimension of the homogeneous space of |
---|
| 2136 | // inviarants of degree d |
---|
| 2137 | if (v) |
---|
| 2138 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
| 2139 | } |
---|
| 2140 | B=invariant_basis_reynolds(REY,d,intvec(cd,6)); // basis of invariants of |
---|
| 2141 | // degree d |
---|
| 2142 | if (B[1]<>0) |
---|
| 2143 | { Pplus=P+B; |
---|
| 2144 | sPplus=std(Pplus); |
---|
| 2145 | newdim=dim(sPplus); |
---|
| 2146 | dif=n-i-newdim; |
---|
| 2147 | } |
---|
| 2148 | else |
---|
| 2149 | { dif=0; |
---|
| 2150 | } |
---|
| 2151 | if (dif<>0) // we have to find dif new primary |
---|
| 2152 | { // invariants |
---|
| 2153 | if (cd<>dif) |
---|
| 2154 | { P,sP,CI,dB=search(n,d,B,cd,P,sP,i,dif,dB,CI); // searching for dif invariants |
---|
| 2155 | } // i.e. we can take all of B |
---|
| 2156 | else |
---|
| 2157 | { for(j=i+1;j>i+dif;j=j+1) |
---|
| 2158 | { CI=CI+ideal(var(j)^d); |
---|
| 2159 | } |
---|
| 2160 | dB=dB+dif*(d-1); |
---|
| 2161 | P=Pplus; |
---|
| 2162 | sP=sPplus; |
---|
| 2163 | } |
---|
| 2164 | if (v) |
---|
| 2165 | { for (j=1;j<=dif;j=j+1) |
---|
| 2166 | { " We find: "+string(P[i+j]); |
---|
| 2167 | } |
---|
| 2168 | } |
---|
| 2169 | i=i+dif; |
---|
| 2170 | if (i==n) // found all primary invariants |
---|
| 2171 | { if (v) |
---|
| 2172 | { ""; |
---|
| 2173 | " We found all primary invariants."; |
---|
| 2174 | ""; |
---|
| 2175 | } |
---|
| 2176 | return(matrix(P)); |
---|
| 2177 | } |
---|
| 2178 | } // done with degree d |
---|
| 2179 | } |
---|
| 2180 | } |
---|
| 2181 | example |
---|
| 2182 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
| 2183 | echo=2; |
---|
| 2184 | ring R=0,(x,y,z),dp; |
---|
| 2185 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 2186 | matrix REY,M=reynolds_molien(A); |
---|
| 2187 | matrix P=primary_char0(REY,M); |
---|
| 2188 | print(P); |
---|
| 2189 | } |
---|
| 2190 | |
---|
| 2191 | proc primary_charp (matrix REY,string ring_name,list #) |
---|
| 2192 | "USAGE: primary_charp(REY,ringname[,v]); |
---|
| 2193 | REY: a <matrix> representing the Reynolds operator, ringname: a |
---|
| 2194 | <string> giving the name of a ring where the Molien series is stored, |
---|
| 2195 | v: an optional <int> |
---|
| 2196 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien and |
---|
| 2197 | ringname gives the name of a ring of characteristic 0 that has been |
---|
| 2198 | created by molien or reynolds_molien |
---|
| 2199 | DISPLAY: information about the various stages of the programme if v does not |
---|
| 2200 | equal 0 |
---|
| 2201 | RETURN: primary invariants (type <matrix>) of the invariant ring |
---|
| 2202 | EXAMPLE: example primary_charp; shows an example |
---|
| 2203 | THEORY: Bases of homogeneous invariants are generated successively and those |
---|
| 2204 | are chosen as primary invariants that lower the dimension of the ideal |
---|
| 2205 | generated by the previously found invariants (see paper \"Generating a |
---|
| 2206 | Noetherian Normalization of the Invariant Ring of a Finite Group\" by |
---|
| 2207 | Decker, Heydtmann, Schreyer (1997) to appear in JSC). |
---|
| 2208 | " |
---|
| 2209 | { degBound=0; |
---|
| 2210 | // ---------------- checking input and setting verbose mode ------------------- |
---|
| 2211 | if (char(basering)==0) |
---|
| 2212 | { "ERROR: primary_charp should only be used with rings of characteristic p>0."; |
---|
| 2213 | return(); |
---|
| 2214 | } |
---|
| 2215 | if (size(#)>1) |
---|
| 2216 | { "ERROR: primary_charp can only have three parameters."; |
---|
| 2217 | return(); |
---|
| 2218 | } |
---|
| 2219 | if (size(#)==1) |
---|
| 2220 | { if (typeof(#[1])<>"int") |
---|
| 2221 | { "ERROR: The third parameter should be of type <int>."; |
---|
| 2222 | return(); |
---|
| 2223 | } |
---|
| 2224 | else |
---|
| 2225 | { int v=#[1]; |
---|
| 2226 | } |
---|
| 2227 | } |
---|
| 2228 | else |
---|
| 2229 | { int v=0; |
---|
| 2230 | } |
---|
| 2231 | def br=basering; |
---|
| 2232 | int n=nvars(br); // n is the number of variables, as well |
---|
| 2233 | // as the size of the matrices, as well |
---|
| 2234 | // as the number of primary invariants, |
---|
| 2235 | // we should get |
---|
| 2236 | if (ncols(REY)<>n) |
---|
| 2237 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
| 2238 | return(); |
---|
| 2239 | } |
---|
| 2240 | if (typeof(`ring_name`)<>"ring") |
---|
[5811fb] | 2241 | { "ERROR: Second parameter ought to the name of a ring where the Molien"; |
---|
[29aa4bf] | 2242 | " is stored."; |
---|
| 2243 | return(); |
---|
| 2244 | } |
---|
| 2245 | //---------------------------------------------------------------------------- |
---|
| 2246 | if (v && voice<>2) |
---|
| 2247 | { " We can start looking for primary invariants..."; |
---|
| 2248 | ""; |
---|
| 2249 | } |
---|
| 2250 | if (v && voice==2) |
---|
| 2251 | { ""; |
---|
| 2252 | } |
---|
| 2253 | //----------------------- initializing variables ----------------------------- |
---|
| 2254 | int dB; |
---|
| 2255 | setring `ring_name`; // the Molien series is stores here - |
---|
| 2256 | poly p(1..2); // p(1) will be used for single terms of |
---|
| 2257 | // the partial expansion, p(2) to store |
---|
| 2258 | p(1..2)=partial_molien(M,1); // the intermediate result - |
---|
| 2259 | poly v1=var(1); // we need v1 to split off coefficients |
---|
| 2260 | // in the partial expansion of M (which |
---|
| 2261 | // is in terms of the first variable) |
---|
| 2262 | setring br; |
---|
| 2263 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
| 2264 | // space of invariants of degree d, |
---|
| 2265 | // newdim: dimension the ideal generated |
---|
| 2266 | // the primary invariants plus basis |
---|
| 2267 | // elements, dif=n-i-newdim, i.e. the |
---|
| 2268 | // number of new primary invairants that |
---|
| 2269 | // should be added in this degree - |
---|
| 2270 | ideal P,Pplus,sPplus,CI,B; // P: will contain primary invariants, |
---|
| 2271 | // Pplus: P+B, CI: a complete |
---|
| 2272 | // intersection with the same Hilbert |
---|
| 2273 | // function as P |
---|
| 2274 | ideal sP=std(P); |
---|
| 2275 | dB=1; // used as degree bound |
---|
| 2276 | int i=0; |
---|
| 2277 | //---------------- loop that searches for primary invariants ----------------- |
---|
| 2278 | while(1) // repeat until n primary invariants are |
---|
| 2279 | { // found |
---|
| 2280 | setring `ring_name`; |
---|
| 2281 | p(1..2)=partial_molien(M,1,p(2)); // next term of the partial expansion - |
---|
| 2282 | d=deg(p(1)); // degree where we'll search - |
---|
| 2283 | cd=int(coef(p(1),v1)[2,1]); // dimension of the homogeneous space of |
---|
| 2284 | // inviarants of degree d |
---|
| 2285 | setring br; |
---|
| 2286 | if (v) |
---|
| 2287 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
| 2288 | } |
---|
| 2289 | B=invariant_basis_reynolds(REY,d,intvec(cd,6)); // basis of invariants of |
---|
| 2290 | // degree d |
---|
| 2291 | if (B[1]<>0) |
---|
| 2292 | { Pplus=P+B; |
---|
| 2293 | sPplus=std(Pplus); |
---|
| 2294 | newdim=dim(sPplus); |
---|
| 2295 | dif=n-i-newdim; |
---|
| 2296 | } |
---|
| 2297 | else |
---|
| 2298 | { dif=0; |
---|
| 2299 | } |
---|
| 2300 | if (dif<>0) // we have to find dif new primary |
---|
| 2301 | { // invariants |
---|
| 2302 | if (cd<>dif) |
---|
| 2303 | { P,sP,CI,dB=p_search(n,d,B,cd,P,sP,i,dif,dB,CI); |
---|
| 2304 | } |
---|
| 2305 | else // i.e. we can take all of B |
---|
| 2306 | { for(j=i+1;j>i+dif;j=j+1) |
---|
| 2307 | { CI=CI+ideal(var(j)^d); |
---|
| 2308 | } |
---|
| 2309 | dB=dB+dif*(d-1); |
---|
| 2310 | P=Pplus; |
---|
| 2311 | sP=sPplus; |
---|
| 2312 | } |
---|
| 2313 | if (v) |
---|
| 2314 | { for (j=1;j<=size(P)-i;j=j+1) |
---|
| 2315 | { " We find: "+string(P[i+j]); |
---|
| 2316 | } |
---|
| 2317 | } |
---|
| 2318 | i=size(P); |
---|
| 2319 | if (i==n) // found all primary invariants |
---|
| 2320 | { if (v) |
---|
| 2321 | { ""; |
---|
| 2322 | " We found all primary invariants."; |
---|
| 2323 | ""; |
---|
| 2324 | } |
---|
| 2325 | return(matrix(P)); |
---|
| 2326 | } |
---|
| 2327 | } // done with degree d |
---|
| 2328 | } |
---|
| 2329 | } |
---|
| 2330 | example |
---|
| 2331 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7 (changed into"; |
---|
| 2332 | " characteristic 3)"; |
---|
| 2333 | echo=2; |
---|
| 2334 | ring R=3,(x,y,z),dp; |
---|
| 2335 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 2336 | list L=group_reynolds(A); |
---|
| 2337 | string newring="alskdfj"; |
---|
| 2338 | molien(L[2..size(L)],newring); |
---|
| 2339 | matrix P=primary_charp(L[1],newring); |
---|
[c67136] | 2340 | if(system("with","Namespaces")) { kill Top::`newring`; } |
---|
[29aa4bf] | 2341 | kill `newring`; |
---|
| 2342 | print(P); |
---|
| 2343 | } |
---|
| 2344 | |
---|
| 2345 | proc primary_char0_no_molien (matrix REY, list #) |
---|
| 2346 | "USAGE: primary_char0_no_molien(REY[,v]); |
---|
| 2347 | REY: a <matrix> representing the Reynolds operator, v: an optional |
---|
| 2348 | <int> |
---|
| 2349 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien |
---|
| 2350 | DISPLAY: information about the various stages of the programme if v does not |
---|
| 2351 | equal 0 |
---|
| 2352 | RETURN: primary invariants (type <matrix>) of the invariant ring and an |
---|
| 2353 | <intvec> listing some of the degrees where no non-trivial homogeneous |
---|
| 2354 | invariants are to be found |
---|
| 2355 | EXAMPLE: example primary_char0_no_molien; shows an example |
---|
| 2356 | THEORY: Bases of homogeneous invariants are generated successively and those |
---|
| 2357 | are chosen as primary invariants that lower the dimension of the ideal |
---|
| 2358 | generated by the previously found invariants (see paper \"Generating a |
---|
| 2359 | Noetherian Normalization of the Invariant Ring of a Finite Group\" by |
---|
| 2360 | Decker, Heydtmann, Schreyer (1997) to appear in JSC). |
---|
| 2361 | " |
---|
| 2362 | { degBound=0; |
---|
| 2363 | //-------------- checking input and setting verbose mode --------------------- |
---|
| 2364 | if (char(basering)<>0) |
---|
| 2365 | { "ERROR: primary_char0_no_molien should only be used with rings of"; |
---|
| 2366 | " characteristic 0."; |
---|
| 2367 | return(); |
---|
| 2368 | } |
---|
| 2369 | if (size(#)>1) |
---|
| 2370 | { "ERROR: primary_char0_no_molien can only have two parameters."; |
---|
| 2371 | return(); |
---|
| 2372 | } |
---|
| 2373 | if (size(#)==1) |
---|
| 2374 | { if (typeof(#[1])<>"int") |
---|
| 2375 | { "ERROR: The second parameter should be of type <int>."; |
---|
| 2376 | return(); |
---|
| 2377 | } |
---|
| 2378 | else |
---|
| 2379 | { int v=#[1]; |
---|
| 2380 | } |
---|
| 2381 | } |
---|
| 2382 | else |
---|
| 2383 | { int v=0; |
---|
| 2384 | } |
---|
| 2385 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 2386 | // as the size of the matrices, as well |
---|
| 2387 | // as the number of primary invariants, |
---|
| 2388 | // we should get |
---|
| 2389 | if (ncols(REY)<>n) |
---|
| 2390 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
| 2391 | return(); |
---|
| 2392 | } |
---|
| 2393 | //---------------------------------------------------------------------------- |
---|
| 2394 | if (v && voice<>2) |
---|
| 2395 | { " We can start looking for primary invariants..."; |
---|
| 2396 | ""; |
---|
| 2397 | } |
---|
| 2398 | if (v && voice==2) |
---|
| 2399 | { ""; |
---|
| 2400 | } |
---|
| 2401 | //----------------------- initializing variables ----------------------------- |
---|
| 2402 | int dB; |
---|
| 2403 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
| 2404 | // space of invariants of degree d, |
---|
| 2405 | // newdim: dimension the ideal generated |
---|
| 2406 | // the primary invariants plus basis |
---|
| 2407 | // elements, dif=n-i-newdim, i.e. the |
---|
| 2408 | // number of new primary invairants that |
---|
| 2409 | // should be added in this degree - |
---|
| 2410 | ideal P,Pplus,CI,B; // P: will contain primary invariants, |
---|
| 2411 | // Pplus: P+B, CI: a complete |
---|
| 2412 | // intersection with the same Hilbert |
---|
| 2413 | // function as P |
---|
| 2414 | ideal sP=std(P); |
---|
| 2415 | dB=1; // used as degree bound - |
---|
| 2416 | d=0; // initializing |
---|
| 2417 | int i=0; |
---|
| 2418 | intvec deg_vector; |
---|
| 2419 | //------------------ loop that searches for primary invariants --------------- |
---|
| 2420 | while(1) // repeat until n primary invariants are |
---|
| 2421 | { // found - |
---|
| 2422 | d=d+1; // degree where we'll search |
---|
| 2423 | if (v) |
---|
| 2424 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
| 2425 | } |
---|
| 2426 | B=invariant_basis_reynolds(REY,d,intvec(-1,6)); // basis of invariants of |
---|
| 2427 | // degree d |
---|
| 2428 | if (B[1]<>0) |
---|
| 2429 | { Pplus=P+B; |
---|
| 2430 | newdim=dim(std(Pplus)); |
---|
| 2431 | dif=n-i-newdim; |
---|
| 2432 | } |
---|
| 2433 | else |
---|
| 2434 | { dif=0; |
---|
| 2435 | deg_vector=deg_vector,d; |
---|
| 2436 | } |
---|
| 2437 | if (dif<>0) // we have to find dif new primary |
---|
| 2438 | { // invariants |
---|
| 2439 | cd=size(B); |
---|
| 2440 | if (cd<>dif) |
---|
| 2441 | { P,sP,CI,dB=search(n,d,B,cd,P,sP,i,dif,dB,CI); |
---|
| 2442 | } |
---|
| 2443 | else // i.e. we can take all of B |
---|
| 2444 | { for(j=i+1;j<=i+dif;j=j+1) |
---|
| 2445 | { CI=CI+ideal(var(j)^d); |
---|
| 2446 | } |
---|
| 2447 | dB=dB+dif*(d-1); |
---|
| 2448 | P=Pplus; |
---|
| 2449 | sP=std(P); |
---|
| 2450 | } |
---|
| 2451 | if (v) |
---|
| 2452 | { for (j=1;j<=dif;j=j+1) |
---|
| 2453 | { " We find: "+string(P[i+j]); |
---|
| 2454 | } |
---|
| 2455 | } |
---|
| 2456 | i=i+dif; |
---|
| 2457 | if (i==n) // found all primary invariants |
---|
| 2458 | { if (v) |
---|
| 2459 | { ""; |
---|
| 2460 | " We found all primary invariants."; |
---|
| 2461 | ""; |
---|
| 2462 | } |
---|
| 2463 | if (deg_vector==0) |
---|
| 2464 | { return(matrix(P)); |
---|
| 2465 | } |
---|
| 2466 | else |
---|
| 2467 | { return(matrix(P),compress(deg_vector)); |
---|
| 2468 | } |
---|
| 2469 | } |
---|
| 2470 | } // done with degree d |
---|
| 2471 | else |
---|
| 2472 | { if (v) |
---|
| 2473 | { " None here..."; |
---|
| 2474 | } |
---|
| 2475 | } |
---|
| 2476 | } |
---|
| 2477 | } |
---|
| 2478 | example |
---|
| 2479 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
| 2480 | echo=2; |
---|
| 2481 | ring R=0,(x,y,z),dp; |
---|
| 2482 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 2483 | list L=group_reynolds(A); |
---|
| 2484 | list l=primary_char0_no_molien(L[1]); |
---|
| 2485 | print(l[1]); |
---|
| 2486 | } |
---|
| 2487 | |
---|
| 2488 | proc primary_charp_no_molien (matrix REY, list #) |
---|
| 2489 | "USAGE: primary_charp_no_molien(REY[,v]); |
---|
| 2490 | REY: a <matrix> representing the Reynolds operator, v: an optional |
---|
| 2491 | <int> |
---|
| 2492 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien |
---|
| 2493 | DISPLAY: information about the various stages of the programme if v does not |
---|
| 2494 | equal 0 |
---|
| 2495 | RETURN: primary invariants (type <matrix>) of the invariant ring and an |
---|
| 2496 | <intvec> listing some of the degrees where no non-trivial homogeneous |
---|
| 2497 | invariants are to be found |
---|
| 2498 | EXAMPLE: example primary_charp_no_molien; shows an example |
---|
| 2499 | THEORY: Bases of homogeneous invariants are generated successively and those |
---|
| 2500 | are chosen as primary invariants that lower the dimension of the ideal |
---|
| 2501 | generated by the previously found invariants (see paper \"Generating a |
---|
| 2502 | Noetherian Normalization of the Invariant Ring of a Finite Group\" by |
---|
| 2503 | Decker, Heydtmann, Schreyer (1997) to appear in JSC). |
---|
| 2504 | " |
---|
| 2505 | { degBound=0; |
---|
| 2506 | //----------------- checking input and setting verbose mode ------------------ |
---|
| 2507 | if (char(basering)==0) |
---|
| 2508 | { "ERROR: primary_charp_no_molien should only be used with rings of"; |
---|
| 2509 | " characteristic p>0."; |
---|
| 2510 | return(); |
---|
| 2511 | } |
---|
| 2512 | if (size(#)>1) |
---|
| 2513 | { "ERROR: primary_charp_no_molien can only have two parameters."; |
---|
| 2514 | return(); |
---|
| 2515 | } |
---|
| 2516 | if (size(#)==1) |
---|
| 2517 | { if (typeof(#[1])<>"int") |
---|
| 2518 | { "ERROR: The second parameter should be of type <int>."; |
---|
| 2519 | return(); |
---|
| 2520 | } |
---|
| 2521 | else |
---|
| 2522 | { int v=#[1]; |
---|
| 2523 | } |
---|
| 2524 | } |
---|
| 2525 | else |
---|
| 2526 | { int v=0; |
---|
| 2527 | } |
---|
| 2528 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 2529 | // as the size of the matrices, as well |
---|
| 2530 | // as the number of primary invariants, |
---|
| 2531 | // we should get |
---|
| 2532 | if (ncols(REY)<>n) |
---|
| 2533 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
| 2534 | return(); |
---|
| 2535 | } |
---|
| 2536 | //---------------------------------------------------------------------------- |
---|
| 2537 | if (v && voice<>2) |
---|
| 2538 | { " We can start looking for primary invariants..."; |
---|
| 2539 | ""; |
---|
| 2540 | } |
---|
| 2541 | if (v && voice==2) |
---|
| 2542 | { ""; |
---|
| 2543 | } |
---|
| 2544 | //-------------------- initializing variables -------------------------------- |
---|
| 2545 | int dB; |
---|
| 2546 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
| 2547 | // space of invariants of degree d, |
---|
| 2548 | // newdim: dimension the ideal generated |
---|
| 2549 | // the primary invariants plus basis |
---|
| 2550 | // elements, dif=n-i-newdim, i.e. the |
---|
| 2551 | // number of new primary invairants that |
---|
| 2552 | // should be added in this degree - |
---|
| 2553 | ideal P,Pplus,sPplus,CI,B; // P: will contain primary invariants, |
---|
| 2554 | // Pplus: P+B, CI: a complete |
---|
| 2555 | // intersection with the same Hilbert |
---|
| 2556 | // function as P |
---|
| 2557 | ideal sP=std(P); |
---|
| 2558 | dB=1; // used as degree bound - |
---|
| 2559 | d=0; // initializing |
---|
| 2560 | int i=0; |
---|
| 2561 | intvec deg_vector; |
---|
| 2562 | //------------------ loop that searches for primary invariants --------------- |
---|
| 2563 | while(1) // repeat until n primary invariants are |
---|
| 2564 | { // found - |
---|
| 2565 | d=d+1; // degree where we'll search |
---|
| 2566 | if (v) |
---|
| 2567 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
| 2568 | } |
---|
| 2569 | B=invariant_basis_reynolds(REY,d,intvec(-1,6)); // basis of invariants of |
---|
| 2570 | // degree d |
---|
| 2571 | if (B[1]<>0) |
---|
| 2572 | { Pplus=P+B; |
---|
| 2573 | sPplus=std(Pplus); |
---|
| 2574 | newdim=dim(sPplus); |
---|
| 2575 | dif=n-i-newdim; |
---|
| 2576 | } |
---|
| 2577 | else |
---|
| 2578 | { dif=0; |
---|
| 2579 | deg_vector=deg_vector,d; |
---|
| 2580 | } |
---|
| 2581 | if (dif<>0) // we have to find dif new primary |
---|
| 2582 | { // invariants |
---|
| 2583 | cd=size(B); |
---|
| 2584 | if (cd<>dif) |
---|
| 2585 | { P,sP,CI,dB=p_search(n,d,B,cd,P,sP,i,dif,dB,CI); |
---|
| 2586 | } |
---|
| 2587 | else // i.e. we can take all of B |
---|
| 2588 | { for(j=i+1;j<=i+dif;j=j+1) |
---|
| 2589 | { CI=CI+ideal(var(j)^d); |
---|
| 2590 | } |
---|
| 2591 | dB=dB+dif*(d-1); |
---|
| 2592 | P=Pplus; |
---|
| 2593 | sP=sPplus; |
---|
| 2594 | } |
---|
| 2595 | if (v) |
---|
| 2596 | { for (j=1;j<=size(P)-i;j=j+1) |
---|
| 2597 | { " We find: "+string(P[i+j]); |
---|
| 2598 | } |
---|
| 2599 | } |
---|
| 2600 | i=size(P); |
---|
| 2601 | if (i==n) // found all primary invariants |
---|
| 2602 | { if (v) |
---|
| 2603 | { ""; |
---|
| 2604 | " We found all primary invariants."; |
---|
| 2605 | ""; |
---|
| 2606 | } |
---|
| 2607 | if (deg_vector==0) |
---|
| 2608 | { return(matrix(P)); |
---|
| 2609 | } |
---|
| 2610 | else |
---|
| 2611 | { return(matrix(P),compress(deg_vector)); |
---|
| 2612 | } |
---|
| 2613 | } |
---|
| 2614 | } // done with degree d |
---|
| 2615 | else |
---|
| 2616 | { if (v) |
---|
| 2617 | { " None here..."; |
---|
| 2618 | } |
---|
| 2619 | } |
---|
| 2620 | } |
---|
| 2621 | } |
---|
| 2622 | example |
---|
| 2623 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7 (changed into"; |
---|
| 2624 | " characteristic 3)"; |
---|
| 2625 | echo=2; |
---|
| 2626 | ring R=3,(x,y,z),dp; |
---|
| 2627 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 2628 | list L=group_reynolds(A); |
---|
| 2629 | list l=primary_charp_no_molien(L[1]); |
---|
| 2630 | print(l[1]); |
---|
| 2631 | } |
---|
| 2632 | |
---|
| 2633 | proc primary_charp_without (list #) |
---|
| 2634 | "USAGE: primary_charp_without(G1,G2,...[,v]); |
---|
| 2635 | G1,G2,...: <matrices> generating a finite matrix group, v: an optional |
---|
| 2636 | <int> |
---|
| 2637 | DISPLAY: information about the various stages of the programme if v does not |
---|
| 2638 | equal 0 |
---|
| 2639 | RETURN: primary invariants (type <matrix>) of the invariant ring |
---|
| 2640 | EXAMPLE: example primary_charp_without; shows an example |
---|
| 2641 | THEORY: Bases of homogeneous invariants are generated successively and those |
---|
| 2642 | are chosen as primary invariants that lower the dimension of the ideal |
---|
| 2643 | generated by the previously found invariants (see paper \"Generating a |
---|
| 2644 | Noetherian Normalization of the Invariant Ring of a Finite Group\" by |
---|
| 2645 | Decker, Heydtmann, Schreyer (1997) to appear in JSC). No Reynolds |
---|
| 2646 | operator or Molien series is used. |
---|
| 2647 | " |
---|
| 2648 | { degBound=0; |
---|
| 2649 | //--------------------- checking input and setting verbose mode -------------- |
---|
| 2650 | if (char(basering)==0) |
---|
| 2651 | { "ERROR: primary_charp_without should only be used with rings of"; |
---|
| 2652 | " characteristic 0."; |
---|
| 2653 | return(); |
---|
| 2654 | } |
---|
| 2655 | if (size(#)==0) |
---|
| 2656 | { "ERROR: There are no parameters."; |
---|
| 2657 | return(); |
---|
| 2658 | } |
---|
| 2659 | if (typeof(#[size(#)])=="int") |
---|
| 2660 | { int v=#[size(#)]; |
---|
| 2661 | int gen_num=size(#)-1; |
---|
| 2662 | if (gen_num==0) |
---|
| 2663 | { "ERROR: There are no generators of a finite matrix group given."; |
---|
| 2664 | return(); |
---|
| 2665 | } |
---|
| 2666 | } |
---|
| 2667 | else |
---|
| 2668 | { int v=0; |
---|
| 2669 | int gen_num=size(#); |
---|
| 2670 | } |
---|
| 2671 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 2672 | // as the size of the matrices, as well |
---|
| 2673 | // as the number of primary invariants, |
---|
| 2674 | // we should get |
---|
| 2675 | for (int i=1;i<=gen_num;i=i+1) |
---|
| 2676 | { if (typeof(#[i])=="matrix") |
---|
| 2677 | { if (nrows(#[i])<>n or ncols(#[i])<>n) |
---|
| 2678 | { "ERROR: The number of variables of the base ring needs to be the same"; |
---|
| 2679 | " as the dimension of the square matrices"; |
---|
| 2680 | return(); |
---|
| 2681 | } |
---|
| 2682 | } |
---|
| 2683 | else |
---|
| 2684 | { "ERROR: The first parameters should be a list of matrices"; |
---|
| 2685 | return(); |
---|
| 2686 | } |
---|
| 2687 | } |
---|
| 2688 | //---------------------------------------------------------------------------- |
---|
| 2689 | if (v && voice==2) |
---|
| 2690 | { ""; |
---|
| 2691 | } |
---|
| 2692 | //---------------------------- initializing variables ------------------------ |
---|
| 2693 | int dB; |
---|
| 2694 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
| 2695 | // space of invariants of degree d, |
---|
| 2696 | // newdim: dimension the ideal generated |
---|
| 2697 | // the primary invariants plus basis |
---|
| 2698 | // elements, dif=n-i-newdim, i.e. the |
---|
| 2699 | // number of new primary invairants that |
---|
| 2700 | // should be added in this degree - |
---|
| 2701 | ideal P,Pplus,sPplus,CI,B; // P: will contain primary invariants, |
---|
| 2702 | // Pplus: P+B, CI: a complete |
---|
| 2703 | // intersection with the same Hilbert |
---|
| 2704 | // function as P |
---|
| 2705 | ideal sP=std(P); |
---|
| 2706 | dB=1; // used as degree bound - |
---|
| 2707 | d=0; // initializing |
---|
| 2708 | i=0; |
---|
| 2709 | intvec deg_vector; |
---|
| 2710 | //-------------------- loop that searches for primary invariants ------------- |
---|
| 2711 | while(1) // repeat until n primary invariants are |
---|
| 2712 | { // found - |
---|
| 2713 | d=d+1; // degree where we'll search |
---|
| 2714 | if (v) |
---|
| 2715 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
| 2716 | } |
---|
| 2717 | B=invariant_basis(d,#[1..gen_num]); // basis of invariants of degree d |
---|
| 2718 | if (B[1]<>0) |
---|
| 2719 | { Pplus=P+B; |
---|
| 2720 | sPplus=std(Pplus); |
---|
| 2721 | newdim=dim(sPplus); |
---|
| 2722 | dif=n-i-newdim; |
---|
| 2723 | } |
---|
| 2724 | else |
---|
| 2725 | { dif=0; |
---|
| 2726 | deg_vector=deg_vector,d; |
---|
| 2727 | } |
---|
| 2728 | if (dif<>0) // we have to find dif new primary |
---|
| 2729 | { // invariants |
---|
| 2730 | cd=size(B); |
---|
| 2731 | if (cd<>dif) |
---|
| 2732 | { P,sP,CI,dB=p_search(n,d,B,cd,P,sP,i,dif,dB,CI); |
---|
| 2733 | } |
---|
| 2734 | else // i.e. we can take all of B |
---|
| 2735 | { for(j=i+1;j<=i+dif;j=j+1) |
---|
| 2736 | { CI=CI+ideal(var(j)^d); |
---|
| 2737 | } |
---|
| 2738 | dB=dB+dif*(d-1); |
---|
| 2739 | P=Pplus; |
---|
| 2740 | sP=sPplus; |
---|
| 2741 | } |
---|
| 2742 | if (v) |
---|
| 2743 | { for (j=1;j<=size(P)-i;j=j+1) |
---|
| 2744 | { " We find: "+string(P[i+j]); |
---|
| 2745 | } |
---|
| 2746 | } |
---|
| 2747 | i=size(P); |
---|
| 2748 | if (i==n) // found all primary invariants |
---|
| 2749 | { if (v) |
---|
| 2750 | { ""; |
---|
| 2751 | " We found all primary invariants."; |
---|
| 2752 | ""; |
---|
| 2753 | } |
---|
| 2754 | return(matrix(P)); |
---|
| 2755 | } |
---|
| 2756 | } // done with degree d |
---|
| 2757 | else |
---|
| 2758 | { if (v) |
---|
| 2759 | { " None here..."; |
---|
| 2760 | } |
---|
| 2761 | } |
---|
| 2762 | } |
---|
| 2763 | } |
---|
| 2764 | example |
---|
| 2765 | { echo=2; |
---|
| 2766 | ring R=2,(x,y,z),dp; |
---|
| 2767 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 2768 | matrix P=primary_charp_without(A); |
---|
| 2769 | print(P); |
---|
| 2770 | } |
---|
| 2771 | |
---|
| 2772 | proc primary_invariants (list #) |
---|
| 2773 | "USAGE: primary_invariants(G1,G2,...[,flags]); |
---|
| 2774 | G1,G2,...: <matrices> generating a finite matrix group, flags: an |
---|
| 2775 | optional <intvec> with three entries, if the first one equals 0 (also |
---|
| 2776 | the default), the programme attempts to compute the Molien series and |
---|
| 2777 | Reynolds operator, if it equals 1, the programme is told that the |
---|
| 2778 | Molien series should not be computed, if it equals -1 characteristic 0 |
---|
| 2779 | is simulated, i.e. the Molien series is computed as if the base field |
---|
| 2780 | were characteristic 0 (the user must choose a field of large prime |
---|
| 2781 | characteristic, e.g. 32003) and if the first one is anything else, it |
---|
| 2782 | means that the characteristic of the base field divides the group |
---|
| 2783 | order, the second component should give the size of intervals between |
---|
| 2784 | canceling common factors in the expansion of the Molien series, 0 (the |
---|
| 2785 | default) means only once after generating all terms, in prime |
---|
| 2786 | characteristic also a negative number can be given to indicate that |
---|
| 2787 | common factors should always be canceled when the expansion is simple |
---|
| 2788 | (the root of the extension field does not occur among the coefficients) |
---|
| 2789 | DISPLAY: information about the various stages of the programme if the third |
---|
| 2790 | flag does not equal 0 |
---|
| 2791 | RETURN: primary invariants (type <matrix>) of the invariant ring and if |
---|
| 2792 | computable Reynolds operator (type <matrix>) and Molien series (type |
---|
| 2793 | <matrix>), if the first flag is 1 and we are in the non-modular case |
---|
| 2794 | then an <intvec> is returned giving some of the degrees where no |
---|
| 2795 | non-trivial homogeneous invariants can be found |
---|
| 2796 | EXAMPLE: example primary_invariants; shows an example |
---|
| 2797 | THEORY: Bases of homogeneous invariants are generated successively and those |
---|
| 2798 | are chosen as primary invariants that lower the dimension of the ideal |
---|
| 2799 | generated by the previously found invariants (see paper \"Generating a |
---|
| 2800 | Noetherian Normalization of the Invariant Ring of a Finite Group\" by |
---|
| 2801 | Decker, Heydtmann, Schreyer (1997) to appear in JSC). |
---|
| 2802 | " |
---|
| 2803 | { |
---|
| 2804 | // ----------------- checking input and setting flags ------------------------ |
---|
| 2805 | if (size(#)==0) |
---|
| 2806 | { "ERROR: There are no parameters."; |
---|
| 2807 | return(); |
---|
| 2808 | } |
---|
| 2809 | int ch=char(basering); // the algorithms depend very much on the |
---|
| 2810 | // characteristic of the ground field |
---|
| 2811 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 2812 | // as the size of the matrices, as well |
---|
| 2813 | // as the number of primary invariants, |
---|
| 2814 | // we should get |
---|
| 2815 | int gen_num; |
---|
| 2816 | int mol_flag,v; |
---|
| 2817 | if (typeof(#[size(#)])=="intvec") |
---|
| 2818 | { if (size(#[size(#)])<>3) |
---|
| 2819 | { "ERROR: <intvec> should have three entries."; |
---|
| 2820 | return(); |
---|
| 2821 | } |
---|
| 2822 | gen_num=size(#)-1; |
---|
| 2823 | mol_flag=#[size(#)][1]; |
---|
| 2824 | if (#[size(#)][2]<0 && (ch==0 or (ch<>0 && mol_flag==-1))) |
---|
| 2825 | { "ERROR: the second component of <intvec> should be >=0"; |
---|
| 2826 | return(); |
---|
| 2827 | } |
---|
| 2828 | int interval=#[size(#)][2]; |
---|
| 2829 | v=#[size(#)][3]; |
---|
| 2830 | if (gen_num==0) |
---|
| 2831 | { "ERROR: There are no generators of a finite matrix group given."; |
---|
| 2832 | return(); |
---|
| 2833 | } |
---|
| 2834 | } |
---|
| 2835 | else |
---|
| 2836 | { gen_num=size(#); |
---|
| 2837 | mol_flag=0; |
---|
| 2838 | int interval=0; |
---|
| 2839 | v=0; |
---|
| 2840 | } |
---|
| 2841 | for (int i=1;i<=gen_num;i=i+1) |
---|
| 2842 | { if (typeof(#[i])=="matrix") |
---|
| 2843 | { if (nrows(#[i])<>n or ncols(#[i])<>n) |
---|
| 2844 | { "ERROR: The number of variables of the base ring needs to be the same"; |
---|
| 2845 | " as the dimension of the square matrices"; |
---|
| 2846 | return(); |
---|
| 2847 | } |
---|
| 2848 | } |
---|
| 2849 | else |
---|
| 2850 | { "ERROR: The first parameters should be a list of matrices"; |
---|
| 2851 | return(); |
---|
| 2852 | } |
---|
| 2853 | } |
---|
| 2854 | //---------------------------------------------------------------------------- |
---|
| 2855 | if (mol_flag==0) |
---|
| 2856 | { if (ch==0) |
---|
| 2857 | { matrix REY,M=reynolds_molien(#[1..gen_num],intvec(mol_flag,interval,v)); |
---|
| 2858 | // one will contain Reynolds operator and |
---|
| 2859 | // the other enumerator and denominator |
---|
| 2860 | // of Molien series |
---|
| 2861 | matrix P=primary_char0(REY,M,v); |
---|
| 2862 | return(P,REY,M); |
---|
| 2863 | } |
---|
| 2864 | else |
---|
| 2865 | { list L=group_reynolds(#[1..gen_num],v); |
---|
| 2866 | if (L[1]<>0) // testing whether we are in the modular |
---|
| 2867 | { string newring="aksldfalkdsflkj"; // case |
---|
| 2868 | if (minpoly==0) |
---|
| 2869 | { if (v) |
---|
| 2870 | { " We are dealing with the non-modular case."; |
---|
| 2871 | } |
---|
| 2872 | if (typeof(L[2])=="int") |
---|
| 2873 | { molien(L[3..size(L)],newring,L[2],intvec(mol_flag,interval,v)); |
---|
| 2874 | } |
---|
| 2875 | else |
---|
| 2876 | { molien(L[2..size(L)],newring,intvec(mol_flag,interval,v)); |
---|
| 2877 | } |
---|
| 2878 | matrix P=primary_charp(L[1],newring,v); |
---|
| 2879 | return(P,L[1],newring); |
---|
| 2880 | } |
---|
| 2881 | else |
---|
| 2882 | { if (v) |
---|
| 2883 | { " Since it is impossible for this programme to calculate the Molien series for"; |
---|
| 2884 | " invariant rings over extension fields of prime characteristic, we have to"; |
---|
| 2885 | " continue without it."; |
---|
| 2886 | ""; |
---|
| 2887 | |
---|
| 2888 | } |
---|
| 2889 | list l=primary_charp_no_molien(L[1],v); |
---|
| 2890 | if (size(l)==2) |
---|
| 2891 | { return(l[1],L[1],l[2]); |
---|
| 2892 | } |
---|
| 2893 | else |
---|
| 2894 | { return(l[1],L[1]); |
---|
| 2895 | } |
---|
| 2896 | } |
---|
| 2897 | } |
---|
| 2898 | else // the modular case |
---|
| 2899 | { if (v) |
---|
| 2900 | { " There is also no Molien series, we can make use of..."; |
---|
| 2901 | ""; |
---|
| 2902 | " We can start looking for primary invariants..."; |
---|
| 2903 | ""; |
---|
| 2904 | } |
---|
| 2905 | return(primary_charp_without(#[1..gen_num],v)); |
---|
| 2906 | } |
---|
| 2907 | } |
---|
| 2908 | } |
---|
| 2909 | if (mol_flag==1) // the user wants no calculation of the |
---|
| 2910 | { list L=group_reynolds(#[1..gen_num],v); // Molien series |
---|
| 2911 | if (ch==0) |
---|
| 2912 | { list l=primary_char0_no_molien(L[1],v); |
---|
| 2913 | if (size(l)==2) |
---|
| 2914 | { return(l[1],L[1],l[2]); |
---|
| 2915 | } |
---|
| 2916 | else |
---|
| 2917 | { return(l[1],L[1]); |
---|
| 2918 | } |
---|
| 2919 | } |
---|
| 2920 | else |
---|
| 2921 | { if (L[1]<>0) // testing whether we are in the modular |
---|
| 2922 | { list l=primary_charp_no_molien(L[1],v); // case |
---|
| 2923 | if (size(l)==2) |
---|
| 2924 | { return(l[1],L[1],l[2]); |
---|
| 2925 | } |
---|
| 2926 | else |
---|
| 2927 | { return(l[1],L[1]); |
---|
| 2928 | } |
---|
| 2929 | } |
---|
| 2930 | else // the modular case |
---|
| 2931 | { if (v) |
---|
| 2932 | { " We can start looking for primary invariants..."; |
---|
| 2933 | ""; |
---|
| 2934 | } |
---|
| 2935 | return(primary_charp_without(#[1..gen_num],v)); |
---|
| 2936 | } |
---|
| 2937 | } |
---|
| 2938 | } |
---|
| 2939 | if (mol_flag==-1) |
---|
| 2940 | { if (ch==0) |
---|
| 2941 | { "ERROR: Characteristic 0 can only be simulated in characteristic p>>0."; |
---|
| 2942 | return(); |
---|
| 2943 | } |
---|
| 2944 | list L=group_reynolds(#[1..gen_num],v); |
---|
| 2945 | string newring="aksldfalkdsflkj"; |
---|
[5811fb] | 2946 | molien(L[2..size(L)],newring,intvec(1,interval,v)); |
---|
[29aa4bf] | 2947 | matrix P=primary_charp(L[1],newring,v); |
---|
| 2948 | return(P,L[1],newring); |
---|
| 2949 | } |
---|
| 2950 | else // the user specified that the |
---|
| 2951 | { if (ch==0) // characteristic divides the group order |
---|
| 2952 | { "ERROR: The characteristic cannot divide the group order when it is 0."; |
---|
| 2953 | return(); |
---|
| 2954 | } |
---|
| 2955 | if (v) |
---|
| 2956 | { ""; |
---|
| 2957 | } |
---|
| 2958 | return(primary_charp_without(#[1..gen_num],v)); |
---|
| 2959 | } |
---|
| 2960 | } |
---|
| 2961 | example |
---|
| 2962 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
| 2963 | echo=2; |
---|
| 2964 | ring R=0,(x,y,z),dp; |
---|
| 2965 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 2966 | list L=primary_invariants(A); |
---|
| 2967 | print(L[1]); |
---|
| 2968 | } |
---|
| 2969 | |
---|
| 2970 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 2971 | // This procedure finds dif primary invariants in degree d. It returns all |
---|
| 2972 | // primary invariants found so far. The coefficients lie in a field of |
---|
| 2973 | // characteristic 0. |
---|
| 2974 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 2975 | proc search_random (int n,int d,ideal B,int cd,ideal P,int i,int dif,int dB,ideal CI,int max) |
---|
| 2976 | { string answer; |
---|
| 2977 | degBound=0; |
---|
| 2978 | int j,k,test_dim,flag; |
---|
| 2979 | matrix test_matrix[1][dif]; // the linear combination to test |
---|
| 2980 | intvec h; // Hilbert series |
---|
| 2981 | for (j=i+1;j<=i+dif;j=j+1) |
---|
| 2982 | { CI=CI+ideal(var(j)^d); // homogeneous polynomial of the same |
---|
| 2983 | // degree as the one we're looking for |
---|
| 2984 | // is added |
---|
| 2985 | } |
---|
| 2986 | ideal TEST; |
---|
| 2987 | // h=hilb(std(CI),1); |
---|
| 2988 | dB=dB+dif*(d-1); // used as degBound |
---|
| 2989 | while (1) |
---|
| 2990 | { test_matrix=matrix(B)*random(max,cd,dif); |
---|
| 2991 | // degBound=dB; |
---|
| 2992 | TEST=P+ideal(test_matrix); |
---|
| 2993 | attrib(TEST,"isSB",1); |
---|
| 2994 | test_dim=dim(TEST); |
---|
| 2995 | // degBound=0; |
---|
| 2996 | if (n-test_dim==i+dif) |
---|
| 2997 | { break; |
---|
| 2998 | } |
---|
| 2999 | // degBound=dB; |
---|
| 3000 | test_dim=dim(std(TEST)); |
---|
| 3001 | // test_dim=dim(std(TEST,h)); // Hilbert driven std-calculation |
---|
| 3002 | // degBound=0; |
---|
| 3003 | if (n-test_dim==i+dif) |
---|
| 3004 | { break; |
---|
| 3005 | } |
---|
| 3006 | else |
---|
| 3007 | { "HELP: The "+string(dif)+" random combination(s) of the "+string(cd)+" basis elements with"; |
---|
| 3008 | " coefficients in the range from -"+string(max)+" to "+string(max)+" did not lower the"; |
---|
| 3009 | " dimension by "+string(dif)+". You can abort, try again or give a new range:"; |
---|
| 3010 | answer=""; |
---|
| 3011 | while (answer<>"n |
---|
| 3012 | " && answer<>"y |
---|
| 3013 | ") |
---|
| 3014 | { " Do you want to abort (y/n)?"; |
---|
| 3015 | answer=read(""); |
---|
| 3016 | } |
---|
| 3017 | if (answer=="y |
---|
| 3018 | ") |
---|
| 3019 | { flag=1; |
---|
| 3020 | break; |
---|
| 3021 | } |
---|
| 3022 | answer=""; |
---|
| 3023 | while (answer<>"n |
---|
| 3024 | " && answer<>"y |
---|
| 3025 | ") |
---|
| 3026 | { " Do you want to try again (y/n)?"; |
---|
| 3027 | answer=read(""); |
---|
| 3028 | } |
---|
| 3029 | if (answer=="n |
---|
| 3030 | ") |
---|
| 3031 | { flag=1; |
---|
| 3032 | while (flag) |
---|
| 3033 | { " Give a new <int> > "+string(max)+" that bounds the range of coefficients:"; |
---|
| 3034 | answer=read(""); |
---|
| 3035 | for (j=1;j<=size(answer)-1;j=j+1) |
---|
| 3036 | { for (k=0;k<=9;k=k+1) |
---|
| 3037 | { if (answer[j]==string(k)) |
---|
| 3038 | { break; |
---|
| 3039 | } |
---|
| 3040 | } |
---|
| 3041 | if (k>9) |
---|
| 3042 | { flag=1; |
---|
| 3043 | break; |
---|
| 3044 | } |
---|
| 3045 | flag=0; |
---|
| 3046 | } |
---|
| 3047 | if (not(flag)) |
---|
| 3048 | { execute "test_dim="+string(answer[1..size(answer)]); |
---|
| 3049 | if (test_dim<=max) |
---|
| 3050 | { flag=1; |
---|
| 3051 | } |
---|
| 3052 | else |
---|
| 3053 | { max=test_dim; |
---|
| 3054 | } |
---|
| 3055 | } |
---|
| 3056 | } |
---|
| 3057 | } |
---|
| 3058 | } |
---|
| 3059 | } |
---|
| 3060 | if (not(flag)) |
---|
| 3061 | { P[(i+1)..(i+dif)]=test_matrix[1,1..dif]; |
---|
| 3062 | } |
---|
| 3063 | return(P,CI,dB); |
---|
| 3064 | } |
---|
| 3065 | |
---|
| 3066 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 3067 | // This procedure finds at most dif primary invariants in degree d. It returns |
---|
| 3068 | // all primary invariants found so far. The coefficients lie in the field of |
---|
| 3069 | // characteristic p>0. |
---|
| 3070 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 3071 | proc p_search_random (int n,int d,ideal B,int cd,ideal P,int i,int dif,int dB,ideal CI,int max) |
---|
| 3072 | { string answer; |
---|
| 3073 | degBound=0; |
---|
| 3074 | int j,k,test_dim,flag; |
---|
| 3075 | matrix test_matrix[1][dif]; // the linear combination to test |
---|
| 3076 | intvec h; // Hilbert series |
---|
| 3077 | ideal TEST; |
---|
| 3078 | while (dif>0) |
---|
| 3079 | { for (j=i+1;j<=i+dif;j=j+1) |
---|
| 3080 | { CI=CI+ideal(var(j)^d); // homogeneous polynomial of the same |
---|
| 3081 | // degree as the one we're looking for |
---|
| 3082 | // is added |
---|
| 3083 | } |
---|
| 3084 | // h=hilb(std(CI),1); |
---|
| 3085 | dB=dB+dif*(d-1); // used as degBound |
---|
| 3086 | test_matrix=matrix(B)*random(max,cd,dif); |
---|
| 3087 | // degBound=dB; |
---|
| 3088 | TEST=P+ideal(test_matrix); |
---|
| 3089 | attrib(TEST,"isSB",1); |
---|
| 3090 | test_dim=dim(TEST); |
---|
| 3091 | // degBound=0; |
---|
| 3092 | if (n-test_dim==i+dif) |
---|
| 3093 | { break; |
---|
| 3094 | } |
---|
| 3095 | // degBound=dB; |
---|
| 3096 | test_dim=dim(std(TEST)); |
---|
| 3097 | // test_dim=dim(std(TEST,h)); // Hilbert driven std-calculation |
---|
| 3098 | // degBound=0; |
---|
| 3099 | if (n-test_dim==i+dif) |
---|
| 3100 | { break; |
---|
| 3101 | } |
---|
| 3102 | else |
---|
| 3103 | { "HELP: The "+string(dif)+" random combination(s) of the "+string(cd)+" basis elements with"; |
---|
| 3104 | " coefficients in the range from -"+string(max)+" to "+string(max)+" did not lower the"; |
---|
| 3105 | " dimension by "+string(dif)+". You can abort, try again, lower the number of"; |
---|
| 3106 | " combinations searched for by 1 or give a larger coefficient range:"; |
---|
| 3107 | answer=""; |
---|
| 3108 | while (answer<>"n |
---|
| 3109 | " && answer<>"y |
---|
| 3110 | ") |
---|
| 3111 | { " Do you want to abort (y/n)?"; |
---|
| 3112 | answer=read(""); |
---|
| 3113 | } |
---|
| 3114 | if (answer=="y |
---|
| 3115 | ") |
---|
| 3116 | { flag=1; |
---|
| 3117 | break; |
---|
| 3118 | } |
---|
| 3119 | answer=""; |
---|
| 3120 | while (answer<>"n |
---|
| 3121 | " && answer<>"y |
---|
| 3122 | ") |
---|
| 3123 | { " Do you want to try again (y/n)?"; |
---|
| 3124 | answer=read(""); |
---|
| 3125 | } |
---|
| 3126 | if (answer=="n |
---|
| 3127 | ") |
---|
| 3128 | { answer=""; |
---|
| 3129 | while (answer<>"n |
---|
| 3130 | " && answer<>"y |
---|
| 3131 | ") |
---|
| 3132 | { " Do you want to lower the number of combinations by 1 (y/n)?"; |
---|
| 3133 | answer=read(""); |
---|
| 3134 | } |
---|
| 3135 | if (answer=="y |
---|
| 3136 | ") |
---|
| 3137 | { dif=dif-1; |
---|
| 3138 | } |
---|
| 3139 | else |
---|
| 3140 | { flag=1; |
---|
| 3141 | while (flag) |
---|
| 3142 | { " Give a new <int> > "+string(max)+" that bounds the range of coefficients:"; |
---|
| 3143 | answer=read(""); |
---|
| 3144 | for (j=1;j<=size(answer)-1;j=j+1) |
---|
| 3145 | { for (k=0;k<=9;k=k+1) |
---|
| 3146 | { if (answer[j]==string(k)) |
---|
| 3147 | { break; |
---|
| 3148 | } |
---|
| 3149 | } |
---|
| 3150 | if (k>9) |
---|
| 3151 | { flag=1; |
---|
| 3152 | break; |
---|
| 3153 | } |
---|
| 3154 | flag=0; |
---|
| 3155 | } |
---|
| 3156 | if (not(flag)) |
---|
| 3157 | { execute "test_dim="+string(answer[1..size(answer)]); |
---|
| 3158 | if (test_dim<=max) |
---|
| 3159 | { flag=1; |
---|
| 3160 | } |
---|
| 3161 | else |
---|
| 3162 | { max=test_dim; |
---|
| 3163 | } |
---|
| 3164 | } |
---|
| 3165 | } |
---|
| 3166 | } |
---|
| 3167 | } |
---|
| 3168 | } |
---|
| 3169 | CI=CI[1..i]; |
---|
| 3170 | dB=dB-dif*(d-1); |
---|
| 3171 | } |
---|
| 3172 | if (dif && not(flag)) |
---|
| 3173 | { P[(i+1)..(i+dif)]=test_matrix[1,1..dif]; |
---|
| 3174 | } |
---|
| 3175 | if (dif && flag) |
---|
| 3176 | { P[n+1]=0; |
---|
| 3177 | } |
---|
| 3178 | return(P,CI,dB); |
---|
| 3179 | } |
---|
| 3180 | |
---|
| 3181 | proc primary_char0_random (matrix REY,matrix M,int max,list #) |
---|
| 3182 | "USAGE: primary_char0_random(REY,M,r[,v]); |
---|
| 3183 | REY: a <matrix> representing the Reynolds operator, M: a 1x2 <matrix> |
---|
| 3184 | representing the Molien series, r: an <int> where -|r| to |r| is the |
---|
| 3185 | range of coefficients of the random combinations of bases elements, |
---|
| 3186 | v: an optional <int> |
---|
| 3187 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien and |
---|
| 3188 | M the one of molien or the second one of reynolds_molien |
---|
| 3189 | DISPLAY: information about the various stages of the programme if v does not |
---|
| 3190 | equal 0 |
---|
| 3191 | RETURN: primary invariants (type <matrix>) of the invariant ring |
---|
| 3192 | EXAMPLE: example primary_char0_random; shows an example |
---|
| 3193 | THEORY: Bases of homogeneous invariants are generated successively and random |
---|
| 3194 | linear combinations are chosen as primary invariants that lower the |
---|
| 3195 | dimension of the ideal generated by the previously found invariants |
---|
| 3196 | (see paper \"Generating a Noetherian Normalization of the Invariant Ring |
---|
| 3197 | of a Finite Group\" by Decker, Heydtmann, Schreyer (1997) to appear in |
---|
| 3198 | JSC). |
---|
| 3199 | " |
---|
| 3200 | { degBound=0; |
---|
| 3201 | if (char(basering)<>0) |
---|
| 3202 | { "ERROR: primary_char0_random should only be used with rings of"; |
---|
| 3203 | " characteristic 0."; |
---|
| 3204 | return(); |
---|
| 3205 | } |
---|
| 3206 | //----------------- checking input and setting verbose mode ------------------ |
---|
| 3207 | if (size(#)>1) |
---|
| 3208 | { "ERROR: primary_char0_random can only have four parameters."; |
---|
| 3209 | return(); |
---|
| 3210 | } |
---|
| 3211 | if (size(#)==1) |
---|
| 3212 | { if (typeof(#[1])<>"int") |
---|
| 3213 | { "ERROR: The fourth parameter should be of type <int>."; |
---|
| 3214 | return(); |
---|
| 3215 | } |
---|
| 3216 | else |
---|
| 3217 | { int v=#[1]; |
---|
| 3218 | } |
---|
| 3219 | } |
---|
| 3220 | else |
---|
| 3221 | { int v=0; |
---|
| 3222 | } |
---|
| 3223 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 3224 | // as the size of the matrices, as well |
---|
| 3225 | // as the number of primary invariants, |
---|
| 3226 | // we should get |
---|
| 3227 | if (ncols(REY)<>n) |
---|
| 3228 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
| 3229 | return(); |
---|
| 3230 | } |
---|
| 3231 | if (ncols(M)<>2 or nrows(M)<>1) |
---|
| 3232 | { "ERROR: Second parameter ought to be the Molien series." |
---|
| 3233 | return(); |
---|
| 3234 | } |
---|
| 3235 | //---------------------------------------------------------------------------- |
---|
| 3236 | if (v && voice<>2) |
---|
| 3237 | { " We can start looking for primary invariants..."; |
---|
| 3238 | ""; |
---|
| 3239 | } |
---|
| 3240 | if (v && voice==2) |
---|
| 3241 | { ""; |
---|
| 3242 | } |
---|
| 3243 | //------------------------- initializing variables --------------------------- |
---|
| 3244 | int dB; |
---|
| 3245 | poly p(1..2); // p(1) will be used for single terms of |
---|
| 3246 | // the partial expansion, p(2) to store |
---|
| 3247 | p(1..2)=partial_molien(M,1); // the intermediate result - |
---|
| 3248 | poly v1=var(1); // we need v1 to split off coefficients |
---|
| 3249 | // in the partial expansion of M (which |
---|
| 3250 | // is in terms of the first variable) - |
---|
| 3251 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
| 3252 | // space of invariants of degree d, |
---|
| 3253 | // newdim: dimension the ideal generated |
---|
| 3254 | // the primary invariants plus basis |
---|
| 3255 | // elements, dif=n-i-newdim, i.e. the |
---|
| 3256 | // number of new primary invairants that |
---|
| 3257 | // should be added in this degree - |
---|
| 3258 | ideal P,Pplus,CI,B; // P: will contain primary invariants, |
---|
| 3259 | // Pplus: P+B,CI: a complete |
---|
| 3260 | // intersection with the same Hilbert |
---|
| 3261 | // function as P - |
---|
| 3262 | dB=1; // used as degree bound |
---|
| 3263 | int i=0; |
---|
| 3264 | //-------------- loop that searches for primary invariants ------------------ |
---|
| 3265 | while(1) // repeat until n primary invariants are |
---|
| 3266 | { // found - |
---|
| 3267 | p(1..2)=partial_molien(M,1,p(2)); // next term of the partial expansion - |
---|
| 3268 | d=deg(p(1)); // degree where we'll search - |
---|
| 3269 | cd=int(coef(p(1),v1)[2,1]); // dimension of the homogeneous space of |
---|
| 3270 | // inviarants of degree d |
---|
| 3271 | if (v) |
---|
| 3272 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
| 3273 | } |
---|
| 3274 | B=invariant_basis_reynolds(REY,d,intvec(cd,6)); // basis of invariants of |
---|
| 3275 | // degree d |
---|
| 3276 | if (B[1]<>0) |
---|
| 3277 | { Pplus=P+B; |
---|
| 3278 | newdim=dim(std(Pplus)); |
---|
| 3279 | dif=n-i-newdim; |
---|
| 3280 | } |
---|
| 3281 | else |
---|
| 3282 | { dif=0; |
---|
| 3283 | } |
---|
| 3284 | if (dif<>0) // we have to find dif new primary |
---|
| 3285 | { // invariants |
---|
| 3286 | if (cd<>dif) |
---|
| 3287 | { P,CI,dB=search_random(n,d,B,cd,P,i,dif,dB,CI,max); // searching for |
---|
| 3288 | } // dif invariants - |
---|
| 3289 | else // i.e. we can take all of B |
---|
| 3290 | { for(j=i+1;j>i+dif;j=j+1) |
---|
| 3291 | { CI=CI+ideal(var(j)^d); |
---|
| 3292 | } |
---|
| 3293 | dB=dB+dif*(d-1); |
---|
| 3294 | P=Pplus; |
---|
| 3295 | } |
---|
| 3296 | if (ncols(P)==i) |
---|
| 3297 | { "WARNING: The return value is not a set of primary invariants, but"; |
---|
| 3298 | " polynomials qualifying as the first "+string(i)+" primary invariants."; |
---|
| 3299 | return(matrix(P)); |
---|
| 3300 | } |
---|
| 3301 | if (v) |
---|
| 3302 | { for (j=1;j<=dif;j=j+1) |
---|
| 3303 | { " We find: "+string(P[i+j]); |
---|
| 3304 | } |
---|
| 3305 | } |
---|
| 3306 | i=i+dif; |
---|
| 3307 | if (i==n) // found all primary invariants |
---|
| 3308 | { if (v) |
---|
| 3309 | { ""; |
---|
| 3310 | " We found all primary invariants."; |
---|
| 3311 | ""; |
---|
| 3312 | } |
---|
| 3313 | return(matrix(P)); |
---|
| 3314 | } |
---|
| 3315 | } // done with degree d |
---|
| 3316 | } |
---|
| 3317 | } |
---|
| 3318 | example |
---|
| 3319 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
| 3320 | echo=2; |
---|
| 3321 | ring R=0,(x,y,z),dp; |
---|
| 3322 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 3323 | matrix REY,M=reynolds_molien(A); |
---|
| 3324 | matrix P=primary_char0_random(REY,M,1); |
---|
| 3325 | print(P); |
---|
| 3326 | } |
---|
| 3327 | |
---|
| 3328 | proc primary_charp_random (matrix REY,string ring_name,int max,list #) |
---|
| 3329 | "USAGE: primary_charp_random(REY,ringname,r[,v]); |
---|
| 3330 | REY: a <matrix> representing the Reynolds operator, ringname: a |
---|
| 3331 | <string> giving the name of a ring where the Molien series is stored, |
---|
| 3332 | r: an <int> where -|r| to |r| is the range of coefficients of the |
---|
| 3333 | random combinations of bases elements, v: an optional <int> |
---|
| 3334 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien and |
---|
| 3335 | ringname gives the name of a ring of characteristic 0 that has been |
---|
| 3336 | created by molien or reynolds_molien |
---|
| 3337 | DISPLAY: information about the various stages of the programme if v does not |
---|
| 3338 | equal 0 |
---|
| 3339 | RETURN: primary invariants (type <matrix>) of the invariant ring |
---|
| 3340 | EXAMPLE: example primary_charp_random; shows an example |
---|
| 3341 | THEORY: Bases of homogeneous invariants are generated successively and random |
---|
| 3342 | linear combinations are chosen as primary invariants that lower the |
---|
| 3343 | dimension of the ideal generated by the previously found invariants |
---|
| 3344 | (see paper \"Generating a Noetherian Normalization of the Invariant Ring |
---|
| 3345 | of a Finite Group\" by Decker, Heydtmann, Schreyer (1997) to appear in |
---|
| 3346 | JSC). |
---|
| 3347 | " |
---|
| 3348 | { degBound=0; |
---|
| 3349 | // ---------------- checking input and setting verbose mode ------------------ |
---|
| 3350 | if (char(basering)==0) |
---|
| 3351 | { "ERROR: primary_charp_random should only be used with rings of"; |
---|
| 3352 | " characteristic p>0."; |
---|
| 3353 | return(); |
---|
| 3354 | } |
---|
| 3355 | if (size(#)>1) |
---|
| 3356 | { "ERROR: primary_charp_random can only have four parameters."; |
---|
| 3357 | return(); |
---|
| 3358 | } |
---|
| 3359 | if (size(#)==1) |
---|
| 3360 | { if (typeof(#[1])<>"int") |
---|
| 3361 | { "ERROR: The fourth parameter should be of type <int>."; |
---|
| 3362 | return(); |
---|
| 3363 | } |
---|
| 3364 | else |
---|
| 3365 | { int v=#[1]; |
---|
| 3366 | } |
---|
| 3367 | } |
---|
| 3368 | else |
---|
| 3369 | { int v=0; |
---|
| 3370 | } |
---|
| 3371 | def br=basering; |
---|
| 3372 | int n=nvars(br); // n is the number of variables, as well |
---|
| 3373 | // as the size of the matrices, as well |
---|
| 3374 | // as the number of primary invariants, |
---|
| 3375 | // we should get |
---|
| 3376 | if (ncols(REY)<>n) |
---|
| 3377 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
| 3378 | return(); |
---|
| 3379 | } |
---|
| 3380 | if (typeof(`ring_name`)<>"ring") |
---|
[5811fb] | 3381 | { "ERROR: Second parameter ought to the name of a ring where the Molien"; |
---|
[29aa4bf] | 3382 | " is stored."; |
---|
| 3383 | return(); |
---|
| 3384 | } |
---|
| 3385 | //---------------------------------------------------------------------------- |
---|
| 3386 | if (v && voice<>2) |
---|
| 3387 | { " We can start looking for primary invariants..."; |
---|
| 3388 | ""; |
---|
| 3389 | } |
---|
| 3390 | if (v && voice==2) |
---|
| 3391 | { ""; |
---|
| 3392 | } |
---|
| 3393 | //----------------------- initializing variables ----------------------------- |
---|
| 3394 | int dB; |
---|
| 3395 | setring `ring_name`; // the Molien series is stores here - |
---|
| 3396 | poly p(1..2); // p(1) will be used for single terms of |
---|
| 3397 | // the partial expansion, p(2) to store |
---|
| 3398 | p(1..2)=partial_molien(M,1); // the intermediate result - |
---|
| 3399 | poly v1=var(1); // we need v1 to split off coefficients |
---|
| 3400 | // in the partial expansion of M (which |
---|
| 3401 | // is in terms of the first variable) |
---|
| 3402 | setring br; |
---|
| 3403 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
| 3404 | // space of invariants of degree d, |
---|
| 3405 | // newdim: dimension the ideal generated |
---|
| 3406 | // the primary invariants plus basis |
---|
| 3407 | // elements, dif=n-i-newdim, i.e. the |
---|
| 3408 | // number of new primary invairants that |
---|
| 3409 | // should be added in this degree - |
---|
| 3410 | ideal P,Pplus,CI,B; // P: will contain primary invariants, |
---|
| 3411 | // Pplus: P+B, CI: a complete |
---|
| 3412 | // intersection with the same Hilbert |
---|
| 3413 | // function as P - |
---|
| 3414 | dB=1; // used as degree bound |
---|
| 3415 | int i=0; |
---|
| 3416 | //---------------- loop that searches for primary invariants ----------------- |
---|
| 3417 | while(1) // repeat until n primary invariants are |
---|
| 3418 | { // found |
---|
| 3419 | setring `ring_name`; |
---|
| 3420 | p(1..2)=partial_molien(M,1,p(2)); // next term of the partial expansion - |
---|
| 3421 | d=deg(p(1)); // degree where we'll search - |
---|
| 3422 | cd=int(coef(p(1),v1)[2,1]); // dimension of the homogeneous space of |
---|
| 3423 | // inviarants of degree d |
---|
| 3424 | setring br; |
---|
| 3425 | if (v) |
---|
| 3426 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
| 3427 | } |
---|
| 3428 | B=invariant_basis_reynolds(REY,d,intvec(cd,6)); // basis of invariants of |
---|
| 3429 | // degree d |
---|
| 3430 | if (B[1]<>0) |
---|
| 3431 | { Pplus=P+B; |
---|
| 3432 | newdim=dim(std(Pplus)); |
---|
| 3433 | dif=n-i-newdim; |
---|
| 3434 | } |
---|
| 3435 | else |
---|
| 3436 | { dif=0; |
---|
| 3437 | } |
---|
| 3438 | if (dif<>0) // we have to find dif new primary |
---|
| 3439 | { // invariants |
---|
| 3440 | if (cd<>dif) |
---|
| 3441 | { P,CI,dB=p_search_random(n,d,B,cd,P,i,dif,dB,CI,max); |
---|
| 3442 | } |
---|
| 3443 | else // i.e. we can take all of B |
---|
| 3444 | { for(j=i+1;j>i+dif;j=j+1) |
---|
| 3445 | { CI=CI+ideal(var(j)^d); |
---|
| 3446 | } |
---|
| 3447 | dB=dB+dif*(d-1); |
---|
| 3448 | P=Pplus; |
---|
| 3449 | } |
---|
| 3450 | if (ncols(P)==n+1) |
---|
| 3451 | { "WARNING: The first return value is not a set of primary invariants,"; |
---|
| 3452 | " but polynomials qualifying as the first "+string(i)+" primary invariants."; |
---|
| 3453 | return(matrix(P)); |
---|
| 3454 | } |
---|
| 3455 | if (v) |
---|
| 3456 | { for (j=1;j<=size(P)-i;j=j+1) |
---|
| 3457 | { " We find: "+string(P[i+j]); |
---|
| 3458 | } |
---|
| 3459 | } |
---|
| 3460 | i=size(P); |
---|
| 3461 | if (i==n) // found all primary invariants |
---|
| 3462 | { if (v) |
---|
| 3463 | { ""; |
---|
| 3464 | " We found all primary invariants."; |
---|
| 3465 | ""; |
---|
| 3466 | } |
---|
| 3467 | return(matrix(P)); |
---|
| 3468 | } |
---|
| 3469 | } // done with degree d |
---|
| 3470 | } |
---|
| 3471 | } |
---|
| 3472 | example |
---|
| 3473 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7 (changed into"; |
---|
| 3474 | " characteristic 3)"; |
---|
| 3475 | echo=2; |
---|
| 3476 | ring R=3,(x,y,z),dp; |
---|
| 3477 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 3478 | list L=group_reynolds(A); |
---|
| 3479 | string newring="alskdfj"; |
---|
| 3480 | molien(L[2..size(L)],newring); |
---|
| 3481 | matrix P=primary_charp_random(L[1],newring,1); |
---|
[c67136] | 3482 | if(system("with","Namespaces")) { kill Top::`newring`; } |
---|
[29aa4bf] | 3483 | kill `newring`; |
---|
| 3484 | print(P); |
---|
| 3485 | } |
---|
| 3486 | |
---|
| 3487 | proc primary_char0_no_molien_random (matrix REY, int max, list #) |
---|
| 3488 | "USAGE: primary_char0_no_molien_random(REY,r[,v]); |
---|
| 3489 | REY: a <matrix> representing the Reynolds operator, r: an <int> where |
---|
| 3490 | -|r| to |r| is the range of coefficients of the random combinations of |
---|
| 3491 | bases elements, v: an optional <int> |
---|
| 3492 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien |
---|
| 3493 | DISPLAY: information about the various stages of the programme if v does not |
---|
| 3494 | equal 0 |
---|
| 3495 | RETURN: primary invariants (type <matrix>) of the invariant ring and an |
---|
| 3496 | <intvec> listing some of the degrees where no non-trivial homogeneous |
---|
| 3497 | invariants are to be found |
---|
| 3498 | EXAMPLE: example primary_char0_no_molien_random; shows an example |
---|
| 3499 | THEORY: Bases of homogeneous invariants are generated successively and random |
---|
| 3500 | linear combinations are chosen as primary invariants that lower the |
---|
| 3501 | dimension of the ideal generated by the previously found invariants |
---|
| 3502 | (see paper \"Generating a Noetherian Normalization of the Invariant Ring |
---|
| 3503 | of a Finite Group\" by Decker, Heydtmann, Schreyer (1997) to appear in |
---|
| 3504 | JSC). |
---|
| 3505 | " |
---|
| 3506 | { degBound=0; |
---|
| 3507 | //-------------- checking input and setting verbose mode --------------------- |
---|
| 3508 | if (char(basering)<>0) |
---|
| 3509 | { "ERROR: primary_char0_no_molien_random should only be used with rings of"; |
---|
| 3510 | " characteristic 0."; |
---|
| 3511 | return(); |
---|
| 3512 | } |
---|
| 3513 | if (size(#)>1) |
---|
| 3514 | { "ERROR: primary_char0_no_molien_random can only have three parameters."; |
---|
| 3515 | return(); |
---|
| 3516 | } |
---|
| 3517 | if (size(#)==1) |
---|
| 3518 | { if (typeof(#[1])<>"int") |
---|
| 3519 | { "ERROR: The third parameter should be of type <int>."; |
---|
| 3520 | return(); |
---|
| 3521 | } |
---|
| 3522 | else |
---|
| 3523 | { int v=#[1]; |
---|
| 3524 | } |
---|
| 3525 | } |
---|
| 3526 | else |
---|
| 3527 | { int v=0; |
---|
| 3528 | } |
---|
| 3529 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 3530 | // as the size of the matrices, as well |
---|
| 3531 | // as the number of primary invariants, |
---|
| 3532 | // we should get |
---|
| 3533 | if (ncols(REY)<>n) |
---|
| 3534 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
| 3535 | return(); |
---|
| 3536 | } |
---|
| 3537 | //---------------------------------------------------------------------------- |
---|
| 3538 | if (v && voice<>2) |
---|
| 3539 | { " We can start looking for primary invariants..."; |
---|
| 3540 | ""; |
---|
| 3541 | } |
---|
| 3542 | if (v && voice==2) |
---|
| 3543 | { ""; |
---|
| 3544 | } |
---|
| 3545 | //----------------------- initializing variables ----------------------------- |
---|
| 3546 | int dB; |
---|
| 3547 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
| 3548 | // space of invariants of degree d, |
---|
| 3549 | // newdim: dimension the ideal generated |
---|
| 3550 | // the primary invariants plus basis |
---|
| 3551 | // elements, dif=n-i-newdim, i.e. the |
---|
| 3552 | // number of new primary invairants that |
---|
| 3553 | // should be added in this degree - |
---|
| 3554 | ideal P,Pplus,CI,B; // P: will contain primary invariants, |
---|
| 3555 | // Pplus: P+B, CI: a complete |
---|
| 3556 | // intersection with the same Hilbert |
---|
| 3557 | // function as P - |
---|
| 3558 | dB=1; // used as degree bound - |
---|
| 3559 | d=0; // initializing |
---|
| 3560 | int i=0; |
---|
| 3561 | intvec deg_vector; |
---|
| 3562 | //------------------ loop that searches for primary invariants --------------- |
---|
| 3563 | while(1) // repeat until n primary invariants are |
---|
| 3564 | { // found - |
---|
| 3565 | d=d+1; // degree where we'll search |
---|
| 3566 | if (v) |
---|
| 3567 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
| 3568 | } |
---|
| 3569 | B=invariant_basis_reynolds(REY,d,intvec(-1,6)); // basis of invariants of |
---|
| 3570 | // degree d |
---|
| 3571 | if (B[1]<>0) |
---|
| 3572 | { Pplus=P+B; |
---|
| 3573 | newdim=dim(std(Pplus)); |
---|
| 3574 | dif=n-i-newdim; |
---|
| 3575 | } |
---|
| 3576 | else |
---|
| 3577 | { dif=0; |
---|
| 3578 | deg_vector=deg_vector,d; |
---|
| 3579 | } |
---|
| 3580 | if (dif<>0) // we have to find dif new primary |
---|
| 3581 | { // invariants |
---|
| 3582 | cd=size(B); |
---|
| 3583 | if (cd<>dif) |
---|
| 3584 | { P,CI,dB=search_random(n,d,B,cd,P,i,dif,dB,CI,max); |
---|
| 3585 | } |
---|
| 3586 | else // i.e. we can take all of B |
---|
| 3587 | { for(j=i+1;j<=i+dif;j=j+1) |
---|
| 3588 | { CI=CI+ideal(var(j)^d); |
---|
| 3589 | } |
---|
| 3590 | dB=dB+dif*(d-1); |
---|
| 3591 | P=Pplus; |
---|
| 3592 | } |
---|
| 3593 | if (ncols(P)==i) |
---|
| 3594 | { "WARNING: The first return value is not a set of primary invariants,"; |
---|
| 3595 | " but polynomials qualifying as the first "+string(i)+" primary invariants."; |
---|
| 3596 | return(matrix(P)); |
---|
| 3597 | } |
---|
| 3598 | if (v) |
---|
| 3599 | { for (j=1;j<=dif;j=j+1) |
---|
| 3600 | { " We find: "+string(P[i+j]); |
---|
| 3601 | } |
---|
| 3602 | } |
---|
| 3603 | i=i+dif; |
---|
| 3604 | if (i==n) // found all primary invariants |
---|
| 3605 | { if (v) |
---|
| 3606 | { ""; |
---|
| 3607 | " We found all primary invariants."; |
---|
| 3608 | ""; |
---|
| 3609 | } |
---|
| 3610 | if (deg_vector==0) |
---|
| 3611 | { return(matrix(P)); |
---|
| 3612 | } |
---|
| 3613 | else |
---|
| 3614 | { return(matrix(P),compress(deg_vector)); |
---|
| 3615 | } |
---|
| 3616 | } |
---|
| 3617 | } // done with degree d |
---|
| 3618 | else |
---|
| 3619 | { if (v) |
---|
| 3620 | { " None here..."; |
---|
| 3621 | } |
---|
| 3622 | } |
---|
| 3623 | } |
---|
| 3624 | } |
---|
| 3625 | example |
---|
| 3626 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
| 3627 | echo=2; |
---|
| 3628 | ring R=0,(x,y,z),dp; |
---|
| 3629 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 3630 | list L=group_reynolds(A); |
---|
| 3631 | list l=primary_char0_no_molien_random(L[1],1); |
---|
| 3632 | print(l[1]); |
---|
| 3633 | } |
---|
| 3634 | |
---|
| 3635 | proc primary_charp_no_molien_random (matrix REY, int max, list #) |
---|
| 3636 | "USAGE: primary_charp_no_molien_random(REY,r[,v]); |
---|
| 3637 | REY: a <matrix> representing the Reynolds operator, r: an <int> where |
---|
| 3638 | -|r| to |r| is the range of coefficients of the random combinations of |
---|
| 3639 | bases elements, v: an optional <int> |
---|
| 3640 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien |
---|
| 3641 | DISPLAY: information about the various stages of the programme if v does not |
---|
| 3642 | equal 0 |
---|
| 3643 | RETURN: primary invariants (type <matrix>) of the invariant ring and an |
---|
| 3644 | <intvec> listing some of the degrees where no non-trivial homogeneous |
---|
| 3645 | invariants are to be found |
---|
| 3646 | EXAMPLE: example primary_charp_no_molien_random; shows an example |
---|
| 3647 | THEORY: Bases of homogeneous invariants are generated successively and random |
---|
| 3648 | linear combinations are chosen as primary invariants that lower the |
---|
| 3649 | dimension of the ideal generated by the previously found invariants |
---|
| 3650 | (see paper \"Generating a Noetherian Normalization of the Invariant Ring |
---|
| 3651 | of a Finite Group\" by Decker, Heydtmann, Schreyer (1997) to appear in |
---|
| 3652 | JSC). |
---|
| 3653 | " |
---|
| 3654 | { degBound=0; |
---|
| 3655 | //----------------- checking input and setting verbose mode ------------------ |
---|
| 3656 | if (char(basering)==0) |
---|
| 3657 | { "ERROR: primary_charp_no_molien_random should only be used with rings of"; |
---|
| 3658 | " characteristic p>0."; |
---|
| 3659 | return(); |
---|
| 3660 | } |
---|
| 3661 | if (size(#)>1) |
---|
| 3662 | { "ERROR: primary_charp_no_molien_random can only have three parameters."; |
---|
| 3663 | return(); |
---|
| 3664 | } |
---|
| 3665 | if (size(#)==1) |
---|
| 3666 | { if (typeof(#[1])<>"int") |
---|
| 3667 | { "ERROR: The third parameter should be of type <int>."; |
---|
| 3668 | return(); |
---|
| 3669 | } |
---|
| 3670 | else |
---|
| 3671 | { int v=#[1]; |
---|
| 3672 | } |
---|
| 3673 | } |
---|
| 3674 | else |
---|
| 3675 | { int v=0; |
---|
| 3676 | } |
---|
| 3677 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 3678 | // as the size of the matrices, as well |
---|
| 3679 | // as the number of primary invariants, |
---|
| 3680 | // we should get |
---|
| 3681 | if (ncols(REY)<>n) |
---|
| 3682 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
| 3683 | return(); |
---|
| 3684 | } |
---|
| 3685 | //---------------------------------------------------------------------------- |
---|
| 3686 | if (v && voice<>2) |
---|
| 3687 | { " We can start looking for primary invariants..."; |
---|
| 3688 | ""; |
---|
| 3689 | } |
---|
| 3690 | if (v && voice==2) |
---|
| 3691 | { ""; |
---|
| 3692 | } |
---|
| 3693 | //-------------------- initializing variables -------------------------------- |
---|
| 3694 | int dB; |
---|
| 3695 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
| 3696 | // space of invariants of degree d, |
---|
| 3697 | // newdim: dimension the ideal generated |
---|
| 3698 | // the primary invariants plus basis |
---|
| 3699 | // elements, dif=n-i-newdim, i.e. the |
---|
| 3700 | // number of new primary invairants that |
---|
| 3701 | // should be added in this degree - |
---|
| 3702 | ideal P,Pplus,CI,B; // P: will contain primary invariants, |
---|
| 3703 | // Pplus: P+B, CI: a complete |
---|
| 3704 | // intersection with the same Hilbert |
---|
| 3705 | // function as P - |
---|
| 3706 | dB=1; // used as degree bound - |
---|
| 3707 | d=0; // initializing |
---|
| 3708 | int i=0; |
---|
| 3709 | intvec deg_vector; |
---|
| 3710 | //------------------ loop that searches for primary invariants --------------- |
---|
| 3711 | while(1) // repeat until n primary invariants are |
---|
| 3712 | { // found - |
---|
| 3713 | d=d+1; // degree where we'll search |
---|
| 3714 | if (v) |
---|
| 3715 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
| 3716 | } |
---|
| 3717 | B=invariant_basis_reynolds(REY,d,intvec(-1,6)); // basis of invariants of |
---|
| 3718 | // degree d |
---|
| 3719 | if (B[1]<>0) |
---|
| 3720 | { Pplus=P+B; |
---|
| 3721 | newdim=dim(std(Pplus)); |
---|
| 3722 | dif=n-i-newdim; |
---|
| 3723 | } |
---|
| 3724 | else |
---|
| 3725 | { dif=0; |
---|
| 3726 | deg_vector=deg_vector,d; |
---|
| 3727 | } |
---|
| 3728 | if (dif<>0) // we have to find dif new primary |
---|
| 3729 | { // invariants |
---|
| 3730 | cd=size(B); |
---|
| 3731 | if (cd<>dif) |
---|
| 3732 | { P,CI,dB=p_search_random(n,d,B,cd,P,i,dif,dB,CI,max); |
---|
| 3733 | } |
---|
| 3734 | else // i.e. we can take all of B |
---|
| 3735 | { for(j=i+1;j<=i+dif;j=j+1) |
---|
| 3736 | { CI=CI+ideal(var(j)^d); |
---|
| 3737 | } |
---|
| 3738 | dB=dB+dif*(d-1); |
---|
| 3739 | P=Pplus; |
---|
| 3740 | } |
---|
| 3741 | if (ncols(P)==n+1) |
---|
| 3742 | { "WARNING: The first return value is not a set of primary invariants,"; |
---|
| 3743 | " but polynomials qualifying as the first "+string(i)+" primary invariants."; |
---|
| 3744 | return(matrix(P)); |
---|
| 3745 | } |
---|
| 3746 | if (v) |
---|
| 3747 | { for (j=1;j<=size(P)-i;j=j+1) |
---|
| 3748 | { " We find: "+string(P[i+j]); |
---|
| 3749 | } |
---|
| 3750 | } |
---|
| 3751 | i=size(P); |
---|
| 3752 | if (i==n) // found all primary invariants |
---|
| 3753 | { if (v) |
---|
| 3754 | { ""; |
---|
| 3755 | " We found all primary invariants."; |
---|
| 3756 | ""; |
---|
| 3757 | } |
---|
| 3758 | if (deg_vector==0) |
---|
| 3759 | { return(matrix(P)); |
---|
| 3760 | } |
---|
| 3761 | else |
---|
| 3762 | { return(matrix(P),compress(deg_vector)); |
---|
| 3763 | } |
---|
| 3764 | } |
---|
| 3765 | } // done with degree d |
---|
| 3766 | else |
---|
| 3767 | { if (v) |
---|
| 3768 | { " None here..."; |
---|
| 3769 | } |
---|
| 3770 | } |
---|
| 3771 | } |
---|
| 3772 | } |
---|
| 3773 | example |
---|
| 3774 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7 (changed into"; |
---|
| 3775 | " characteristic 3)"; |
---|
| 3776 | echo=2; |
---|
| 3777 | ring R=3,(x,y,z),dp; |
---|
| 3778 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 3779 | list L=group_reynolds(A); |
---|
| 3780 | list l=primary_charp_no_molien_random(L[1],1); |
---|
| 3781 | print(l[1]); |
---|
| 3782 | } |
---|
| 3783 | |
---|
| 3784 | proc primary_charp_without_random (list #) |
---|
| 3785 | "USAGE: primary_charp_without_random(G1,G2,...,r[,v]); |
---|
| 3786 | G1,G2,...: <matrices> generating a finite matrix group, r: an <int> |
---|
| 3787 | where -|r| to |r| is the range of coefficients of the random |
---|
| 3788 | combinations of bases elements, v: an optional <int> |
---|
| 3789 | DISPLAY: information about the various stages of the programme if v does not |
---|
| 3790 | equal 0 |
---|
| 3791 | RETURN: primary invariants (type <matrix>) of the invariant ring |
---|
| 3792 | EXAMPLE: example primary_charp_without_random; shows an example |
---|
| 3793 | THEORY: Bases of homogeneous invariants are generated successively and random |
---|
| 3794 | linear combinations are chosen as primary invariants that lower the |
---|
| 3795 | dimension of the ideal generated by the previously found invariants |
---|
| 3796 | (see paper \"Generating a Noetherian Normalization of the Invariant Ring |
---|
| 3797 | of a Finite Group\" by Decker, Heydtmann, Schreyer (1997) to appear in |
---|
| 3798 | JSC). No Reynolds operator or Molien series is used. |
---|
| 3799 | " |
---|
| 3800 | { degBound=0; |
---|
| 3801 | //--------------------- checking input and setting verbose mode -------------- |
---|
| 3802 | if (char(basering)==0) |
---|
| 3803 | { "ERROR: primary_charp_without_random should only be used with rings of"; |
---|
| 3804 | " characteristic 0."; |
---|
| 3805 | return(); |
---|
| 3806 | } |
---|
| 3807 | if (size(#)<2) |
---|
| 3808 | { "ERROR: There are too few parameters."; |
---|
| 3809 | return(); |
---|
| 3810 | } |
---|
| 3811 | if (typeof(#[size(#)])=="int" && typeof(#[size(#)-1])=="int") |
---|
| 3812 | { int v=#[size(#)]; |
---|
| 3813 | int max=#[size(#)-1]; |
---|
| 3814 | int gen_num=size(#)-2; |
---|
| 3815 | if (gen_num==0) |
---|
| 3816 | { "ERROR: There are no generators of a finite matrix group given."; |
---|
| 3817 | return(); |
---|
| 3818 | } |
---|
| 3819 | } |
---|
| 3820 | else |
---|
| 3821 | { if (typeof(#[size(#)])=="int") |
---|
| 3822 | { int max=#[size(#)]; |
---|
| 3823 | int v=0; |
---|
| 3824 | int gen_num=size(#)-1; |
---|
| 3825 | } |
---|
| 3826 | else |
---|
| 3827 | { "ERROR: The last parameter should be an <int>."; |
---|
| 3828 | return(); |
---|
| 3829 | } |
---|
| 3830 | } |
---|
| 3831 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 3832 | // as the size of the matrices, as well |
---|
| 3833 | // as the number of primary invariants, |
---|
| 3834 | // we should get |
---|
| 3835 | for (int i=1;i<=gen_num;i=i+1) |
---|
| 3836 | { if (typeof(#[i])=="matrix") |
---|
| 3837 | { if (nrows(#[i])<>n or ncols(#[i])<>n) |
---|
| 3838 | { "ERROR: The number of variables of the base ring needs to be the same"; |
---|
| 3839 | " as the dimension of the square matrices"; |
---|
| 3840 | return(); |
---|
| 3841 | } |
---|
| 3842 | } |
---|
| 3843 | else |
---|
| 3844 | { "ERROR: The first parameters should be a list of matrices"; |
---|
| 3845 | return(); |
---|
| 3846 | } |
---|
| 3847 | } |
---|
| 3848 | //---------------------------------------------------------------------------- |
---|
| 3849 | if (v && voice==2) |
---|
| 3850 | { ""; |
---|
| 3851 | } |
---|
| 3852 | //---------------------------- initializing variables ------------------------ |
---|
| 3853 | int dB; |
---|
| 3854 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
| 3855 | // space of invariants of degree d, |
---|
| 3856 | // newdim: dimension the ideal generated |
---|
| 3857 | // the primary invariants plus basis |
---|
| 3858 | // elements, dif=n-i-newdim, i.e. the |
---|
| 3859 | // number of new primary invairants that |
---|
| 3860 | // should be added in this degree - |
---|
| 3861 | ideal P,Pplus,CI,B; // P: will contain primary invariants, |
---|
| 3862 | // Pplus: P+B, CI: a complete |
---|
| 3863 | // intersection with the same Hilbert |
---|
| 3864 | // function as P - |
---|
| 3865 | dB=1; // used as degree bound - |
---|
| 3866 | d=0; // initializing |
---|
| 3867 | i=0; |
---|
| 3868 | intvec deg_vector; |
---|
| 3869 | //-------------------- loop that searches for primary invariants ------------- |
---|
| 3870 | while(1) // repeat until n primary invariants are |
---|
| 3871 | { // found - |
---|
| 3872 | d=d+1; // degree where we'll search |
---|
| 3873 | if (v) |
---|
| 3874 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
| 3875 | } |
---|
| 3876 | B=invariant_basis(d,#[1..gen_num]); // basis of invariants of degree d |
---|
| 3877 | if (B[1]<>0) |
---|
| 3878 | { Pplus=P+B; |
---|
| 3879 | newdim=dim(std(Pplus)); |
---|
| 3880 | dif=n-i-newdim; |
---|
| 3881 | } |
---|
| 3882 | else |
---|
| 3883 | { dif=0; |
---|
| 3884 | deg_vector=deg_vector,d; |
---|
| 3885 | } |
---|
| 3886 | if (dif<>0) // we have to find dif new primary |
---|
| 3887 | { // invariants |
---|
| 3888 | cd=size(B); |
---|
| 3889 | if (cd<>dif) |
---|
| 3890 | { P,CI,dB=p_search_random(n,d,B,cd,P,i,dif,dB,CI,max); |
---|
| 3891 | } |
---|
| 3892 | else // i.e. we can take all of B |
---|
| 3893 | { for(j=i+1;j<=i+dif;j=j+1) |
---|
| 3894 | { CI=CI+ideal(var(j)^d); |
---|
| 3895 | } |
---|
| 3896 | dB=dB+dif*(d-1); |
---|
| 3897 | P=Pplus; |
---|
| 3898 | } |
---|
| 3899 | if (ncols(P)==n+1) |
---|
| 3900 | { "WARNING: The first return value is not a set of primary invariants,"; |
---|
| 3901 | " but polynomials qualifying as the first "+string(i)+" primary invariants."; |
---|
| 3902 | return(matrix(P)); |
---|
| 3903 | } |
---|
| 3904 | if (v) |
---|
| 3905 | { for (j=1;j<=size(P)-i;j=j+1) |
---|
| 3906 | { " We find: "+string(P[i+j]); |
---|
| 3907 | } |
---|
| 3908 | } |
---|
| 3909 | i=size(P); |
---|
| 3910 | if (i==n) // found all primary invariants |
---|
| 3911 | { if (v) |
---|
| 3912 | { ""; |
---|
| 3913 | " We found all primary invariants."; |
---|
| 3914 | ""; |
---|
| 3915 | } |
---|
| 3916 | return(matrix(P)); |
---|
| 3917 | } |
---|
| 3918 | } // done with degree d |
---|
| 3919 | else |
---|
| 3920 | { if (v) |
---|
| 3921 | { " None here..."; |
---|
| 3922 | } |
---|
| 3923 | } |
---|
| 3924 | } |
---|
| 3925 | } |
---|
| 3926 | example |
---|
| 3927 | { echo=2; |
---|
| 3928 | ring R=2,(x,y,z),dp; |
---|
| 3929 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 3930 | matrix P=primary_charp_without_random(A,1); |
---|
| 3931 | print(P); |
---|
| 3932 | } |
---|
| 3933 | |
---|
| 3934 | proc primary_invariants_random (list #) |
---|
| 3935 | "USAGE: primary_invariants_random(G1,G2,...,r[,flags]); |
---|
| 3936 | G1,G2,...: <matrices> generating a finite matrix group, r: an <int> |
---|
| 3937 | where -|r| to |r| is the range of coefficients of the random |
---|
| 3938 | combinations of bases elements, flags: an optional <intvec> with three |
---|
| 3939 | entries, if the first one equals 0 (also the default), the programme |
---|
| 3940 | attempts to compute the Molien series and Reynolds operator, if it |
---|
| 3941 | equals 1, the programme is told that the Molien series should not be |
---|
| 3942 | computed, if it equals -1 characteristic 0 is simulated, i.e. the |
---|
| 3943 | Molien series is computed as if the base field were characteristic 0 |
---|
| 3944 | (the user must choose a field of large prime characteristic, e.g. |
---|
| 3945 | 32003) and if the first one is anything else, it means that the |
---|
| 3946 | characteristic of the base field divides the group order, the second |
---|
| 3947 | component should give the size of intervals between canceling common |
---|
| 3948 | factors in the expansion of the Molien series, 0 (the default) means |
---|
| 3949 | only once after generating all terms, in prime characteristic also a |
---|
| 3950 | negative number can be given to indicate that common factors should |
---|
| 3951 | always be canceled when the expansion is simple (the root of the |
---|
| 3952 | extension field does not occur among the coefficients) |
---|
| 3953 | DISPLAY: information about the various stages of the programme if the third |
---|
| 3954 | flag does not equal 0 |
---|
| 3955 | RETURN: primary invariants (type <matrix>) of the invariant ring and if |
---|
| 3956 | computable Reynolds operator (type <matrix>) and Molien series (type |
---|
| 3957 | <matrix>), if the first flag is 1 and we are in the non-modular case |
---|
| 3958 | then an <intvec> is returned giving some of the degrees where no |
---|
| 3959 | non-trivial homogeneous invariants can be found |
---|
| 3960 | EXAMPLE: example primary_invariants_random; shows an example |
---|
| 3961 | THEORY: Bases of homogeneous invariants are generated successively and random |
---|
| 3962 | linear combinations are chosen as primary invariants that lower the |
---|
| 3963 | dimension of the ideal generated by the previously found invariants |
---|
| 3964 | (see paper \"Generating a Noetherian Normalization of the Invariant Ring |
---|
| 3965 | of a Finite Group\" by Decker, Heydtmann, Schreyer (1997) to appear in |
---|
| 3966 | JSC). |
---|
| 3967 | " |
---|
| 3968 | { |
---|
| 3969 | // ----------------- checking input and setting flags ------------------------ |
---|
| 3970 | if (size(#)<2) |
---|
| 3971 | { "ERROR: There are too few parameters."; |
---|
| 3972 | return(); |
---|
| 3973 | } |
---|
| 3974 | int ch=char(basering); // the algorithms depend very much on the |
---|
| 3975 | // characteristic of the ground field |
---|
| 3976 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 3977 | // as the size of the matrices, as well |
---|
| 3978 | // as the number of primary invariants, |
---|
| 3979 | // we should get |
---|
| 3980 | int gen_num; |
---|
| 3981 | int mol_flag,v; |
---|
| 3982 | if (typeof(#[size(#)])=="intvec" && typeof(#[size(#)-1])=="int") |
---|
| 3983 | { if (size(#[size(#)])<>3) |
---|
| 3984 | { "ERROR: <intvec> should have three entries."; |
---|
| 3985 | return(); |
---|
| 3986 | } |
---|
| 3987 | gen_num=size(#)-2; |
---|
| 3988 | mol_flag=#[size(#)][1]; |
---|
| 3989 | if (#[size(#)][2]<0 && (ch==0 or (ch<>0 && mol_flag<>0))) |
---|
| 3990 | { "ERROR: the second component of <intvec> should be >=0"; |
---|
| 3991 | return(); |
---|
| 3992 | } |
---|
| 3993 | int interval=#[size(#)][2]; |
---|
| 3994 | v=#[size(#)][3]; |
---|
| 3995 | int max=#[size(#)-1]; |
---|
| 3996 | if (gen_num==0) |
---|
| 3997 | { "ERROR: There are no generators of a finite matrix group given."; |
---|
| 3998 | return(); |
---|
| 3999 | } |
---|
| 4000 | } |
---|
| 4001 | else |
---|
| 4002 | { if (typeof(#[size(#)])=="int") |
---|
| 4003 | { gen_num=size(#)-1; |
---|
| 4004 | mol_flag=0; |
---|
| 4005 | int interval=0; |
---|
| 4006 | v=0; |
---|
| 4007 | int max=#[size(#)]; |
---|
| 4008 | } |
---|
| 4009 | else |
---|
| 4010 | { "ERROR: If the two last parameters are not <int> and <intvec>, the last"; |
---|
| 4011 | " parameter should be an <int>."; |
---|
| 4012 | return(); |
---|
| 4013 | } |
---|
| 4014 | } |
---|
| 4015 | for (int i=1;i<=gen_num;i=i+1) |
---|
| 4016 | { if (typeof(#[i])=="matrix") |
---|
| 4017 | { if (nrows(#[i])<>n or ncols(#[i])<>n) |
---|
| 4018 | { "ERROR: The number of variables of the base ring needs to be the same"; |
---|
| 4019 | " as the dimension of the square matrices"; |
---|
| 4020 | return(); |
---|
| 4021 | } |
---|
| 4022 | } |
---|
| 4023 | else |
---|
| 4024 | { "ERROR: The first parameters should be a list of matrices"; |
---|
| 4025 | return(); |
---|
| 4026 | } |
---|
| 4027 | } |
---|
| 4028 | //---------------------------------------------------------------------------- |
---|
| 4029 | if (mol_flag==0) |
---|
| 4030 | { if (ch==0) |
---|
| 4031 | { matrix REY,M=reynolds_molien(#[1..gen_num],intvec(0,interval,v)); |
---|
| 4032 | // one will contain Reynolds operator and |
---|
| 4033 | // the other enumerator and denominator |
---|
| 4034 | // of Molien series |
---|
| 4035 | matrix P=primary_char0_random(REY,M,max,v); |
---|
| 4036 | return(P,REY,M); |
---|
| 4037 | } |
---|
| 4038 | else |
---|
| 4039 | { list L=group_reynolds(#[1..gen_num],v); |
---|
| 4040 | if (L[1]<>0) // testing whether we are in the modular |
---|
| 4041 | { string newring="aksldfalkdsflkj"; // case |
---|
| 4042 | if (minpoly==0) |
---|
| 4043 | { if (v) |
---|
| 4044 | { " We are dealing with the non-modular case."; |
---|
| 4045 | } |
---|
[5811fb] | 4046 | molien(L[2..size(L)],newring,intvec(0,interval,v)); |
---|
[29aa4bf] | 4047 | matrix P=primary_charp_random(L[1],newring,max,v); |
---|
| 4048 | return(P,L[1],newring); |
---|
| 4049 | } |
---|
| 4050 | else |
---|
| 4051 | { if (v) |
---|
| 4052 | { " Since it is impossible for this programme to calculate the Molien series for"; |
---|
| 4053 | " invariant rings over extension fields of prime characteristic, we have to"; |
---|
| 4054 | " continue without it."; |
---|
| 4055 | ""; |
---|
| 4056 | |
---|
| 4057 | } |
---|
| 4058 | list l=primary_charp_no_molien_random(L[1],max,v); |
---|
| 4059 | if (size(l)==2) |
---|
| 4060 | { return(l[1],L[1],l[2]); |
---|
| 4061 | } |
---|
| 4062 | else |
---|
| 4063 | { return(l[1],L[1]); |
---|
| 4064 | } |
---|
| 4065 | } |
---|
| 4066 | } |
---|
| 4067 | else // the modular case |
---|
| 4068 | { if (v) |
---|
| 4069 | { " There is also no Molien series, we can make use of..."; |
---|
| 4070 | ""; |
---|
| 4071 | " We can start looking for primary invariants..."; |
---|
| 4072 | ""; |
---|
| 4073 | } |
---|
| 4074 | return(primary_charp_without_random(#[1..gen_num],max,v)); |
---|
| 4075 | } |
---|
| 4076 | } |
---|
| 4077 | } |
---|
| 4078 | if (mol_flag==1) // the user wants no calculation of the |
---|
| 4079 | { list L=group_reynolds(#[1..gen_num],v); // Molien series |
---|
| 4080 | if (ch==0) |
---|
| 4081 | { list l=primary_char0_no_molien_random(L[1],max,v); |
---|
| 4082 | if (size(l)==2) |
---|
| 4083 | { return(l[1],L[1],l[2]); |
---|
| 4084 | } |
---|
| 4085 | else |
---|
| 4086 | { return(l[1],L[1]); |
---|
| 4087 | } |
---|
| 4088 | } |
---|
| 4089 | else |
---|
| 4090 | { if (L[1]<>0) // testing whether we are in the modular |
---|
| 4091 | { list l=primary_charp_no_molien_random(L[1],max,v); // case |
---|
| 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 // the modular case |
---|
| 4100 | { if (v) |
---|
| 4101 | { " We can start looking for primary invariants..."; |
---|
| 4102 | ""; |
---|
| 4103 | } |
---|
| 4104 | return(primary_charp_without_random(#[1..gen_num],max,v)); |
---|
| 4105 | } |
---|
| 4106 | } |
---|
| 4107 | } |
---|
| 4108 | if (mol_flag==-1) |
---|
| 4109 | { if (ch==0) |
---|
| 4110 | { "ERROR: Characteristic 0 can only be simulated in characteristic p>>0."; |
---|
| 4111 | return(); |
---|
| 4112 | } |
---|
| 4113 | list L=group_reynolds(#[1..gen_num],v); |
---|
| 4114 | string newring="aksldfalkdsflkj"; |
---|
[5811fb] | 4115 | molien(L[2..size(L)],newring,intvec(1,interval,v)); |
---|
[29aa4bf] | 4116 | matrix P=primary_charp_random(L[1],newring,max,v); |
---|
| 4117 | return(P,L[1],newring); |
---|
| 4118 | } |
---|
| 4119 | else // the user specified that the |
---|
| 4120 | { if (ch==0) // characteristic divides the group order |
---|
| 4121 | { "ERROR: The characteristic cannot divide the group order when it is 0."; |
---|
| 4122 | return(); |
---|
| 4123 | } |
---|
| 4124 | if (v) |
---|
| 4125 | { ""; |
---|
| 4126 | } |
---|
| 4127 | return(primary_charp_without_random(#[1..gen_num],max,v)); |
---|
| 4128 | } |
---|
| 4129 | } |
---|
| 4130 | example |
---|
| 4131 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
| 4132 | echo=2; |
---|
| 4133 | ring R=0,(x,y,z),dp; |
---|
| 4134 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 4135 | list L=primary_invariants_random(A,1); |
---|
| 4136 | print(L[1]); |
---|
| 4137 | } |
---|
| 4138 | |
---|
| 4139 | proc concat_intmat(intmat A,intmat B) |
---|
| 4140 | { int n=nrows(A); |
---|
| 4141 | int m1=ncols(A); |
---|
| 4142 | int m2=ncols(B); |
---|
| 4143 | intmat C[n][m1+m2]; |
---|
| 4144 | C[1..n,1..m1]=A[1..n,1..m1]; |
---|
| 4145 | C[1..n,m1+1..m1+m2]=B[1..n,1..m2]; |
---|
| 4146 | return(C); |
---|
| 4147 | } |
---|
| 4148 | |
---|
| 4149 | proc power_products(intvec deg_vec,int d) |
---|
| 4150 | "USAGE: power_products(dv,d); |
---|
| 4151 | dv: an <intvec> giving the degrees of homogeneous polynomials, d: the |
---|
| 4152 | degree of the desired power products |
---|
| 4153 | RETURN: a size(dv)*m <intmat> where each column ought to be interpreted as |
---|
| 4154 | containing the exponents of the corresponding polynomials. The product |
---|
| 4155 | of the powers is then homogeneous of degree d. |
---|
| 4156 | EXAMPLE: example power_products; gives an example |
---|
| 4157 | " |
---|
| 4158 | { ring R=0,x,dp; |
---|
| 4159 | if (d<=0) |
---|
| 4160 | { "ERROR: The <int> may not be <= 0"; |
---|
| 4161 | return(); |
---|
| 4162 | } |
---|
| 4163 | int d_neu,j,nc; |
---|
| 4164 | int s=size(deg_vec); |
---|
| 4165 | intmat PP[s][1]; |
---|
| 4166 | intmat TEST[s][1]; |
---|
| 4167 | for (int i=1;i<=s;i=i+1) |
---|
| 4168 | { if (i<0) |
---|
| 4169 | { "ERROR: The entries of <intvec> may not be <= 0"; |
---|
| 4170 | return(); |
---|
| 4171 | } |
---|
| 4172 | d_neu=d-deg_vec[i]; |
---|
| 4173 | if (d_neu>0) |
---|
| 4174 | { intmat PPd_neu=power_products(intvec(deg_vec[i..s]),d_neu); |
---|
| 4175 | if (size(ideal(PPd_neu))<>0) |
---|
| 4176 | { nc=ncols(PPd_neu); |
---|
| 4177 | intmat PPd_neu_gross[s][nc]; |
---|
| 4178 | PPd_neu_gross[i..s,1..nc]=PPd_neu[1..s-i+1,1..nc]; |
---|
| 4179 | for (j=1;j<=nc;j=j+1) |
---|
| 4180 | { PPd_neu_gross[i,j]=PPd_neu_gross[i,j]+1; |
---|
| 4181 | } |
---|
| 4182 | PP=concat_intmat(PP,PPd_neu_gross); |
---|
| 4183 | kill PPd_neu_gross; |
---|
| 4184 | } |
---|
| 4185 | kill PPd_neu; |
---|
| 4186 | } |
---|
| 4187 | if (d_neu==0) |
---|
| 4188 | { intmat PPd_neu[s][1]; |
---|
| 4189 | PPd_neu[i,1]=1; |
---|
| 4190 | PP=concat_intmat(PP,PPd_neu); |
---|
| 4191 | kill PPd_neu; |
---|
| 4192 | } |
---|
| 4193 | } |
---|
| 4194 | if (matrix(PP)<>matrix(TEST)) |
---|
| 4195 | { PP=compress(PP); |
---|
| 4196 | } |
---|
| 4197 | return(PP); |
---|
| 4198 | } |
---|
| 4199 | example |
---|
| 4200 | { echo=2; |
---|
| 4201 | intvec dv=5,5,5,10,10; |
---|
| 4202 | print(power_products(dv,10)); |
---|
| 4203 | print(power_products(dv,7)); |
---|
| 4204 | } |
---|
| 4205 | |
---|
| 4206 | proc secondary_char0 (matrix P, matrix REY, matrix M, list #) |
---|
| 4207 | "USAGE: secondary_char0(P,REY,M[,v]); |
---|
| 4208 | P: a 1xn <matrix> with primary invariants, REY: a gxn <matrix> |
---|
| 4209 | representing the Reynolds operator, M: a 1x2 <matrix> giving enumerator |
---|
| 4210 | and denominator of the Molien series, v: an optional <int> |
---|
| 4211 | ASSUME: n is the number of variables of the basering, g the size of the group, |
---|
| 4212 | REY is the 1st return value of group_reynolds(), reynolds_molien() or |
---|
| 4213 | the second one of primary_invariants(), M the return value of molien() |
---|
| 4214 | or the second one of reynolds_molien() or the third one of |
---|
| 4215 | primary_invariants() |
---|
| 4216 | RETURN: secondary invariants of the invariant ring (type <matrix>) and |
---|
| 4217 | irreducible secondary invariants (type <matrix>) |
---|
| 4218 | DISPLAY: information if v does not equal 0 |
---|
| 4219 | EXAMPLE: example secondary_char0; shows an example |
---|
| 4220 | THEORY: The secondary invariants are calculated by finding a basis (in terms of |
---|
| 4221 | monomials) of the basering modulo the primary invariants, mapping those |
---|
| 4222 | to invariants with the Reynolds operator and using these images or |
---|
| 4223 | their power products such that they are linearly independent modulo the |
---|
| 4224 | primary invariants (see paper \"Some Algorithms in Invariant Theory of |
---|
| 4225 | Finite Groups\" by Kemper and Steel (1997)). |
---|
| 4226 | " |
---|
| 4227 | { def br=basering; |
---|
| 4228 | degBound=0; |
---|
| 4229 | //----------------- checking input and setting verbose mode ------------------ |
---|
| 4230 | if (char(br)<>0) |
---|
| 4231 | { "ERROR: secondary_char0 should only be used with rings of characteristic 0."; |
---|
| 4232 | return(); |
---|
| 4233 | } |
---|
| 4234 | int i; |
---|
| 4235 | if (size(#)>0) |
---|
| 4236 | { if (typeof(#[size(#)])=="int") |
---|
| 4237 | { int v=#[size(#)]; |
---|
| 4238 | } |
---|
| 4239 | else |
---|
| 4240 | { int v=0; |
---|
| 4241 | } |
---|
| 4242 | } |
---|
| 4243 | else |
---|
| 4244 | { int v=0; |
---|
| 4245 | } |
---|
| 4246 | int n=nvars(br); // n is the number of variables, as well |
---|
| 4247 | // as the size of the matrices, as well |
---|
| 4248 | // as the number of primary invariants, |
---|
| 4249 | // we should get |
---|
| 4250 | if (ncols(P)<>n) |
---|
| 4251 | { "ERROR: The first parameter ought to be the matrix of the primary"; |
---|
| 4252 | " invariants." |
---|
| 4253 | return(); |
---|
| 4254 | } |
---|
| 4255 | if (ncols(REY)<>n) |
---|
| 4256 | { "ERROR: The second parameter ought to be the Reynolds operator." |
---|
| 4257 | return(); |
---|
| 4258 | } |
---|
| 4259 | if (ncols(M)<>2 or nrows(M)<>1) |
---|
| 4260 | { "ERROR: The third parameter ought to be the Molien series." |
---|
| 4261 | return(); |
---|
| 4262 | } |
---|
| 4263 | if (v && voice==2) |
---|
| 4264 | { ""; |
---|
| 4265 | } |
---|
| 4266 | int j, m, counter; |
---|
| 4267 | //- finding the polynomial giving number and degrees of secondary invariants - |
---|
| 4268 | poly p=1; |
---|
| 4269 | for (j=1;j<=n;j=j+1) // calculating the denominator of the |
---|
| 4270 | { p=p*(1-var(1)^deg(P[j])); // Hilbert series of the ring generated |
---|
| 4271 | } // by the primary invariants - |
---|
| 4272 | matrix s[1][2]=M[1,1]*p,M[1,2]; // s is used for canceling |
---|
| 4273 | s=matrix(syz(ideal(s))); |
---|
| 4274 | p=s[2,1]; // the polynomial telling us where to |
---|
| 4275 | // search for secondary invariants |
---|
| 4276 | map slead=br,ideal(0); |
---|
| 4277 | p=1/slead(p)*p; // smallest term of p needs to be 1 |
---|
| 4278 | if (v) |
---|
| 4279 | { " Polynomial telling us where to look for secondary invariants:"; |
---|
| 4280 | " "+string(p); |
---|
| 4281 | ""; |
---|
| 4282 | } |
---|
| 4283 | matrix dimmat=coeffs(p,var(1)); // dimmat will contain the number of |
---|
| 4284 | // secondary invariants, we need to find |
---|
| 4285 | // of a certain degree - |
---|
| 4286 | m=nrows(dimmat); // m-1 is the highest degree |
---|
| 4287 | if (v) |
---|
| 4288 | { " In degree 0 we have: 1"; |
---|
| 4289 | ""; |
---|
| 4290 | } |
---|
| 4291 | //-------------------------- initializing variables -------------------------- |
---|
| 4292 | intmat PP; |
---|
| 4293 | poly pp; |
---|
| 4294 | int k; |
---|
| 4295 | intvec deg_vec; |
---|
| 4296 | ideal sP=std(ideal(P)); |
---|
| 4297 | ideal TEST,B,IS; |
---|
| 4298 | ideal S=1; // 1 is the first secondary invariant - |
---|
| 4299 | //--------------------- generating secondary invariants ---------------------- |
---|
| 4300 | for (i=2;i<=m;i=i+1) // going through dimmat - |
---|
| 4301 | { if (int(dimmat[i,1])<>0) // when it is == 0 we need to find 0 |
---|
| 4302 | { // elements in the current degree (i-1) |
---|
| 4303 | if (v) |
---|
| 4304 | { " Searching in degree "+string(i-1)+", we need to find "+string(int(dimmat[i,1]))+" invariant(s)..."; |
---|
| 4305 | } |
---|
| 4306 | TEST=sP; |
---|
| 4307 | counter=0; // we'll count up to degvec[i] |
---|
| 4308 | if (IS[1]<>0) |
---|
| 4309 | { PP=power_products(deg_vec,i-1); // finding power products of irreducible |
---|
| 4310 | } // secondary invariants |
---|
| 4311 | if (size(ideal(PP))<>0) |
---|
| 4312 | { for (j=1;j<=ncols(PP);j=j+1) // going through all the power products |
---|
| 4313 | { pp=1; |
---|
| 4314 | for (k=1;k<=nrows(PP);k=k+1) |
---|
| 4315 | { pp=pp*IS[1,k]^PP[k,j]; |
---|
| 4316 | } |
---|
| 4317 | if (reduce(pp,TEST)<>0) |
---|
| 4318 | { S=S,pp; |
---|
| 4319 | counter=counter+1; |
---|
| 4320 | if (v) |
---|
| 4321 | { " We find: "+string(pp); |
---|
| 4322 | } |
---|
| 4323 | if (int(dimmat[i,1])<>counter) |
---|
| 4324 | { TEST=std(TEST+ideal(NF(pp,TEST))); // should be replaced by next |
---|
| 4325 | // line soon |
---|
| 4326 | // TEST=std(TEST,NF(pp,TEST)); |
---|
| 4327 | } |
---|
| 4328 | else |
---|
| 4329 | { break; |
---|
| 4330 | } |
---|
| 4331 | } |
---|
| 4332 | } |
---|
| 4333 | } |
---|
| 4334 | if (int(dimmat[i,1])<>counter) |
---|
| 4335 | { B=sort_of_invariant_basis(sP,REY,i-1,int(dimmat[i,1])*6); // B contains |
---|
| 4336 | // images of kbase(sP,i-1) under the |
---|
| 4337 | // Reynolds operator that are linearly |
---|
| 4338 | // independent and that don't reduce to |
---|
| 4339 | // 0 modulo sP - |
---|
| 4340 | if (counter==0 && ncols(B)==int(dimmat[i,1])) // then we can take all of |
---|
| 4341 | { S=S,B; // B |
---|
| 4342 | IS=IS+B; |
---|
| 4343 | if (deg_vec[1]==0) |
---|
| 4344 | { deg_vec=i-1; |
---|
| 4345 | if (v) |
---|
| 4346 | { " We find: "+string(B[1]); |
---|
| 4347 | } |
---|
| 4348 | for (j=2;j<=int(dimmat[i,1]);j=j+1) |
---|
| 4349 | { deg_vec=deg_vec,i-1; |
---|
| 4350 | if (v) |
---|
| 4351 | { " We find: "+string(B[j]); |
---|
| 4352 | } |
---|
| 4353 | } |
---|
| 4354 | } |
---|
| 4355 | else |
---|
| 4356 | { for (j=1;j<=int(dimmat[i,1]);j=j+1) |
---|
| 4357 | { deg_vec=deg_vec,i-1; |
---|
| 4358 | if (v) |
---|
| 4359 | { " We find: "+string(B[j]); |
---|
| 4360 | } |
---|
| 4361 | } |
---|
| 4362 | } |
---|
| 4363 | } |
---|
| 4364 | else |
---|
| 4365 | { j=0; // j goes through all of B - |
---|
| 4366 | while (int(dimmat[i,1])<>counter) // need to find dimmat[i,1] |
---|
| 4367 | { // invariants that are linearly |
---|
| 4368 | // independent modulo TEST |
---|
| 4369 | j=j+1; |
---|
| 4370 | if (reduce(B[j],TEST)<>0) // B[j] should be added |
---|
| 4371 | { S=S,B[j]; |
---|
| 4372 | IS=IS+ideal(B[j]); |
---|
| 4373 | if (deg_vec[1]==0) |
---|
| 4374 | { deg_vec[1]=i-1; |
---|
| 4375 | } |
---|
| 4376 | else |
---|
| 4377 | { deg_vec=deg_vec,i-1; |
---|
| 4378 | } |
---|
| 4379 | counter=counter+1; |
---|
| 4380 | if (v) |
---|
| 4381 | { " We find: "+string(B[j]); |
---|
| 4382 | } |
---|
| 4383 | if (int(dimmat[i,1])<>counter) |
---|
| 4384 | { TEST=std(TEST+ideal(NF(B[j],TEST))); // should be replaced by |
---|
| 4385 | // next line |
---|
| 4386 | // TEST=std(TEST,NF(B[j],TEST)); |
---|
| 4387 | } |
---|
| 4388 | } |
---|
| 4389 | } |
---|
| 4390 | } |
---|
| 4391 | } |
---|
| 4392 | if (v) |
---|
| 4393 | { ""; |
---|
| 4394 | } |
---|
| 4395 | } |
---|
| 4396 | } |
---|
| 4397 | if (v) |
---|
| 4398 | { " We're done!"; |
---|
| 4399 | ""; |
---|
| 4400 | } |
---|
| 4401 | return(matrix(S),matrix(IS)); |
---|
| 4402 | } |
---|
| 4403 | example |
---|
| 4404 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
| 4405 | echo=2; |
---|
| 4406 | ring R=0,(x,y,z),dp; |
---|
| 4407 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 4408 | list L=primary_invariants(A); |
---|
| 4409 | matrix S,IS=secondary_char0(L[1..3]); |
---|
| 4410 | print(S); |
---|
| 4411 | print(IS); |
---|
| 4412 | } |
---|
| 4413 | |
---|
| 4414 | proc secondary_charp (matrix P, matrix REY, string ring_name, list #) |
---|
| 4415 | "USAGE: secondary_charp(P,REY,ringname[,v]); |
---|
| 4416 | P: a 1xn <matrix> with primary invariants, REY: a gxn <matrix> |
---|
| 4417 | representing the Reynolds operator, ringname: a <string> giving the |
---|
| 4418 | name of a ring of characteristic 0 where the Molien series is stored, |
---|
| 4419 | v: an optional <int> |
---|
| 4420 | ASSUME: n is the number of variables of the basering, g the size of the group, |
---|
| 4421 | REY is the 1st return value of group_reynolds(), reynolds_molien() or |
---|
| 4422 | the second one of primary_invariants(), `ringname` is ring of |
---|
| 4423 | characteristic 0 that has been created by molien() or reynolds_molien() |
---|
| 4424 | or primary_invariants() |
---|
| 4425 | RETURN: secondary invariants of the invariant ring (type <matrix>) and |
---|
| 4426 | irreducible secondary invariants (type <matrix>) |
---|
| 4427 | DISPLAY: information if v does not equal 0 |
---|
| 4428 | EXAMPLE: example secondary_charp; shows an example |
---|
| 4429 | THEORY: The secondary invariants are calculated by finding a basis (in terms of |
---|
| 4430 | monomials) of the basering modulo the primary invariants, mapping those |
---|
| 4431 | to invariants with the Reynolds operator and using these images or |
---|
| 4432 | their power products such that they are linearly independent modulo the |
---|
| 4433 | primary invariants (see paper \"Some Algorithms in Invariant Theory of |
---|
| 4434 | Finite Groups\" by Kemper and Steel (1997)). |
---|
| 4435 | " |
---|
| 4436 | { def br=basering; |
---|
| 4437 | degBound=0; |
---|
| 4438 | //---------------- checking input and setting verbose mode ------------------- |
---|
| 4439 | if (char(br)==0) |
---|
| 4440 | { "ERROR: secondary_charp should only be used with rings of characteristic p>0."; |
---|
| 4441 | return(); |
---|
| 4442 | } |
---|
| 4443 | int i; |
---|
| 4444 | if (size(#)>0) |
---|
| 4445 | { if (typeof(#[size(#)])=="int") |
---|
| 4446 | { int v=#[size(#)]; |
---|
| 4447 | } |
---|
| 4448 | else |
---|
| 4449 | { int v=0; |
---|
| 4450 | } |
---|
| 4451 | } |
---|
| 4452 | else |
---|
| 4453 | { int v=0; |
---|
| 4454 | } |
---|
| 4455 | int n=nvars(br); // n is the number of variables, as well |
---|
| 4456 | // as the size of the matrices, as well |
---|
| 4457 | // as the number of primary invariants, |
---|
| 4458 | // we should get |
---|
| 4459 | if (ncols(P)<>n) |
---|
| 4460 | { "ERROR: The first parameter ought to be the matrix of the primary"; |
---|
| 4461 | " invariants." |
---|
| 4462 | return(); |
---|
| 4463 | } |
---|
| 4464 | if (ncols(REY)<>n) |
---|
| 4465 | { "ERROR: The second parameter ought to be the Reynolds operator." |
---|
| 4466 | return(); |
---|
| 4467 | } |
---|
| 4468 | if (typeof(`ring_name`)<>"ring") |
---|
| 4469 | { "ERROR: The <string> should give the name of the ring where the Molien." |
---|
| 4470 | " series is stored."; |
---|
| 4471 | return(); |
---|
| 4472 | } |
---|
| 4473 | if (v && voice==2) |
---|
| 4474 | { ""; |
---|
| 4475 | } |
---|
| 4476 | int j, m, counter, d; |
---|
| 4477 | intvec deg_dim_vec; |
---|
| 4478 | //- finding the polynomial giving number and degrees of secondary invariants - |
---|
| 4479 | for (j=1;j<=n;j=j+1) |
---|
| 4480 | { deg_dim_vec[j]=deg(P[j]); |
---|
| 4481 | } |
---|
| 4482 | setring `ring_name`; |
---|
| 4483 | poly p=1; |
---|
| 4484 | for (j=1;j<=n;j=j+1) // calculating the denominator of the |
---|
| 4485 | { p=p*(1-var(1)^deg_dim_vec[j]); // Hilbert series of the ring generated |
---|
| 4486 | } // by the primary invariants - |
---|
| 4487 | matrix s[1][2]=M[1,1]*p,M[1,2]; // s is used for canceling |
---|
| 4488 | s=matrix(syz(ideal(s))); |
---|
| 4489 | p=s[2,1]; // the polynomial telling us where to |
---|
| 4490 | // search for secondary invariants |
---|
| 4491 | map slead=basering,ideal(0); |
---|
| 4492 | p=1/slead(p)*p; // smallest term of p needs to be 1 |
---|
| 4493 | if (v) |
---|
| 4494 | { " Polynomial telling us where to look for secondary invariants:"; |
---|
| 4495 | " "+string(p); |
---|
| 4496 | ""; |
---|
| 4497 | } |
---|
| 4498 | matrix dimmat=coeffs(p,var(1)); // dimmat will contain the number of |
---|
| 4499 | // secondary invariants, we need to find |
---|
| 4500 | // of a certain degree - |
---|
| 4501 | m=nrows(dimmat); // m-1 is the highest degree |
---|
| 4502 | deg_dim_vec=1; |
---|
| 4503 | for (j=2;j<=m;j=j+1) |
---|
| 4504 | { deg_dim_vec=deg_dim_vec,int(dimmat[j,1]); |
---|
| 4505 | } |
---|
| 4506 | if (v) |
---|
| 4507 | { " In degree 0 we have: 1"; |
---|
| 4508 | ""; |
---|
| 4509 | } |
---|
| 4510 | //------------------------ initializing variables ---------------------------- |
---|
| 4511 | setring br; |
---|
| 4512 | intmat PP; |
---|
| 4513 | poly pp; |
---|
| 4514 | int k; |
---|
| 4515 | intvec deg_vec; |
---|
| 4516 | ideal sP=std(ideal(P)); |
---|
| 4517 | ideal TEST,B,IS; |
---|
| 4518 | ideal S=1; // 1 is the first secondary invariant |
---|
| 4519 | //------------------- generating secondary invariants ------------------------ |
---|
| 4520 | for (i=2;i<=m;i=i+1) // going through deg_dim_vec - |
---|
| 4521 | { if (deg_dim_vec[i]<>0) // when it is == 0 we need to find 0 |
---|
| 4522 | { // elements in the current degree (i-1) |
---|
| 4523 | if (v) |
---|
| 4524 | { " Searching in degree "+string(i-1)+", we need to find "+string(deg_dim_vec[i])+" invariant(s)..."; |
---|
| 4525 | } |
---|
| 4526 | TEST=sP; |
---|
| 4527 | counter=0; // we'll count up to degvec[i] |
---|
| 4528 | if (IS[1]<>0) |
---|
| 4529 | { PP=power_products(deg_vec,i-1); // generating power products of |
---|
| 4530 | } // irreducible secondary invariants |
---|
| 4531 | if (size(ideal(PP))<>0) |
---|
| 4532 | { for (j=1;j<=ncols(PP);j=j+1) // going through all of those |
---|
| 4533 | { pp=1; |
---|
| 4534 | for (k=1;k<=nrows(PP);k=k+1) |
---|
| 4535 | { pp=pp*IS[1,k]^PP[k,j]; |
---|
| 4536 | } |
---|
| 4537 | if (reduce(pp,TEST)<>0) |
---|
| 4538 | { S=S,pp; |
---|
| 4539 | counter=counter+1; |
---|
| 4540 | if (v) |
---|
| 4541 | { " We find: "+string(pp); |
---|
| 4542 | } |
---|
| 4543 | if (deg_dim_vec[i]<>counter) |
---|
| 4544 | { TEST=std(TEST+ideal(NF(pp,TEST))); // should be soon replaced by |
---|
| 4545 | // next line |
---|
| 4546 | // TEST=std(TEST,NF(pp,TEST)); |
---|
| 4547 | } |
---|
| 4548 | else |
---|
| 4549 | { break; |
---|
| 4550 | } |
---|
| 4551 | } |
---|
| 4552 | } |
---|
| 4553 | } |
---|
| 4554 | if (deg_dim_vec[i]<>counter) |
---|
| 4555 | { B=sort_of_invariant_basis(sP,REY,i-1,deg_dim_vec[i]*6); // B contains |
---|
| 4556 | // images of kbase(sP,i-1) under the |
---|
| 4557 | // Reynolds operator that are linearly |
---|
| 4558 | // independent and that don't reduce to |
---|
| 4559 | // 0 modulo sP - |
---|
| 4560 | if (counter==0 && ncols(B)==deg_dim_vec[i]) // then we can add all of B |
---|
| 4561 | { S=S,B; |
---|
| 4562 | IS=IS+B; |
---|
| 4563 | if (deg_vec[1]==0) |
---|
| 4564 | { deg_vec=i-1; |
---|
| 4565 | if (v) |
---|
| 4566 | { " We find: "+string(B[1]); |
---|
| 4567 | } |
---|
| 4568 | for (j=2;j<=deg_dim_vec[i];j=j+1) |
---|
| 4569 | { deg_vec=deg_vec,i-1; |
---|
| 4570 | if (v) |
---|
| 4571 | { " We find: "+string(B[j]); |
---|
| 4572 | } |
---|
| 4573 | } |
---|
| 4574 | } |
---|
| 4575 | else |
---|
| 4576 | { for (j=1;j<=deg_dim_vec[i];j=j+1) |
---|
| 4577 | { deg_vec=deg_vec,i-1; |
---|
| 4578 | if (v) |
---|
| 4579 | { " We find: "+string(B[j]); |
---|
| 4580 | } |
---|
| 4581 | } |
---|
| 4582 | } |
---|
| 4583 | } |
---|
| 4584 | else |
---|
| 4585 | { j=0; // j goes through all of B - |
---|
| 4586 | while (deg_dim_vec[i]<>counter) // need to find deg_dim_vec[i] |
---|
| 4587 | { // invariants that are linearly |
---|
| 4588 | // independent modulo TEST |
---|
| 4589 | j=j+1; |
---|
| 4590 | if (reduce(B[j],TEST)<>0) // B[j] should be added |
---|
| 4591 | { S=S,B[j]; |
---|
| 4592 | IS=IS+ideal(B[j]); |
---|
| 4593 | if (deg_vec[1]==0) |
---|
| 4594 | { deg_vec[1]=i-1; |
---|
| 4595 | } |
---|
| 4596 | else |
---|
| 4597 | { deg_vec=deg_vec,i-1; |
---|
| 4598 | } |
---|
| 4599 | counter=counter+1; |
---|
| 4600 | if (v) |
---|
| 4601 | { " We find: "+string(B[j]); |
---|
| 4602 | } |
---|
| 4603 | if (deg_dim_vec[i]<>counter) |
---|
| 4604 | { TEST=std(TEST+ideal(NF(B[j],TEST))); // should be soon replaced |
---|
| 4605 | // by next line |
---|
| 4606 | // TEST=std(TEST,NF(B[j],TEST)); |
---|
| 4607 | } |
---|
| 4608 | } |
---|
| 4609 | } |
---|
| 4610 | } |
---|
| 4611 | } |
---|
| 4612 | if (v) |
---|
| 4613 | { ""; |
---|
| 4614 | } |
---|
| 4615 | } |
---|
| 4616 | } |
---|
| 4617 | if (v) |
---|
| 4618 | { " We're done!"; |
---|
| 4619 | ""; |
---|
| 4620 | } |
---|
| 4621 | if (ring_name=="aksldfalkdsflkj") |
---|
| 4622 | { kill `ring_name`; |
---|
| 4623 | } |
---|
| 4624 | return(matrix(S),matrix(IS)); |
---|
| 4625 | } |
---|
| 4626 | example |
---|
| 4627 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7 (changed into"; |
---|
| 4628 | " characteristic 3)"; |
---|
| 4629 | echo=2; |
---|
| 4630 | ring R=3,(x,y,z),dp; |
---|
| 4631 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 4632 | list L=primary_invariants(A); |
---|
| 4633 | matrix S,IS=secondary_charp(L[1..size(L)]); |
---|
| 4634 | print(S); |
---|
| 4635 | print(IS); |
---|
| 4636 | } |
---|
| 4637 | |
---|
| 4638 | proc secondary_no_molien (matrix P, matrix REY, list #) |
---|
| 4639 | "USAGE: secondary_no_molien(P,REY[,deg_vec,v]); |
---|
| 4640 | P: a 1xn <matrix> with primary invariants, REY: a gxn <matrix> |
---|
| 4641 | representing the Reynolds operator, deg_vec: an optional <intvec> |
---|
| 4642 | listing some degrees where no non-trivial homogeneous invariants can be |
---|
| 4643 | found, v: an optional <int> |
---|
| 4644 | ASSUME: n is the number of variables of the basering, g the size of the group, |
---|
| 4645 | REY is the 1st return value of group_reynolds(), reynolds_molien() or |
---|
| 4646 | the second one of primary_invariants(), deg_vec is the second return |
---|
| 4647 | value of primary_char0_no_molien(), primary_charp_no_molien(), |
---|
| 4648 | primary_char0_no_molien_random() or primary_charp_no_molien_random() |
---|
| 4649 | RETURN: secondary invariants of the invariant ring (type <matrix>) |
---|
| 4650 | DISPLAY: information if v does not equal 0 |
---|
| 4651 | EXAMPLE: example secondary_no_molien; shows an example |
---|
| 4652 | THEORY: The secondary invariants are calculated by finding a basis (in terms of |
---|
| 4653 | monomials) of the basering modulo the primary invariants, mapping those |
---|
| 4654 | to invariants with the Reynolds operator and using these images as |
---|
| 4655 | candidates for secondary invariants. |
---|
| 4656 | " |
---|
| 4657 | { int i; |
---|
| 4658 | degBound=0; |
---|
| 4659 | //------------------ checking input and setting verbose ---------------------- |
---|
| 4660 | if (size(#)==1 or size(#)==2) |
---|
| 4661 | { if (typeof(#[size(#)])=="int") |
---|
| 4662 | { if (size(#)==2) |
---|
| 4663 | { if (typeof(#[size(#)-1])=="intvec") |
---|
| 4664 | { intvec deg_vec=#[size(#)-1]; |
---|
| 4665 | } |
---|
| 4666 | else |
---|
| 4667 | { "ERROR: the third parameter should be an <intvec>"; |
---|
| 4668 | return(); |
---|
| 4669 | } |
---|
| 4670 | } |
---|
| 4671 | int v=#[size(#)]; |
---|
| 4672 | } |
---|
| 4673 | else |
---|
| 4674 | { if (size(#)==1) |
---|
| 4675 | { if (typeof(#[size(#)])=="intvec") |
---|
| 4676 | { intvec deg_vec=#[size(#)]; |
---|
| 4677 | int v=0; |
---|
| 4678 | } |
---|
| 4679 | else |
---|
| 4680 | { "ERROR: the third parameter should be an <intvec>"; |
---|
| 4681 | return(); |
---|
| 4682 | } |
---|
| 4683 | } |
---|
| 4684 | else |
---|
| 4685 | { "ERROR: wrong list of parameters"; |
---|
| 4686 | return(); |
---|
| 4687 | } |
---|
| 4688 | } |
---|
| 4689 | } |
---|
| 4690 | else |
---|
| 4691 | { if (size(#)>2) |
---|
| 4692 | { "ERROR: there are too many parameters"; |
---|
| 4693 | return(); |
---|
| 4694 | } |
---|
| 4695 | int v=0; |
---|
| 4696 | } |
---|
| 4697 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 4698 | // as the size of the matrices, as well |
---|
| 4699 | // as the number of primary invariants, |
---|
| 4700 | // we should get |
---|
| 4701 | if (ncols(P)<>n) |
---|
| 4702 | { "ERROR: The first parameter ought to be the matrix of the primary"; |
---|
| 4703 | " invariants." |
---|
| 4704 | return(); |
---|
| 4705 | } |
---|
| 4706 | if (ncols(REY)<>n) |
---|
| 4707 | { "ERROR: The second parameter ought to be the Reynolds operator." |
---|
| 4708 | return(); |
---|
| 4709 | } |
---|
| 4710 | if (v && voice==2) |
---|
| 4711 | { ""; |
---|
| 4712 | } |
---|
| 4713 | int j, m, d; |
---|
| 4714 | int max=1; |
---|
| 4715 | for (j=1;j<=n;j=j+1) |
---|
| 4716 | { max=max*deg(P[j]); |
---|
| 4717 | } |
---|
| 4718 | max=max/nrows(REY); |
---|
| 4719 | if (v) |
---|
| 4720 | { " We need to find "+string(max)+" secondary invariants."; |
---|
| 4721 | ""; |
---|
| 4722 | " In degree 0 we have: 1"; |
---|
| 4723 | ""; |
---|
| 4724 | } |
---|
| 4725 | //------------------------- initializing variables --------------------------- |
---|
| 4726 | ideal sP=std(ideal(P)); |
---|
| 4727 | ideal B, TEST; |
---|
| 4728 | ideal S=1; // 1 is the first secondary invariant |
---|
| 4729 | int counter=1; |
---|
| 4730 | i=0; |
---|
| 4731 | if (defined(deg_vec)<>voice) |
---|
| 4732 | { intvec deg_vec; |
---|
| 4733 | } |
---|
| 4734 | int k=1; |
---|
| 4735 | //--------------------- generating secondary invariants ---------------------- |
---|
| 4736 | while (counter<>max) |
---|
| 4737 | { i=i+1; |
---|
| 4738 | if (deg_vec[k]<>i) |
---|
| 4739 | { if (v) |
---|
| 4740 | { " Searching in degree "+string(i)+"..."; |
---|
| 4741 | } |
---|
| 4742 | B=sort_of_invariant_basis(sP,REY,i,max); // B contains images of |
---|
| 4743 | // kbase(sP,i) under the Reynolds |
---|
| 4744 | // operator that are linearly independent |
---|
| 4745 | // and that don't reduce to 0 modulo sP |
---|
| 4746 | TEST=sP; |
---|
| 4747 | for (j=1;j<=ncols(B);j=j+1) |
---|
| 4748 | { // that are linearly independent modulo |
---|
| 4749 | // TEST |
---|
| 4750 | if (reduce(B[j],TEST)<>0) // B[j] should be added |
---|
| 4751 | { S=S,B[j]; |
---|
| 4752 | counter=counter+1; |
---|
| 4753 | if (v) |
---|
| 4754 | { " We find: "+string(B[j]); |
---|
| 4755 | } |
---|
| 4756 | if (counter==max) |
---|
| 4757 | { break; |
---|
| 4758 | } |
---|
| 4759 | else |
---|
| 4760 | { if (j<>ncols(B)) |
---|
| 4761 | { TEST=std(TEST+ideal(NF(B[j],TEST))); // should soon be replaced by |
---|
| 4762 | // next line |
---|
| 4763 | // TEST=std(TEST,NF(B[j],TEST)); |
---|
| 4764 | } |
---|
| 4765 | } |
---|
| 4766 | } |
---|
| 4767 | } |
---|
| 4768 | } |
---|
| 4769 | else |
---|
| 4770 | { if (size(deg_vec)==k) |
---|
| 4771 | { k=1; |
---|
| 4772 | } |
---|
| 4773 | else |
---|
| 4774 | { k=k+1; |
---|
| 4775 | } |
---|
| 4776 | } |
---|
| 4777 | } |
---|
| 4778 | if (v) |
---|
| 4779 | { ""; |
---|
| 4780 | } |
---|
| 4781 | if (v) |
---|
| 4782 | { " We're done!"; |
---|
| 4783 | ""; |
---|
| 4784 | } |
---|
| 4785 | return(matrix(S)); |
---|
| 4786 | } |
---|
| 4787 | example |
---|
| 4788 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
| 4789 | echo=2; |
---|
| 4790 | ring R=0,(x,y,z),dp; |
---|
| 4791 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 4792 | list L=primary_invariants(A,intvec(1,1,0)); |
---|
| 4793 | matrix S=secondary_no_molien(L[1..3]); |
---|
| 4794 | print(S); |
---|
| 4795 | } |
---|
| 4796 | |
---|
[5811fb] | 4797 | proc secondary_and_irreducibles_no_molien (matrix P, matrix REY, list #) |
---|
| 4798 | "USAGE: secondary_and_irreducibles_no_molien(P,REY[,v]); |
---|
[29aa4bf] | 4799 | P: a 1xn <matrix> with primary invariants, REY: a gxn <matrix> |
---|
| 4800 | representing the Reynolds operator, v: an optional <int> |
---|
| 4801 | ASSUME: n is the number of variables of the basering, g the size of the group, |
---|
| 4802 | REY is the 1st return value of group_reynolds(), reynolds_molien() or |
---|
| 4803 | the second one of primary_invariants() |
---|
| 4804 | RETURN: secondary invariants of the invariant ring (type <matrix>) and |
---|
| 4805 | irreducible secondary invariants (type <matrix>) |
---|
| 4806 | DISPLAY: information if v does not equal 0 |
---|
[5811fb] | 4807 | EXAMPLE: example secondary_and_irreducibles_no_molien; shows an example |
---|
[29aa4bf] | 4808 | THEORY: The secondary invariants are calculated by finding a basis (in terms of |
---|
| 4809 | monomials) of the basering modulo the primary invariants, mapping those |
---|
| 4810 | to invariants with the Reynolds operator and using these images or |
---|
| 4811 | their power products such that they are linearly independent modulo the |
---|
| 4812 | primary invariants (see paper \"Some Algorithms in Invariant Theory of |
---|
| 4813 | Finite Groups\" by Kemper and Steel (1997)). |
---|
| 4814 | " |
---|
| 4815 | { int i; |
---|
| 4816 | degBound=0; |
---|
| 4817 | //--------------------- checking input and setting verbose mode -------------- |
---|
| 4818 | if (size(#)==1 or size(#)==2) |
---|
| 4819 | { if (typeof(#[size(#)])=="int") |
---|
| 4820 | { if (size(#)==2) |
---|
| 4821 | { if (typeof(#[size(#)-1])=="intvec") |
---|
| 4822 | { intvec deg_vec=#[size(#)-1]; |
---|
| 4823 | } |
---|
| 4824 | else |
---|
| 4825 | { "ERROR: the third parameter should be an <intvec>"; |
---|
| 4826 | return(); |
---|
| 4827 | } |
---|
| 4828 | } |
---|
| 4829 | int v=#[size(#)]; |
---|
| 4830 | } |
---|
| 4831 | else |
---|
| 4832 | { if (size(#)==1) |
---|
| 4833 | { if (typeof(#[size(#)])=="intvec") |
---|
| 4834 | { intvec deg_vec=#[size(#)]; |
---|
| 4835 | int v=0; |
---|
| 4836 | } |
---|
| 4837 | else |
---|
| 4838 | { "ERROR: the third parameter should be an <intvec>"; |
---|
| 4839 | return(); |
---|
| 4840 | } |
---|
| 4841 | } |
---|
| 4842 | else |
---|
| 4843 | { "ERROR: wrong list of parameters"; |
---|
| 4844 | return(); |
---|
| 4845 | } |
---|
| 4846 | } |
---|
| 4847 | } |
---|
| 4848 | else |
---|
| 4849 | { if (size(#)>2) |
---|
| 4850 | { "ERROR: there are too many parameters"; |
---|
| 4851 | return(); |
---|
| 4852 | } |
---|
| 4853 | int v=0; |
---|
| 4854 | } |
---|
| 4855 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 4856 | // as the size of the matrices, as well |
---|
| 4857 | // as the number of primary invariants, |
---|
| 4858 | // we should get |
---|
| 4859 | if (ncols(P)<>n) |
---|
| 4860 | { "ERROR: The first parameter ought to be the matrix of the primary"; |
---|
| 4861 | " invariants." |
---|
| 4862 | return(); |
---|
| 4863 | } |
---|
| 4864 | if (ncols(REY)<>n) |
---|
| 4865 | { "ERROR: The second parameter ought to be the Reynolds operator." |
---|
| 4866 | return(); |
---|
| 4867 | } |
---|
| 4868 | if (v && voice==2) |
---|
| 4869 | { ""; |
---|
| 4870 | } |
---|
| 4871 | int j, m, d; |
---|
| 4872 | int max=1; |
---|
| 4873 | for (j=1;j<=n;j=j+1) |
---|
| 4874 | { max=max*deg(P[j]); |
---|
| 4875 | } |
---|
| 4876 | max=max/nrows(REY); |
---|
| 4877 | if (v) |
---|
| 4878 | { " We need to find "+string(max)+" secondary invariants."; |
---|
| 4879 | ""; |
---|
| 4880 | " In degree 0 we have: 1"; |
---|
| 4881 | ""; |
---|
| 4882 | } |
---|
| 4883 | //------------------------ initializing variables ---------------------------- |
---|
| 4884 | intmat PP; |
---|
| 4885 | poly pp; |
---|
| 4886 | int k; |
---|
| 4887 | intvec irreducible_deg_vec; |
---|
| 4888 | ideal sP=std(ideal(P)); |
---|
| 4889 | ideal B,TEST,IS; |
---|
| 4890 | ideal S=1; // 1 is the first secondary invariant |
---|
| 4891 | int counter=1; |
---|
| 4892 | i=0; |
---|
| 4893 | if (defined(deg_vec)<>voice) |
---|
| 4894 | { intvec deg_vec; |
---|
| 4895 | } |
---|
| 4896 | int l=1; |
---|
| 4897 | //------------------- generating secondary invariants ------------------------ |
---|
| 4898 | while (counter<>max) |
---|
| 4899 | { i=i+1; |
---|
| 4900 | if (deg_vec[l]<>i) |
---|
| 4901 | { if (v) |
---|
| 4902 | { " Searching in degree "+string(i)+"..."; |
---|
| 4903 | } |
---|
| 4904 | TEST=sP; |
---|
| 4905 | if (IS[1]<>0) |
---|
| 4906 | { PP=power_products(irreducible_deg_vec,i); // generating all power |
---|
| 4907 | } // products of irreducible secondary |
---|
| 4908 | // invariants |
---|
| 4909 | if (size(ideal(PP))<>0) |
---|
| 4910 | { for (j=1;j<=ncols(PP);j=j+1) // going through all those power products |
---|
| 4911 | { pp=1; |
---|
| 4912 | for (k=1;k<=nrows(PP);k=k+1) |
---|
| 4913 | { pp=pp*IS[1,k]^PP[k,j]; |
---|
| 4914 | } |
---|
| 4915 | if (reduce(pp,TEST)<>0) |
---|
| 4916 | { S=S,pp; |
---|
| 4917 | counter=counter+1; |
---|
| 4918 | if (v) |
---|
| 4919 | { " We find: "+string(pp); |
---|
| 4920 | } |
---|
| 4921 | if (counter<>max) |
---|
| 4922 | { TEST=std(TEST+ideal(NF(pp,TEST))); // should soon be replaced by |
---|
| 4923 | // next line |
---|
| 4924 | // TEST=std(TEST,NF(pp,TEST)); |
---|
| 4925 | } |
---|
| 4926 | else |
---|
| 4927 | { break; |
---|
| 4928 | } |
---|
| 4929 | } |
---|
| 4930 | } |
---|
| 4931 | } |
---|
| 4932 | if (max<>counter) |
---|
| 4933 | { B=sort_of_invariant_basis(sP,REY,i,max); // B contains images of |
---|
| 4934 | // kbase(sP,i) under the Reynolds |
---|
| 4935 | // operator that are linearly independent |
---|
| 4936 | // and that don't reduce to 0 modulo sP |
---|
| 4937 | for (j=1;j<=ncols(B);j=j+1) |
---|
| 4938 | { if (reduce(B[j],TEST)<>0) // B[j] should be added |
---|
| 4939 | { S=S,B[j]; |
---|
| 4940 | IS=IS+ideal(B[j]); |
---|
| 4941 | if (irreducible_deg_vec[1]==0) |
---|
| 4942 | { irreducible_deg_vec[1]=i; |
---|
| 4943 | } |
---|
| 4944 | else |
---|
| 4945 | { irreducible_deg_vec=irreducible_deg_vec,i; |
---|
| 4946 | } |
---|
| 4947 | counter=counter+1; |
---|
| 4948 | if (v) |
---|
| 4949 | { " We find: "+string(B[j]); |
---|
| 4950 | } |
---|
| 4951 | if (counter==max) |
---|
| 4952 | { break; |
---|
| 4953 | } |
---|
| 4954 | else |
---|
| 4955 | { if (j<>ncols(B)) |
---|
| 4956 | { TEST=std(TEST+ideal(NF(B[j],TEST))); // should soon be replaced |
---|
| 4957 | // by next line |
---|
| 4958 | // TEST=std(TEST,NF(B[j],TEST)); |
---|
| 4959 | } |
---|
| 4960 | } |
---|
| 4961 | } |
---|
| 4962 | } |
---|
| 4963 | } |
---|
| 4964 | } |
---|
| 4965 | else |
---|
| 4966 | { if (size(deg_vec)==l) |
---|
| 4967 | { l=1; |
---|
| 4968 | } |
---|
| 4969 | else |
---|
| 4970 | { l=l+1; |
---|
| 4971 | } |
---|
| 4972 | } |
---|
| 4973 | } |
---|
| 4974 | if (v) |
---|
| 4975 | { ""; |
---|
| 4976 | } |
---|
| 4977 | if (v) |
---|
| 4978 | { " We're done!"; |
---|
| 4979 | ""; |
---|
| 4980 | } |
---|
| 4981 | return(matrix(S),matrix(IS)); |
---|
| 4982 | } |
---|
| 4983 | example |
---|
| 4984 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
| 4985 | echo=2; |
---|
| 4986 | ring R=0,(x,y,z),dp; |
---|
| 4987 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 4988 | list L=primary_invariants(A,intvec(1,1,0)); |
---|
[5811fb] | 4989 | matrix S,IS=secondary_and_irreducibles_no_molien(L[1..2]); |
---|
[29aa4bf] | 4990 | print(S); |
---|
| 4991 | print(IS); |
---|
| 4992 | } |
---|
| 4993 | |
---|
| 4994 | proc secondary_not_cohen_macaulay (matrix P, list #) |
---|
| 4995 | "USAGE: secondary_not_cohen_macaulay(P,G1,G2,...[,v]); |
---|
| 4996 | P: a 1xn <matrix> with primary invariants, G1,G2,...: nxn <matrices> |
---|
| 4997 | generating a finite matrix group, v: an optional <int> |
---|
| 4998 | ASSUME: n is the number of variables of the basering |
---|
| 4999 | RETURN: secondary invariants of the invariant ring (type <matrix>) |
---|
| 5000 | DISPLAY: information if v does not equal 0 |
---|
| 5001 | EXAMPLE: example secondary_not_cohen_macaulay; shows an example |
---|
| 5002 | THEORY: The secondary invariants are generated following \"Generating Invariant |
---|
| 5003 | Rings of Finite Groups over Arbitrary Fields\" by Kemper (1996, to |
---|
| 5004 | appear in JSC). |
---|
| 5005 | " |
---|
| 5006 | { int i, j; |
---|
| 5007 | degBound=0; |
---|
| 5008 | def br=basering; |
---|
| 5009 | int n=nvars(br); // n is the number of variables, as well |
---|
| 5010 | // as the size of the matrices, as well |
---|
| 5011 | // as the number of primary invariants, |
---|
| 5012 | // we should get - |
---|
| 5013 | if (size(#)>0) // checking input and setting verbose |
---|
| 5014 | { if (typeof(#[size(#)])=="int") |
---|
| 5015 | { int gen_num=size(#)-1; |
---|
| 5016 | if (gen_num==0) |
---|
| 5017 | { "ERROR: There are no generators of the finite matrix group given."; |
---|
| 5018 | return(); |
---|
| 5019 | } |
---|
| 5020 | int v=#[size(#)]; |
---|
| 5021 | for (i=1;i<=gen_num;i=i+1) |
---|
| 5022 | { if (typeof(#[i])<>"matrix") |
---|
| 5023 | { "ERROR: These parameters should be generators of the finite matrix group."; |
---|
| 5024 | return(); |
---|
| 5025 | } |
---|
| 5026 | if ((n<>nrows(#[i])) or (n<>ncols(#[i]))) |
---|
| 5027 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
| 5028 | return(); |
---|
| 5029 | } |
---|
| 5030 | } |
---|
| 5031 | } |
---|
| 5032 | else |
---|
| 5033 | { int v=0; |
---|
| 5034 | int gen_num=size(#); |
---|
| 5035 | for (i=1;i<=gen_num;i=i+1) |
---|
| 5036 | { if (typeof(#[i])<>"matrix") |
---|
| 5037 | { "ERROR: These parameters should be generators of the finite matrix group."; |
---|
| 5038 | return(); |
---|
| 5039 | } |
---|
| 5040 | if ((n<>nrows(#[i])) or (n<>ncols(#[i]))) |
---|
| 5041 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
| 5042 | return(); |
---|
| 5043 | } |
---|
| 5044 | } |
---|
| 5045 | } |
---|
| 5046 | } |
---|
| 5047 | else |
---|
| 5048 | { "ERROR: There are no generators of the finite matrix group given."; |
---|
| 5049 | return(); |
---|
| 5050 | } |
---|
| 5051 | if (ncols(P)<>n) |
---|
| 5052 | { "ERROR: The first parameter ought to be the matrix of the primary"; |
---|
| 5053 | " invariants." |
---|
| 5054 | return(); |
---|
| 5055 | } |
---|
| 5056 | if (v && voice==2) |
---|
| 5057 | { ""; |
---|
| 5058 | } |
---|
| 5059 | ring alskdfalkdsj=0,x,dp; |
---|
| 5060 | matrix M[1][2]=1,(1-x)^n; // we look at our primary invariants as |
---|
| 5061 | export alskdfalkdsj; |
---|
| 5062 | export M; |
---|
| 5063 | setring br; // such of the subgroup that only |
---|
| 5064 | matrix REY=matrix(maxideal(1)); // contains the identity, this means that |
---|
| 5065 | // ch does not divide the order anymore, |
---|
| 5066 | // this means that we can make use of the |
---|
| 5067 | // Molien series again - M[1,1]/M[1,2] is |
---|
| 5068 | // the Molien series of that group, we |
---|
| 5069 | // now calculate the secondary invariants |
---|
| 5070 | // of this subgroup in the usual fashion |
---|
| 5071 | // where the primary invariants are the |
---|
| 5072 | // ones from the bigger group |
---|
| 5073 | if (v) |
---|
| 5074 | { " The procedure secondary_charp() is called to calculate secondary invariants"; |
---|
| 5075 | " of the invariant ring of the trivial group with respect to the primary"; |
---|
| 5076 | " invariants found previously."; |
---|
| 5077 | ""; |
---|
| 5078 | } |
---|
| 5079 | matrix trivialS=secondary_charp(P,REY,"alskdfalkdsj",v); |
---|
| 5080 | kill alskdfalkdsj; |
---|
| 5081 | // now we have those secondary invariants |
---|
| 5082 | int k=ncols(trivialS); // k is the number of the secondary |
---|
| 5083 | // invariants, we just calculated |
---|
| 5084 | if (v) |
---|
| 5085 | { " We calculate secondary invariants from the ones found for the trivial"; |
---|
| 5086 | " subgroup."; |
---|
| 5087 | ""; |
---|
| 5088 | } |
---|
| 5089 | map f; // used to let generators act on |
---|
| 5090 | // secondary invariants with respect to |
---|
| 5091 | // the trivial group - |
---|
| 5092 | matrix M(1)[gen_num][k]; // M(1) will contain a module |
---|
| 5093 | ideal B; |
---|
| 5094 | for (i=1;i<=gen_num;i=i+1) |
---|
| 5095 | { B=ideal(matrix(maxideal(1))*transpose(#[i])); // image of the various |
---|
| 5096 | // variables under the i-th generator - |
---|
| 5097 | f=br,B; // the corresponding mapping - |
---|
| 5098 | B=f(trivialS)-trivialS; // these relations should be 0 - |
---|
| 5099 | M(1)[i,1..k]=B[1..k]; // we will look for the syzygies of M(1) |
---|
| 5100 | } |
---|
| 5101 | module M(2)=nres(M(1),2)[2]; |
---|
| 5102 | int m=ncols(M(2)); // number of generators of the module |
---|
| 5103 | // M(2) - |
---|
| 5104 | // the following steps calculates the intersection of the module M(2) with |
---|
| 5105 | // the algebra A^k where A denote the subalgebra of the usual polynomial |
---|
| 5106 | // ring, generated by the primary invariants |
---|
| 5107 | string mp=string(minpoly); // generating a ring where we can do |
---|
| 5108 | // elimination |
---|
| 5109 | execute "ring R=("+charstr(br)+"),(x(1..n),y(1..n),h),dp;"; |
---|
| 5110 | execute "minpoly=number("+mp+");"; |
---|
| 5111 | map f=br,maxideal(1); // canonical mapping |
---|
| 5112 | matrix M[k][m+k*n]; |
---|
| 5113 | M[1..k,1..m]=matrix(f(M(2))); // will contain a module - |
---|
| 5114 | matrix P=f(P); // primary invariants in the new ring |
---|
| 5115 | for (i=1;i<=n;i=i+1) |
---|
| 5116 | { for (j=1;j<=k;j=j+1) |
---|
| 5117 | { M[j,m+(i-1)*k+j]=y(i)-P[1,i]; |
---|
| 5118 | } |
---|
| 5119 | } |
---|
| 5120 | M=elim(module(M),1,n); // eliminating x(1..n), std-calculation |
---|
| 5121 | // is done internally - |
---|
| 5122 | M=homog(module(M),h); // homogenize for 'minbase' |
---|
| 5123 | M=minbase(module(M)); |
---|
| 5124 | setring br; |
---|
| 5125 | ideal substitute=maxideal(1),ideal(P),1; |
---|
| 5126 | f=R,substitute; // replacing y(1..n) by primary |
---|
| 5127 | // invariants - |
---|
| 5128 | M(2)=f(M); // M(2) is the new module |
---|
| 5129 | m=ncols(M(2)); |
---|
| 5130 | matrix S[1][m]; |
---|
| 5131 | S=matrix(trivialS)*matrix(M(2)); // S now contains the secondary |
---|
| 5132 | // invariants |
---|
| 5133 | for (i=1; i<=m;i=i+1) |
---|
| 5134 | { S[1,i]=S[1,i]/leadcoef(S[1,i]); // making elements nice |
---|
| 5135 | } |
---|
| 5136 | S=sort(ideal(S))[1]; |
---|
| 5137 | if (v) |
---|
| 5138 | { " These are the secondary invariants: "; |
---|
| 5139 | for (i=1;i<=m;i=i+1) |
---|
| 5140 | { " "+string(S[1,i]); |
---|
| 5141 | } |
---|
| 5142 | ""; |
---|
| 5143 | " We're done!"; |
---|
| 5144 | ""; |
---|
| 5145 | } |
---|
| 5146 | if ((v or (voice==2)) && (m>1)) |
---|
| 5147 | { " WARNING: The invariant ring might not have a Hironaka decomposition"; |
---|
| 5148 | " if the characteristic of the coefficient field divides the"; |
---|
| 5149 | " group order."; |
---|
| 5150 | } |
---|
| 5151 | return(S); |
---|
| 5152 | } |
---|
| 5153 | example |
---|
| 5154 | { echo=2; |
---|
| 5155 | ring R=2,(x,y,z),dp; |
---|
| 5156 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 5157 | list L=primary_invariants(A); |
---|
| 5158 | matrix S=secondary_not_cohen_macaulay(L[1],A); |
---|
| 5159 | print(S); |
---|
| 5160 | } |
---|
| 5161 | |
---|
| 5162 | proc invariant_ring (list #) |
---|
| 5163 | "USAGE: invariant_ring(G1,G2,...[,flags]); |
---|
| 5164 | G1,G2,...: <matrices> generating a finite matrix group, flags: an |
---|
| 5165 | optional <intvec> with three entries: if the first one equals 0, the |
---|
| 5166 | program attempts to compute the Molien series and Reynolds operator, |
---|
| 5167 | if it equals 1, the program is told that the Molien series should not |
---|
| 5168 | be computed, if it equals -1 characteristic 0 is simulated, i.e. the |
---|
| 5169 | Molien series is computed as if the base field were characteristic 0 |
---|
| 5170 | (the user must choose a field of large prime characteristic, e.g. |
---|
| 5171 | 32003) and if the first one is anything else, it means that the |
---|
| 5172 | characteristic of the base field divides the group order (i.e. it will |
---|
| 5173 | not even be attempted to compute the Reynolds operator or Molien |
---|
| 5174 | series), the second component should give the size of intervals |
---|
| 5175 | between canceling common factors in the expansion of the Molien series, |
---|
| 5176 | 0 (the default) means only once after generating all terms, in prime |
---|
| 5177 | characteristic also a negative number can be given to indicate that |
---|
| 5178 | common factors should always be canceled when the expansion is simple |
---|
| 5179 | (the root of the extension field does not occur among the coefficients) |
---|
| 5180 | RETURN: primary and secondary invariants (both of type <matrix>) generating the |
---|
| 5181 | invariant ring with respect to the matrix group generated by the |
---|
| 5182 | matrices in the input and irreducible secondary invariants (type |
---|
| 5183 | <matrix>) if the Molien series was available |
---|
| 5184 | DISPLAY: information about the various stages of the program if the third flag |
---|
| 5185 | does not equal 0 |
---|
| 5186 | EXAMPLE: example invariant_ring; shows an example |
---|
| 5187 | THEORY: Bases of homogeneous invariants are generated successively and those |
---|
| 5188 | are chosen as primary invariants that lower the dimension of the ideal |
---|
| 5189 | generated by the previously found invariants (see paper \"Generating a |
---|
| 5190 | Noetherian Normalization of the Invariant Ring of a Finite Group\" by |
---|
| 5191 | Decker, Heydtmann, Schreyer (1997) to appear in JSC). In the |
---|
| 5192 | non-modular case secondary invariants are calculated by finding a |
---|
| 5193 | basis (in terms of monomials) of the basering modulo the primary |
---|
| 5194 | invariants, mapping to invariants with the Reynolds operator and using |
---|
| 5195 | those or their power products such that they are linearly independent |
---|
| 5196 | modulo the primary invariants (see paper \"Some Algorithms in Invariant |
---|
| 5197 | Theory of Finite Groups\" by Kemper and Steel (1997)). In the modular |
---|
| 5198 | case they are generated according to \"Generating Invariant Rings of |
---|
| 5199 | Finite Groups over Arbitrary Fields\" by Kemper (1996, to appear in |
---|
| 5200 | JSC). |
---|
| 5201 | " |
---|
| 5202 | { if (size(#)==0) |
---|
| 5203 | { "ERROR: There are no generators given."; |
---|
| 5204 | return(); |
---|
| 5205 | } |
---|
| 5206 | int ch=char(basering); // the algorithms depend very much on the |
---|
| 5207 | // characteristic of the ground field - |
---|
| 5208 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 5209 | // as the size of the matrices, as well |
---|
| 5210 | // as the number of primary invariants, |
---|
| 5211 | // we should get |
---|
| 5212 | int gen_num; |
---|
| 5213 | int mol_flag, v; |
---|
| 5214 | //------------------- checking input and setting flags ----------------------- |
---|
| 5215 | if (typeof(#[size(#)])=="intvec") |
---|
| 5216 | { if (size(#[size(#)])<>3) |
---|
| 5217 | { "ERROR: The <intvec> should have three entries."; |
---|
| 5218 | return(); |
---|
| 5219 | } |
---|
| 5220 | gen_num=size(#)-1; |
---|
| 5221 | mol_flag=#[size(#)][1]; |
---|
| 5222 | if (#[size(#)][2]<0 && (ch==0 or (ch<>0 && mol_flag<>0))) |
---|
| 5223 | { "ERROR: the second component of <intvec> should be >=0"; |
---|
| 5224 | return(); |
---|
| 5225 | } |
---|
| 5226 | int interval=#[size(#)][2]; |
---|
| 5227 | v=#[size(#)][3]; |
---|
| 5228 | } |
---|
| 5229 | else |
---|
| 5230 | { gen_num=size(#); |
---|
| 5231 | mol_flag=0; |
---|
| 5232 | int interval=0; |
---|
| 5233 | v=0; |
---|
| 5234 | } |
---|
| 5235 | //---------------------------------------------------------------------------- |
---|
| 5236 | if (mol_flag==0) // calculation Molien series will be |
---|
| 5237 | { if (ch==0) // attempted - |
---|
| 5238 | { matrix REY,M=reynolds_molien(#[1..gen_num],intvec(0,interval,v)); // one |
---|
| 5239 | // will contain Reynolds operator and the |
---|
| 5240 | // other enumerator and denominator of |
---|
| 5241 | // Molien series |
---|
| 5242 | matrix P=primary_char0(REY,M,v); |
---|
| 5243 | matrix S,IS=secondary_char0(P,REY,M,v); |
---|
| 5244 | return(P,S,IS); |
---|
| 5245 | } |
---|
| 5246 | else |
---|
| 5247 | { list L=group_reynolds(#[1..gen_num],v); |
---|
| 5248 | if (L[1]<>0) // testing whether we are in the modular |
---|
| 5249 | { string newring="aksldfalkdsflkj"; // case |
---|
| 5250 | if (minpoly==0) |
---|
| 5251 | { if (v) |
---|
| 5252 | { " We are dealing with the non-modular case."; |
---|
| 5253 | } |
---|
[5811fb] | 5254 | molien(L[3..size(L)],newring,L[2],intvec(0,interval,v)); |
---|
[29aa4bf] | 5255 | matrix P=primary_charp(L[1],newring,v); |
---|
| 5256 | matrix S,IS=secondary_charp(P,L[1],newring,v); |
---|
| 5257 | if (defined(aksldfalkdsflkj)==2) |
---|
| 5258 | { kill aksldfalkdsflkj; |
---|
| 5259 | } |
---|
| 5260 | return(P,S,IS); |
---|
| 5261 | } |
---|
| 5262 | else |
---|
| 5263 | { if (v) |
---|
| 5264 | { " Since it is impossible for this programme to calculate the Molien |
---|
| 5265 | series for"; |
---|
| 5266 | " invariant rings over extension fields of prime characteristic, we |
---|
| 5267 | have to"; |
---|
| 5268 | " continue without it."; |
---|
| 5269 | ""; |
---|
| 5270 | |
---|
| 5271 | } |
---|
| 5272 | list l=primary_charp_no_molien(L[1],v); |
---|
| 5273 | if (size(l)==2) |
---|
| 5274 | { matrix S=secondary_no_molien(l[1],L[1],l[2],v); |
---|
| 5275 | } |
---|
| 5276 | else |
---|
| 5277 | { matix S=secondary_no_molien(l[1],L[1],v); |
---|
| 5278 | } |
---|
| 5279 | return(l[1],S); |
---|
| 5280 | } |
---|
| 5281 | } |
---|
| 5282 | else // the modular case |
---|
| 5283 | { if (v) |
---|
| 5284 | { " There is also no Molien series or Reynolds operator, we can make use of..."; |
---|
| 5285 | ""; |
---|
| 5286 | " We can start looking for primary invariants..."; |
---|
| 5287 | ""; |
---|
| 5288 | } |
---|
| 5289 | matrix P=primary_charp_without(#[1..gen_num],v); |
---|
| 5290 | matrix S=secondary_not_cohen_macaulay(P,#[1..gen_num],v); |
---|
| 5291 | return(P,S); |
---|
| 5292 | } |
---|
| 5293 | } |
---|
| 5294 | } |
---|
| 5295 | if (mol_flag==1) // the user wants no calculation of the |
---|
| 5296 | { list L=group_reynolds(#[1..gen_num],v); // Molien series |
---|
| 5297 | if (ch==0) |
---|
| 5298 | { list l=primary_char0_no_molien(L[1],v); |
---|
| 5299 | if (size(l)==2) |
---|
| 5300 | { matix S=secondary_no_molien(l[1],L[1],l[2],v); |
---|
| 5301 | } |
---|
| 5302 | else |
---|
| 5303 | { matix S=secondary_no_molien(l[1],L[1],v); |
---|
| 5304 | } |
---|
| 5305 | return(l[1],S); |
---|
| 5306 | } |
---|
| 5307 | else |
---|
| 5308 | { if (L[1]<>0) // testing whether we are in the modular |
---|
| 5309 | { list l=primary_charp_no_molien(L[1],v); // case |
---|
| 5310 | if (size(l)==2) |
---|
| 5311 | { matix S=secondary_no_molien(l[1],L[1],l[2],v); |
---|
| 5312 | } |
---|
| 5313 | else |
---|
| 5314 | { matix S=secondary_no_molien(l[1],L[1],v); |
---|
| 5315 | } |
---|
| 5316 | return(l[1],S); |
---|
| 5317 | } |
---|
| 5318 | else // the modular case |
---|
| 5319 | { if (v) |
---|
| 5320 | { " We can start looking for primary invariants..."; |
---|
| 5321 | ""; |
---|
| 5322 | } |
---|
| 5323 | matrix P=primary_charp_without(#[1..gen_num],v); |
---|
| 5324 | matrix S=secondary_not_cohen_macaulay(P,#[1..gen_num],v); |
---|
| 5325 | return(L[1],S); |
---|
| 5326 | } |
---|
| 5327 | } |
---|
| 5328 | } |
---|
| 5329 | if (mol_flag==-1) |
---|
| 5330 | { if (ch==0) |
---|
| 5331 | { "ERROR: Characteristic 0 can only be simulated in characteristic p>>0. |
---|
| 5332 | "; |
---|
| 5333 | return(); |
---|
| 5334 | } |
---|
| 5335 | list L=group_reynolds(#[1..gen_num],v); |
---|
| 5336 | string newring="aksldfalkdsflkj"; |
---|
[5811fb] | 5337 | molien(L[2..size(L)],newring,intvec(1,interval,v)); |
---|
[29aa4bf] | 5338 | matrix P=primary_charp(L[1],newring,v); |
---|
| 5339 | matrix S,IS=secondary_charp(P,L[1],newring,v); |
---|
| 5340 | kill aksldfalkdsflkj; |
---|
| 5341 | return(P,S,IS); |
---|
| 5342 | } |
---|
| 5343 | else // the user specified that the |
---|
| 5344 | { if (ch==0) // characteristic divides the group order |
---|
| 5345 | { "ERROR: The characteristic cannot divide the group order when it is 0. |
---|
| 5346 | "; |
---|
| 5347 | return(); |
---|
| 5348 | } |
---|
| 5349 | if (v) |
---|
| 5350 | { ""; |
---|
| 5351 | } |
---|
| 5352 | matrix P=primary_charp_without(#[1..gen_num],v); |
---|
| 5353 | matrix S=secondary_not_cohen_macaulay(P,#[1..gen_num],v); |
---|
| 5354 | return(L[1],S); |
---|
| 5355 | } |
---|
| 5356 | } |
---|
| 5357 | example |
---|
| 5358 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
| 5359 | echo=2; |
---|
| 5360 | ring R=0,(x,y,z),dp; |
---|
| 5361 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 5362 | matrix P,S,IS=invariant_ring(A); |
---|
| 5363 | print(P); |
---|
| 5364 | print(S); |
---|
| 5365 | print(IS); |
---|
| 5366 | } |
---|
| 5367 | |
---|
| 5368 | proc invariant_ring_random (list #) |
---|
| 5369 | "USAGE: invariant_ring_random(G1,G2,...,r[,flags]); |
---|
| 5370 | G1,G2,...: <matrices> generating a finite matrix group, r: an <int> |
---|
| 5371 | where -|r| to |r| is the range of coefficients of random |
---|
| 5372 | combinations of bases elements that serve as primary invariants, |
---|
| 5373 | flags: an optional <intvec> with three entries: if the first one equals |
---|
| 5374 | 0, the program attempts to compute the Molien series and Reynolds |
---|
| 5375 | operator, if it equals 1, the program is told that the Molien series |
---|
| 5376 | should not be computed, if it equals -1 characteristic 0 is simulated, |
---|
| 5377 | i.e. the Molien series is computed as if the base field were |
---|
| 5378 | characteristic 0 (the user must choose a field of large prime |
---|
| 5379 | characteristic, e.g. 32003) and if the first one is anything else, it |
---|
| 5380 | means that the characteristic of the base field divides the group order |
---|
| 5381 | (i.e. it will not even be attempted to compute the Reynolds operator or |
---|
| 5382 | Molien series), the second component should give the size of intervals |
---|
| 5383 | between canceling common factors in the expansion of the Molien series, |
---|
| 5384 | 0 (the default) means only once after generating all terms, in prime |
---|
| 5385 | characteristic also a negative number can be given to indicate that |
---|
| 5386 | common factors should always be canceled when the expansion is simple |
---|
| 5387 | (the root of the extension field does not occur among the coefficients) |
---|
| 5388 | RETURN: primary and secondary invariants (both of type <matrix>) generating the |
---|
| 5389 | invariant ring with respect to the matrix group generated by the |
---|
| 5390 | matrices in the input and irreducible secondary invariants (type |
---|
| 5391 | <matrix>) if the Molien series was available |
---|
| 5392 | DISPLAY: information about the various stages of the program if the third flag |
---|
| 5393 | does not equal 0 |
---|
| 5394 | EXAMPLE: example invariant_ring_random; shows an example |
---|
| 5395 | THEORY: is the same as for invariant_ring except that random combinations of |
---|
| 5396 | basis elements are chosen as candidates for primary invariants and |
---|
| 5397 | hopefully they lower the dimension of the previously found primary |
---|
| 5398 | invariants by the right amount. |
---|
| 5399 | " |
---|
| 5400 | { if (size(#)<2) |
---|
| 5401 | { "ERROR: There are too few parameters."; |
---|
| 5402 | return(); |
---|
| 5403 | } |
---|
| 5404 | int ch=char(basering); // the algorithms depend very much on the |
---|
| 5405 | // characteristic of the ground field |
---|
| 5406 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 5407 | // as the size of the matrices, as well |
---|
| 5408 | // as the number of primary invariants, |
---|
| 5409 | // we should get |
---|
| 5410 | int gen_num; |
---|
| 5411 | int mol_flag, v; |
---|
| 5412 | //------------------- checking input and setting flags ----------------------- |
---|
| 5413 | if (typeof(#[size(#)])=="intvec" && typeof(#[size(#)-1])=="int") |
---|
| 5414 | { if (size(#[size(#)])<>3) |
---|
| 5415 | { "ERROR: <intvec> should have three entries."; |
---|
| 5416 | return(); |
---|
| 5417 | } |
---|
| 5418 | gen_num=size(#)-2; |
---|
| 5419 | mol_flag=#[size(#)][1]; |
---|
| 5420 | if (#[size(#)][2]<0 && (ch==0 or (ch<>0 && mol_flag<>0))) |
---|
| 5421 | { "ERROR: the second component of <intvec> should be >=0"; |
---|
| 5422 | return(); |
---|
| 5423 | } |
---|
| 5424 | int interval=#[size(#)][2]; |
---|
| 5425 | v=#[size(#)][3]; |
---|
| 5426 | int max=#[size(#)-1]; |
---|
| 5427 | if (gen_num==0) |
---|
| 5428 | { "ERROR: There are no generators of a finite matrix group given."; |
---|
| 5429 | return(); |
---|
| 5430 | } |
---|
| 5431 | } |
---|
| 5432 | else |
---|
| 5433 | { if (typeof(#[size(#)])=="int") |
---|
| 5434 | { gen_num=size(#)-1; |
---|
| 5435 | mol_flag=0; |
---|
| 5436 | int interval=0; |
---|
| 5437 | v=0; |
---|
| 5438 | int max=#[size(#)]; |
---|
| 5439 | } |
---|
| 5440 | else |
---|
| 5441 | { "ERROR: If the two last parameters are not <int> and <intvec>, the last"; |
---|
| 5442 | " parameter should be an <int>."; |
---|
| 5443 | return(); |
---|
| 5444 | } |
---|
| 5445 | } |
---|
| 5446 | for (int i=1;i<=gen_num;i=i+1) |
---|
| 5447 | { if (typeof(#[i])=="matrix") |
---|
| 5448 | { if (nrows(#[i])<>n or ncols(#[i])<>n) |
---|
| 5449 | { "ERROR: The number of variables of the base ring needs to be the same"; |
---|
| 5450 | " as the dimension of the square matrices"; |
---|
| 5451 | return(); |
---|
| 5452 | } |
---|
| 5453 | } |
---|
| 5454 | else |
---|
| 5455 | { "ERROR: The first parameters should be a list of matrices"; |
---|
| 5456 | return(); |
---|
| 5457 | } |
---|
| 5458 | } |
---|
| 5459 | //---------------------------------------------------------------------------- |
---|
| 5460 | if (mol_flag==0) |
---|
| 5461 | { if (ch==0) |
---|
| 5462 | { matrix REY,M=reynolds_molien(#[1..gen_num],intvec(0,interval,v)); // one |
---|
| 5463 | // will contain Reynolds operator and the |
---|
| 5464 | // other enumerator and denominator of |
---|
| 5465 | // Molien series |
---|
| 5466 | matrix P=primary_char0_random(REY,M,max,v); |
---|
| 5467 | matrix S,IS=secondary_char0(P,REY,M,v); |
---|
| 5468 | return(P,S,IS); |
---|
| 5469 | } |
---|
| 5470 | else |
---|
| 5471 | { list L=group_reynolds(#[1..gen_num],v); |
---|
| 5472 | if (L[1]<>0) // testing whether we are in the modular |
---|
| 5473 | { string newring="aksldfalkdsflkj"; // case |
---|
| 5474 | if (minpoly==0) |
---|
| 5475 | { if (v) |
---|
| 5476 | { " We are dealing with the non-modular case."; |
---|
| 5477 | } |
---|
[5811fb] | 5478 | molien(L[3..size(L)],newring,L[2],intvec(0,interval,v)); |
---|
[29aa4bf] | 5479 | matrix P=primary_charp_random(L[1],newring,max,v); |
---|
| 5480 | matrix S,IS=secondary_charp(P,L[1],newring,v); |
---|
| 5481 | if (voice==2) |
---|
| 5482 | { kill aksldfalkdsflkj; |
---|
| 5483 | } |
---|
| 5484 | return(P,S,IS); |
---|
| 5485 | } |
---|
| 5486 | else |
---|
| 5487 | { if (v) |
---|
| 5488 | { " Since it is impossible for this programme to calculate the Molien |
---|
| 5489 | series for"; |
---|
| 5490 | " invariant rings over extension fields of prime characteristic, we |
---|
| 5491 | have to"; |
---|
| 5492 | " continue without it."; |
---|
| 5493 | ""; |
---|
| 5494 | |
---|
| 5495 | } |
---|
| 5496 | list l=primary_charp_no_molien_random(L[1],max,v); |
---|
| 5497 | if (size(l)==2) |
---|
| 5498 | { matix S=secondary_no_molien(l[1],L[1],l[2],v); |
---|
| 5499 | } |
---|
| 5500 | else |
---|
| 5501 | { matix S=secondary_no_molien(l[1],L[1],v); |
---|
| 5502 | } |
---|
| 5503 | return(l[1],S); |
---|
| 5504 | } |
---|
| 5505 | } |
---|
| 5506 | else // the modular case |
---|
| 5507 | { if (v) |
---|
| 5508 | { " There is also no Molien series, we can make use of..."; |
---|
| 5509 | ""; |
---|
| 5510 | " We can start looking for primary invariants..."; |
---|
| 5511 | ""; |
---|
| 5512 | } |
---|
| 5513 | matrix P=primary_charp_without_random(#[1..gen_num],max,v); |
---|
| 5514 | matrix S=secondary_not_cohen_macaulay(P,#[1..gen_num],v); |
---|
| 5515 | return(L[1],S); |
---|
| 5516 | } |
---|
| 5517 | } |
---|
| 5518 | } |
---|
| 5519 | if (mol_flag==1) // the user wants no calculation of the |
---|
| 5520 | { list L=group_reynolds(#[1..gen_num],v); // Molien series |
---|
| 5521 | if (ch==0) |
---|
| 5522 | { list l=primary_char0_no_molien_random(L[1],max,v); |
---|
| 5523 | if (size(l)==2) |
---|
| 5524 | { matix S=secondary_no_molien(l[1],L[1],l[2],v); |
---|
| 5525 | } |
---|
| 5526 | else |
---|
| 5527 | { matix S=secondary_no_molien(l[1],L[1],v); |
---|
| 5528 | } |
---|
| 5529 | return(l[1],S); |
---|
| 5530 | } |
---|
| 5531 | else |
---|
| 5532 | { if (L[1]<>0) // testing whether we are in the modular |
---|
| 5533 | { list l=primary_charp_no_molien_random(L[1],max,v); // case |
---|
| 5534 | if (size(l)==2) |
---|
| 5535 | { matix S=secondary_no_molien(l[1],L[1],l[2],v); |
---|
| 5536 | } |
---|
| 5537 | else |
---|
| 5538 | { matix S=secondary_no_molien(l[1],L[1],v); |
---|
| 5539 | } |
---|
| 5540 | return(l[1],S); |
---|
| 5541 | } |
---|
| 5542 | else // the modular case |
---|
| 5543 | { if (v) |
---|
| 5544 | { " We can start looking for primary invariants..."; |
---|
| 5545 | ""; |
---|
| 5546 | } |
---|
| 5547 | matrix P=primary_charp_without_random(#[1..gen_num],max,v); |
---|
| 5548 | matrix S=secondary_not_cohen_macaulay(P,#[1..gen_num],v); |
---|
| 5549 | return(L[1],S); |
---|
| 5550 | } |
---|
| 5551 | } |
---|
| 5552 | } |
---|
| 5553 | if (mol_flag==-1) |
---|
| 5554 | { if (ch==0) |
---|
| 5555 | { "ERROR: Characteristic 0 can only be simulated in characteristic p>>0. |
---|
| 5556 | "; |
---|
| 5557 | return(); |
---|
| 5558 | } |
---|
| 5559 | list L=group_reynolds(#[1..gen_num],v); |
---|
| 5560 | string newring="aksldfalkdsflkj"; |
---|
[5811fb] | 5561 | molien(L[2..size(L)],newring,intvec(1,v)); |
---|
[29aa4bf] | 5562 | matrix P=primary_charp_random(L[1],newring,max,v); |
---|
| 5563 | matrix S,IS=secondary_charp(P,L[1],newring,v); |
---|
| 5564 | kill aksldfalkdsflkj; |
---|
| 5565 | return(P,S,IS); |
---|
| 5566 | } |
---|
| 5567 | else // the user specified that the |
---|
| 5568 | { if (ch==0) // characteristic divides the group order |
---|
| 5569 | { "ERROR: The characteristic cannot divide the group order when it is 0. |
---|
| 5570 | "; |
---|
| 5571 | return(); |
---|
| 5572 | } |
---|
| 5573 | if (v) |
---|
| 5574 | { ""; |
---|
| 5575 | } |
---|
| 5576 | matrix P=primary_charp_without_random(#[1..gen_num],max,v); |
---|
| 5577 | matrix S=secondary_not_cohen_macaulay(P,#[1..gen_num],v); |
---|
| 5578 | return(L[1],S); |
---|
| 5579 | } |
---|
| 5580 | } |
---|
| 5581 | example |
---|
| 5582 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
| 5583 | echo=2; |
---|
| 5584 | ring R=0,(x,y,z),dp; |
---|
| 5585 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 5586 | matrix P,S,IS=invariant_ring_random(A,1); |
---|
| 5587 | print(P); |
---|
| 5588 | print(S); |
---|
| 5589 | print(IS); |
---|
| 5590 | } |
---|
| 5591 | |
---|
| 5592 | proc algebra_containment (poly p, matrix A) |
---|
| 5593 | "USAGE: algebra_containment(p,A); |
---|
| 5594 | p: arbitrary <poly>, A: a 1xm <matrix> giving generators of a |
---|
| 5595 | subalgebra of the basering |
---|
| 5596 | RETURN: 1 (TRUE) (type <int>) if p is contained in the subalgebra |
---|
| 5597 | 0 (FALSE) (type <int>) if <poly> is not contained |
---|
| 5598 | DISPLAY: a representation of p in terms of algebra generators A[1,i]=y(i) if p |
---|
| 5599 | is contained in the subalgebra |
---|
| 5600 | EXAMPLE: example algebra_containment; shows an example |
---|
| 5601 | THEORY: The ideal of algebraic relations of the algebra generators f1,...,fm |
---|
| 5602 | given by A is computed introducing new variables y(i) and the product |
---|
| 5603 | order: x^a*y^b > y^d*y^e if x^a > x^d or else if y^b > y^e. p reduces |
---|
| 5604 | to a polynomial only in the y(i) <=> p is contained in the subring |
---|
| 5605 | generated by the polynomials in A. |
---|
| 5606 | " |
---|
| 5607 | { degBound=0; |
---|
| 5608 | if (nrows(A)==1) |
---|
| 5609 | { def br=basering; |
---|
| 5610 | int n=nvars(br); |
---|
| 5611 | int m=ncols(A); |
---|
| 5612 | string mp=string(minpoly); |
---|
| 5613 | execute "ring R=("+charstr(br)+"),(x(1..n),y(1..m)),(dp(n),dp(m));"; |
---|
| 5614 | execute "minpoly=number("+mp+");"; |
---|
| 5615 | ideal vars=x(1..n); |
---|
| 5616 | map emb=br,vars; |
---|
| 5617 | ideal A=ideal(emb(A)); |
---|
| 5618 | ideal check=emb(p); |
---|
| 5619 | for (int i=1;i<=m;i=i+1) |
---|
| 5620 | { A[i]=A[i]-y(i); |
---|
| 5621 | } |
---|
| 5622 | A=std(A); |
---|
| 5623 | check[1]=reduce(check[1],A); |
---|
| 5624 | A=elim(check,1,n); |
---|
| 5625 | if (A[1]<>0) |
---|
| 5626 | { "\/\/ "+string(check); |
---|
| 5627 | return(1); |
---|
| 5628 | } |
---|
| 5629 | else |
---|
| 5630 | { return(0); |
---|
| 5631 | } |
---|
| 5632 | } |
---|
| 5633 | else |
---|
| 5634 | { "ERROR: <matrix> may only have one row"; |
---|
| 5635 | return(); |
---|
| 5636 | } |
---|
| 5637 | } |
---|
| 5638 | example |
---|
| 5639 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
| 5640 | echo=2; |
---|
| 5641 | ring R=0,(x,y,z),dp; |
---|
| 5642 | matrix A[1][7]=x2+y2,z2,x4+y4,1,x2z-1y2z,xyz,x3y-1xy3; |
---|
| 5643 | poly p1=x10z3-x8y2z3+2x6y4z3-2x4y6z3+x2y8z3-y10z3+x6z4+3x4y2z4+3x2y4z4+y6z4; |
---|
| 5644 | algebra_containment(p1,A); |
---|
| 5645 | poly p2=z; |
---|
| 5646 | algebra_containment(p2,A); |
---|
| 5647 | } |
---|
| 5648 | |
---|
| 5649 | proc module_containment(poly p, matrix P, matrix S) |
---|
| 5650 | "USAGE: module_containment(p,P,S); |
---|
| 5651 | p: arbitrary <poly>, P: a 1xn <matrix> giving generators of an algebra, |
---|
| 5652 | S: a 1xt <matrix> giving generators of a module over the algebra |
---|
| 5653 | generated by P |
---|
| 5654 | ASSUME: n is the number of variables in the basering and the generators in P |
---|
| 5655 | are algebraically independent |
---|
| 5656 | RETURNS: 1 (TRUE) (type <int>) if p is contained in the ring |
---|
| 5657 | 0 (FALSE) type <int>) if p is not contained |
---|
| 5658 | DISPLAY: the representation of p in terms of algebra generators P[1,i]=z(i) and |
---|
| 5659 | module generators S[1,j]=y(j) if p is contained in the module |
---|
| 5660 | EXAMPLE: example module_containment; shows an example |
---|
| 5661 | THEORY: The ideal of algebraic relations of all the generators p1,...,pn, |
---|
| 5662 | s1,...,st given by P and S is computed introducing new variables y(j), |
---|
| 5663 | z(i) and the product order: x^a*y^b*z^c > x^d*y^e*z^f if x^a > x^d |
---|
| 5664 | with respect to the lp ordering or else if z^c > z^f with respect to |
---|
| 5665 | the dp ordering or else if y^b > y^e with respect to the lp ordering |
---|
| 5666 | again. p reduces to a polynomial only in the y(j) and z(i) linear in |
---|
| 5667 | the z(i)) <=> p is contained in the module. |
---|
| 5668 | " |
---|
| 5669 | { def br=basering; |
---|
| 5670 | degBound=0; |
---|
| 5671 | int n=nvars(br); |
---|
| 5672 | if (ncols(P)==n and nrows(P)==1 and nrows(S)==1) |
---|
| 5673 | { int m=ncols(S); |
---|
| 5674 | string mp=string(minpoly); |
---|
| 5675 | execute "ring R=("+charstr(br)+"),(x(1..n),y(1..m),z(1..n)),(lp(n),dp(m),lp(n));"; |
---|
| 5676 | execute "minpoly=number("+mp+");"; |
---|
| 5677 | ideal vars=x(1..n); |
---|
| 5678 | map emb=br,vars; |
---|
| 5679 | matrix P=emb(P); |
---|
| 5680 | matrix S=emb(S); |
---|
| 5681 | ideal check=emb(p); |
---|
| 5682 | ideal I; |
---|
| 5683 | for (int i=1;i<=m;i=i+1) |
---|
| 5684 | { I[i]=S[1,i]-y(i); |
---|
| 5685 | } |
---|
| 5686 | for (i=1;i<=n;i=i+1) |
---|
| 5687 | { I[m+i]=P[1,i]-z(i); |
---|
| 5688 | } |
---|
| 5689 | I=std(I); |
---|
| 5690 | check[1]=reduce(check[1],I); |
---|
| 5691 | I=elim(check,1,n); // checking whether all variables from |
---|
| 5692 | if (I[1]<>0) // the former ring have disappeared |
---|
| 5693 | { "\/\/ "+string(check); |
---|
| 5694 | return(1); |
---|
| 5695 | } |
---|
| 5696 | else |
---|
| 5697 | { return(0); |
---|
| 5698 | } |
---|
| 5699 | } |
---|
| 5700 | else |
---|
| 5701 | { "ERROR: the first <matrix> must have the same number of columns as the"; |
---|
| 5702 | " basering and both <matrices> may only have one row"; |
---|
| 5703 | return(); |
---|
| 5704 | } |
---|
| 5705 | } |
---|
| 5706 | example |
---|
| 5707 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
| 5708 | echo=2; |
---|
| 5709 | ring R=0,(x,y,z),dp; |
---|
| 5710 | matrix P[1][3]=x2+y2,z2,x4+y4; |
---|
| 5711 | matrix S[1][4]=1,x2z-1y2z,xyz,x3y-1xy3; |
---|
| 5712 | poly p1=x10z3-x8y2z3+2x6y4z3-2x4y6z3+x2y8z3-y10z3+x6z4+3x4y2z4+3x2y4z4+y6z4; |
---|
| 5713 | module_containment(p1,P,S); |
---|
| 5714 | poly p2=z; |
---|
| 5715 | module_containment(p2,P,S); |
---|
| 5716 | } |
---|
| 5717 | |
---|
| 5718 | proc orbit_variety (matrix F,string newring) |
---|
| 5719 | "USAGE: orbit_variety(F,s); |
---|
| 5720 | F: a 1xm <matrix> defing an invariant ring, s: a <string> giving the |
---|
| 5721 | name for a new ring |
---|
| 5722 | RETURN: a Groebner basis (type <ideal>, named G) for the ideal defining the |
---|
| 5723 | orbit variety (i.e. the syzygy ideal) in the new ring (named `s`) |
---|
| 5724 | EXAMPLE: example orbit_variety; shows an example |
---|
| 5725 | THEORY: The ideal of algebraic relations of the invariant ring generators is |
---|
| 5726 | calculated, then the variables of the original ring are eliminated and |
---|
| 5727 | the polynomials that are left over define the orbit variety |
---|
| 5728 | " |
---|
| 5729 | { if (newring=="") |
---|
| 5730 | { "ERROR: the second parameter may not be an empty <string>"; |
---|
| 5731 | return(); |
---|
| 5732 | } |
---|
| 5733 | if (nrows(F)==1) |
---|
| 5734 | { def br=basering; |
---|
| 5735 | int n=nvars(br); |
---|
| 5736 | int m=ncols(F); |
---|
| 5737 | string mp=string(minpoly); |
---|
| 5738 | execute "ring R=("+charstr(br)+"),("+varstr(br)+",y(1..m)),dp;"; |
---|
| 5739 | execute "minpoly=number("+mp+");"; |
---|
| 5740 | ideal I=ideal(imap(br,F)); |
---|
| 5741 | for (int i=1;i<=m;i=i+1) |
---|
| 5742 | { I[i]=I[i]-y(i); |
---|
| 5743 | } |
---|
| 5744 | I=elim(I,1,n); |
---|
| 5745 | execute "ring "+newring+"=("+charstr(br)+"),(y(1..m)),dp(m);"; |
---|
| 5746 | execute "minpoly=number("+mp+");"; |
---|
| 5747 | ideal vars; |
---|
| 5748 | for (i=2;i<=n;i=i+1) |
---|
| 5749 | { vars[i]=0; |
---|
| 5750 | } |
---|
| 5751 | vars=vars,y(1..m); |
---|
| 5752 | map emb=R,vars; |
---|
| 5753 | ideal G=emb(I); |
---|
| 5754 | kill emb, vars, R; |
---|
| 5755 | keepring `newring`; |
---|
| 5756 | // execute "keepring "+newring+";"; |
---|
| 5757 | return(); |
---|
| 5758 | } |
---|
| 5759 | else |
---|
| 5760 | { "ERROR: the <matrix> may only have one row"; |
---|
| 5761 | return(); |
---|
| 5762 | } |
---|
| 5763 | } |
---|
| 5764 | example |
---|
| 5765 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
| 5766 | echo=2; |
---|
| 5767 | ring R=0,(x,y,z),dp; |
---|
| 5768 | matrix F[1][7]=x2+y2,z2,x4+y4,1,x2z-1y2z,xyz,x3y-1xy3; |
---|
| 5769 | string newring="E"; |
---|
| 5770 | orbit_variety(F,newring); |
---|
| 5771 | print(G); |
---|
| 5772 | basering; |
---|
| 5773 | } |
---|
| 5774 | |
---|
| 5775 | proc relative_orbit_variety(ideal I,matrix F,string newring) |
---|
| 5776 | "USAGE: relative_orbit_variety(I,F,s); |
---|
| 5777 | I: an <ideal> invariant under the action of a group, F: a 1xm |
---|
| 5778 | <matrix> defining the invariant ring of this group, s: a <string> |
---|
| 5779 | giving a name for a new ring |
---|
| 5780 | RETURN: a Groebner basis (type <ideal>, named G) for the ideal defining the |
---|
| 5781 | relative orbit variety with respect to I in the new ring (named s) |
---|
| 5782 | EXAMPLE: example relative_orbit_variety; shows an example |
---|
| 5783 | THEORY: A Groebner basis of the ideal of algebraic relations of the invariant |
---|
| 5784 | ring generators is calculated, then one of the basis elements plus the |
---|
| 5785 | ideal generators. The variables of the original ring are eliminated and |
---|
| 5786 | the polynomials that are left over define thecrelative orbit variety |
---|
| 5787 | with respect to I. |
---|
| 5788 | " |
---|
| 5789 | { if (newring=="") |
---|
| 5790 | { "ERROR: the third parameter may not be empty a <string>"; |
---|
| 5791 | return(); |
---|
| 5792 | } |
---|
| 5793 | degBound=0; |
---|
| 5794 | if (nrows(F)==1) |
---|
| 5795 | { def br=basering; |
---|
| 5796 | int n=nvars(br); |
---|
| 5797 | int m=ncols(F); |
---|
| 5798 | string mp=string(minpoly); |
---|
| 5799 | execute "ring R=("+charstr(br)+"),("+varstr(br)+",y(1..m)),lp;"; |
---|
| 5800 | execute "minpoly=number("+mp+");"; |
---|
| 5801 | ideal J=ideal(imap(br,F)); |
---|
| 5802 | ideal I=imap(br,I); |
---|
| 5803 | for (int i=1;i<=m;i=i+1) |
---|
| 5804 | { J[i]=J[i]-y(i); |
---|
| 5805 | } |
---|
| 5806 | J=std(J); |
---|
| 5807 | J=J,I; |
---|
| 5808 | option(redSB); |
---|
| 5809 | J=std(J); |
---|
| 5810 | ideal vars; |
---|
| 5811 | //for (i=1;i<=n;i=i+1) |
---|
| 5812 | //{ vars[i]=0; |
---|
| 5813 | //} |
---|
| 5814 | vars[n]=0; |
---|
| 5815 | vars=vars,y(1..m); |
---|
| 5816 | map emb=R,vars; |
---|
| 5817 | ideal G=emb(J); |
---|
| 5818 | J=J-G; |
---|
| 5819 | for (i=1;i<=ncols(G);i=i+1) |
---|
| 5820 | { if (J[i]<>0) |
---|
| 5821 | { G[i]=0; |
---|
| 5822 | } |
---|
| 5823 | } |
---|
| 5824 | G=compress(G); |
---|
| 5825 | execute "ring "+newring+"=("+charstr(br)+"),(y(1..m)),lp;"; |
---|
| 5826 | execute "minpoly=number("+mp+");"; |
---|
| 5827 | ideal vars; |
---|
| 5828 | for (i=2;i<=n;i=i+1) |
---|
| 5829 | { vars[i]=0; |
---|
| 5830 | } |
---|
| 5831 | vars=vars,y(1..m); |
---|
| 5832 | map emb=R,vars; |
---|
| 5833 | ideal G=emb(G); |
---|
| 5834 | kill vars, emb; |
---|
| 5835 | keepring `newring`; |
---|
| 5836 | // execute "keepring "+newring+";"; |
---|
| 5837 | return(); |
---|
| 5838 | } |
---|
| 5839 | else |
---|
| 5840 | { "ERROR: the <matrix> may only have one row"; |
---|
| 5841 | return(); |
---|
| 5842 | } |
---|
| 5843 | } |
---|
| 5844 | example |
---|
| 5845 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.6.3:"; |
---|
| 5846 | echo=2; |
---|
| 5847 | ring R=0,(x,y,z),dp; |
---|
| 5848 | matrix F[1][3]=x+y+z,xy+xz+yz,xyz; |
---|
| 5849 | ideal I=x2+y2+z2-1,x2y+y2z+z2x-2x-2y-2z,xy2+yz2+zx2-2x-2y-2z; |
---|
| 5850 | string newring="E"; |
---|
| 5851 | relative_orbit_variety(I,F,newring); |
---|
| 5852 | print(G); |
---|
| 5853 | basering; |
---|
| 5854 | } |
---|
| 5855 | |
---|
| 5856 | proc image_of_variety(ideal I,matrix F) |
---|
| 5857 | "USAGE: image_of_variety(I,F); |
---|
| 5858 | I: an arbitray <ideal>, F: a 1xm <matrix> defining an invariant ring |
---|
| 5859 | of a some matrix group |
---|
| 5860 | RETURN: the <ideal> defining the image under that group of the variety defined |
---|
| 5861 | by I |
---|
| 5862 | EXAMPLE: example image_of_variety; shows an example |
---|
| 5863 | THEORY: relative_orbit_variety(I,F,s) is called and the newly introduced |
---|
| 5864 | variables in the output are replaced by the generators of the |
---|
| 5865 | invariant ring. This ideal in the original variables defines the image |
---|
| 5866 | of the variety defined by I |
---|
| 5867 | " |
---|
| 5868 | { if (nrows(F)==1) |
---|
| 5869 | { def br=basering; |
---|
| 5870 | int n=nvars(br); |
---|
| 5871 | string newring="E"; |
---|
| 5872 | relative_orbit_variety(I,F,newring); |
---|
| 5873 | execute "ring R=("+charstr(br)+"),("+varstr(br)+","+varstr(E)+"),lp;"; |
---|
| 5874 | ideal F=imap(br,F); |
---|
| 5875 | for (int i=1;i<=n;i=i+1) |
---|
| 5876 | { F=0,F; |
---|
| 5877 | } |
---|
| 5878 | setring br; |
---|
| 5879 | map emb2=E,F; |
---|
| 5880 | return(compress(emb2(G))); |
---|
| 5881 | } |
---|
| 5882 | else |
---|
| 5883 | { "ERROR: the <matrix> may only have one row"; |
---|
| 5884 | return(); |
---|
| 5885 | } |
---|
| 5886 | } |
---|
| 5887 | example |
---|
| 5888 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.6.8:"; |
---|
| 5889 | echo=2; |
---|
| 5890 | ring R=0,(x,y,z),dp; |
---|
| 5891 | matrix F[1][3]=x+y+z,xy+xz+yz,xyz; |
---|
| 5892 | ideal I=xy; |
---|
| 5893 | print(image_of_variety(I,F)); |
---|
| 5894 | } |
---|