[f34c37c] | 1 | // last change: 98/11/05 |
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[9f79aa] | 2 | /////////////////////////////////////////////////////////////////////////////// |
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[f54c83] | 3 | version="$Id: finvar.lib,v 1.41 2006-03-16 15:50:37 Singular Exp $" |
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[fd3fb7] | 4 | category="Invariant theory"; |
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[f34c37c] | 5 | info=" |
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[558eb2] | 6 | LIBRARY: finvar.lib Invariant Rings of Finite Groups |
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[3684b5] | 7 | AUTHOR: Agnes E. Heydtmann, email: agnes@math.uni-sb.de |
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[f34c37c] | 8 | |
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[558eb2] | 9 | OVERVIEW: |
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[f34c37c] | 10 | A library for computing polynomial invariants of finite matrix groups and |
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| 11 | generators of related varieties. The algorithms are based on B. Sturmfels, |
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[68e678] | 12 | G. Kemper and W. Decker et al.. |
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[f34c37c] | 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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| 19 | |
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[558eb2] | 20 | AUXILIARY PROCEDURES: |
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[f34c37c] | 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 | orbit_variety() ideal of the orbit variety |
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| 46 | relative_orbit_variety() ideal of a relative orbit variety |
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| 47 | image_of_variety() ideal of the image of a variety |
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| 48 | "; |
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[68e678] | 49 | /////////////////////////////////////////////////////////////////////////////// |
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[f34c37c] | 50 | // perhaps useful procedures (no help provided): |
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| 51 | // unique() is a matrix among other matrices? |
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| 52 | // exponent() gives the exponent of a number |
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| 53 | // sort_of_invariant_basis() lin. ind. invariants of a degree mod p.i. |
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| 54 | // next_vector lists all of Z^n with first nonzero entry 1 |
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| 55 | // int_number_map integers 1..q are maped to q field elements |
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| 56 | // search searches a number of p.i., char 0 |
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| 57 | // p_search searches a number of p.i., char p |
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| 58 | // search_random searches a # of p.i., char 0, randomized |
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| 59 | // p_search_random searches a # of p.i., char p, randomized |
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| 60 | // concat_intmat concatenates two integer matrices |
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[68e678] | 61 | /////////////////////////////////////////////////////////////////////////////// |
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| 62 | |
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[f34c37c] | 63 | LIB "matrix.lib"; |
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| 64 | LIB "elim.lib"; |
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| 65 | LIB "general.lib"; |
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[e53306f] | 66 | LIB "algebra.lib"; |
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[9f79aa] | 67 | |
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[68e678] | 68 | /////////////////////////////////////////////////////////////////////////////// |
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[f34c37c] | 69 | // Checks whether the last parameter, being a matrix, is among the previous |
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| 70 | // parameters, also being matrices |
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[68e678] | 71 | /////////////////////////////////////////////////////////////////////////////// |
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[f34c37c] | 72 | proc unique (list #) |
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[4370f3] | 73 | { int s=size(#); def m=#[s]; |
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| 74 | for (int i=1;i<s;i++) |
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| 75 | { if (#[i]==m) |
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| 76 | { return(0); } |
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[f34c37c] | 77 | } |
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| 78 | return(1); |
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| 79 | } |
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[9f79aa] | 80 | /////////////////////////////////////////////////////////////////////////////// |
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[f34c37c] | 81 | |
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| 82 | proc cyclotomic (int i) |
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[68e678] | 83 | "USAGE: cyclotomic(i); i integer > 0 |
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[f34c37c] | 84 | RETURNS: the i-th cyclotomic polynomial (type <poly>) as one in the first ring |
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| 85 | variable |
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[18bd9c] | 86 | THEORY: x^i-1 is divided by the j-th cyclotomic polynomial where j takes on |
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[68e678] | 87 | the value of proper divisors of i |
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[f34c37c] | 88 | EXAMPLE: example cyclotomic; shows an example |
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| 89 | " |
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| 90 | { if (i<=0) |
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| 91 | { "ERROR: the input should be > 0."; |
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| 92 | return(); |
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| 93 | } |
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| 94 | poly v1=var(1); |
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| 95 | if (i==1) |
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| 96 | { return(v1-1); // 1-st cyclotomic polynomial |
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| 97 | } |
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| 98 | poly min=v1^i-1; |
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| 99 | matrix s[1][2]; |
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| 100 | min=min/(v1-1); // dividing by the 1-st cyclotomic |
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| 101 | // polynomial |
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| 102 | int j=2; |
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| 103 | int n; |
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| 104 | poly c; |
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| 105 | int flag=1; |
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| 106 | while(2*j<=i) // there are no proper divisors of i |
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| 107 | { if ((i%j)==0) // greater than i/2 |
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| 108 | { if (flag==1) |
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| 109 | { n=j; // n stores the first proper divisor of |
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| 110 | } // i > 1 |
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| 111 | flag=0; |
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| 112 | c=cyclotomic(j); // recursive computation |
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| 113 | s=min,c; |
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| 114 | s=matrix(syz(ideal(s))); // dividing |
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| 115 | min=s[2,1]; |
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| 116 | } |
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| 117 | if (n*j==i) // the earliest possible point to break |
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| 118 | { break; |
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| 119 | } |
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[18bd9c] | 120 | j++; |
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[f34c37c] | 121 | } |
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| 122 | min=min/leadcoef(min); // making sure that the leading |
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| 123 | return(min); // coefficient is 1 |
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| 124 | } |
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| 125 | example |
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[68e678] | 126 | { "EXAMPLE:"; echo=2; |
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[f34c37c] | 127 | ring R=0,(x,y,z),dp; |
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| 128 | print(cyclotomic(25)); |
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| 129 | } |
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| 130 | |
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| 131 | proc group_reynolds (list #) |
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| 132 | "USAGE: group_reynolds(G1,G2,...[,v]); |
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| 133 | G1,G2,...: nxn <matrices> generating a finite matrix group, v: an |
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| 134 | optional <int> |
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| 135 | ASSUME: n is the number of variables of the basering, g the number of group |
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| 136 | elements |
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| 137 | RETURN: a <list>, the first list element will be a gxn <matrix> representing |
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| 138 | the Reynolds operator if we are in the non-modular case; if the |
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| 139 | characteristic is >0, minpoly==0 and the finite group non-cyclic the |
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| 140 | second list element is an <int> giving the lowest common multiple of |
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[9f79aa] | 141 | the matrix group elements' order (used in molien); in general all |
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| 142 | other list elements are nxn <matrices> listing all elements of the |
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| 143 | finite group |
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[f34c37c] | 144 | DISPLAY: information if v does not equal 0 |
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| 145 | THEORY: The entire matrix group is generated by getting all left products of |
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[68e678] | 146 | generators with the new elements from the last run through the loop |
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[f34c37c] | 147 | (or the generators themselves during the first run). All the ones that |
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| 148 | have been generated before are thrown out and the program terminates |
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[9f79aa] | 149 | when no new elements found in one run. Additionally each time a new |
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| 150 | group element is found the corresponding ring mapping of which the |
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| 151 | Reynolds operator is made up is generated. They are stored in the rows |
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| 152 | of the first return value. |
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[68e678] | 153 | EXAMPLE: example group_reynolds; shows an example |
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[f34c37c] | 154 | " |
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[558eb2] | 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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[f34c37c] | 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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[9f79aa] | 173 | { "ERROR: The parameters must be a list of matrices and maybe an <int>"; |
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[f34c37c] | 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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[b9b906] | 192 | if (ch<>0 && minpoly==0 && gen_num<>1) // finding out of which order the |
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[558eb2] | 193 | { matrix I=diag(1,n); // group element is |
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[f34c37c] | 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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[18bd9c] | 199 | o1++; |
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[f34c37c] | 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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[18bd9c] | 204 | for (int j=2;j<=gen_num;j++) // this loop adds the parameters to the |
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[f34c37c] | 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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[9f79aa] | 210 | { "ERROR: The parameters must be a list of matrices and maybe an <int>"; |
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[f34c37c] | 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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[18bd9c] | 218 | { i++; |
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[f34c37c] | 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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[18bd9c] | 225 | o2++; |
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[f34c37c] | 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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[18bd9c] | 247 | for (m=g-j+1;m<=g;m++) |
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| 248 | { for (k=1;k<=i;k++) |
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[9f79aa] | 249 | { l=l+1; |
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[f34c37c] | 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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[18bd9c] | 254 | for (k=1;k<=l;k++) |
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[f34c37c] | 255 | { if (unique(G(1..g),P(k))) |
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[18bd9c] | 256 | { j++; // a new factor for next run |
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| 257 | g++; |
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[f34c37c] | 258 | matrix G(g)=P(k); // a new group element - |
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[b9b906] | 259 | if (ch<>0 && minpoly==0 && i<>1) // finding out of which order the |
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[558eb2] | 260 | { TEST=G(g); //group element is |
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[f34c37c] | 261 | o2=1; |
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| 262 | while (TEST<>I) |
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| 263 | { TEST=TEST*G(g); |
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[18bd9c] | 264 | o2++; |
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[f34c37c] | 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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[558eb2] | 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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[f34c37c] | 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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[9f79aa] | 319 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
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[f34c37c] | 320 | ring R=0,(x,y,z),dp; |
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| 321 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
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| 322 | list L=group_reynolds(A); |
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| 323 | print(L[1]); |
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| 324 | print(L[2..size(L)]); |
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| 325 | } |
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| 326 | |
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[68e678] | 327 | /////////////////////////////////////////////////////////////////////////////// |
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[f34c37c] | 328 | // Returns i such that root^i==n, i.e. it heavily relies on the right input. |
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[68e678] | 329 | /////////////////////////////////////////////////////////////////////////////// |
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[f34c37c] | 330 | proc exponent(number n, number root) |
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| 331 | { int i=0; |
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| 332 | while((n/root^i)<>1) |
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[18bd9c] | 333 | { i++; |
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[f34c37c] | 334 | } |
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| 335 | return(i); |
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| 336 | } |
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[9f79aa] | 337 | /////////////////////////////////////////////////////////////////////////////// |
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[f34c37c] | 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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[9f79aa] | 341 | G1,G2,...: nxn <matrices>, all elements of a finite matrix group, |
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| 342 | ringname: a <string> giving a name for a new ring of characteristic 0 |
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| 343 | for the Molien series in case of prime characteristic, lcm: an <int> |
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| 344 | giving the lowest common multiple of the elements' orders in case of |
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| 345 | prime characteristic, minpoly==0 and a non-cyclic group, flags: an |
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| 346 | optional <intvec> with three components: if the first element is not |
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| 347 | equal to 0 characteristic 0 is simulated, i.e. the Molien series is |
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| 348 | computed as if the base field were characteristic 0 (the user must |
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| 349 | choose a field of large prime characteristic, e.g. 32003), the second |
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| 350 | component should give the size of intervals between canceling common |
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| 351 | factors in the expansion of the Molien series, 0 (the default) means |
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| 352 | only once after generating all terms, in prime characteristic also a |
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| 353 | negative number can be given to indicate that common factors should |
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| 354 | always be canceled when the expansion is simple (the root of the |
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| 355 | extension field does not occur among the coefficients) |
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[f34c37c] | 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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[18bd9c] | 364 | THEORY: In characteristic 0 the terms 1/det(1-xE) for all group elements of |
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[68e678] | 365 | the Molien series are computed in a straight forward way. In prime |
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[f34c37c] | 366 | characteristic a Brauer lift is involved. The returned matrix gives |
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[18bd9c] | 367 | enumerator and denominator of the expanded version where common |
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[68e678] | 368 | factors have been canceled. |
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| 369 | EXAMPLE: example molien; shows an example |
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[f34c37c] | 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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[9f79aa] | 394 | { "ERROR: In characteristic p>0 a <string> must be given for the name of a new"; |
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[f34c37c] | 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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[9f79aa] | 409 | { "ERROR: In characteristic p>0 a <string> must be given for the name of a new"; |
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[f34c37c] | 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(); |
---|
| 417 | } |
---|
| 418 | string newring=#[size(#)-1]; |
---|
| 419 | g=size(#)-2; |
---|
| 420 | int r=g; |
---|
| 421 | } |
---|
| 422 | } |
---|
| 423 | } |
---|
| 424 | else // then <string> ist not needed |
---|
| 425 | { g=size(#)-1; |
---|
| 426 | } |
---|
| 427 | } |
---|
| 428 | else // last parameter is not <intvec> |
---|
| 429 | { int v=0; // no information is default |
---|
| 430 | int mol_flag=0; // computing of Molien series is default |
---|
| 431 | int interval=0; |
---|
| 432 | if (ch<>0) |
---|
| 433 | { if (typeof(#[size(#)])=="int") |
---|
| 434 | { int r=#[size(#)]; |
---|
| 435 | if (typeof(#[size(#)-1])<>"string") |
---|
| 436 | { "ERROR: in characteristic p>0 a <string> must be given for the name of a new"; |
---|
| 437 | " ring where the Molien series can be stored"; |
---|
| 438 | return(); |
---|
| 439 | } |
---|
| 440 | else |
---|
| 441 | { if (#[size(#)-1]=="") |
---|
| 442 | { "ERROR: <string> may not be empty"; |
---|
| 443 | return(); |
---|
| 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 ------------------ |
---|
[18bd9c] | 539 | for (int j=1;j<=g;j++) |
---|
[f34c37c] | 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); |
---|
[558eb2] | 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 |
---|
[f34c37c] | 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 --------- |
---|
[558eb2] | 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 |
---|
[f34c37c] | 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); |
---|
[558eb2] | 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 |
---|
[f34c37c] | 633 | int i, j, k; |
---|
| 634 | // -------------- finding all the terms of the Molien series ----------------- |
---|
[18bd9c] | 635 | for (i=1;i<=g;i++) |
---|
[f34c37c] | 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; |
---|
[558eb2] | 646 | p=det(xI-f2(G(i))); // characteristic polynomial of G(i) |
---|
[f34c37c] | 647 | L=factorize(p); |
---|
| 648 | F=L[1]; |
---|
| 649 | C=L[2]; |
---|
[18bd9c] | 650 | for (j=2;j<=ncols(F);j++) |
---|
[558eb2] | 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 |
---|
[f34c37c] | 655 | setring `newring`; |
---|
[558eb2] | 656 | p=p*(1-x*(e^k))^C[j]; // building the denominator of the new |
---|
| 657 | setring bre; // term |
---|
[f34c37c] | 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 | // } |
---|
[9f79aa] | 673 | // k=k+1; |
---|
[f34c37c] | 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); |
---|
[18bd9c] | 681 | for (j=1;j<=size(Mstring);j++) |
---|
[f34c37c] | 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); |
---|
[558eb2] | 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 |
---|
[f34c37c] | 734 | M[1,2]=1/s[1,2]*M[1,2]; |
---|
| 735 | kill slead; |
---|
| 736 | kill s; |
---|
| 737 | kill p; |
---|
| 738 | } |
---|
[558eb2] | 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 |
---|
[f34c37c] | 742 | } |
---|
[1fc5ce] | 743 | exportto(Top,`newring`); // we keep the ring where we computed the |
---|
[558eb2] | 744 | export M; // Molien series in such that we can |
---|
| 745 | setring br; // keep it |
---|
[f34c37c] | 746 | if (v) |
---|
| 747 | { " We are done calculating the Molien series."; |
---|
| 748 | ""; |
---|
| 749 | } |
---|
| 750 | } |
---|
[558eb2] | 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 |
---|
[f34c37c] | 757 | //----------------------- simulating characteristic 0 ------------------------ |
---|
| 758 | string chst=charstr(br); |
---|
[18bd9c] | 759 | for (int i=1;i<=size(chst);i++) |
---|
[f34c37c] | 760 | { if (chst[i]==",") |
---|
| 761 | { break; |
---|
| 762 | } |
---|
| 763 | } |
---|
| 764 | //----------------- generating ring of characteristic 0 ---------------------- |
---|
| 765 | if (minpoly==0) |
---|
| 766 | { if (i>size(chst)) |
---|
[034ce1] | 767 | { execute("ring "+newring+"=0,("+varstr(br)+"),("+ordstr(br)+")"); |
---|
[f34c37c] | 768 | } |
---|
| 769 | else |
---|
| 770 | { chst=chst[i..size(chst)]; |
---|
[034ce1] | 771 | execute |
---|
| 772 | ("ring "+newring+"=(0"+chst+"),("+varstr(br)+"),("+ordstr(br)+")"); |
---|
[f34c37c] | 773 | } |
---|
| 774 | } |
---|
| 775 | else |
---|
| 776 | { string minp=string(minpoly); |
---|
| 777 | minp=minp[2..size(minp)-1]; |
---|
| 778 | chst=chst[i..size(chst)]; |
---|
[034ce1] | 779 | execute("ring "+newring+"=(0"+chst+"),("+varstr(br)+"),("+ordstr(br)+")"); |
---|
| 780 | execute("minpoly="+minp); |
---|
[f34c37c] | 781 | } |
---|
| 782 | matrix I=diag(1,n); |
---|
| 783 | poly v1=maxideal(1)[1]; // the Molien series will be in terms of |
---|
| 784 | // the first variable of the current |
---|
| 785 | // ring - |
---|
| 786 | matrix M[1][2]; // M will contain the Molien series - |
---|
| 787 | M[1,1]=0; // M[1,1] will be the numerator - |
---|
| 788 | M[1,2]=1; // M[1,2] will be the denominator - |
---|
| 789 | matrix s; // will help us canceling in the |
---|
| 790 | // fraction |
---|
| 791 | poly p; // will contain the denominator of the |
---|
| 792 | // new term of the Molien series |
---|
| 793 | int j; |
---|
| 794 | string links, rechts; |
---|
| 795 | //----------------- finding all terms of the Molien series ------------------- |
---|
[18bd9c] | 796 | for (i=1;i<=g;i++) |
---|
[f34c37c] | 797 | { setring br; |
---|
| 798 | if (not(typeof(#[i])=="matrix")) |
---|
| 799 | { "ERROR: the parameters must be a list of matrices and maybe an <intvec>"; |
---|
| 800 | return(); |
---|
| 801 | } |
---|
| 802 | if ((n<>nrows(#[i])) or (n<>ncols(#[i]))) |
---|
| 803 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
| 804 | return(); |
---|
| 805 | } |
---|
| 806 | string stM(i)=string(#[i]); |
---|
[18bd9c] | 807 | for (j=1;j<=size(stM(i));j++) |
---|
[f34c37c] | 808 | { if (stM(i)[j]==" |
---|
| 809 | ") |
---|
| 810 | { links=stM(i)[1..j-1]; |
---|
| 811 | rechts=stM(i)[j+1..size(stM(i))]; |
---|
| 812 | stM(i)=links+rechts; |
---|
| 813 | } |
---|
| 814 | } |
---|
| 815 | setring `newring`; |
---|
[034ce1] | 816 | execute("matrix G(i)["+string(n)+"]["+string(n)+"]="+stM(i)); |
---|
[f34c37c] | 817 | p=det(I-v1*G(i)); // denominator of new term - |
---|
| 818 | M[1,1]=M[1,1]*p+M[1,2]; // expanding M[1,1]/M[1,2] + 1/p |
---|
| 819 | M[1,2]=M[1,2]*p; |
---|
| 820 | if (interval<>0) // canceling common terms of denominator |
---|
| 821 | { if ((i/interval)*interval==i or i==g) // and enumerator |
---|
| 822 | { |
---|
| 823 | s=matrix(syz(ideal(M))); // once gcd() is faster than syz() these |
---|
| 824 | M[1,1]=-s[2,1]; // three lines should be replaced by the |
---|
| 825 | M[1,2]=s[1,1]; // following three |
---|
| 826 | // p=gcd(M[1,1],M[1,2]); |
---|
| 827 | // M[1,1]=M[1,1]/p; |
---|
| 828 | // M[1,2]=M[1,2]/p; |
---|
| 829 | } |
---|
| 830 | } |
---|
| 831 | if (v) |
---|
| 832 | { " Term "+string(i)+" of the Molien series has been computed."; |
---|
| 833 | } |
---|
| 834 | } |
---|
| 835 | if (interval==0) // canceling common terms of denominator |
---|
| 836 | { // and enumerator - |
---|
| 837 | s=matrix(syz(ideal(M))); // once gcd() is faster than syz() these |
---|
| 838 | M[1,1]=-s[2,1]; // three lines should be replaced by the |
---|
| 839 | M[1,2]=s[1,1]; // following three |
---|
| 840 | // p=gcd(M[1,1],M[1,2]); |
---|
| 841 | // M[1,1]=M[1,1]/p; |
---|
| 842 | // M[1,2]=M[1,2]/p; |
---|
| 843 | } |
---|
| 844 | map slead=`newring`,ideal(0); |
---|
| 845 | s=slead(M); |
---|
[558eb2] | 846 | M[1,1]=1/s[1,1]*M[1,1]; // numerator and denominator have to have |
---|
| 847 | M[1,2]=1/s[1,2]*M[1,2]; // a constant term of 1 |
---|
[f34c37c] | 848 | if (v) |
---|
| 849 | { ""; |
---|
| 850 | " We are done calculating the Molien series."; |
---|
| 851 | ""; |
---|
| 852 | } |
---|
| 853 | kill G(1..g), s, slead, p, v1, I; |
---|
[558eb2] | 854 | export `newring`; // we keep the ring where we computed the |
---|
| 855 | export M; // the Molien series such that we can |
---|
| 856 | setring br; // keep it |
---|
[f34c37c] | 857 | } |
---|
| 858 | } |
---|
| 859 | example |
---|
| 860 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
[68e678] | 861 | " note the case of prime characteristic"; echo=2; |
---|
[f34c37c] | 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 | } |
---|
[68e678] | 877 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 878 | |
---|
| 879 | proc reynolds_molien (list #) |
---|
| 880 | "USAGE: reynolds_molien(G1,G2,...[,ringname,flags]); |
---|
| 881 | G1,G2,...: nxn <matrices> generating a finite matrix group, ringname: |
---|
| 882 | a <string> giving a name for a new ring of characteristic 0 for the |
---|
| 883 | Molien series in case of prime characteristic, flags: an optional |
---|
| 884 | <intvec> with three components: if the first element is not equal to 0 |
---|
| 885 | characteristic 0 is simulated, i.e. the Molien series is computed as |
---|
| 886 | if the base field were characteristic 0 (the user must choose a field |
---|
| 887 | of large prime characteristic, e.g. 32003) the second component should |
---|
| 888 | give the size of intervals between canceling common factors in the |
---|
| 889 | expansion of the Molien series, 0 (the default) means only once after |
---|
| 890 | generating all terms, in prime characteristic also a negative number |
---|
| 891 | can be given to indicate that common factors should always be canceled |
---|
| 892 | when the expansion is simple (the root of the extension field does not |
---|
| 893 | occur among the coefficients) |
---|
| 894 | ASSUME: n is the number of variables of the basering, G1,G2... are the group |
---|
| 895 | elements generated by group_reynolds(), g is the size of the group |
---|
| 896 | RETURN: a gxn <matrix> representing the Reynolds operator is the first return |
---|
| 897 | value and in case of characteristic 0 a 1x2 <matrix> giving enumerator |
---|
| 898 | and denominator of Molien series is the second one; in case of prime |
---|
| 899 | characteristic a ring with the name `ringname` of characteristic 0 is |
---|
| 900 | created where the same Molien series (named M) is stored |
---|
| 901 | DISPLAY: information if the third component of flags does not equal 0 |
---|
| 902 | THEORY: The entire matrix group is generated by getting all left products of |
---|
[68e678] | 903 | the generators with new elements from the last run through the loop |
---|
[f34c37c] | 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 |
---|
[9f79aa] | 906 | when are no new elements found in one run. Additionally each time a |
---|
| 907 | new group element is found the corresponding ring mapping of which the |
---|
| 908 | Reynolds operator is made up is generated. They are stored in the rows |
---|
| 909 | of the first return value. In characteristic 0 the terms 1/det(1-xE) |
---|
| 910 | is computed whenever a new element E is found. In prime characteristic |
---|
| 911 | a Brauer lift is involved and the terms are only computed after the |
---|
| 912 | entire matrix group is generated (to avoid the modular case). The |
---|
| 913 | returned matrix gives enumerator and denominator of the expanded |
---|
| 914 | version where common factors have been canceled. |
---|
[68e678] | 915 | EXAMPLE: example reynolds_molien; shows an example |
---|
[f34c37c] | 916 | " |
---|
| 917 | { def br=basering; // the Molien series depends on the |
---|
| 918 | int ch=char(br); // characteristic of the coefficient |
---|
| 919 | // field |
---|
| 920 | int gen_num; |
---|
| 921 | //------------------- making sure the input is okay -------------------------- |
---|
| 922 | if (typeof(#[size(#)])=="intvec") |
---|
| 923 | { if (size(#[size(#)])==3) |
---|
| 924 | { int mol_flag=#[size(#)][1]; |
---|
| 925 | if (#[size(#)][2]<0 && (ch==0 or (ch<>0 && mol_flag<>0))) |
---|
| 926 | { "ERROR: the second component of the <intvec> should be >=0"; |
---|
| 927 | return(); |
---|
| 928 | } |
---|
| 929 | int interval=#[size(#)][2]; |
---|
| 930 | int v=#[size(#)][3]; |
---|
| 931 | } |
---|
| 932 | else |
---|
| 933 | { "ERROR: <intvec> should have three components"; |
---|
| 934 | return(); |
---|
| 935 | } |
---|
| 936 | if (ch<>0) |
---|
| 937 | { if (typeof(#[size(#)-1])<>"string") |
---|
| 938 | { "ERROR: in characteristic p a <string> must be given for the name"; |
---|
| 939 | " of a new ring where the Molien series can be stored"; |
---|
| 940 | return(); |
---|
| 941 | } |
---|
| 942 | else |
---|
| 943 | { if (#[size(#)-1]=="") |
---|
| 944 | { "ERROR: <string> may not be empty"; |
---|
| 945 | return(); |
---|
| 946 | } |
---|
| 947 | string newring=#[size(#)-1]; |
---|
| 948 | gen_num=size(#)-2; |
---|
| 949 | } |
---|
| 950 | } |
---|
| 951 | else // then <string> ist not needed |
---|
| 952 | { gen_num=size(#)-1; |
---|
| 953 | } |
---|
| 954 | } |
---|
| 955 | else // last parameter is not <intvec> |
---|
| 956 | { int v=0; // no information is default |
---|
| 957 | int interval; |
---|
| 958 | int mol_flag=0; // computing of Molien series is default |
---|
| 959 | if (ch<>0) |
---|
| 960 | { if (typeof(#[size(#)])<>"string") |
---|
| 961 | { "ERROR: in characteristic p a <string> must be given for the name"; |
---|
| 962 | " of a new ring where the Molien series can be stored"; |
---|
| 963 | return(); |
---|
| 964 | } |
---|
| 965 | else |
---|
| 966 | { if (#[size(#)]=="") |
---|
| 967 | { "ERROR: <string> may not be empty"; |
---|
| 968 | return(); |
---|
| 969 | } |
---|
| 970 | string newring=#[size(#)]; |
---|
| 971 | gen_num=size(#)-1; |
---|
| 972 | } |
---|
| 973 | } |
---|
| 974 | else |
---|
| 975 | { gen_num=size(#); |
---|
| 976 | } |
---|
| 977 | } |
---|
| 978 | // ----------------- computing the terms with Brauer lift -------------------- |
---|
| 979 | if (ch<>0 && mol_flag==0) |
---|
| 980 | { list L=group_reynolds(#[1..gen_num],v); |
---|
| 981 | if (L[1]==0) |
---|
| 982 | { if (voice==2) |
---|
| 983 | { "WARNING: The characteristic of the coefficient field divides the group order."; |
---|
| 984 | " Proceed without the Reynolds operator or the Molien series!"; |
---|
| 985 | return(); |
---|
| 986 | } |
---|
| 987 | if (v) |
---|
| 988 | { " The characteristic of the base field divides the group order."; |
---|
| 989 | " We have to continue without Reynolds operator or the Molien series..."; |
---|
| 990 | return(); |
---|
| 991 | } |
---|
| 992 | } |
---|
| 993 | if (minpoly<>0) |
---|
| 994 | { if (voice==2) |
---|
| 995 | { "WARNING: It is impossible for this program to calculate the Molien series"; |
---|
| 996 | " for finite groups over extension fields of prime characteristic."; |
---|
| 997 | return(L[1]); |
---|
| 998 | } |
---|
| 999 | else |
---|
| 1000 | { if (v) |
---|
| 1001 | { " Since it is impossible for this program to calculate the Molien series for"; |
---|
| 1002 | " invariant rings over extension fields of prime characteristic, we have to"; |
---|
| 1003 | " continue without it."; |
---|
| 1004 | return(L[1]); |
---|
| 1005 | } |
---|
| 1006 | } |
---|
| 1007 | } |
---|
| 1008 | if (typeof(L[2])=="int") |
---|
| 1009 | { molien(L[3..size(L)],newring,L[2],intvec(mol_flag,interval,v)); |
---|
| 1010 | } |
---|
| 1011 | else |
---|
| 1012 | { molien(L[2..size(L)],newring,intvec(mol_flag,interval,v)); |
---|
| 1013 | } |
---|
| 1014 | return(L[1]); |
---|
| 1015 | } |
---|
| 1016 | //----------- computing Molien series in the straight forward way ------------ |
---|
| 1017 | if (ch==0) |
---|
| 1018 | { if (typeof(#[1])<>"matrix") |
---|
| 1019 | { "ERROR: the parameters must be a list of matrices and maybe an <intvec>"; |
---|
| 1020 | return(); |
---|
| 1021 | } |
---|
| 1022 | int n=nrows(#[1]); |
---|
| 1023 | if (n<>nvars(br)) |
---|
| 1024 | { "ERROR: the number of variables of the basering needs to be the same"; |
---|
| 1025 | " as the dimension of the matrices"; |
---|
| 1026 | return(); |
---|
| 1027 | } |
---|
| 1028 | if (n<>ncols(#[1])) |
---|
| 1029 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
| 1030 | return(); |
---|
| 1031 | } |
---|
| 1032 | matrix vars=matrix(maxideal(1)); // creating an nx1-matrix containing the |
---|
| 1033 | vars=transpose(vars); // variables of the ring - |
---|
| 1034 | matrix A(1)=#[1]*vars; // calculating the first ring mapping - |
---|
| 1035 | // A(1) will contain the Reynolds |
---|
| 1036 | // operator - |
---|
| 1037 | poly v1=vars[1,1]; // the Molien series will be in terms of |
---|
| 1038 | // the first variable of the current |
---|
| 1039 | // ring |
---|
| 1040 | matrix I=diag(1,n); |
---|
| 1041 | matrix A(2)[1][2]; // A(2) will contain the Molien series - |
---|
| 1042 | A(2)[1,1]=1; // A(2)[1,1] will be the numerator |
---|
| 1043 | matrix G(1)=#[1]; // G(k) are elements of the group - |
---|
| 1044 | A(2)[1,2]=det(I-v1*(G(1))); // A(2)[1,2] will be the denominator - |
---|
| 1045 | matrix s; // will help us canceling in the |
---|
| 1046 | // fraction |
---|
| 1047 | poly p; // will contain the denominator of the |
---|
| 1048 | // new term of the Molien series |
---|
| 1049 | int i=1; |
---|
[18bd9c] | 1050 | for (int j=2;j<=gen_num;j++) // this loop adds the parameters to the |
---|
[f34c37c] | 1051 | { // group, leaving out doubles and |
---|
| 1052 | // checking whether the parameters are |
---|
| 1053 | // compatible with the task of the |
---|
| 1054 | // procedure |
---|
| 1055 | if (not(typeof(#[j])=="matrix")) |
---|
| 1056 | { "ERROR: the parameters must be a list of matrices and maybe an <intvec>"; |
---|
| 1057 | return(); |
---|
| 1058 | } |
---|
| 1059 | if ((n!=nrows(#[j])) or (n!=ncols(#[j]))) |
---|
| 1060 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
| 1061 | return(); |
---|
| 1062 | } |
---|
| 1063 | if (unique(G(1..i),#[j])) |
---|
[18bd9c] | 1064 | { i++; |
---|
[f34c37c] | 1065 | matrix G(i)=#[j]; |
---|
| 1066 | A(1)=concat(A(1),#[j]*vars); // adding ring homomorphisms to A(1) - |
---|
| 1067 | p=det(I-v1*#[j]); // denominator of new term - |
---|
| 1068 | A(2)[1,1]=A(2)[1,1]*p+A(2)[1,2]; // expanding A(2)[1,1]/A(2)[1,2] +1/p |
---|
| 1069 | A(2)[1,2]=A(2)[1,2]*p; |
---|
| 1070 | if (interval<>0) // canceling common terms of denominator |
---|
| 1071 | { if ((i/interval)*interval==i) // and enumerator |
---|
| 1072 | { |
---|
| 1073 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() these |
---|
| 1074 | A(2)[1,1]=-s[2,1]; // three lines should be replaced by the |
---|
| 1075 | A(2)[1,2]=s[1,1]; // following three |
---|
| 1076 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
| 1077 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
| 1078 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
| 1079 | } |
---|
| 1080 | } |
---|
| 1081 | } |
---|
| 1082 | } |
---|
| 1083 | int g=i; // G(1)..G(i) are generators without |
---|
| 1084 | // doubles - g generally is the number |
---|
| 1085 | // of elements in the group so far - |
---|
| 1086 | j=i; // j is the number of new elements that |
---|
| 1087 | // we use as factors |
---|
| 1088 | int k, m, l; |
---|
| 1089 | if (v) |
---|
| 1090 | { ""; |
---|
| 1091 | " Generating the entire matrix group. Whenever a new group element is found,"; |
---|
[425a9d] | 1092 | " the corresponding ring homomorphism of the Reynolds operator and the"; |
---|
[f34c37c] | 1093 | " corresponding term of the Molien series is generated."; |
---|
| 1094 | ""; |
---|
| 1095 | } |
---|
| 1096 | //----------- computing 1/det(I-xE) whenever a new element E is found -------- |
---|
| 1097 | while (1) |
---|
| 1098 | { l=0; // l is the number of products we get in |
---|
| 1099 | // one going |
---|
[9f79aa] | 1100 | for (m=g-j+1;m<=g;m=m+1) |
---|
[18bd9c] | 1101 | { for (k=1;k<=i;k++) |
---|
| 1102 | { l++; |
---|
[f34c37c] | 1103 | matrix P(l)=G(k)*G(m); // possible new element |
---|
| 1104 | } |
---|
| 1105 | } |
---|
| 1106 | j=0; |
---|
[18bd9c] | 1107 | for (k=1;k<=l;k++) |
---|
[f34c37c] | 1108 | { if (unique(G(1..g),P(k))) |
---|
[18bd9c] | 1109 | { j++; // a new factor for next run |
---|
| 1110 | g++; |
---|
[f34c37c] | 1111 | matrix G(g)=P(k); // a new group element - |
---|
| 1112 | A(1)=concat(A(1),P(k)*vars); // adding new mapping to A(1) |
---|
| 1113 | p=det(I-v1*P(k)); // denominator of new term |
---|
| 1114 | A(2)[1,1]=A(2)[1,1]*p+A(2)[1,2]; |
---|
| 1115 | A(2)[1,2]=A(2)[1,2]*p; // expanding A(2)[1,1]/A(2)[1,2] + 1/p - |
---|
| 1116 | if (interval<>0) // canceling common terms of denominator |
---|
| 1117 | { if ((g/interval)*interval==g) // and enumerator |
---|
| 1118 | { |
---|
| 1119 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() |
---|
| 1120 | A(2)[1,1]=-s[2,1]; // these three lines should be replaced |
---|
| 1121 | A(2)[1,2]=s[1,1]; // by the following three |
---|
| 1122 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
| 1123 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
| 1124 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
| 1125 | } |
---|
| 1126 | } |
---|
| 1127 | if (v) |
---|
| 1128 | { " Group element "+string(g)+" has been found."; |
---|
| 1129 | } |
---|
| 1130 | } |
---|
| 1131 | kill P(k); |
---|
| 1132 | } |
---|
| 1133 | if (j==0) // when we didn't add any new elements |
---|
| 1134 | { break; // in one run through the while loop |
---|
| 1135 | } // we are done |
---|
| 1136 | } |
---|
| 1137 | if (v) |
---|
| 1138 | { if (g<=i) |
---|
| 1139 | { " There are only "+string(g)+" group elements."; |
---|
| 1140 | } |
---|
| 1141 | ""; |
---|
| 1142 | } |
---|
[558eb2] | 1143 | A(1)=transpose(A(1)); // when we evaluate the Reynolds operator |
---|
| 1144 | // later on, we actually want 1xn |
---|
| 1145 | // matrices |
---|
| 1146 | if (interval==0) // canceling common terms of denominator |
---|
| 1147 | { // and enumerator - |
---|
| 1148 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() |
---|
| 1149 | A(2)[1,1]=-s[2,1]; // these three lines should be replaced |
---|
| 1150 | A(2)[1,2]=s[1,1]; // by the following three |
---|
[f34c37c] | 1151 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
| 1152 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
| 1153 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
| 1154 | } |
---|
| 1155 | if (interval<>0) // canceling common terms of denominator |
---|
| 1156 | { if ((g/interval)*interval<>g) // and enumerator |
---|
| 1157 | { |
---|
| 1158 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() |
---|
| 1159 | A(2)[1,1]=-s[2,1]; // these three lines should be replaced |
---|
| 1160 | A(2)[1,2]=s[1,1]; // by the following three |
---|
| 1161 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
| 1162 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
| 1163 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
| 1164 | } |
---|
| 1165 | } |
---|
| 1166 | map slead=br,ideal(0); |
---|
| 1167 | s=slead(A(2)); |
---|
[558eb2] | 1168 | A(2)[1,1]=1/s[1,1]*A(2)[1,1]; // numerator and denominator have to have |
---|
| 1169 | A(2)[1,2]=1/s[1,2]*A(2)[1,2]; // a constant term of 1 |
---|
[f34c37c] | 1170 | if (v) |
---|
| 1171 | { " Now we are done calculating Molien series and Reynolds operator."; |
---|
| 1172 | ""; |
---|
| 1173 | } |
---|
| 1174 | return(A(1..2)); |
---|
| 1175 | } |
---|
| 1176 | //------------------------ simulating characteristic 0 ----------------------- |
---|
| 1177 | else // if ch<>0 and mol_flag<>0 |
---|
| 1178 | { if (typeof(#[1])<>"matrix") |
---|
| 1179 | { "ERROR: the parameters must be a list of matrices and maybe an <intvec>"; |
---|
| 1180 | return(); |
---|
| 1181 | } |
---|
| 1182 | int n=nrows(#[1]); |
---|
| 1183 | if (n<>nvars(br)) |
---|
| 1184 | { "ERROR: the number of variables of the basering needs to be the same"; |
---|
| 1185 | " as the dimension of the matrices"; |
---|
| 1186 | return(); |
---|
| 1187 | } |
---|
| 1188 | if (n<>ncols(#[1])) |
---|
| 1189 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
| 1190 | return(); |
---|
| 1191 | } |
---|
| 1192 | matrix vars=matrix(maxideal(1)); // creating an nx1-matrix containing the |
---|
| 1193 | vars=transpose(vars); // variables of the ring - |
---|
| 1194 | matrix A(1)=#[1]*vars; // calculating the first ring mapping - |
---|
| 1195 | // A(1) will contain the Reynolds |
---|
| 1196 | // operator |
---|
| 1197 | string chst=charstr(br); |
---|
[18bd9c] | 1198 | for (int i=1;i<=size(chst);i++) |
---|
[f34c37c] | 1199 | { if (chst[i]==",") |
---|
| 1200 | { break; |
---|
| 1201 | } |
---|
| 1202 | } |
---|
| 1203 | if (minpoly==0) |
---|
| 1204 | { if (i>size(chst)) |
---|
[034ce1] | 1205 | { execute("ring "+newring+"=0,("+varstr(br)+"),("+ordstr(br)+")"); |
---|
[f34c37c] | 1206 | } |
---|
| 1207 | else |
---|
| 1208 | { chst=chst[i..size(chst)]; |
---|
[034ce1] | 1209 | execute |
---|
[3c4dcc] | 1210 | ("ring "+newring+"=(0"+chst+"),("+varstr(br)+"),("+ordstr(br)+")"); |
---|
[f34c37c] | 1211 | } |
---|
| 1212 | } |
---|
| 1213 | else |
---|
| 1214 | { string minp=string(minpoly); |
---|
| 1215 | minp=minp[2..size(minp)-1]; |
---|
| 1216 | chst=chst[i..size(chst)]; |
---|
[034ce1] | 1217 | execute("ring "+newring+"=(0"+chst+"),("+varstr(br)+"),("+ordstr(br)+")"); |
---|
| 1218 | execute("minpoly="+minp); |
---|
[f34c37c] | 1219 | } |
---|
| 1220 | poly v1=var(1); // the Molien series will be in terms of |
---|
| 1221 | // the first variable of the current |
---|
| 1222 | // ring |
---|
| 1223 | matrix I=diag(1,n); |
---|
| 1224 | int o; |
---|
| 1225 | setring br; |
---|
| 1226 | matrix G(1)=#[1]; |
---|
| 1227 | string links, rechts; |
---|
| 1228 | string stM(1)=string(#[1]); |
---|
[18bd9c] | 1229 | for (o=1;o<=size(stM(1));o++) |
---|
[f34c37c] | 1230 | { if (stM(1)[o]==" |
---|
| 1231 | ") |
---|
| 1232 | { links=stM(1)[1..o-1]; |
---|
| 1233 | rechts=stM(1)[o+1..size(stM(1))]; |
---|
| 1234 | stM(1)=links+rechts; |
---|
| 1235 | } |
---|
| 1236 | } |
---|
| 1237 | setring `newring`; |
---|
[034ce1] | 1238 | execute("matrix G(1)["+string(n)+"]["+string(n)+"]="+stM(1)); |
---|
[f34c37c] | 1239 | matrix A(2)[1][2]; // A(2) will contain the Molien series - |
---|
| 1240 | A(2)[1,1]=1; // A(2)[1,1] will be the numerator |
---|
| 1241 | A(2)[1,2]=det(I-v1*(G(1))); // A(2)[1,2] will be the denominator - |
---|
| 1242 | matrix s; // will help us canceling in the |
---|
| 1243 | // fraction |
---|
| 1244 | poly p; // will contain the denominator of the |
---|
| 1245 | // new term of the Molien series |
---|
| 1246 | i=1; |
---|
[9f79aa] | 1247 | for (int j=2;j<=gen_num;j++) // this loop adds the parameters to the |
---|
[f34c37c] | 1248 | { // group, leaving out doubles and |
---|
| 1249 | // checking whether the parameters are |
---|
| 1250 | // compatible with the task of the |
---|
| 1251 | // procedure |
---|
| 1252 | setring br; |
---|
| 1253 | if (not(typeof(#[j])=="matrix")) |
---|
| 1254 | { "ERROR: the parameters must be a list of matrices and maybe an <intvec>"; |
---|
| 1255 | return(); |
---|
| 1256 | } |
---|
| 1257 | if ((n!=nrows(#[j])) or (n!=ncols(#[j]))) |
---|
| 1258 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
| 1259 | return(); |
---|
| 1260 | } |
---|
| 1261 | if (unique(G(1..i),#[j])) |
---|
[18bd9c] | 1262 | { i++; |
---|
[f34c37c] | 1263 | matrix G(i)=#[j]; |
---|
| 1264 | A(1)=concat(A(1),G(i)*vars); // adding ring homomorphisms to A(1) |
---|
| 1265 | string stM(i)=string(G(i)); |
---|
[18bd9c] | 1266 | for (o=1;o<=size(stM(i));o++) |
---|
[f34c37c] | 1267 | { if (stM(i)[o]==" |
---|
| 1268 | ") |
---|
| 1269 | { links=stM(i)[1..o-1]; |
---|
| 1270 | rechts=stM(i)[o+1..size(stM(i))]; |
---|
| 1271 | stM(i)=links+rechts; |
---|
| 1272 | } |
---|
| 1273 | } |
---|
| 1274 | setring `newring`; |
---|
[034ce1] | 1275 | execute("matrix G(i)["+string(n)+"]["+string(n)+"]="+stM(i)); |
---|
[f34c37c] | 1276 | p=det(I-v1*G(i)); // denominator of new term - |
---|
| 1277 | A(2)[1,1]=A(2)[1,1]*p+A(2)[1,2]; // expanding A(2)[1,1]/A(2)[1,2] +1/p |
---|
| 1278 | A(2)[1,2]=A(2)[1,2]*p; |
---|
| 1279 | if (interval<>0) // canceling common terms of denominator |
---|
| 1280 | { if ((i/interval)*interval==i) // and enumerator |
---|
| 1281 | { |
---|
| 1282 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() these |
---|
| 1283 | A(2)[1,1]=-s[2,1]; // three lines should be replaced by the |
---|
| 1284 | A(2)[1,2]=s[1,1]; // following three |
---|
| 1285 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
| 1286 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
| 1287 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
| 1288 | } |
---|
| 1289 | } |
---|
| 1290 | setring br; |
---|
| 1291 | } |
---|
| 1292 | } |
---|
| 1293 | int g=i; // G(1)..G(i) are generators without |
---|
| 1294 | // doubles - g generally is the number |
---|
| 1295 | // of elements in the group so far - |
---|
| 1296 | j=i; // j is the number of new elements that |
---|
| 1297 | // we use as factors |
---|
| 1298 | int k, m, l; |
---|
| 1299 | if (v) |
---|
| 1300 | { ""; |
---|
| 1301 | " Generating the entire matrix group. Whenever a new group element is found,"; |
---|
[425a9d] | 1302 | " the corresponding ring homomorphism of the Reynolds operator and the"; |
---|
[f34c37c] | 1303 | " corresponding term of the Molien series is generated."; |
---|
| 1304 | ""; |
---|
| 1305 | } |
---|
| 1306 | // taking all elements in a ring of characteristic 0 and computing the terms |
---|
| 1307 | // of the Molien series there |
---|
| 1308 | while (1) |
---|
| 1309 | { l=0; // l is the number of products we get in |
---|
| 1310 | // one going |
---|
[18bd9c] | 1311 | for (m=g-j+1;m<=g;m++) |
---|
| 1312 | { for (k=1;k<=i;k++) |
---|
| 1313 | { l++; |
---|
[f34c37c] | 1314 | matrix P(l)=G(k)*G(m); // possible new element |
---|
| 1315 | } |
---|
| 1316 | } |
---|
| 1317 | j=0; |
---|
[18bd9c] | 1318 | for (k=1;k<=l;k++) |
---|
[f34c37c] | 1319 | { if (unique(G(1..g),P(k))) |
---|
[18bd9c] | 1320 | { j++; // a new factor for next run |
---|
| 1321 | g++; |
---|
[f34c37c] | 1322 | matrix G(g)=P(k); // a new group element - |
---|
| 1323 | A(1)=concat(A(1),G(g)*vars); // adding new mapping to A(1) |
---|
| 1324 | string stM(g)=string(G(g)); |
---|
[18bd9c] | 1325 | for (o=1;o<=size(stM(g));o++) |
---|
[f34c37c] | 1326 | { if (stM(g)[o]==" |
---|
| 1327 | ") |
---|
| 1328 | { links=stM(g)[1..o-1]; |
---|
| 1329 | rechts=stM(g)[o+1..size(stM(g))]; |
---|
| 1330 | stM(g)=links+rechts; |
---|
| 1331 | } |
---|
| 1332 | } |
---|
| 1333 | setring `newring`; |
---|
[034ce1] | 1334 | execute("matrix G(g)["+string(n)+"]["+string(n)+"]="+stM(g)); |
---|
[f34c37c] | 1335 | p=det(I-v1*G(g)); // denominator of new term |
---|
| 1336 | A(2)[1,1]=A(2)[1,1]*p+A(2)[1,2]; |
---|
| 1337 | A(2)[1,2]=A(2)[1,2]*p; // expanding A(2)[1,1]/A(2)[1,2] + 1/p - |
---|
| 1338 | if (interval<>0) // canceling common terms of denominator |
---|
| 1339 | { if ((g/interval)*interval==g) // and enumerator |
---|
| 1340 | { |
---|
| 1341 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() |
---|
| 1342 | A(2)[1,1]=-s[2,1]; // these three lines should be replaced |
---|
| 1343 | A(2)[1,2]=s[1,1]; // by the following three |
---|
| 1344 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
| 1345 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
| 1346 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
| 1347 | } |
---|
| 1348 | } |
---|
| 1349 | if (v) |
---|
| 1350 | { " Group element "+string(g)+" has been found."; |
---|
| 1351 | } |
---|
| 1352 | setring br; |
---|
| 1353 | } |
---|
| 1354 | kill P(k); |
---|
| 1355 | } |
---|
| 1356 | if (j==0) // when we didn't add any new elements |
---|
| 1357 | { break; // in one run through the while loop |
---|
| 1358 | } // we are done |
---|
| 1359 | } |
---|
| 1360 | if (v) |
---|
| 1361 | { if (g<=i) |
---|
| 1362 | { " There are only "+string(g)+" group elements."; |
---|
| 1363 | } |
---|
| 1364 | ""; |
---|
| 1365 | } |
---|
| 1366 | A(1)=transpose(A(1)); // when we evaluate the Reynolds operator |
---|
| 1367 | // later on, we actually want 1xn |
---|
| 1368 | // matrices |
---|
| 1369 | setring `newring`; |
---|
| 1370 | if (interval==0) // canceling common terms of denominator |
---|
| 1371 | { // and enumerator - |
---|
| 1372 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() |
---|
| 1373 | A(2)[1,1]=-s[2,1]; // these three lines should be replaced |
---|
| 1374 | A(2)[1,2]=s[1,1]; // by the following three |
---|
| 1375 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
| 1376 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
| 1377 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
| 1378 | } |
---|
| 1379 | if (interval<>0) // canceling common terms of denominator |
---|
| 1380 | { if ((g/interval)*interval<>g) // and enumerator |
---|
| 1381 | { |
---|
| 1382 | s=matrix(syz(ideal(A(2)))); // once gcd() is faster than syz() |
---|
| 1383 | A(2)[1,1]=-s[2,1]; // these three lines should be replaced |
---|
| 1384 | A(2)[1,2]=s[1,1]; // by the following three |
---|
| 1385 | // p=gcd(A(2)[1,1],A(2)[1,2]); |
---|
| 1386 | // A(2)[1,1]=A(2)[1,1]/p; |
---|
| 1387 | // A(2)[1,2]=A(2)[1,2]/p; |
---|
| 1388 | } |
---|
| 1389 | } |
---|
| 1390 | map slead=`newring`,ideal(0); |
---|
| 1391 | s=slead(A(2)); |
---|
[558eb2] | 1392 | A(2)[1,1]=1/s[1,1]*A(2)[1,1]; // numerator and denominator have to have |
---|
| 1393 | A(2)[1,2]=1/s[1,2]*A(2)[1,2]; // a constant term of 1 |
---|
[f34c37c] | 1394 | if (v) |
---|
| 1395 | { " Now we are done calculating Molien series and Reynolds operator."; |
---|
| 1396 | ""; |
---|
| 1397 | } |
---|
| 1398 | matrix M=A(2); |
---|
| 1399 | kill G(1..g), s, slead, p, v1, I, A(2); |
---|
[558eb2] | 1400 | export `newring`; // we keep the ring where we computed the |
---|
| 1401 | export M; // the Molien series such that we can |
---|
| 1402 | setring br; // keep it |
---|
[f34c37c] | 1403 | return(A(1)); |
---|
| 1404 | } |
---|
| 1405 | } |
---|
| 1406 | example |
---|
| 1407 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
[68e678] | 1408 | " note the case of prime characteristic"; echo=2; |
---|
[f34c37c] | 1409 | ring R=0,(x,y,z),dp; |
---|
| 1410 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 1411 | matrix REY,M=reynolds_molien(A); |
---|
| 1412 | print(REY); |
---|
| 1413 | print(M); |
---|
| 1414 | ring S=3,(x,y,z),dp; |
---|
| 1415 | string newring="Qadjoint"; |
---|
| 1416 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 1417 | matrix REY=reynolds_molien(A,newring); |
---|
| 1418 | print(REY); |
---|
| 1419 | setring Qadjoint; |
---|
| 1420 | print(M); |
---|
| 1421 | setring S; |
---|
| 1422 | kill Qadjoint; |
---|
| 1423 | } |
---|
[68e678] | 1424 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 1425 | |
---|
| 1426 | proc partial_molien (matrix M, int n, list #) |
---|
| 1427 | "USAGE: partial_molien(M,n[,p]); |
---|
| 1428 | M: a 1x2 <matrix>, n: an <int> indicating number of terms in the |
---|
| 1429 | expansion, p: an optional <poly> |
---|
| 1430 | ASSUME: M is the return value of molien or the second return value of |
---|
| 1431 | reynolds_molien, p ought to be the second return value of a previous |
---|
| 1432 | run of partial_molien and avoids recalculating known terms |
---|
| 1433 | RETURN: n terms (type <poly>) of the partial expansion of the Molien series |
---|
| 1434 | (first n if there is no third parameter given, otherwise the next n |
---|
| 1435 | terms depending on a previous calculation) and an intermediate result |
---|
[18bd9c] | 1436 | (type <poly>) of the calculation to be used as third parameter in a |
---|
[68e678] | 1437 | next run of partial_molien |
---|
[f34c37c] | 1438 | THEORY: The following calculation is implemented: |
---|
[51d95b] | 1439 | @format |
---|
| 1440 | (1+a1x+a2x^2+...+anx^n)/(1+b1x+b2x^2+...+bmx^m)=(1+(a1-b1)x+... |
---|
| 1441 | (1+b1x+b2x^2+...+bmx^m) |
---|
| 1442 | ----------------------- |
---|
| 1443 | (a1-b1)x+(a2-b2)x^2+... |
---|
| 1444 | (a1-b1)x+b1(a1-b1)x^2+... |
---|
| 1445 | @end format |
---|
[f34c37c] | 1446 | EXAMPLE: example partial_molien; shows an example |
---|
| 1447 | " |
---|
| 1448 | { poly A(2); // A(2) will contain the return value of |
---|
| 1449 | // the intermediate result |
---|
| 1450 | if (char(basering)<>0) |
---|
| 1451 | { "ERROR: you have to change to a basering of characteristic 0, one in"; |
---|
| 1452 | " which the Molien series is defined"; |
---|
| 1453 | } |
---|
| 1454 | if (ncols(M)==2 && nrows(M)==1 && n>0 && size(#)<2) |
---|
| 1455 | { def br=basering; // keeping track of the old ring |
---|
| 1456 | map slead=br,ideal(0); |
---|
| 1457 | matrix s=slead(M); |
---|
| 1458 | if (s[1,1]<>1 || s[1,2]<>1) |
---|
| 1459 | { "ERROR: the constant terms of enumerator and denominator are not 1"; |
---|
| 1460 | return(); |
---|
| 1461 | } |
---|
| 1462 | |
---|
| 1463 | if (size(#)==0) |
---|
[558eb2] | 1464 | { A(2)=M[1,1]; // if a third parameter is not given, the |
---|
| 1465 | // intermediate result from the last run |
---|
| 1466 | // corresponds to the numerator - we need |
---|
| 1467 | } // its smallest term |
---|
[f34c37c] | 1468 | else |
---|
| 1469 | { if (typeof(#[1])=="poly") |
---|
[558eb2] | 1470 | { A(2)=#[1]; // if a third term is given we 'start' |
---|
| 1471 | } // with its smallest term |
---|
[f34c37c] | 1472 | else |
---|
| 1473 | { "ERROR: <poly> as third parameter expected"; |
---|
| 1474 | return(); |
---|
| 1475 | } |
---|
| 1476 | } |
---|
[558eb2] | 1477 | poly A(1)=M[1,2]; // denominator of Molien series (for now) |
---|
[f34c37c] | 1478 | string mp=string(minpoly); |
---|
[034ce1] | 1479 | execute("ring R=("+charstr(br)+"),("+varstr(br)+"),ds;"); |
---|
| 1480 | execute("minpoly=number("+mp+");"); |
---|
[558eb2] | 1481 | poly A(1)=0; // A(1) will contain the sum of n terms - |
---|
| 1482 | poly min; // min will be our smallest term - |
---|
| 1483 | poly A(2)=fetch(br,A(2)); // fetching A(2) and M[1,2] into R |
---|
[f34c37c] | 1484 | poly den=fetch(br,A(1)); |
---|
[f875fae] | 1485 | for (int i=1; i<=n; i++) // getting n terms and adding them up |
---|
[f34c37c] | 1486 | { min=lead(A(2)); |
---|
| 1487 | A(1)=A(1)+min; |
---|
| 1488 | A(2)=A(2)-min*den; |
---|
| 1489 | } |
---|
[558eb2] | 1490 | setring br; // moving A(1) and A(2) back in the |
---|
| 1491 | A(1)=fetch(R,A(1)); // actual ring for output |
---|
[f34c37c] | 1492 | A(2)=fetch(R,A(2)); |
---|
| 1493 | return(A(1..2)); |
---|
| 1494 | } |
---|
| 1495 | else |
---|
| 1496 | { "ERROR: the first parameter has to be a 1x2-matrix, i.e. the matrix"; |
---|
| 1497 | " returned by the procedure 'reynolds_molien', the second one"; |
---|
| 1498 | " should be > 0 and there should be no more than 3 parameters;" |
---|
| 1499 | return(); |
---|
| 1500 | } |
---|
| 1501 | } |
---|
| 1502 | example |
---|
[68e678] | 1503 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
[f34c37c] | 1504 | ring R=0,(x,y,z),dp; |
---|
| 1505 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 1506 | matrix REY,M=reynolds_molien(A); |
---|
| 1507 | poly p(1..2); |
---|
| 1508 | p(1..2)=partial_molien(M,5); |
---|
| 1509 | p(1); |
---|
| 1510 | p(1..2)=partial_molien(M,5,p(2)); |
---|
| 1511 | p(1); |
---|
| 1512 | } |
---|
[558eb2] | 1513 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 1514 | |
---|
| 1515 | proc evaluate_reynolds (matrix REY, ideal I) |
---|
| 1516 | "USAGE: evaluate_reynolds(REY,I); |
---|
| 1517 | REY: a <matrix> representing the Reynolds operator, I: an arbitrary |
---|
| 1518 | <ideal> |
---|
| 1519 | ASSUME: REY is the first return value of group_reynolds() or reynolds_molien() |
---|
| 1520 | RETURNS: image of the polynomials defining I under the Reynolds operator |
---|
| 1521 | (type <ideal>) |
---|
| 1522 | NOTE: the characteristic of the coefficient field of the polynomial ring |
---|
| 1523 | should not divide the order of the finite matrix group |
---|
| 1524 | THEORY: REY has been constructed in such a way that each row serves as a ring |
---|
| 1525 | mapping of which the Reynolds operator is made up. |
---|
[68e678] | 1526 | EXAMPLE: example evaluate_reynolds; shows an example |
---|
[f34c37c] | 1527 | " |
---|
| 1528 | { def br=basering; |
---|
| 1529 | int n=nvars(br); |
---|
| 1530 | if (ncols(REY)==n) |
---|
| 1531 | { int m=nrows(REY); // we need m to 'cut' the ring |
---|
| 1532 | // homomorphisms 'out' of REY and to |
---|
| 1533 | // divide by the group order in the end |
---|
| 1534 | int num_poly=ncols(I); |
---|
| 1535 | matrix MI=matrix(I); |
---|
| 1536 | matrix MiI[1][num_poly]; |
---|
| 1537 | map pREY; |
---|
| 1538 | matrix rowREY[1][n]; |
---|
[18bd9c] | 1539 | for (int i=1;i<=m;i++) |
---|
[f34c37c] | 1540 | { rowREY=REY[i,1..n]; |
---|
| 1541 | pREY=br,ideal(rowREY); // f is now the i-th ring homomorphism |
---|
| 1542 | MiI=pREY(MI)+MiI; |
---|
| 1543 | } |
---|
| 1544 | MiI=(1/number(m))*MiI; |
---|
| 1545 | return(ideal(MiI)); |
---|
| 1546 | } |
---|
| 1547 | else |
---|
| 1548 | { "ERROR: the number of columns in the <matrix> should be the same as the"; |
---|
| 1549 | " number of variables in the basering; in fact it should be first"; |
---|
| 1550 | " return value of group_reynolds() or reynolds_molien()."; |
---|
| 1551 | return(); |
---|
| 1552 | } |
---|
| 1553 | } |
---|
| 1554 | example |
---|
[68e678] | 1555 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
[f34c37c] | 1556 | ring R=0,(x,y,z),dp; |
---|
| 1557 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 1558 | list L=group_reynolds(A); |
---|
| 1559 | ideal I=x2,y2,z2; |
---|
| 1560 | print(evaluate_reynolds(L[1],I)); |
---|
| 1561 | } |
---|
[558eb2] | 1562 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 1563 | |
---|
| 1564 | proc invariant_basis (int g,list #) |
---|
| 1565 | "USAGE: invariant_basis(g,G1,G2,...); |
---|
| 1566 | g: an <int> indicating of which degree (>0) the homogeneous basis |
---|
| 1567 | shoud be, G1,G2,...: <matrices> generating a finite matrix group |
---|
| 1568 | RETURNS: the basis (type <ideal>) of the space of invariants of degree g |
---|
[18bd9c] | 1569 | THEORY: A general polynomial of degree g is generated and the generators of |
---|
[68e678] | 1570 | the matrix group applied. The difference ought to be 0 and this way a |
---|
[f34c37c] | 1571 | system of linear equations is created. It is solved by computing |
---|
| 1572 | syzygies. |
---|
[68e678] | 1573 | EXAMPLE: example invariant_basis; shows an example |
---|
[f34c37c] | 1574 | " |
---|
| 1575 | { if (g<=0) |
---|
| 1576 | { "ERROR: the first parameter should be > 0"; |
---|
| 1577 | return(); |
---|
| 1578 | } |
---|
| 1579 | def br=basering; |
---|
| 1580 | ideal mon=sort(maxideal(g))[1]; // needed for constructing a general |
---|
| 1581 | int m=ncols(mon); // homogeneous polynomial of degree g |
---|
| 1582 | mon=sort(mon,intvec(m..1))[1]; |
---|
| 1583 | int a=size(#); |
---|
| 1584 | int i; |
---|
| 1585 | int n=nvars(br); |
---|
| 1586 | //---------------------- checking that the input is ok ----------------------- |
---|
[18bd9c] | 1587 | for (i=1;i<=a;i++) |
---|
[f34c37c] | 1588 | { if (typeof(#[i])=="matrix") |
---|
| 1589 | { if (nrows(#[i])==n && ncols(#[i])==n) |
---|
| 1590 | { matrix G(i)=#[i]; |
---|
| 1591 | } |
---|
| 1592 | else |
---|
| 1593 | { "ERROR: the number of variables of the base ring needs to be the same"; |
---|
| 1594 | " as the dimension of the square matrices"; |
---|
| 1595 | return(); |
---|
| 1596 | } |
---|
| 1597 | } |
---|
| 1598 | else |
---|
| 1599 | { "ERROR: the last parameters should be a list of matrices"; |
---|
| 1600 | return(); |
---|
| 1601 | } |
---|
| 1602 | } |
---|
| 1603 | //---------------------------------------------------------------------------- |
---|
[034ce1] | 1604 | execute("ring T=("+charstr(br)+"),("+varstr(br)+",p(1..m)),lp;"); |
---|
[f34c37c] | 1605 | // p(1..m) are the general coefficients of the general polynomial of degree g |
---|
[034ce1] | 1606 | execute("ideal vars="+varstr(br)+";"); |
---|
[f34c37c] | 1607 | map f; |
---|
| 1608 | ideal mon=imap(br,mon); |
---|
| 1609 | poly P=0; |
---|
[9f79aa] | 1610 | for (i=m;i>=1;i--) |
---|
[f34c37c] | 1611 | { P=P+p(i)*mon[i]; // P is the general polynomial |
---|
| 1612 | } |
---|
| 1613 | ideal I; // will help substituting variables in P |
---|
| 1614 | // by linear combinations of variables - |
---|
| 1615 | poly Pnew,temp; // Pnew is P with substitutions - |
---|
| 1616 | matrix S[m*a][m]; // will contain system of linear |
---|
| 1617 | // equations |
---|
| 1618 | int j,k; |
---|
| 1619 | //------------------- building the system of linear equations ---------------- |
---|
[18bd9c] | 1620 | for (i=1;i<=a;i++) |
---|
[f34c37c] | 1621 | { I=ideal(matrix(vars)*transpose(imap(br,G(i)))); |
---|
| 1622 | I=I,p(1..m); |
---|
| 1623 | f=T,I; |
---|
| 1624 | Pnew=f(P); |
---|
[18bd9c] | 1625 | for (j=1;j<=m;j++) |
---|
[f34c37c] | 1626 | { temp=P/mon[j]-Pnew/mon[j]; |
---|
[18bd9c] | 1627 | for (k=1;k<=m;k++) |
---|
[f34c37c] | 1628 | { S[m*(i-1)+j,k]=temp/p(k); |
---|
| 1629 | } |
---|
| 1630 | } |
---|
| 1631 | } |
---|
| 1632 | //---------------------------------------------------------------------------- |
---|
| 1633 | setring br; |
---|
| 1634 | map f=T,ideal(0); |
---|
| 1635 | matrix S=f(S); |
---|
| 1636 | matrix s=matrix(syz(S)); // s contains a basis of the space of |
---|
| 1637 | // solutions - |
---|
| 1638 | ideal I=ideal(matrix(mon)*s); // I contains a basis of homogeneous |
---|
| 1639 | if (I[1]<>0) // invariants of degree d |
---|
[18bd9c] | 1640 | { for (i=1;i<=ncols(I);i++) |
---|
[f34c37c] | 1641 | { I[i]=I[i]/leadcoef(I[i]); // setting leading coefficients to 1 |
---|
| 1642 | } |
---|
| 1643 | } |
---|
| 1644 | return(I); |
---|
| 1645 | } |
---|
| 1646 | example |
---|
[68e678] | 1647 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
[f34c37c] | 1648 | ring R=0,(x,y,z),dp; |
---|
| 1649 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 1650 | print(invariant_basis(2,A)); |
---|
| 1651 | } |
---|
[68e678] | 1652 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 1653 | |
---|
| 1654 | proc invariant_basis_reynolds (matrix REY,int d,list #) |
---|
| 1655 | "USAGE: invariant_basis_reynolds(REY,d[,flags]); |
---|
| 1656 | REY: a <matrix> representing the Reynolds operator, d: an <int> |
---|
| 1657 | indicating of which degree (>0) the homogeneous basis shoud be, flags: |
---|
| 1658 | an optional <intvec> with two entries: its first component gives the |
---|
| 1659 | dimension of the space (default <0 meaning unknown) and its second |
---|
| 1660 | component is used as the number of polynomials that should be mapped |
---|
| 1661 | to invariants during one call of evaluate_reynolds if the dimension of |
---|
| 1662 | the space is unknown or the number such that number x dimension |
---|
| 1663 | polynomials are mapped to invariants during one call of |
---|
| 1664 | evaluate_reynolds |
---|
| 1665 | ASSUME: REY is the first return value of group_reynolds() or reynolds_molien() |
---|
| 1666 | and flags[1] given by partial_molien |
---|
| 1667 | RETURN: the basis (type <ideal>) of the space of invariants of degree d |
---|
| 1668 | THEORY: Monomials of degree d are mapped to invariants with the Reynolds |
---|
| 1669 | operator. A linearly independent set is generated with the help of |
---|
| 1670 | minbase. |
---|
[68e678] | 1671 | EXAMPLE: example invariant_basis_reynolds; shows an example |
---|
[f34c37c] | 1672 | " |
---|
| 1673 | { |
---|
| 1674 | //---------------------- checking that the input is ok ----------------------- |
---|
| 1675 | if (d<=0) |
---|
| 1676 | { " ERROR: the second parameter should be > 0"; |
---|
| 1677 | return(); |
---|
| 1678 | } |
---|
| 1679 | if (size(#)>1) |
---|
| 1680 | { " ERROR: there should be at most three parameters"; |
---|
| 1681 | return(); |
---|
| 1682 | } |
---|
| 1683 | if (size(#)==1) |
---|
| 1684 | { if (typeof(#[1])<>"intvec") |
---|
| 1685 | { " ERROR: the third parameter should be of type <intvec>"; |
---|
| 1686 | return(); |
---|
| 1687 | } |
---|
| 1688 | if (size(#[1])<>2) |
---|
| 1689 | { " ERROR: there should be two components in <intvec>"; |
---|
| 1690 | return(); |
---|
| 1691 | } |
---|
| 1692 | else |
---|
| 1693 | { int cd=#[1][1]; |
---|
| 1694 | int step_fac=#[1][2]; |
---|
| 1695 | } |
---|
| 1696 | if (step_fac<=0) |
---|
| 1697 | { " ERROR: the second component of <intvec> should be > 0"; |
---|
| 1698 | return(); |
---|
| 1699 | } |
---|
| 1700 | if (cd==0) |
---|
| 1701 | { return(ideal(0)); |
---|
| 1702 | } |
---|
| 1703 | } |
---|
| 1704 | else |
---|
| 1705 | { int step_fac=1; |
---|
| 1706 | int cd=-1; |
---|
| 1707 | } |
---|
| 1708 | if (ncols(REY)<>nvars(basering)) |
---|
| 1709 | { "ERROR: the number of columns in the <matrix> should be the same as the"; |
---|
| 1710 | " number of variables in the basering; in fact it should be first"; |
---|
| 1711 | " return value of group_reynolds() or reynolds_molien()."; |
---|
| 1712 | return(); |
---|
| 1713 | } |
---|
| 1714 | //---------------------------------------------------------------------------- |
---|
| 1715 | ideal mon=sort(maxideal(d))[1]; |
---|
[68e678] | 1716 | int DEGB = degBound; |
---|
[f34c37c] | 1717 | degBound=d; |
---|
| 1718 | int j=ncols(mon); |
---|
| 1719 | mon=sort(mon,intvec(j..1))[1]; |
---|
| 1720 | ideal B; // will contain the basis |
---|
| 1721 | if (cd<0) |
---|
| 1722 | { if (step_fac>j) // all of mon will be mapped to |
---|
| 1723 | { B=evaluate_reynolds(REY,mon); // invariants at once |
---|
| 1724 | B=minbase(B); |
---|
[68e678] | 1725 | degBound=DEGB; |
---|
[f34c37c] | 1726 | return(B); |
---|
| 1727 | } |
---|
| 1728 | } |
---|
| 1729 | else |
---|
| 1730 | { if (step_fac*cd>j) // all of mon will be mapped to |
---|
| 1731 | { B=evaluate_reynolds(REY,mon); // invariants at once |
---|
| 1732 | B=minbase(B); |
---|
[68e678] | 1733 | degBound=DEGB; |
---|
[f34c37c] | 1734 | return(B); |
---|
| 1735 | } |
---|
| 1736 | } |
---|
| 1737 | int i,k; |
---|
| 1738 | int upper_bound=0; |
---|
| 1739 | int lower_bound=0; |
---|
| 1740 | ideal part_mon; // a part of mon of size step_fac*cd |
---|
| 1741 | while (1) |
---|
| 1742 | { lower_bound=upper_bound+1; |
---|
| 1743 | if (cd<0) |
---|
| 1744 | { upper_bound=upper_bound+step_fac; |
---|
| 1745 | } |
---|
| 1746 | else |
---|
| 1747 | { upper_bound=upper_bound+step_fac*cd; |
---|
| 1748 | } |
---|
| 1749 | if (upper_bound>j) |
---|
| 1750 | { upper_bound=j; |
---|
| 1751 | } |
---|
| 1752 | part_mon=mon[lower_bound..upper_bound]; |
---|
| 1753 | B=minbase(B+evaluate_reynolds(REY,part_mon)); |
---|
| 1754 | if ((ncols(B)==cd and B[1]<>0) or upper_bound==j) |
---|
[68e678] | 1755 | { degBound=DEGB; |
---|
[f34c37c] | 1756 | return(B); |
---|
| 1757 | } |
---|
| 1758 | } |
---|
| 1759 | } |
---|
| 1760 | example |
---|
[68e678] | 1761 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
[9f79aa] | 1762 | ring R=0,(x,y,z),dp; |
---|
| 1763 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 1764 | intvec flags=0,1,0; |
---|
| 1765 | matrix REY,M=reynolds_molien(A,flags); |
---|
| 1766 | flags=8,6; |
---|
| 1767 | print(invariant_basis_reynolds(REY,6,flags)); |
---|
[f34c37c] | 1768 | } |
---|
| 1769 | |
---|
[68e678] | 1770 | /////////////////////////////////////////////////////////////////////////////// |
---|
[9f79aa] | 1771 | // This procedure generates linearly independent invariant polynomials of |
---|
| 1772 | // degree d that do not reduce to 0 modulo the primary invariants. It does this |
---|
| 1773 | // by applying the Reynolds operator to the monomials returned by kbase(sP,d). |
---|
| 1774 | // The result is used when computing secondary invariants. |
---|
[68e678] | 1775 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 1776 | proc sort_of_invariant_basis (ideal sP,matrix REY,int d,int step_fac) |
---|
| 1777 | { ideal mon=kbase(sP,d); |
---|
[9f79aa] | 1778 | int DEGB=degBound; |
---|
[f34c37c] | 1779 | degBound=d; |
---|
| 1780 | int j=ncols(mon); |
---|
| 1781 | int i; |
---|
| 1782 | mon=sort(mon,intvec(j..1))[1]; |
---|
| 1783 | ideal B; // will contain the "sort of basis" |
---|
| 1784 | if (step_fac>j) |
---|
| 1785 | { B=compress(evaluate_reynolds(REY,mon)); |
---|
[18bd9c] | 1786 | for (i=1;i<=ncols(B);i++) // those are taken our that are o mod sP |
---|
[f34c37c] | 1787 | { if (reduce(B[i],sP)==0) |
---|
| 1788 | { B[i]=0; |
---|
| 1789 | } |
---|
| 1790 | } |
---|
[558eb2] | 1791 | B=minbase(B); // here are the linearly independent ones |
---|
[68e678] | 1792 | degBound=DEGB; |
---|
[f34c37c] | 1793 | return(B); |
---|
| 1794 | } |
---|
| 1795 | int upper_bound=0; |
---|
| 1796 | int lower_bound=0; |
---|
| 1797 | ideal part_mon; // parts of mon |
---|
| 1798 | while (1) |
---|
| 1799 | { lower_bound=upper_bound+1; |
---|
| 1800 | upper_bound=upper_bound+step_fac; |
---|
| 1801 | if (upper_bound>j) |
---|
| 1802 | { upper_bound=j; |
---|
| 1803 | } |
---|
| 1804 | part_mon=mon[lower_bound..upper_bound]; |
---|
| 1805 | part_mon=compress(evaluate_reynolds(REY,part_mon)); |
---|
[18bd9c] | 1806 | for (i=1;i<=ncols(part_mon);i++) |
---|
[f34c37c] | 1807 | { if (reduce(part_mon[i],sP)==0) |
---|
| 1808 | { part_mon[i]=0; |
---|
| 1809 | } |
---|
| 1810 | } |
---|
[558eb2] | 1811 | B=minbase(B+part_mon); // here are the linearly independent ones |
---|
[f34c37c] | 1812 | if (upper_bound==j) |
---|
[68e678] | 1813 | { degBound=DEGB; |
---|
[f34c37c] | 1814 | return(B); |
---|
| 1815 | } |
---|
| 1816 | } |
---|
| 1817 | } |
---|
| 1818 | |
---|
[68e678] | 1819 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 1820 | // Procedure returning the succeeding vector after vec. It is used to list |
---|
| 1821 | // all the vectors of Z^n with first nonzero entry 1. They are listed by |
---|
| 1822 | // increasing sum of the absolute value of their entries. |
---|
[68e678] | 1823 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 1824 | proc next_vector(intmat vec) |
---|
| 1825 | { int n=ncols(vec); // p: >0, n: <0, p0: >=0, n0: <=0 |
---|
[18bd9c] | 1826 | for (int i=1;i<=n;i++) // finding out which is the first |
---|
[f34c37c] | 1827 | { if (vec[1,i]<>0) // component <>0 |
---|
| 1828 | { break; |
---|
| 1829 | } |
---|
| 1830 | } |
---|
| 1831 | intmat new[1][n]; |
---|
| 1832 | if (i>n) // 0,...,0 --> 1,0....,0 |
---|
| 1833 | { new[1,1]=1; |
---|
| 1834 | return(new); |
---|
| 1835 | } |
---|
| 1836 | if (i==n) // 0,...,1 --> 1,1,0,...,0 |
---|
| 1837 | { new[1,1..2]=1,1; |
---|
| 1838 | return(new); |
---|
| 1839 | } |
---|
| 1840 | if (i==n-1) |
---|
| 1841 | { if (vec[1,n]==0) // 0,...,0,1,0 --> 0,...,0,1 |
---|
| 1842 | { new[1,n]=1; |
---|
| 1843 | return(new); |
---|
| 1844 | } |
---|
| 1845 | if (vec[1,n]>0) // 0,..,0,1,p --> 0,...,0,1,-p |
---|
| 1846 | { new[1,1..n]=vec[1,1..n-1],-vec[1,n]; |
---|
| 1847 | return(new); |
---|
| 1848 | } |
---|
| 1849 | new[1,1..2]=1,1-vec[1,n]; // 0,..,0,1,n --> 1,1-n,0,..,0 |
---|
| 1850 | return(new); |
---|
| 1851 | } |
---|
| 1852 | if (i>1) |
---|
| 1853 | { intmat temp[1][n-i+1]=vec[1,i..n]; // 0,...,0,1,*,...,* --> 1,*,...,* |
---|
| 1854 | temp=next_vector(temp); |
---|
| 1855 | new[1,i..n]=temp[1,1..n-i+1]; |
---|
| 1856 | return(new); |
---|
| 1857 | } // case left: 1,*,...,* |
---|
[18bd9c] | 1858 | for (i=2;i<=n;i++) |
---|
[f34c37c] | 1859 | { if (vec[1,i]>0) // make first positive negative and |
---|
| 1860 | { vec[1,i]=-vec[1,i]; // return |
---|
| 1861 | return(vec); |
---|
| 1862 | } |
---|
| 1863 | else |
---|
[558eb2] | 1864 | { vec[1,i]=-vec[1,i]; // make all negatives before positives |
---|
| 1865 | } // positive |
---|
[f34c37c] | 1866 | } |
---|
[558eb2] | 1867 | for (i=2;i<=n-1;i++) // case: 1,p,...,p after 1,n,...,n |
---|
[f34c37c] | 1868 | { if (vec[1,i]>0) |
---|
[558eb2] | 1869 | { vec[1,2]=vec[1,i]-1; // shuffleing things around... |
---|
| 1870 | if (i>2) // same sum of absolute values of entries |
---|
[f34c37c] | 1871 | { vec[1,i]=0; |
---|
| 1872 | } |
---|
| 1873 | vec[1,i+1]=vec[1,i+1]+1; |
---|
| 1874 | return(vec); |
---|
| 1875 | } |
---|
| 1876 | } // case left: 1,0,...,0 --> 1,1,0,...,0 |
---|
| 1877 | new[1,2..3]=1,vec[1,n]; // and: 1,0,...,0,1 --> 0,1,1,0,...,0 |
---|
| 1878 | return(new); |
---|
| 1879 | } |
---|
| 1880 | |
---|
[68e678] | 1881 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 1882 | // Maps integers to elements of the base field. It is only called if the base |
---|
[18bd9c] | 1883 | // field is of prime characteristic. If the base field has q elements |
---|
[68e678] | 1884 | // (depending on minpoly) 1..q is mapped to those q elements. |
---|
| 1885 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 1886 | proc int_number_map (int i) |
---|
| 1887 | { int p=char(basering); |
---|
| 1888 | if (minpoly==0) // if no minpoly is given, we have p |
---|
| 1889 | { i=i%p; // elements in the field |
---|
| 1890 | return(number(i)); |
---|
| 1891 | } |
---|
| 1892 | int d=pardeg(minpoly); |
---|
| 1893 | if (i<0) |
---|
| 1894 | { int bool=1; |
---|
| 1895 | i=(-1)*i; |
---|
| 1896 | } |
---|
[558eb2] | 1897 | i=i%p^d; // base field has p^d elements - |
---|
| 1898 | number a=par(1); // a is the root of the minpoly - we have |
---|
| 1899 | number out=0; // to construct a linear combination of |
---|
| 1900 | int j=1; // a^k |
---|
[f34c37c] | 1901 | int k; |
---|
| 1902 | while (1) |
---|
| 1903 | { if (i<p^j) // finding an upper bound on i |
---|
[18bd9c] | 1904 | { for (k=0;k<j-1;k++) |
---|
[f34c37c] | 1905 | { out=out+((i/p^k)%p)*a^k; // finding how often p^k is contained in |
---|
| 1906 | } // i |
---|
| 1907 | out=out+(i/p^(j-1))*a^(j-1); |
---|
| 1908 | if (defined(bool)==voice) |
---|
| 1909 | { return((-1)*out); |
---|
| 1910 | } |
---|
| 1911 | return(out); |
---|
| 1912 | } |
---|
[18bd9c] | 1913 | j++; |
---|
[f34c37c] | 1914 | } |
---|
| 1915 | } |
---|
| 1916 | |
---|
[68e678] | 1917 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 1918 | // This procedure finds dif primary invariants in degree d. It returns all |
---|
| 1919 | // primary invariants found so far. The coefficients lie in a field of |
---|
| 1920 | // characteristic 0. |
---|
[68e678] | 1921 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 1922 | proc search (int n,int d,ideal B,int cd,ideal P,ideal sP,int i,int dif,int dB,ideal CI) |
---|
| 1923 | { intmat vec[1][cd]; // the coefficients for the next |
---|
| 1924 | // combination - |
---|
| 1925 | degBound=0; |
---|
| 1926 | poly test_poly; // the linear combination to test |
---|
| 1927 | int test_dim; |
---|
| 1928 | intvec h; // Hilbert series |
---|
| 1929 | int j=i+1; |
---|
| 1930 | matrix tB=transpose(B); |
---|
| 1931 | ideal TEST; |
---|
| 1932 | while(j<=i+dif) |
---|
| 1933 | { CI=CI+ideal(var(j)^d); // homogeneous polynomial of the same |
---|
| 1934 | // degree as the one we're looking for is |
---|
| 1935 | // added |
---|
| 1936 | // h=hilb(std(CI),1); |
---|
| 1937 | dB=dB+d-1; // used as degBound |
---|
| 1938 | while(1) |
---|
| 1939 | { vec=next_vector(vec); // next vector |
---|
| 1940 | test_poly=(vec*tB)[1,1]; |
---|
| 1941 | // degBound=dB; |
---|
| 1942 | TEST=sP+ideal(test_poly); |
---|
| 1943 | attrib(TEST,"isSB",1); |
---|
| 1944 | test_dim=dim(TEST); |
---|
| 1945 | // degBound=0; |
---|
| 1946 | if (n-test_dim==j) // the dimension has been lowered by one |
---|
| 1947 | { sP=TEST; |
---|
| 1948 | break; |
---|
| 1949 | } |
---|
| 1950 | // degBound=dB; |
---|
[73054bb] | 1951 | //TEST=std(sP+ideal(test_poly)); // should soon be replaced by next line |
---|
| 1952 | TEST=std(sP,test_poly); // or, better: |
---|
| 1953 | //TEST=std(sP,test_poly,h); // Hilbert driven std-calculation |
---|
[f34c37c] | 1954 | test_dim=dim(TEST); |
---|
| 1955 | // degBound=0; |
---|
| 1956 | if (n-test_dim==j) // the dimension has been lowered by one |
---|
| 1957 | { sP=TEST; |
---|
| 1958 | break; |
---|
| 1959 | } |
---|
| 1960 | } |
---|
| 1961 | P[j]=test_poly; // test_poly ist added to primary |
---|
[18bd9c] | 1962 | j++; // invariants |
---|
[f34c37c] | 1963 | } |
---|
| 1964 | return(P,sP,CI,dB); |
---|
| 1965 | } |
---|
| 1966 | |
---|
[68e678] | 1967 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 1968 | // This procedure finds at most dif primary invariants in degree d. It returns |
---|
| 1969 | // all primary invariants found so far. The coefficients lie in the field of |
---|
| 1970 | // characteristic p>0. |
---|
[68e678] | 1971 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 1972 | proc p_search (int n,int d,ideal B,int cd,ideal P,ideal sP,int i,int dif,int dB,ideal CI) |
---|
| 1973 | { def br=basering; |
---|
| 1974 | degBound=0; |
---|
| 1975 | matrix vec(1)[1][cd]; // starting with 0-vector - |
---|
| 1976 | intmat new[1][cd]; // the coefficients for the next |
---|
| 1977 | // combination - |
---|
| 1978 | matrix pnew[1][cd]; // new needs to be mapped into br - |
---|
| 1979 | int counter=1; // counts the vectors |
---|
| 1980 | int j; |
---|
| 1981 | int p=char(br); |
---|
| 1982 | if (minpoly<>0) |
---|
| 1983 | { int ext_deg=pardeg(minpoly); // field has p^d elements |
---|
| 1984 | } |
---|
| 1985 | else |
---|
| 1986 | { int ext_deg=1; // field has p^d elements |
---|
| 1987 | } |
---|
| 1988 | poly test_poly; // the linear combination to test |
---|
| 1989 | int test_dim; |
---|
| 1990 | ring R=0,x,dp; // just to calculate next variable |
---|
| 1991 | // bound - |
---|
[558eb2] | 1992 | number bound=(number(p)^(ext_deg*cd)-1)/(number(p)^ext_deg-1)+1; |
---|
| 1993 | // this is how many linearly independent |
---|
| 1994 | // vectors of size cd exist having |
---|
| 1995 | // entries in the base field of br |
---|
[f34c37c] | 1996 | setring br; |
---|
| 1997 | intvec h; // Hilbert series |
---|
| 1998 | int k=i+1; |
---|
[425a9d] | 1999 | if (ncols(B)<cd) { B[cd]=0; } |
---|
[f34c37c] | 2000 | matrix tB=transpose(B); |
---|
| 2001 | ideal TEST; |
---|
| 2002 | while (k<=i+dif) |
---|
| 2003 | { CI=CI+ideal(var(k)^d); // homogeneous polynomial of the same |
---|
[558eb2] | 2004 | //degree as the one we're looking for is |
---|
[f34c37c] | 2005 | // added |
---|
| 2006 | // h=hilb(std(CI),1); |
---|
| 2007 | dB=dB+d-1; // used as degBound |
---|
| 2008 | setring R; |
---|
| 2009 | while (number(counter)<>bound) // otherwise, we are done |
---|
| 2010 | { setring br; |
---|
| 2011 | new=next_vector(new); |
---|
[18bd9c] | 2012 | for (j=1;j<=cd;j++) |
---|
[f34c37c] | 2013 | { pnew[1,j]=int_number_map(new[1,j]); // mapping an integer into br |
---|
| 2014 | } |
---|
[558eb2] | 2015 | if (unique(vec(1..counter),pnew)) //checking whether we tried pnew before |
---|
[18bd9c] | 2016 | { counter++; |
---|
[f34c37c] | 2017 | matrix vec(counter)=pnew; // keeping track of the ones we tried - |
---|
| 2018 | test_poly=(vec(counter)*tB)[1,1]; // linear combination - |
---|
| 2019 | // degBound=dB; |
---|
| 2020 | TEST=sP+ideal(test_poly); |
---|
| 2021 | attrib(TEST,"isSB",1); |
---|
| 2022 | test_dim=dim(TEST); |
---|
| 2023 | // degBound=0; |
---|
| 2024 | if (n-test_dim==k) // the dimension has been lowered by one |
---|
| 2025 | { sP=TEST; |
---|
| 2026 | setring R; |
---|
| 2027 | break; |
---|
| 2028 | } |
---|
| 2029 | // degBound=dB; |
---|
[73054bb] | 2030 | //TEST=std(sP+ideal(test_poly)); // should soon to be replaced by next |
---|
[f34c37c] | 2031 | // line |
---|
[73054bb] | 2032 | TEST=std(sP,test_poly); // or, better: |
---|
[f34c37c] | 2033 | // TEST=std(sP,test_poly,h); // Hilbert driven std-calculation |
---|
| 2034 | test_dim=dim(TEST); |
---|
| 2035 | // degBound=0; |
---|
| 2036 | if (n-test_dim==k) // the dimension has been lowered by one |
---|
| 2037 | { sP=TEST; |
---|
| 2038 | setring R; |
---|
| 2039 | break; |
---|
| 2040 | } |
---|
| 2041 | } |
---|
| 2042 | setring R; |
---|
| 2043 | } |
---|
| 2044 | if (number(counter)<=bound) |
---|
| 2045 | { setring br; |
---|
| 2046 | P[k]=test_poly; // test_poly ist added to primary |
---|
| 2047 | } // invariants |
---|
| 2048 | else |
---|
| 2049 | { setring br; |
---|
| 2050 | CI=CI[1..size(CI)-1]; |
---|
| 2051 | return(P,sP,CI,dB-d+1); |
---|
| 2052 | } |
---|
[18bd9c] | 2053 | k++; |
---|
[f34c37c] | 2054 | } |
---|
| 2055 | return(P,sP,CI,dB); |
---|
| 2056 | } |
---|
[68e678] | 2057 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 2058 | |
---|
| 2059 | proc primary_char0 (matrix REY,matrix M,list #) |
---|
| 2060 | "USAGE: primary_char0(REY,M[,v]); |
---|
| 2061 | REY: a <matrix> representing the Reynolds operator, M: a 1x2 <matrix> |
---|
| 2062 | representing the Molien series, v: an optional <int> |
---|
| 2063 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien and |
---|
| 2064 | M the one of molien or the second one of reynolds_molien |
---|
| 2065 | DISPLAY: information about the various stages of the programme if v does not |
---|
| 2066 | equal 0 |
---|
| 2067 | RETURN: primary invariants (type <matrix>) of the invariant ring |
---|
| 2068 | THEORY: Bases of homogeneous invariants are generated successively and those |
---|
| 2069 | are chosen as primary invariants that lower the dimension of the ideal |
---|
| 2070 | generated by the previously found invariants (see paper \"Generating a |
---|
| 2071 | Noetherian Normalization of the Invariant Ring of a Finite Group\" by |
---|
[9f79aa] | 2072 | Decker, Heydtmann, Schreyer (1998)). |
---|
[68e678] | 2073 | EXAMPLE: example primary_char0; shows an example |
---|
[f34c37c] | 2074 | " |
---|
| 2075 | { degBound=0; |
---|
| 2076 | if (char(basering)<>0) |
---|
[9f79aa] | 2077 | { "ERROR: primary_char0 should only be used with rings of characteristic 0."; |
---|
[f34c37c] | 2078 | return(); |
---|
| 2079 | } |
---|
| 2080 | //----------------- checking input and setting verbose mode ------------------ |
---|
| 2081 | if (size(#)>1) |
---|
| 2082 | { "ERROR: primary_char0 can only have three parameters."; |
---|
| 2083 | return(); |
---|
| 2084 | } |
---|
| 2085 | if (size(#)==1) |
---|
| 2086 | { if (typeof(#[1])<>"int") |
---|
| 2087 | { "ERROR: The third parameter should be of type <int>."; |
---|
| 2088 | return(); |
---|
| 2089 | } |
---|
| 2090 | else |
---|
| 2091 | { int v=#[1]; |
---|
| 2092 | } |
---|
| 2093 | } |
---|
| 2094 | else |
---|
| 2095 | { int v=0; |
---|
| 2096 | } |
---|
| 2097 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 2098 | // as the size of the matrices, as well |
---|
| 2099 | // as the number of primary invariants, |
---|
| 2100 | // we should get |
---|
| 2101 | if (ncols(REY)<>n) |
---|
| 2102 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
| 2103 | return(); |
---|
| 2104 | } |
---|
| 2105 | if (ncols(M)<>2 or nrows(M)<>1) |
---|
| 2106 | { "ERROR: Second parameter ought to be the Molien series." |
---|
| 2107 | return(); |
---|
| 2108 | } |
---|
| 2109 | //---------------------------------------------------------------------------- |
---|
| 2110 | if (v && voice<>2) |
---|
| 2111 | { " We can start looking for primary invariants..."; |
---|
| 2112 | ""; |
---|
| 2113 | } |
---|
| 2114 | if (v && voice==2) |
---|
| 2115 | { ""; |
---|
| 2116 | } |
---|
| 2117 | //------------------------- initializing variables --------------------------- |
---|
| 2118 | int dB; |
---|
| 2119 | poly p(1..2); // p(1) will be used for single terms of |
---|
| 2120 | // the partial expansion, p(2) to store |
---|
| 2121 | p(1..2)=partial_molien(M,1); // the intermediate result - |
---|
| 2122 | poly v1=var(1); // we need v1 to split off coefficients |
---|
| 2123 | // in the partial expansion of M (which |
---|
| 2124 | // is in terms of the first variable) - |
---|
| 2125 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
| 2126 | // space of invariants of degree d, |
---|
| 2127 | // newdim: dimension the ideal generated |
---|
| 2128 | // the primary invariants plus basis |
---|
| 2129 | // elements, dif=n-i-newdim, i.e. the |
---|
| 2130 | // number of new primary invairants that |
---|
| 2131 | // should be added in this degree - |
---|
| 2132 | ideal P,Pplus,sPplus,CI,B; // P: will contain primary invariants, |
---|
| 2133 | // Pplus: P+B, CI: a complete |
---|
| 2134 | // intersection with the same Hilbert |
---|
| 2135 | // function as P |
---|
[f54c83] | 2136 | ideal sP=groebner(P); |
---|
[f34c37c] | 2137 | dB=1; // used as degree bound |
---|
| 2138 | int i=0; |
---|
| 2139 | //-------------- loop that searches for primary invariants ------------------ |
---|
| 2140 | while(1) // repeat until n primary invariants are |
---|
| 2141 | { // found - |
---|
| 2142 | p(1..2)=partial_molien(M,1,p(2)); // next term of the partial expansion - |
---|
| 2143 | d=deg(p(1)); // degree where we'll search - |
---|
| 2144 | cd=int(coef(p(1),v1)[2,1]); // dimension of the homogeneous space of |
---|
| 2145 | // inviarants of degree d |
---|
| 2146 | if (v) |
---|
| 2147 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
| 2148 | } |
---|
| 2149 | B=invariant_basis_reynolds(REY,d,intvec(cd,6)); // basis of invariants of |
---|
| 2150 | // degree d |
---|
| 2151 | if (B[1]<>0) |
---|
| 2152 | { Pplus=P+B; |
---|
[f54c83] | 2153 | sPplus=groebner(Pplus); |
---|
[f34c37c] | 2154 | newdim=dim(sPplus); |
---|
| 2155 | dif=n-i-newdim; |
---|
| 2156 | } |
---|
| 2157 | else |
---|
| 2158 | { dif=0; |
---|
| 2159 | } |
---|
| 2160 | if (dif<>0) // we have to find dif new primary |
---|
| 2161 | { // invariants |
---|
| 2162 | if (cd<>dif) |
---|
| 2163 | { P,sP,CI,dB=search(n,d,B,cd,P,sP,i,dif,dB,CI); // searching for dif invariants |
---|
| 2164 | } // i.e. we can take all of B |
---|
| 2165 | else |
---|
[18bd9c] | 2166 | { for(j=i+1;j>i+dif;j++) |
---|
[f34c37c] | 2167 | { CI=CI+ideal(var(j)^d); |
---|
| 2168 | } |
---|
| 2169 | dB=dB+dif*(d-1); |
---|
| 2170 | P=Pplus; |
---|
| 2171 | sP=sPplus; |
---|
| 2172 | } |
---|
| 2173 | if (v) |
---|
[18bd9c] | 2174 | { for (j=1;j<=dif;j++) |
---|
[f34c37c] | 2175 | { " We find: "+string(P[i+j]); |
---|
| 2176 | } |
---|
| 2177 | } |
---|
| 2178 | i=i+dif; |
---|
| 2179 | if (i==n) // found all primary invariants |
---|
| 2180 | { if (v) |
---|
| 2181 | { ""; |
---|
| 2182 | " We found all primary invariants."; |
---|
| 2183 | ""; |
---|
| 2184 | } |
---|
| 2185 | return(matrix(P)); |
---|
| 2186 | } |
---|
| 2187 | } // done with degree d |
---|
| 2188 | } |
---|
| 2189 | } |
---|
| 2190 | example |
---|
[68e678] | 2191 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
[f34c37c] | 2192 | ring R=0,(x,y,z),dp; |
---|
| 2193 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 2194 | matrix REY,M=reynolds_molien(A); |
---|
| 2195 | matrix P=primary_char0(REY,M); |
---|
| 2196 | print(P); |
---|
| 2197 | } |
---|
[68e678] | 2198 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 2199 | |
---|
| 2200 | proc primary_charp (matrix REY,string ring_name,list #) |
---|
| 2201 | "USAGE: primary_charp(REY,ringname[,v]); |
---|
| 2202 | REY: a <matrix> representing the Reynolds operator, ringname: a |
---|
| 2203 | <string> giving the name of a ring where the Molien series is stored, |
---|
| 2204 | v: an optional <int> |
---|
| 2205 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien and |
---|
| 2206 | ringname gives the name of a ring of characteristic 0 that has been |
---|
| 2207 | created by molien or reynolds_molien |
---|
| 2208 | DISPLAY: information about the various stages of the programme if v does not |
---|
| 2209 | equal 0 |
---|
| 2210 | RETURN: primary invariants (type <matrix>) of the invariant ring |
---|
| 2211 | THEORY: Bases of homogeneous invariants are generated successively and those |
---|
| 2212 | are chosen as primary invariants that lower the dimension of the ideal |
---|
| 2213 | generated by the previously found invariants (see paper \"Generating a |
---|
| 2214 | Noetherian Normalization of the Invariant Ring of a Finite Group\" by |
---|
[9f79aa] | 2215 | Decker, Heydtmann, Schreyer (1998)). |
---|
[68e678] | 2216 | EXAMPLE: example primary_charp; shows an example |
---|
[f34c37c] | 2217 | " |
---|
| 2218 | { degBound=0; |
---|
| 2219 | // ---------------- checking input and setting verbose mode ------------------- |
---|
| 2220 | if (char(basering)==0) |
---|
| 2221 | { "ERROR: primary_charp should only be used with rings of characteristic p>0."; |
---|
| 2222 | return(); |
---|
| 2223 | } |
---|
| 2224 | if (size(#)>1) |
---|
| 2225 | { "ERROR: primary_charp can only have three parameters."; |
---|
| 2226 | return(); |
---|
| 2227 | } |
---|
| 2228 | if (size(#)==1) |
---|
| 2229 | { if (typeof(#[1])<>"int") |
---|
| 2230 | { "ERROR: The third parameter should be of type <int>."; |
---|
| 2231 | return(); |
---|
| 2232 | } |
---|
| 2233 | else |
---|
| 2234 | { int v=#[1]; |
---|
| 2235 | } |
---|
| 2236 | } |
---|
| 2237 | else |
---|
| 2238 | { int v=0; |
---|
| 2239 | } |
---|
| 2240 | def br=basering; |
---|
| 2241 | int n=nvars(br); // n is the number of variables, as well |
---|
| 2242 | // as the size of the matrices, as well |
---|
| 2243 | // as the number of primary invariants, |
---|
| 2244 | // we should get |
---|
| 2245 | if (ncols(REY)<>n) |
---|
| 2246 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
| 2247 | return(); |
---|
| 2248 | } |
---|
| 2249 | if (typeof(`ring_name`)<>"ring") |
---|
| 2250 | { "ERROR: Second parameter ought to the name of a ring where the Molien"; |
---|
| 2251 | " is stored."; |
---|
| 2252 | return(); |
---|
| 2253 | } |
---|
| 2254 | //---------------------------------------------------------------------------- |
---|
| 2255 | if (v && voice<>2) |
---|
| 2256 | { " We can start looking for primary invariants..."; |
---|
| 2257 | ""; |
---|
| 2258 | } |
---|
| 2259 | if (v && voice==2) |
---|
| 2260 | { ""; |
---|
| 2261 | } |
---|
| 2262 | //----------------------- initializing variables ----------------------------- |
---|
| 2263 | int dB; |
---|
| 2264 | setring `ring_name`; // the Molien series is stores here - |
---|
| 2265 | poly p(1..2); // p(1) will be used for single terms of |
---|
| 2266 | // the partial expansion, p(2) to store |
---|
| 2267 | p(1..2)=partial_molien(M,1); // the intermediate result - |
---|
| 2268 | poly v1=var(1); // we need v1 to split off coefficients |
---|
| 2269 | // in the partial expansion of M (which |
---|
| 2270 | // is in terms of the first variable) |
---|
| 2271 | setring br; |
---|
| 2272 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
| 2273 | // space of invariants of degree d, |
---|
| 2274 | // newdim: dimension the ideal generated |
---|
| 2275 | // the primary invariants plus basis |
---|
| 2276 | // elements, dif=n-i-newdim, i.e. the |
---|
| 2277 | // number of new primary invairants that |
---|
| 2278 | // should be added in this degree - |
---|
| 2279 | ideal P,Pplus,sPplus,CI,B; // P: will contain primary invariants, |
---|
| 2280 | // Pplus: P+B, CI: a complete |
---|
| 2281 | // intersection with the same Hilbert |
---|
| 2282 | // function as P |
---|
[f54c83] | 2283 | ideal sP=groebner(P); |
---|
[f34c37c] | 2284 | dB=1; // used as degree bound |
---|
| 2285 | int i=0; |
---|
| 2286 | //---------------- loop that searches for primary invariants ----------------- |
---|
| 2287 | while(1) // repeat until n primary invariants are |
---|
| 2288 | { // found |
---|
| 2289 | setring `ring_name`; |
---|
| 2290 | p(1..2)=partial_molien(M,1,p(2)); // next term of the partial expansion - |
---|
| 2291 | d=deg(p(1)); // degree where we'll search - |
---|
| 2292 | cd=int(coef(p(1),v1)[2,1]); // dimension of the homogeneous space of |
---|
| 2293 | // inviarants of degree d |
---|
| 2294 | setring br; |
---|
| 2295 | if (v) |
---|
| 2296 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
| 2297 | } |
---|
| 2298 | B=invariant_basis_reynolds(REY,d,intvec(cd,6)); // basis of invariants of |
---|
| 2299 | // degree d |
---|
[425a9d] | 2300 | if (ncols(B)<cd) |
---|
| 2301 | { |
---|
| 2302 | " warning: expected ",cd," invars, found ",ncols(B); |
---|
| 2303 | } |
---|
[f34c37c] | 2304 | if (B[1]<>0) |
---|
| 2305 | { Pplus=P+B; |
---|
[f54c83] | 2306 | sPplus=groebner(Pplus); |
---|
[f34c37c] | 2307 | newdim=dim(sPplus); |
---|
| 2308 | dif=n-i-newdim; |
---|
| 2309 | } |
---|
| 2310 | else |
---|
| 2311 | { dif=0; |
---|
| 2312 | } |
---|
| 2313 | if (dif<>0) // we have to find dif new primary |
---|
| 2314 | { // invariants |
---|
| 2315 | if (cd<>dif) |
---|
| 2316 | { P,sP,CI,dB=p_search(n,d,B,cd,P,sP,i,dif,dB,CI); |
---|
| 2317 | } |
---|
| 2318 | else // i.e. we can take all of B |
---|
[18bd9c] | 2319 | { for(j=i+1;j>i+dif;j++) |
---|
[f34c37c] | 2320 | { CI=CI+ideal(var(j)^d); |
---|
| 2321 | } |
---|
| 2322 | dB=dB+dif*(d-1); |
---|
| 2323 | P=Pplus; |
---|
| 2324 | sP=sPplus; |
---|
| 2325 | } |
---|
| 2326 | if (v) |
---|
[18bd9c] | 2327 | { for (j=1;j<=size(P)-i;j++) |
---|
[f34c37c] | 2328 | { " We find: "+string(P[i+j]); |
---|
| 2329 | } |
---|
| 2330 | } |
---|
| 2331 | i=size(P); |
---|
| 2332 | if (i==n) // found all primary invariants |
---|
| 2333 | { if (v) |
---|
| 2334 | { ""; |
---|
| 2335 | " We found all primary invariants."; |
---|
| 2336 | ""; |
---|
| 2337 | } |
---|
| 2338 | return(matrix(P)); |
---|
| 2339 | } |
---|
| 2340 | } // done with degree d |
---|
| 2341 | } |
---|
| 2342 | } |
---|
| 2343 | example |
---|
[9f79aa] | 2344 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7 (changed into char 3)"; echo=2; |
---|
[f34c37c] | 2345 | ring R=3,(x,y,z),dp; |
---|
| 2346 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 2347 | list L=group_reynolds(A); |
---|
| 2348 | string newring="alskdfj"; |
---|
| 2349 | molien(L[2..size(L)],newring); |
---|
| 2350 | matrix P=primary_charp(L[1],newring); |
---|
| 2351 | kill `newring`; |
---|
| 2352 | print(P); |
---|
| 2353 | } |
---|
[68e678] | 2354 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 2355 | |
---|
| 2356 | proc primary_char0_no_molien (matrix REY, list #) |
---|
| 2357 | "USAGE: primary_char0_no_molien(REY[,v]); |
---|
| 2358 | REY: a <matrix> representing the Reynolds operator, v: an optional |
---|
| 2359 | <int> |
---|
| 2360 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien |
---|
| 2361 | DISPLAY: information about the various stages of the programme if v does not |
---|
| 2362 | equal 0 |
---|
| 2363 | RETURN: primary invariants (type <matrix>) of the invariant ring and an |
---|
| 2364 | <intvec> listing some of the degrees where no non-trivial homogeneous |
---|
| 2365 | invariants are to be found |
---|
| 2366 | THEORY: Bases of homogeneous invariants are generated successively and those |
---|
| 2367 | are chosen as primary invariants that lower the dimension of the ideal |
---|
| 2368 | generated by the previously found invariants (see paper \"Generating a |
---|
| 2369 | Noetherian Normalization of the Invariant Ring of a Finite Group\" by |
---|
[9f79aa] | 2370 | Decker, Heydtmann, Schreyer (1998)). |
---|
[68e678] | 2371 | EXAMPLE: example primary_char0_no_molien; shows an example |
---|
[f34c37c] | 2372 | " |
---|
| 2373 | { degBound=0; |
---|
| 2374 | //-------------- checking input and setting verbose mode --------------------- |
---|
| 2375 | if (char(basering)<>0) |
---|
| 2376 | { "ERROR: primary_char0_no_molien should only be used with rings of"; |
---|
| 2377 | " characteristic 0."; |
---|
| 2378 | return(); |
---|
| 2379 | } |
---|
| 2380 | if (size(#)>1) |
---|
| 2381 | { "ERROR: primary_char0_no_molien can only have two parameters."; |
---|
| 2382 | return(); |
---|
| 2383 | } |
---|
| 2384 | if (size(#)==1) |
---|
| 2385 | { if (typeof(#[1])<>"int") |
---|
| 2386 | { "ERROR: The second parameter should be of type <int>."; |
---|
| 2387 | return(); |
---|
| 2388 | } |
---|
| 2389 | else |
---|
| 2390 | { int v=#[1]; |
---|
| 2391 | } |
---|
| 2392 | } |
---|
| 2393 | else |
---|
| 2394 | { int v=0; |
---|
| 2395 | } |
---|
| 2396 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 2397 | // as the size of the matrices, as well |
---|
| 2398 | // as the number of primary invariants, |
---|
| 2399 | // we should get |
---|
| 2400 | if (ncols(REY)<>n) |
---|
| 2401 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
| 2402 | return(); |
---|
| 2403 | } |
---|
| 2404 | //---------------------------------------------------------------------------- |
---|
| 2405 | if (v && voice<>2) |
---|
| 2406 | { " We can start looking for primary invariants..."; |
---|
| 2407 | ""; |
---|
| 2408 | } |
---|
| 2409 | if (v && voice==2) |
---|
| 2410 | { ""; |
---|
| 2411 | } |
---|
| 2412 | //----------------------- initializing variables ----------------------------- |
---|
| 2413 | int dB; |
---|
| 2414 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
| 2415 | // space of invariants of degree d, |
---|
| 2416 | // newdim: dimension the ideal generated |
---|
| 2417 | // the primary invariants plus basis |
---|
| 2418 | // elements, dif=n-i-newdim, i.e. the |
---|
| 2419 | // number of new primary invairants that |
---|
| 2420 | // should be added in this degree - |
---|
| 2421 | ideal P,Pplus,CI,B; // P: will contain primary invariants, |
---|
| 2422 | // Pplus: P+B, CI: a complete |
---|
| 2423 | // intersection with the same Hilbert |
---|
| 2424 | // function as P |
---|
[f54c83] | 2425 | ideal sP=groebner(P); |
---|
[f34c37c] | 2426 | dB=1; // used as degree bound - |
---|
| 2427 | d=0; // initializing |
---|
| 2428 | int i=0; |
---|
| 2429 | intvec deg_vector; |
---|
| 2430 | //------------------ loop that searches for primary invariants --------------- |
---|
| 2431 | while(1) // repeat until n primary invariants are |
---|
| 2432 | { // found - |
---|
[18bd9c] | 2433 | d++; // degree where we'll search |
---|
[f34c37c] | 2434 | if (v) |
---|
| 2435 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
| 2436 | } |
---|
| 2437 | B=invariant_basis_reynolds(REY,d,intvec(-1,6)); // basis of invariants of |
---|
| 2438 | // degree d |
---|
| 2439 | if (B[1]<>0) |
---|
| 2440 | { Pplus=P+B; |
---|
[f54c83] | 2441 | newdim=dim(groebner(Pplus)); |
---|
[f34c37c] | 2442 | dif=n-i-newdim; |
---|
| 2443 | } |
---|
| 2444 | else |
---|
| 2445 | { dif=0; |
---|
| 2446 | deg_vector=deg_vector,d; |
---|
| 2447 | } |
---|
| 2448 | if (dif<>0) // we have to find dif new primary |
---|
| 2449 | { // invariants |
---|
| 2450 | cd=size(B); |
---|
| 2451 | if (cd<>dif) |
---|
| 2452 | { P,sP,CI,dB=search(n,d,B,cd,P,sP,i,dif,dB,CI); |
---|
| 2453 | } |
---|
| 2454 | else // i.e. we can take all of B |
---|
[18bd9c] | 2455 | { for(j=i+1;j<=i+dif;j++) |
---|
[f34c37c] | 2456 | { CI=CI+ideal(var(j)^d); |
---|
| 2457 | } |
---|
| 2458 | dB=dB+dif*(d-1); |
---|
| 2459 | P=Pplus; |
---|
[f54c83] | 2460 | sP=groebner(P); |
---|
[f34c37c] | 2461 | } |
---|
| 2462 | if (v) |
---|
[18bd9c] | 2463 | { for (j=1;j<=dif;j++) |
---|
[f34c37c] | 2464 | { " We find: "+string(P[i+j]); |
---|
| 2465 | } |
---|
| 2466 | } |
---|
| 2467 | i=i+dif; |
---|
| 2468 | if (i==n) // found all primary invariants |
---|
| 2469 | { if (v) |
---|
| 2470 | { ""; |
---|
| 2471 | " We found all primary invariants."; |
---|
| 2472 | ""; |
---|
| 2473 | } |
---|
| 2474 | if (deg_vector==0) |
---|
| 2475 | { return(matrix(P)); |
---|
| 2476 | } |
---|
| 2477 | else |
---|
| 2478 | { return(matrix(P),compress(deg_vector)); |
---|
| 2479 | } |
---|
| 2480 | } |
---|
| 2481 | } // done with degree d |
---|
| 2482 | else |
---|
| 2483 | { if (v) |
---|
| 2484 | { " None here..."; |
---|
| 2485 | } |
---|
| 2486 | } |
---|
| 2487 | } |
---|
| 2488 | } |
---|
| 2489 | example |
---|
[9f79aa] | 2490 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
[f34c37c] | 2491 | ring R=0,(x,y,z),dp; |
---|
| 2492 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 2493 | list L=group_reynolds(A); |
---|
| 2494 | list l=primary_char0_no_molien(L[1]); |
---|
| 2495 | print(l[1]); |
---|
| 2496 | } |
---|
[558eb2] | 2497 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 2498 | |
---|
| 2499 | proc primary_charp_no_molien (matrix REY, list #) |
---|
| 2500 | "USAGE: primary_charp_no_molien(REY[,v]); |
---|
| 2501 | REY: a <matrix> representing the Reynolds operator, v: an optional |
---|
| 2502 | <int> |
---|
| 2503 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien |
---|
| 2504 | DISPLAY: information about the various stages of the programme if v does not |
---|
| 2505 | equal 0 |
---|
| 2506 | RETURN: primary invariants (type <matrix>) of the invariant ring and an |
---|
| 2507 | <intvec> listing some of the degrees where no non-trivial homogeneous |
---|
| 2508 | invariants are to be found |
---|
| 2509 | THEORY: Bases of homogeneous invariants are generated successively and those |
---|
| 2510 | are chosen as primary invariants that lower the dimension of the ideal |
---|
| 2511 | generated by the previously found invariants (see paper \"Generating a |
---|
| 2512 | Noetherian Normalization of the Invariant Ring of a Finite Group\" by |
---|
[9f79aa] | 2513 | Decker, Heydtmann, Schreyer (1998)). |
---|
[68e678] | 2514 | EXAMPLE: example primary_charp_no_molien; shows an example |
---|
[f34c37c] | 2515 | " |
---|
| 2516 | { degBound=0; |
---|
| 2517 | //----------------- checking input and setting verbose mode ------------------ |
---|
| 2518 | if (char(basering)==0) |
---|
| 2519 | { "ERROR: primary_charp_no_molien should only be used with rings of"; |
---|
| 2520 | " characteristic p>0."; |
---|
| 2521 | return(); |
---|
| 2522 | } |
---|
| 2523 | if (size(#)>1) |
---|
| 2524 | { "ERROR: primary_charp_no_molien can only have two parameters."; |
---|
| 2525 | return(); |
---|
| 2526 | } |
---|
| 2527 | if (size(#)==1) |
---|
| 2528 | { if (typeof(#[1])<>"int") |
---|
| 2529 | { "ERROR: The second parameter should be of type <int>."; |
---|
| 2530 | return(); |
---|
| 2531 | } |
---|
| 2532 | else |
---|
[73054bb] | 2533 | { int v=#[1]; } |
---|
[f34c37c] | 2534 | } |
---|
| 2535 | else |
---|
| 2536 | { int v=0; |
---|
| 2537 | } |
---|
| 2538 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 2539 | // as the size of the matrices, as well |
---|
| 2540 | // as the number of primary invariants, |
---|
| 2541 | // we should get |
---|
| 2542 | if (ncols(REY)<>n) |
---|
| 2543 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
| 2544 | return(); |
---|
| 2545 | } |
---|
| 2546 | //---------------------------------------------------------------------------- |
---|
| 2547 | if (v && voice<>2) |
---|
| 2548 | { " We can start looking for primary invariants..."; |
---|
| 2549 | ""; |
---|
| 2550 | } |
---|
| 2551 | if (v && voice==2) |
---|
[73054bb] | 2552 | { ""; } |
---|
[f34c37c] | 2553 | //-------------------- initializing variables -------------------------------- |
---|
| 2554 | int dB; |
---|
| 2555 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
| 2556 | // space of invariants of degree d, |
---|
| 2557 | // newdim: dimension the ideal generated |
---|
| 2558 | // the primary invariants plus basis |
---|
| 2559 | // elements, dif=n-i-newdim, i.e. the |
---|
| 2560 | // number of new primary invairants that |
---|
| 2561 | // should be added in this degree - |
---|
| 2562 | ideal P,Pplus,sPplus,CI,B; // P: will contain primary invariants, |
---|
| 2563 | // Pplus: P+B, CI: a complete |
---|
| 2564 | // intersection with the same Hilbert |
---|
| 2565 | // function as P |
---|
[f54c83] | 2566 | ideal sP=groebner(P); |
---|
[f34c37c] | 2567 | dB=1; // used as degree bound - |
---|
| 2568 | d=0; // initializing |
---|
| 2569 | int i=0; |
---|
| 2570 | intvec deg_vector; |
---|
| 2571 | //------------------ loop that searches for primary invariants --------------- |
---|
| 2572 | while(1) // repeat until n primary invariants are |
---|
| 2573 | { // found - |
---|
[18bd9c] | 2574 | d++; // degree where we'll search |
---|
[f34c37c] | 2575 | if (v) |
---|
| 2576 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
| 2577 | } |
---|
| 2578 | B=invariant_basis_reynolds(REY,d,intvec(-1,6)); // basis of invariants of |
---|
| 2579 | // degree d |
---|
| 2580 | if (B[1]<>0) |
---|
| 2581 | { Pplus=P+B; |
---|
[f54c83] | 2582 | sPplus=groebner(Pplus); |
---|
[f34c37c] | 2583 | newdim=dim(sPplus); |
---|
| 2584 | dif=n-i-newdim; |
---|
| 2585 | } |
---|
| 2586 | else |
---|
| 2587 | { dif=0; |
---|
| 2588 | deg_vector=deg_vector,d; |
---|
| 2589 | } |
---|
| 2590 | if (dif<>0) // we have to find dif new primary |
---|
| 2591 | { // invariants |
---|
| 2592 | cd=size(B); |
---|
| 2593 | if (cd<>dif) |
---|
| 2594 | { P,sP,CI,dB=p_search(n,d,B,cd,P,sP,i,dif,dB,CI); |
---|
| 2595 | } |
---|
| 2596 | else // i.e. we can take all of B |
---|
[18bd9c] | 2597 | { for(j=i+1;j<=i+dif;j++) |
---|
[f34c37c] | 2598 | { CI=CI+ideal(var(j)^d); |
---|
| 2599 | } |
---|
| 2600 | dB=dB+dif*(d-1); |
---|
| 2601 | P=Pplus; |
---|
| 2602 | sP=sPplus; |
---|
| 2603 | } |
---|
| 2604 | if (v) |
---|
[18bd9c] | 2605 | { for (j=1;j<=size(P)-i;j++) |
---|
[f34c37c] | 2606 | { " We find: "+string(P[i+j]); |
---|
| 2607 | } |
---|
| 2608 | } |
---|
| 2609 | i=size(P); |
---|
| 2610 | if (i==n) // found all primary invariants |
---|
| 2611 | { if (v) |
---|
| 2612 | { ""; |
---|
| 2613 | " We found all primary invariants."; |
---|
| 2614 | ""; |
---|
| 2615 | } |
---|
| 2616 | if (deg_vector==0) |
---|
| 2617 | { return(matrix(P)); |
---|
| 2618 | } |
---|
| 2619 | else |
---|
| 2620 | { return(matrix(P),compress(deg_vector)); |
---|
| 2621 | } |
---|
| 2622 | } |
---|
| 2623 | } // done with degree d |
---|
| 2624 | else |
---|
| 2625 | { if (v) |
---|
| 2626 | { " None here..."; |
---|
| 2627 | } |
---|
| 2628 | } |
---|
| 2629 | } |
---|
| 2630 | } |
---|
| 2631 | example |
---|
[9f79aa] | 2632 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7 (changed into char 3)"; echo=2; |
---|
[f34c37c] | 2633 | ring R=3,(x,y,z),dp; |
---|
| 2634 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 2635 | list L=group_reynolds(A); |
---|
| 2636 | list l=primary_charp_no_molien(L[1]); |
---|
| 2637 | print(l[1]); |
---|
| 2638 | } |
---|
[68e678] | 2639 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 2640 | |
---|
| 2641 | proc primary_charp_without (list #) |
---|
| 2642 | "USAGE: primary_charp_without(G1,G2,...[,v]); |
---|
| 2643 | G1,G2,...: <matrices> generating a finite matrix group, v: an optional |
---|
| 2644 | <int> |
---|
| 2645 | DISPLAY: information about the various stages of the programme if v does not |
---|
| 2646 | equal 0 |
---|
| 2647 | RETURN: primary invariants (type <matrix>) of the invariant ring |
---|
| 2648 | THEORY: Bases of homogeneous invariants are generated successively and those |
---|
| 2649 | are chosen as primary invariants that lower the dimension of the ideal |
---|
| 2650 | generated by the previously found invariants (see paper \"Generating a |
---|
| 2651 | Noetherian Normalization of the Invariant Ring of a Finite Group\" by |
---|
[9f79aa] | 2652 | Decker, Heydtmann, Schreyer (1998)). No Reynolds |
---|
[f34c37c] | 2653 | operator or Molien series is used. |
---|
[68e678] | 2654 | EXAMPLE: example primary_charp_without; shows an example |
---|
[f34c37c] | 2655 | " |
---|
| 2656 | { degBound=0; |
---|
| 2657 | //--------------------- checking input and setting verbose mode -------------- |
---|
| 2658 | if (char(basering)==0) |
---|
| 2659 | { "ERROR: primary_charp_without should only be used with rings of"; |
---|
| 2660 | " characteristic 0."; |
---|
| 2661 | return(); |
---|
| 2662 | } |
---|
| 2663 | if (size(#)==0) |
---|
| 2664 | { "ERROR: There are no parameters."; |
---|
| 2665 | return(); |
---|
| 2666 | } |
---|
| 2667 | if (typeof(#[size(#)])=="int") |
---|
| 2668 | { int v=#[size(#)]; |
---|
| 2669 | int gen_num=size(#)-1; |
---|
| 2670 | if (gen_num==0) |
---|
| 2671 | { "ERROR: There are no generators of a finite matrix group given."; |
---|
| 2672 | return(); |
---|
| 2673 | } |
---|
| 2674 | } |
---|
| 2675 | else |
---|
| 2676 | { int v=0; |
---|
| 2677 | int gen_num=size(#); |
---|
| 2678 | } |
---|
| 2679 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 2680 | // as the size of the matrices, as well |
---|
| 2681 | // as the number of primary invariants, |
---|
| 2682 | // we should get |
---|
[18bd9c] | 2683 | for (int i=1;i<=gen_num;i++) |
---|
[f34c37c] | 2684 | { if (typeof(#[i])=="matrix") |
---|
| 2685 | { if (nrows(#[i])<>n or ncols(#[i])<>n) |
---|
| 2686 | { "ERROR: The number of variables of the base ring needs to be the same"; |
---|
| 2687 | " as the dimension of the square matrices"; |
---|
| 2688 | return(); |
---|
| 2689 | } |
---|
| 2690 | } |
---|
| 2691 | else |
---|
| 2692 | { "ERROR: The first parameters should be a list of matrices"; |
---|
| 2693 | return(); |
---|
| 2694 | } |
---|
| 2695 | } |
---|
| 2696 | //---------------------------------------------------------------------------- |
---|
| 2697 | if (v && voice==2) |
---|
| 2698 | { ""; |
---|
| 2699 | } |
---|
| 2700 | //---------------------------- initializing variables ------------------------ |
---|
| 2701 | int dB; |
---|
| 2702 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
| 2703 | // space of invariants of degree d, |
---|
| 2704 | // newdim: dimension the ideal generated |
---|
| 2705 | // the primary invariants plus basis |
---|
| 2706 | // elements, dif=n-i-newdim, i.e. the |
---|
| 2707 | // number of new primary invairants that |
---|
| 2708 | // should be added in this degree - |
---|
| 2709 | ideal P,Pplus,sPplus,CI,B; // P: will contain primary invariants, |
---|
| 2710 | // Pplus: P+B, CI: a complete |
---|
| 2711 | // intersection with the same Hilbert |
---|
| 2712 | // function as P |
---|
[f54c83] | 2713 | ideal sP=groebner(P); |
---|
[f34c37c] | 2714 | dB=1; // used as degree bound - |
---|
| 2715 | d=0; // initializing |
---|
| 2716 | i=0; |
---|
| 2717 | intvec deg_vector; |
---|
| 2718 | //-------------------- loop that searches for primary invariants ------------- |
---|
| 2719 | while(1) // repeat until n primary invariants are |
---|
| 2720 | { // found - |
---|
[18bd9c] | 2721 | d++; // degree where we'll search |
---|
[f34c37c] | 2722 | if (v) |
---|
| 2723 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
| 2724 | } |
---|
| 2725 | B=invariant_basis(d,#[1..gen_num]); // basis of invariants of degree d |
---|
| 2726 | if (B[1]<>0) |
---|
| 2727 | { Pplus=P+B; |
---|
[f54c83] | 2728 | sPplus=groebner(Pplus); |
---|
[f34c37c] | 2729 | newdim=dim(sPplus); |
---|
| 2730 | dif=n-i-newdim; |
---|
| 2731 | } |
---|
| 2732 | else |
---|
| 2733 | { dif=0; |
---|
| 2734 | deg_vector=deg_vector,d; |
---|
| 2735 | } |
---|
| 2736 | if (dif<>0) // we have to find dif new primary |
---|
| 2737 | { // invariants |
---|
| 2738 | cd=size(B); |
---|
| 2739 | if (cd<>dif) |
---|
| 2740 | { P,sP,CI,dB=p_search(n,d,B,cd,P,sP,i,dif,dB,CI); |
---|
| 2741 | } |
---|
| 2742 | else // i.e. we can take all of B |
---|
[18bd9c] | 2743 | { for(j=i+1;j<=i+dif;j++) |
---|
[f34c37c] | 2744 | { CI=CI+ideal(var(j)^d); |
---|
| 2745 | } |
---|
| 2746 | dB=dB+dif*(d-1); |
---|
| 2747 | P=Pplus; |
---|
| 2748 | sP=sPplus; |
---|
| 2749 | } |
---|
| 2750 | if (v) |
---|
[18bd9c] | 2751 | { for (j=1;j<=size(P)-i;j++) |
---|
[f34c37c] | 2752 | { " We find: "+string(P[i+j]); |
---|
| 2753 | } |
---|
| 2754 | } |
---|
| 2755 | i=size(P); |
---|
| 2756 | if (i==n) // found all primary invariants |
---|
| 2757 | { if (v) |
---|
| 2758 | { ""; |
---|
| 2759 | " We found all primary invariants."; |
---|
| 2760 | ""; |
---|
| 2761 | } |
---|
| 2762 | return(matrix(P)); |
---|
| 2763 | } |
---|
| 2764 | } // done with degree d |
---|
| 2765 | else |
---|
| 2766 | { if (v) |
---|
| 2767 | { " None here..."; |
---|
| 2768 | } |
---|
| 2769 | } |
---|
| 2770 | } |
---|
| 2771 | } |
---|
| 2772 | example |
---|
[8632ac] | 2773 | { "EXAMPLE:"; echo=2; |
---|
[f34c37c] | 2774 | ring R=2,(x,y,z),dp; |
---|
| 2775 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 2776 | matrix P=primary_charp_without(A); |
---|
| 2777 | print(P); |
---|
| 2778 | } |
---|
[68e678] | 2779 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 2780 | |
---|
| 2781 | proc primary_invariants (list #) |
---|
| 2782 | "USAGE: primary_invariants(G1,G2,...[,flags]); |
---|
| 2783 | G1,G2,...: <matrices> generating a finite matrix group, flags: an |
---|
| 2784 | optional <intvec> with three entries, if the first one equals 0 (also |
---|
| 2785 | the default), the programme attempts to compute the Molien series and |
---|
| 2786 | Reynolds operator, if it equals 1, the programme is told that the |
---|
| 2787 | Molien series should not be computed, if it equals -1 characteristic 0 |
---|
| 2788 | is simulated, i.e. the Molien series is computed as if the base field |
---|
| 2789 | were characteristic 0 (the user must choose a field of large prime |
---|
| 2790 | characteristic, e.g. 32003) and if the first one is anything else, it |
---|
| 2791 | means that the characteristic of the base field divides the group |
---|
| 2792 | order, the second component should give the size of intervals between |
---|
| 2793 | canceling common factors in the expansion of the Molien series, 0 (the |
---|
| 2794 | default) means only once after generating all terms, in prime |
---|
| 2795 | characteristic also a negative number can be given to indicate that |
---|
| 2796 | common factors should always be canceled when the expansion is simple |
---|
[68e678] | 2797 | (the root of the extension field occurs not among the coefficients) |
---|
[f34c37c] | 2798 | DISPLAY: information about the various stages of the programme if the third |
---|
| 2799 | flag does not equal 0 |
---|
| 2800 | RETURN: primary invariants (type <matrix>) of the invariant ring and if |
---|
| 2801 | computable Reynolds operator (type <matrix>) and Molien series (type |
---|
[9f79aa] | 2802 | <matrix>) or ring name (type string) where the Molien series |
---|
| 2803 | can be found in the char p case; if the first flag is 1 and we are in |
---|
| 2804 | the non-modular case then an <intvec> is returned giving some of the |
---|
| 2805 | degrees where no non-trivial homogeneous invariants can be found |
---|
[f34c37c] | 2806 | THEORY: Bases of homogeneous invariants are generated successively and those |
---|
| 2807 | are chosen as primary invariants that lower the dimension of the ideal |
---|
| 2808 | generated by the previously found invariants (see paper \"Generating a |
---|
| 2809 | Noetherian Normalization of the Invariant Ring of a Finite Group\" by |
---|
[9f79aa] | 2810 | Decker, Heydtmann, Schreyer (1998)). |
---|
[68e678] | 2811 | EXAMPLE: example primary_invariants; shows an example |
---|
[f34c37c] | 2812 | " |
---|
| 2813 | { |
---|
| 2814 | // ----------------- checking input and setting flags ------------------------ |
---|
| 2815 | if (size(#)==0) |
---|
| 2816 | { "ERROR: There are no parameters."; |
---|
| 2817 | return(); |
---|
| 2818 | } |
---|
| 2819 | int ch=char(basering); // the algorithms depend very much on the |
---|
| 2820 | // characteristic of the ground field |
---|
| 2821 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 2822 | // as the size of the matrices, as well |
---|
| 2823 | // as the number of primary invariants, |
---|
| 2824 | // we should get |
---|
| 2825 | int gen_num; |
---|
| 2826 | int mol_flag,v; |
---|
| 2827 | if (typeof(#[size(#)])=="intvec") |
---|
| 2828 | { if (size(#[size(#)])<>3) |
---|
| 2829 | { "ERROR: <intvec> should have three entries."; |
---|
| 2830 | return(); |
---|
| 2831 | } |
---|
| 2832 | gen_num=size(#)-1; |
---|
| 2833 | mol_flag=#[size(#)][1]; |
---|
| 2834 | if (#[size(#)][2]<0 && (ch==0 or (ch<>0 && mol_flag==-1))) |
---|
| 2835 | { "ERROR: the second component of <intvec> should be >=0"; |
---|
| 2836 | return(); |
---|
| 2837 | } |
---|
| 2838 | int interval=#[size(#)][2]; |
---|
| 2839 | v=#[size(#)][3]; |
---|
| 2840 | if (gen_num==0) |
---|
| 2841 | { "ERROR: There are no generators of a finite matrix group given."; |
---|
| 2842 | return(); |
---|
| 2843 | } |
---|
| 2844 | } |
---|
| 2845 | else |
---|
| 2846 | { gen_num=size(#); |
---|
| 2847 | mol_flag=0; |
---|
| 2848 | int interval=0; |
---|
| 2849 | v=0; |
---|
| 2850 | } |
---|
[18bd9c] | 2851 | for (int i=1;i<=gen_num;i++) |
---|
[f34c37c] | 2852 | { if (typeof(#[i])=="matrix") |
---|
| 2853 | { if (nrows(#[i])<>n or ncols(#[i])<>n) |
---|
| 2854 | { "ERROR: The number of variables of the base ring needs to be the same"; |
---|
| 2855 | " as the dimension of the square matrices"; |
---|
| 2856 | return(); |
---|
| 2857 | } |
---|
| 2858 | } |
---|
| 2859 | else |
---|
| 2860 | { "ERROR: The first parameters should be a list of matrices"; |
---|
| 2861 | return(); |
---|
| 2862 | } |
---|
| 2863 | } |
---|
| 2864 | //---------------------------------------------------------------------------- |
---|
| 2865 | if (mol_flag==0) |
---|
| 2866 | { if (ch==0) |
---|
| 2867 | { matrix REY,M=reynolds_molien(#[1..gen_num],intvec(mol_flag,interval,v)); |
---|
| 2868 | // one will contain Reynolds operator and |
---|
| 2869 | // the other enumerator and denominator |
---|
| 2870 | // of Molien series |
---|
| 2871 | matrix P=primary_char0(REY,M,v); |
---|
| 2872 | return(P,REY,M); |
---|
| 2873 | } |
---|
| 2874 | else |
---|
| 2875 | { list L=group_reynolds(#[1..gen_num],v); |
---|
| 2876 | if (L[1]<>0) // testing whether we are in the modular |
---|
| 2877 | { string newring="aksldfalkdsflkj"; // case |
---|
| 2878 | if (minpoly==0) |
---|
| 2879 | { if (v) |
---|
| 2880 | { " We are dealing with the non-modular case."; |
---|
| 2881 | } |
---|
| 2882 | if (typeof(L[2])=="int") |
---|
| 2883 | { molien(L[3..size(L)],newring,L[2],intvec(mol_flag,interval,v)); |
---|
| 2884 | } |
---|
| 2885 | else |
---|
| 2886 | { molien(L[2..size(L)],newring,intvec(mol_flag,interval,v)); |
---|
| 2887 | } |
---|
| 2888 | matrix P=primary_charp(L[1],newring,v); |
---|
| 2889 | return(P,L[1],newring); |
---|
| 2890 | } |
---|
| 2891 | else |
---|
| 2892 | { if (v) |
---|
| 2893 | { " Since it is impossible for this programme to calculate the Molien series for"; |
---|
| 2894 | " invariant rings over extension fields of prime characteristic, we have to"; |
---|
| 2895 | " continue without it."; |
---|
| 2896 | ""; |
---|
| 2897 | |
---|
| 2898 | } |
---|
| 2899 | list l=primary_charp_no_molien(L[1],v); |
---|
| 2900 | if (size(l)==2) |
---|
| 2901 | { return(l[1],L[1],l[2]); |
---|
| 2902 | } |
---|
| 2903 | else |
---|
| 2904 | { return(l[1],L[1]); |
---|
| 2905 | } |
---|
| 2906 | } |
---|
| 2907 | } |
---|
| 2908 | else // the modular case |
---|
| 2909 | { if (v) |
---|
| 2910 | { " There is also no Molien series, we can make use of..."; |
---|
| 2911 | ""; |
---|
| 2912 | " We can start looking for primary invariants..."; |
---|
| 2913 | ""; |
---|
| 2914 | } |
---|
| 2915 | return(primary_charp_without(#[1..gen_num],v)); |
---|
| 2916 | } |
---|
| 2917 | } |
---|
| 2918 | } |
---|
| 2919 | if (mol_flag==1) // the user wants no calculation of the |
---|
| 2920 | { list L=group_reynolds(#[1..gen_num],v); // Molien series |
---|
| 2921 | if (ch==0) |
---|
| 2922 | { list l=primary_char0_no_molien(L[1],v); |
---|
| 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 |
---|
| 2931 | { if (L[1]<>0) // testing whether we are in the modular |
---|
| 2932 | { list l=primary_charp_no_molien(L[1],v); // case |
---|
| 2933 | if (size(l)==2) |
---|
| 2934 | { return(l[1],L[1],l[2]); |
---|
| 2935 | } |
---|
| 2936 | else |
---|
| 2937 | { return(l[1],L[1]); |
---|
| 2938 | } |
---|
| 2939 | } |
---|
| 2940 | else // the modular case |
---|
| 2941 | { if (v) |
---|
| 2942 | { " We can start looking for primary invariants..."; |
---|
| 2943 | ""; |
---|
| 2944 | } |
---|
| 2945 | return(primary_charp_without(#[1..gen_num],v)); |
---|
| 2946 | } |
---|
| 2947 | } |
---|
| 2948 | } |
---|
| 2949 | if (mol_flag==-1) |
---|
| 2950 | { if (ch==0) |
---|
| 2951 | { "ERROR: Characteristic 0 can only be simulated in characteristic p>>0."; |
---|
| 2952 | return(); |
---|
| 2953 | } |
---|
| 2954 | list L=group_reynolds(#[1..gen_num],v); |
---|
| 2955 | string newring="aksldfalkdsflkj"; |
---|
| 2956 | molien(L[2..size(L)],newring,intvec(1,interval,v)); |
---|
| 2957 | matrix P=primary_charp(L[1],newring,v); |
---|
| 2958 | return(P,L[1],newring); |
---|
| 2959 | } |
---|
| 2960 | else // the user specified that the |
---|
| 2961 | { if (ch==0) // characteristic divides the group order |
---|
| 2962 | { "ERROR: The characteristic cannot divide the group order when it is 0."; |
---|
| 2963 | return(); |
---|
| 2964 | } |
---|
| 2965 | if (v) |
---|
| 2966 | { ""; |
---|
| 2967 | } |
---|
| 2968 | return(primary_charp_without(#[1..gen_num],v)); |
---|
| 2969 | } |
---|
| 2970 | } |
---|
| 2971 | example |
---|
| 2972 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; |
---|
| 2973 | echo=2; |
---|
| 2974 | ring R=0,(x,y,z),dp; |
---|
| 2975 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 2976 | list L=primary_invariants(A); |
---|
| 2977 | print(L[1]); |
---|
| 2978 | } |
---|
| 2979 | |
---|
[68e678] | 2980 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 2981 | // This procedure finds dif primary invariants in degree d. It returns all |
---|
| 2982 | // primary invariants found so far. The coefficients lie in a field of |
---|
| 2983 | // characteristic 0. |
---|
[68e678] | 2984 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 2985 | proc search_random (int n,int d,ideal B,int cd,ideal P,int i,int dif,int dB,ideal CI,int max) |
---|
| 2986 | { string answer; |
---|
| 2987 | degBound=0; |
---|
| 2988 | int j,k,test_dim,flag; |
---|
| 2989 | matrix test_matrix[1][dif]; // the linear combination to test |
---|
| 2990 | intvec h; // Hilbert series |
---|
[18bd9c] | 2991 | for (j=i+1;j<=i+dif;j++) |
---|
[f34c37c] | 2992 | { CI=CI+ideal(var(j)^d); // homogeneous polynomial of the same |
---|
| 2993 | // degree as the one we're looking for |
---|
| 2994 | // is added |
---|
| 2995 | } |
---|
| 2996 | ideal TEST; |
---|
| 2997 | // h=hilb(std(CI),1); |
---|
| 2998 | dB=dB+dif*(d-1); // used as degBound |
---|
| 2999 | while (1) |
---|
| 3000 | { test_matrix=matrix(B)*random(max,cd,dif); |
---|
| 3001 | // degBound=dB; |
---|
| 3002 | TEST=P+ideal(test_matrix); |
---|
| 3003 | attrib(TEST,"isSB",1); |
---|
| 3004 | test_dim=dim(TEST); |
---|
| 3005 | // degBound=0; |
---|
| 3006 | if (n-test_dim==i+dif) |
---|
| 3007 | { break; |
---|
| 3008 | } |
---|
| 3009 | // degBound=dB; |
---|
[f54c83] | 3010 | test_dim=dim(groebner(TEST)); |
---|
[f34c37c] | 3011 | // test_dim=dim(std(TEST,h)); // Hilbert driven std-calculation |
---|
| 3012 | // degBound=0; |
---|
| 3013 | if (n-test_dim==i+dif) |
---|
| 3014 | { break; |
---|
| 3015 | } |
---|
| 3016 | else |
---|
| 3017 | { "HELP: The "+string(dif)+" random combination(s) of the "+string(cd)+" basis elements with"; |
---|
| 3018 | " coefficients in the range from -"+string(max)+" to "+string(max)+" did not lower the"; |
---|
| 3019 | " dimension by "+string(dif)+". You can abort, try again or give a new range:"; |
---|
| 3020 | answer=""; |
---|
| 3021 | while (answer<>"n |
---|
| 3022 | " && answer<>"y |
---|
| 3023 | ") |
---|
| 3024 | { " Do you want to abort (y/n)?"; |
---|
| 3025 | answer=read(""); |
---|
| 3026 | } |
---|
| 3027 | if (answer=="y |
---|
| 3028 | ") |
---|
| 3029 | { flag=1; |
---|
| 3030 | break; |
---|
| 3031 | } |
---|
| 3032 | answer=""; |
---|
| 3033 | while (answer<>"n |
---|
| 3034 | " && answer<>"y |
---|
| 3035 | ") |
---|
| 3036 | { " Do you want to try again (y/n)?"; |
---|
| 3037 | answer=read(""); |
---|
| 3038 | } |
---|
| 3039 | if (answer=="n |
---|
| 3040 | ") |
---|
| 3041 | { flag=1; |
---|
| 3042 | while (flag) |
---|
| 3043 | { " Give a new <int> > "+string(max)+" that bounds the range of coefficients:"; |
---|
| 3044 | answer=read(""); |
---|
[18bd9c] | 3045 | for (j=1;j<=size(answer)-1;j++) |
---|
| 3046 | { for (k=0;k<=9;k++) |
---|
[f34c37c] | 3047 | { if (answer[j]==string(k)) |
---|
| 3048 | { break; |
---|
| 3049 | } |
---|
| 3050 | } |
---|
| 3051 | if (k>9) |
---|
| 3052 | { flag=1; |
---|
| 3053 | break; |
---|
| 3054 | } |
---|
| 3055 | flag=0; |
---|
| 3056 | } |
---|
| 3057 | if (not(flag)) |
---|
[034ce1] | 3058 | { execute("test_dim="+string(answer[1..size(answer)])); |
---|
[f34c37c] | 3059 | if (test_dim<=max) |
---|
| 3060 | { flag=1; |
---|
| 3061 | } |
---|
| 3062 | else |
---|
| 3063 | { max=test_dim; |
---|
| 3064 | } |
---|
| 3065 | } |
---|
| 3066 | } |
---|
| 3067 | } |
---|
| 3068 | } |
---|
| 3069 | } |
---|
| 3070 | if (not(flag)) |
---|
| 3071 | { P[(i+1)..(i+dif)]=test_matrix[1,1..dif]; |
---|
| 3072 | } |
---|
| 3073 | return(P,CI,dB); |
---|
| 3074 | } |
---|
| 3075 | |
---|
[68e678] | 3076 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 3077 | // This procedure finds at most dif primary invariants in degree d. It returns |
---|
| 3078 | // all primary invariants found so far. The coefficients lie in the field of |
---|
| 3079 | // characteristic p>0. |
---|
[68e678] | 3080 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 3081 | proc p_search_random (int n,int d,ideal B,int cd,ideal P,int i,int dif,int dB,ideal CI,int max) |
---|
| 3082 | { string answer; |
---|
| 3083 | degBound=0; |
---|
| 3084 | int j,k,test_dim,flag; |
---|
| 3085 | matrix test_matrix[1][dif]; // the linear combination to test |
---|
| 3086 | intvec h; // Hilbert series |
---|
| 3087 | ideal TEST; |
---|
| 3088 | while (dif>0) |
---|
[18bd9c] | 3089 | { for (j=i+1;j<=i+dif;j++) |
---|
[f34c37c] | 3090 | { CI=CI+ideal(var(j)^d); // homogeneous polynomial of the same |
---|
| 3091 | // degree as the one we're looking for |
---|
| 3092 | // is added |
---|
| 3093 | } |
---|
| 3094 | // h=hilb(std(CI),1); |
---|
| 3095 | dB=dB+dif*(d-1); // used as degBound |
---|
| 3096 | test_matrix=matrix(B)*random(max,cd,dif); |
---|
| 3097 | // degBound=dB; |
---|
| 3098 | TEST=P+ideal(test_matrix); |
---|
| 3099 | attrib(TEST,"isSB",1); |
---|
| 3100 | test_dim=dim(TEST); |
---|
| 3101 | // degBound=0; |
---|
| 3102 | if (n-test_dim==i+dif) |
---|
| 3103 | { break; |
---|
| 3104 | } |
---|
| 3105 | // degBound=dB; |
---|
[f54c83] | 3106 | test_dim=dim(groebner(TEST)); |
---|
[f34c37c] | 3107 | // test_dim=dim(std(TEST,h)); // Hilbert driven std-calculation |
---|
| 3108 | // degBound=0; |
---|
| 3109 | if (n-test_dim==i+dif) |
---|
| 3110 | { break; |
---|
| 3111 | } |
---|
| 3112 | else |
---|
| 3113 | { "HELP: The "+string(dif)+" random combination(s) of the "+string(cd)+" basis elements with"; |
---|
| 3114 | " coefficients in the range from -"+string(max)+" to "+string(max)+" did not lower the"; |
---|
| 3115 | " dimension by "+string(dif)+". You can abort, try again, lower the number of"; |
---|
| 3116 | " combinations searched for by 1 or give a larger coefficient range:"; |
---|
| 3117 | answer=""; |
---|
| 3118 | while (answer<>"n |
---|
| 3119 | " && answer<>"y |
---|
| 3120 | ") |
---|
| 3121 | { " Do you want to abort (y/n)?"; |
---|
| 3122 | answer=read(""); |
---|
| 3123 | } |
---|
| 3124 | if (answer=="y |
---|
| 3125 | ") |
---|
| 3126 | { flag=1; |
---|
| 3127 | break; |
---|
| 3128 | } |
---|
| 3129 | answer=""; |
---|
| 3130 | while (answer<>"n |
---|
| 3131 | " && answer<>"y |
---|
| 3132 | ") |
---|
| 3133 | { " Do you want to try again (y/n)?"; |
---|
| 3134 | answer=read(""); |
---|
| 3135 | } |
---|
| 3136 | if (answer=="n |
---|
| 3137 | ") |
---|
| 3138 | { answer=""; |
---|
| 3139 | while (answer<>"n |
---|
| 3140 | " && answer<>"y |
---|
| 3141 | ") |
---|
| 3142 | { " Do you want to lower the number of combinations by 1 (y/n)?"; |
---|
| 3143 | answer=read(""); |
---|
| 3144 | } |
---|
| 3145 | if (answer=="y |
---|
| 3146 | ") |
---|
| 3147 | { dif=dif-1; |
---|
| 3148 | } |
---|
| 3149 | else |
---|
| 3150 | { flag=1; |
---|
| 3151 | while (flag) |
---|
| 3152 | { " Give a new <int> > "+string(max)+" that bounds the range of coefficients:"; |
---|
| 3153 | answer=read(""); |
---|
[18bd9c] | 3154 | for (j=1;j<=size(answer)-1;j++) |
---|
| 3155 | { for (k=0;k<=9;k++) |
---|
[f34c37c] | 3156 | { if (answer[j]==string(k)) |
---|
| 3157 | { break; |
---|
| 3158 | } |
---|
| 3159 | } |
---|
| 3160 | if (k>9) |
---|
| 3161 | { flag=1; |
---|
| 3162 | break; |
---|
| 3163 | } |
---|
| 3164 | flag=0; |
---|
| 3165 | } |
---|
| 3166 | if (not(flag)) |
---|
[034ce1] | 3167 | { execute("test_dim="+string(answer[1..size(answer)])); |
---|
[f34c37c] | 3168 | if (test_dim<=max) |
---|
| 3169 | { flag=1; |
---|
| 3170 | } |
---|
| 3171 | else |
---|
| 3172 | { max=test_dim; |
---|
| 3173 | } |
---|
| 3174 | } |
---|
| 3175 | } |
---|
| 3176 | } |
---|
| 3177 | } |
---|
| 3178 | } |
---|
| 3179 | CI=CI[1..i]; |
---|
| 3180 | dB=dB-dif*(d-1); |
---|
| 3181 | } |
---|
| 3182 | if (dif && not(flag)) |
---|
| 3183 | { P[(i+1)..(i+dif)]=test_matrix[1,1..dif]; |
---|
| 3184 | } |
---|
| 3185 | if (dif && flag) |
---|
| 3186 | { P[n+1]=0; |
---|
| 3187 | } |
---|
| 3188 | return(P,CI,dB); |
---|
| 3189 | } |
---|
[68e678] | 3190 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 3191 | |
---|
| 3192 | proc primary_char0_random (matrix REY,matrix M,int max,list #) |
---|
| 3193 | "USAGE: primary_char0_random(REY,M,r[,v]); |
---|
| 3194 | REY: a <matrix> representing the Reynolds operator, M: a 1x2 <matrix> |
---|
| 3195 | representing the Molien series, r: an <int> where -|r| to |r| is the |
---|
| 3196 | range of coefficients of the random combinations of bases elements, |
---|
| 3197 | v: an optional <int> |
---|
| 3198 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien and |
---|
| 3199 | M the one of molien or the second one of reynolds_molien |
---|
| 3200 | DISPLAY: information about the various stages of the programme if v does not |
---|
| 3201 | equal 0 |
---|
| 3202 | RETURN: primary invariants (type <matrix>) of the invariant ring |
---|
| 3203 | THEORY: Bases of homogeneous invariants are generated successively and random |
---|
| 3204 | linear combinations are chosen as primary invariants that lower the |
---|
| 3205 | dimension of the ideal generated by the previously found invariants |
---|
[9f79aa] | 3206 | (see \"Generating a Noetherian Normalization of the Invariant Ring of |
---|
| 3207 | a Finite Group\" by Decker, Heydtmann, Schreyer (1998)). |
---|
[68e678] | 3208 | EXAMPLE: example primary_char0_random; shows an example |
---|
[f34c37c] | 3209 | " |
---|
| 3210 | { degBound=0; |
---|
| 3211 | if (char(basering)<>0) |
---|
| 3212 | { "ERROR: primary_char0_random should only be used with rings of"; |
---|
| 3213 | " characteristic 0."; |
---|
| 3214 | return(); |
---|
| 3215 | } |
---|
| 3216 | //----------------- checking input and setting verbose mode ------------------ |
---|
| 3217 | if (size(#)>1) |
---|
| 3218 | { "ERROR: primary_char0_random can only have four parameters."; |
---|
| 3219 | return(); |
---|
| 3220 | } |
---|
| 3221 | if (size(#)==1) |
---|
| 3222 | { if (typeof(#[1])<>"int") |
---|
| 3223 | { "ERROR: The fourth parameter should be of type <int>."; |
---|
| 3224 | return(); |
---|
| 3225 | } |
---|
| 3226 | else |
---|
| 3227 | { int v=#[1]; |
---|
| 3228 | } |
---|
| 3229 | } |
---|
| 3230 | else |
---|
| 3231 | { int v=0; |
---|
| 3232 | } |
---|
| 3233 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 3234 | // as the size of the matrices, as well |
---|
| 3235 | // as the number of primary invariants, |
---|
| 3236 | // we should get |
---|
| 3237 | if (ncols(REY)<>n) |
---|
| 3238 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
| 3239 | return(); |
---|
| 3240 | } |
---|
| 3241 | if (ncols(M)<>2 or nrows(M)<>1) |
---|
| 3242 | { "ERROR: Second parameter ought to be the Molien series." |
---|
| 3243 | return(); |
---|
| 3244 | } |
---|
| 3245 | //---------------------------------------------------------------------------- |
---|
| 3246 | if (v && voice<>2) |
---|
| 3247 | { " We can start looking for primary invariants..."; |
---|
| 3248 | ""; |
---|
| 3249 | } |
---|
| 3250 | if (v && voice==2) |
---|
| 3251 | { ""; |
---|
| 3252 | } |
---|
| 3253 | //------------------------- initializing variables --------------------------- |
---|
| 3254 | int dB; |
---|
| 3255 | poly p(1..2); // p(1) will be used for single terms of |
---|
| 3256 | // the partial expansion, p(2) to store |
---|
| 3257 | p(1..2)=partial_molien(M,1); // the intermediate result - |
---|
| 3258 | poly v1=var(1); // we need v1 to split off coefficients |
---|
| 3259 | // in the partial expansion of M (which |
---|
| 3260 | // is in terms of the first variable) - |
---|
| 3261 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
| 3262 | // space of invariants of degree d, |
---|
| 3263 | // newdim: dimension the ideal generated |
---|
| 3264 | // the primary invariants plus basis |
---|
| 3265 | // elements, dif=n-i-newdim, i.e. the |
---|
| 3266 | // number of new primary invairants that |
---|
| 3267 | // should be added in this degree - |
---|
| 3268 | ideal P,Pplus,CI,B; // P: will contain primary invariants, |
---|
| 3269 | // Pplus: P+B,CI: a complete |
---|
| 3270 | // intersection with the same Hilbert |
---|
| 3271 | // function as P - |
---|
| 3272 | dB=1; // used as degree bound |
---|
| 3273 | int i=0; |
---|
| 3274 | //-------------- loop that searches for primary invariants ------------------ |
---|
| 3275 | while(1) // repeat until n primary invariants are |
---|
| 3276 | { // found - |
---|
| 3277 | p(1..2)=partial_molien(M,1,p(2)); // next term of the partial expansion - |
---|
| 3278 | d=deg(p(1)); // degree where we'll search - |
---|
| 3279 | cd=int(coef(p(1),v1)[2,1]); // dimension of the homogeneous space of |
---|
| 3280 | // inviarants of degree d |
---|
| 3281 | if (v) |
---|
| 3282 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
| 3283 | } |
---|
| 3284 | B=invariant_basis_reynolds(REY,d,intvec(cd,6)); // basis of invariants of |
---|
| 3285 | // degree d |
---|
| 3286 | if (B[1]<>0) |
---|
| 3287 | { Pplus=P+B; |
---|
[f54c83] | 3288 | newdim=dim(groebner(Pplus)); |
---|
[f34c37c] | 3289 | dif=n-i-newdim; |
---|
| 3290 | } |
---|
| 3291 | else |
---|
| 3292 | { dif=0; |
---|
| 3293 | } |
---|
| 3294 | if (dif<>0) // we have to find dif new primary |
---|
| 3295 | { // invariants |
---|
| 3296 | if (cd<>dif) |
---|
| 3297 | { P,CI,dB=search_random(n,d,B,cd,P,i,dif,dB,CI,max); // searching for |
---|
| 3298 | } // dif invariants - |
---|
| 3299 | else // i.e. we can take all of B |
---|
[18bd9c] | 3300 | { for(j=i+1;j>i+dif;j++) |
---|
[f34c37c] | 3301 | { CI=CI+ideal(var(j)^d); |
---|
| 3302 | } |
---|
| 3303 | dB=dB+dif*(d-1); |
---|
| 3304 | P=Pplus; |
---|
| 3305 | } |
---|
| 3306 | if (ncols(P)==i) |
---|
| 3307 | { "WARNING: The return value is not a set of primary invariants, but"; |
---|
| 3308 | " polynomials qualifying as the first "+string(i)+" primary invariants."; |
---|
| 3309 | return(matrix(P)); |
---|
| 3310 | } |
---|
| 3311 | if (v) |
---|
[18bd9c] | 3312 | { for (j=1;j<=dif;j++) |
---|
[f34c37c] | 3313 | { " We find: "+string(P[i+j]); |
---|
| 3314 | } |
---|
| 3315 | } |
---|
| 3316 | i=i+dif; |
---|
| 3317 | if (i==n) // found all primary invariants |
---|
| 3318 | { if (v) |
---|
| 3319 | { ""; |
---|
| 3320 | " We found all primary invariants."; |
---|
| 3321 | ""; |
---|
| 3322 | } |
---|
| 3323 | return(matrix(P)); |
---|
| 3324 | } |
---|
| 3325 | } // done with degree d |
---|
| 3326 | } |
---|
| 3327 | } |
---|
| 3328 | example |
---|
[9f79aa] | 3329 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
[f34c37c] | 3330 | ring R=0,(x,y,z),dp; |
---|
| 3331 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 3332 | matrix REY,M=reynolds_molien(A); |
---|
| 3333 | matrix P=primary_char0_random(REY,M,1); |
---|
| 3334 | print(P); |
---|
| 3335 | } |
---|
[68e678] | 3336 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 3337 | |
---|
| 3338 | proc primary_charp_random (matrix REY,string ring_name,int max,list #) |
---|
| 3339 | "USAGE: primary_charp_random(REY,ringname,r[,v]); |
---|
| 3340 | REY: a <matrix> representing the Reynolds operator, ringname: a |
---|
| 3341 | <string> giving the name of a ring where the Molien series is stored, |
---|
| 3342 | r: an <int> where -|r| to |r| is the range of coefficients of the |
---|
| 3343 | random combinations of bases elements, v: an optional <int> |
---|
| 3344 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien and |
---|
| 3345 | ringname gives the name of a ring of characteristic 0 that has been |
---|
| 3346 | created by molien or reynolds_molien |
---|
| 3347 | DISPLAY: information about the various stages of the programme if v does not |
---|
| 3348 | equal 0 |
---|
| 3349 | RETURN: primary invariants (type <matrix>) of the invariant ring |
---|
| 3350 | THEORY: Bases of homogeneous invariants are generated successively and random |
---|
| 3351 | linear combinations are chosen as primary invariants that lower the |
---|
| 3352 | dimension of the ideal generated by the previously found invariants |
---|
[9f79aa] | 3353 | (see \"Generating a Noetherian Normalization of the Invariant Ring of |
---|
| 3354 | a Finite Group\" by Decker, Heydtmann, Schreyer (1998)). |
---|
[68e678] | 3355 | EXAMPLE: example primary_charp_random; shows an example |
---|
[f34c37c] | 3356 | " |
---|
| 3357 | { degBound=0; |
---|
| 3358 | // ---------------- checking input and setting verbose mode ------------------ |
---|
| 3359 | if (char(basering)==0) |
---|
| 3360 | { "ERROR: primary_charp_random should only be used with rings of"; |
---|
| 3361 | " characteristic p>0."; |
---|
| 3362 | return(); |
---|
| 3363 | } |
---|
| 3364 | if (size(#)>1) |
---|
| 3365 | { "ERROR: primary_charp_random can only have four parameters."; |
---|
| 3366 | return(); |
---|
| 3367 | } |
---|
| 3368 | if (size(#)==1) |
---|
| 3369 | { if (typeof(#[1])<>"int") |
---|
| 3370 | { "ERROR: The fourth parameter should be of type <int>."; |
---|
| 3371 | return(); |
---|
| 3372 | } |
---|
| 3373 | else |
---|
| 3374 | { int v=#[1]; |
---|
| 3375 | } |
---|
| 3376 | } |
---|
| 3377 | else |
---|
| 3378 | { int v=0; |
---|
| 3379 | } |
---|
| 3380 | def br=basering; |
---|
| 3381 | int n=nvars(br); // n is the number of variables, as well |
---|
| 3382 | // as the size of the matrices, as well |
---|
| 3383 | // as the number of primary invariants, |
---|
| 3384 | // we should get |
---|
| 3385 | if (ncols(REY)<>n) |
---|
| 3386 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
| 3387 | return(); |
---|
| 3388 | } |
---|
| 3389 | if (typeof(`ring_name`)<>"ring") |
---|
| 3390 | { "ERROR: Second parameter ought to the name of a ring where the Molien"; |
---|
| 3391 | " is stored."; |
---|
| 3392 | return(); |
---|
| 3393 | } |
---|
| 3394 | //---------------------------------------------------------------------------- |
---|
| 3395 | if (v && voice<>2) |
---|
| 3396 | { " We can start looking for primary invariants..."; |
---|
| 3397 | ""; |
---|
| 3398 | } |
---|
| 3399 | if (v && voice==2) |
---|
| 3400 | { ""; |
---|
| 3401 | } |
---|
| 3402 | //----------------------- initializing variables ----------------------------- |
---|
| 3403 | int dB; |
---|
| 3404 | setring `ring_name`; // the Molien series is stores here - |
---|
| 3405 | poly p(1..2); // p(1) will be used for single terms of |
---|
| 3406 | // the partial expansion, p(2) to store |
---|
| 3407 | p(1..2)=partial_molien(M,1); // the intermediate result - |
---|
| 3408 | poly v1=var(1); // we need v1 to split off coefficients |
---|
| 3409 | // in the partial expansion of M (which |
---|
| 3410 | // is in terms of the first variable) |
---|
| 3411 | setring br; |
---|
| 3412 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
| 3413 | // space of invariants of degree d, |
---|
| 3414 | // newdim: dimension the ideal generated |
---|
| 3415 | // the primary invariants plus basis |
---|
| 3416 | // elements, dif=n-i-newdim, i.e. the |
---|
| 3417 | // number of new primary invairants that |
---|
| 3418 | // should be added in this degree - |
---|
| 3419 | ideal P,Pplus,CI,B; // P: will contain primary invariants, |
---|
| 3420 | // Pplus: P+B, CI: a complete |
---|
| 3421 | // intersection with the same Hilbert |
---|
| 3422 | // function as P - |
---|
| 3423 | dB=1; // used as degree bound |
---|
| 3424 | int i=0; |
---|
| 3425 | //---------------- loop that searches for primary invariants ----------------- |
---|
| 3426 | while(1) // repeat until n primary invariants are |
---|
| 3427 | { // found |
---|
| 3428 | setring `ring_name`; |
---|
| 3429 | p(1..2)=partial_molien(M,1,p(2)); // next term of the partial expansion - |
---|
| 3430 | d=deg(p(1)); // degree where we'll search - |
---|
| 3431 | cd=int(coef(p(1),v1)[2,1]); // dimension of the homogeneous space of |
---|
| 3432 | // inviarants of degree d |
---|
| 3433 | setring br; |
---|
| 3434 | if (v) |
---|
| 3435 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
| 3436 | } |
---|
| 3437 | B=invariant_basis_reynolds(REY,d,intvec(cd,6)); // basis of invariants of |
---|
| 3438 | // degree d |
---|
| 3439 | if (B[1]<>0) |
---|
| 3440 | { Pplus=P+B; |
---|
[f54c83] | 3441 | newdim=dim(groebner(Pplus)); |
---|
[f34c37c] | 3442 | dif=n-i-newdim; |
---|
| 3443 | } |
---|
| 3444 | else |
---|
| 3445 | { dif=0; |
---|
| 3446 | } |
---|
| 3447 | if (dif<>0) // we have to find dif new primary |
---|
| 3448 | { // invariants |
---|
| 3449 | if (cd<>dif) |
---|
| 3450 | { P,CI,dB=p_search_random(n,d,B,cd,P,i,dif,dB,CI,max); |
---|
| 3451 | } |
---|
| 3452 | else // i.e. we can take all of B |
---|
[18bd9c] | 3453 | { for(j=i+1;j>i+dif;j++) |
---|
[f34c37c] | 3454 | { CI=CI+ideal(var(j)^d); |
---|
| 3455 | } |
---|
| 3456 | dB=dB+dif*(d-1); |
---|
| 3457 | P=Pplus; |
---|
| 3458 | } |
---|
| 3459 | if (ncols(P)==n+1) |
---|
| 3460 | { "WARNING: The first return value is not a set of primary invariants,"; |
---|
| 3461 | " but polynomials qualifying as the first "+string(i)+" primary invariants."; |
---|
| 3462 | return(matrix(P)); |
---|
| 3463 | } |
---|
| 3464 | if (v) |
---|
[18bd9c] | 3465 | { for (j=1;j<=size(P)-i;j++) |
---|
[f34c37c] | 3466 | { " We find: "+string(P[i+j]); |
---|
| 3467 | } |
---|
| 3468 | } |
---|
| 3469 | i=size(P); |
---|
| 3470 | if (i==n) // found all primary invariants |
---|
| 3471 | { if (v) |
---|
| 3472 | { ""; |
---|
| 3473 | " We found all primary invariants."; |
---|
| 3474 | ""; |
---|
| 3475 | } |
---|
| 3476 | return(matrix(P)); |
---|
| 3477 | } |
---|
| 3478 | } // done with degree d |
---|
| 3479 | } |
---|
| 3480 | } |
---|
| 3481 | example |
---|
[9f79aa] | 3482 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7 (changed into char 3)"; echo=2; |
---|
[f34c37c] | 3483 | ring R=3,(x,y,z),dp; |
---|
| 3484 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 3485 | list L=group_reynolds(A); |
---|
| 3486 | string newring="alskdfj"; |
---|
| 3487 | molien(L[2..size(L)],newring); |
---|
| 3488 | matrix P=primary_charp_random(L[1],newring,1); |
---|
| 3489 | kill `newring`; |
---|
| 3490 | print(P); |
---|
| 3491 | } |
---|
[68e678] | 3492 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 3493 | |
---|
| 3494 | proc primary_char0_no_molien_random (matrix REY, int max, list #) |
---|
| 3495 | "USAGE: primary_char0_no_molien_random(REY,r[,v]); |
---|
| 3496 | REY: a <matrix> representing the Reynolds operator, r: an <int> where |
---|
| 3497 | -|r| to |r| is the range of coefficients of the random combinations of |
---|
| 3498 | bases elements, v: an optional <int> |
---|
| 3499 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien |
---|
| 3500 | DISPLAY: information about the various stages of the programme if v does not |
---|
| 3501 | equal 0 |
---|
| 3502 | RETURN: primary invariants (type <matrix>) of the invariant ring and an |
---|
| 3503 | <intvec> listing some of the degrees where no non-trivial homogeneous |
---|
| 3504 | invariants are to be found |
---|
| 3505 | THEORY: Bases of homogeneous invariants are generated successively and random |
---|
| 3506 | linear combinations are chosen as primary invariants that lower the |
---|
| 3507 | dimension of the ideal generated by the previously found invariants |
---|
[9f79aa] | 3508 | (see \"Generating a Noetherian Normalization of the Invariant Ring of |
---|
| 3509 | a Finite Group\" by Decker, Heydtmann, Schreyer (1998)). |
---|
[68e678] | 3510 | EXAMPLE: example primary_char0_no_molien_random; shows an example |
---|
[f34c37c] | 3511 | " |
---|
| 3512 | { degBound=0; |
---|
| 3513 | //-------------- checking input and setting verbose mode --------------------- |
---|
| 3514 | if (char(basering)<>0) |
---|
| 3515 | { "ERROR: primary_char0_no_molien_random should only be used with rings of"; |
---|
| 3516 | " characteristic 0."; |
---|
| 3517 | return(); |
---|
| 3518 | } |
---|
| 3519 | if (size(#)>1) |
---|
| 3520 | { "ERROR: primary_char0_no_molien_random can only have three parameters."; |
---|
| 3521 | return(); |
---|
| 3522 | } |
---|
| 3523 | if (size(#)==1) |
---|
| 3524 | { if (typeof(#[1])<>"int") |
---|
| 3525 | { "ERROR: The third parameter should be of type <int>."; |
---|
| 3526 | return(); |
---|
| 3527 | } |
---|
| 3528 | else |
---|
| 3529 | { int v=#[1]; |
---|
| 3530 | } |
---|
| 3531 | } |
---|
| 3532 | else |
---|
| 3533 | { int v=0; |
---|
| 3534 | } |
---|
| 3535 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 3536 | // as the size of the matrices, as well |
---|
| 3537 | // as the number of primary invariants, |
---|
| 3538 | // we should get |
---|
| 3539 | if (ncols(REY)<>n) |
---|
| 3540 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
| 3541 | return(); |
---|
| 3542 | } |
---|
| 3543 | //---------------------------------------------------------------------------- |
---|
| 3544 | if (v && voice<>2) |
---|
| 3545 | { " We can start looking for primary invariants..."; |
---|
| 3546 | ""; |
---|
| 3547 | } |
---|
| 3548 | if (v && voice==2) |
---|
| 3549 | { ""; |
---|
| 3550 | } |
---|
| 3551 | //----------------------- initializing variables ----------------------------- |
---|
| 3552 | int dB; |
---|
| 3553 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
| 3554 | // space of invariants of degree d, |
---|
| 3555 | // newdim: dimension the ideal generated |
---|
| 3556 | // the primary invariants plus basis |
---|
| 3557 | // elements, dif=n-i-newdim, i.e. the |
---|
| 3558 | // number of new primary invairants that |
---|
| 3559 | // should be added in this degree - |
---|
| 3560 | ideal P,Pplus,CI,B; // P: will contain primary invariants, |
---|
| 3561 | // Pplus: P+B, CI: a complete |
---|
| 3562 | // intersection with the same Hilbert |
---|
| 3563 | // function as P - |
---|
| 3564 | dB=1; // used as degree bound - |
---|
| 3565 | d=0; // initializing |
---|
| 3566 | int i=0; |
---|
| 3567 | intvec deg_vector; |
---|
| 3568 | //------------------ loop that searches for primary invariants --------------- |
---|
| 3569 | while(1) // repeat until n primary invariants are |
---|
| 3570 | { // found - |
---|
[18bd9c] | 3571 | d++; // degree where we'll search |
---|
[f34c37c] | 3572 | if (v) |
---|
| 3573 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
| 3574 | } |
---|
| 3575 | B=invariant_basis_reynolds(REY,d,intvec(-1,6)); // basis of invariants of |
---|
| 3576 | // degree d |
---|
| 3577 | if (B[1]<>0) |
---|
| 3578 | { Pplus=P+B; |
---|
[f54c83] | 3579 | newdim=dim(groebner(Pplus)); |
---|
[f34c37c] | 3580 | dif=n-i-newdim; |
---|
| 3581 | } |
---|
| 3582 | else |
---|
| 3583 | { dif=0; |
---|
| 3584 | deg_vector=deg_vector,d; |
---|
| 3585 | } |
---|
| 3586 | if (dif<>0) // we have to find dif new primary |
---|
| 3587 | { // invariants |
---|
| 3588 | cd=size(B); |
---|
| 3589 | if (cd<>dif) |
---|
| 3590 | { P,CI,dB=search_random(n,d,B,cd,P,i,dif,dB,CI,max); |
---|
| 3591 | } |
---|
| 3592 | else // i.e. we can take all of B |
---|
[18bd9c] | 3593 | { for(j=i+1;j<=i+dif;j++) |
---|
[f34c37c] | 3594 | { CI=CI+ideal(var(j)^d); |
---|
| 3595 | } |
---|
| 3596 | dB=dB+dif*(d-1); |
---|
| 3597 | P=Pplus; |
---|
| 3598 | } |
---|
| 3599 | if (ncols(P)==i) |
---|
| 3600 | { "WARNING: The first return value is not a set of primary invariants,"; |
---|
| 3601 | " but polynomials qualifying as the first "+string(i)+" primary invariants."; |
---|
| 3602 | return(matrix(P)); |
---|
| 3603 | } |
---|
| 3604 | if (v) |
---|
[18bd9c] | 3605 | { for (j=1;j<=dif;j++) |
---|
[f34c37c] | 3606 | { " We find: "+string(P[i+j]); |
---|
| 3607 | } |
---|
| 3608 | } |
---|
| 3609 | i=i+dif; |
---|
| 3610 | if (i==n) // found all primary invariants |
---|
| 3611 | { if (v) |
---|
| 3612 | { ""; |
---|
| 3613 | " We found all primary invariants."; |
---|
| 3614 | ""; |
---|
| 3615 | } |
---|
| 3616 | if (deg_vector==0) |
---|
| 3617 | { return(matrix(P)); |
---|
| 3618 | } |
---|
| 3619 | else |
---|
| 3620 | { return(matrix(P),compress(deg_vector)); |
---|
| 3621 | } |
---|
| 3622 | } |
---|
| 3623 | } // done with degree d |
---|
| 3624 | else |
---|
| 3625 | { if (v) |
---|
| 3626 | { " None here..."; |
---|
| 3627 | } |
---|
| 3628 | } |
---|
| 3629 | } |
---|
| 3630 | } |
---|
| 3631 | example |
---|
[9f79aa] | 3632 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
[f34c37c] | 3633 | ring R=0,(x,y,z),dp; |
---|
| 3634 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 3635 | list L=group_reynolds(A); |
---|
| 3636 | list l=primary_char0_no_molien_random(L[1],1); |
---|
| 3637 | print(l[1]); |
---|
| 3638 | } |
---|
[68e678] | 3639 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 3640 | |
---|
| 3641 | proc primary_charp_no_molien_random (matrix REY, int max, list #) |
---|
| 3642 | "USAGE: primary_charp_no_molien_random(REY,r[,v]); |
---|
| 3643 | REY: a <matrix> representing the Reynolds operator, r: an <int> where |
---|
| 3644 | -|r| to |r| is the range of coefficients of the random combinations of |
---|
| 3645 | bases elements, v: an optional <int> |
---|
| 3646 | ASSUME: REY is the first return value of group_reynolds or reynolds_molien |
---|
| 3647 | DISPLAY: information about the various stages of the programme if v does not |
---|
| 3648 | equal 0 |
---|
| 3649 | RETURN: primary invariants (type <matrix>) of the invariant ring and an |
---|
| 3650 | <intvec> listing some of the degrees where no non-trivial homogeneous |
---|
| 3651 | invariants are to be found |
---|
| 3652 | THEORY: Bases of homogeneous invariants are generated successively and random |
---|
| 3653 | linear combinations are chosen as primary invariants that lower the |
---|
| 3654 | dimension of the ideal generated by the previously found invariants |
---|
[9f79aa] | 3655 | (see \"Generating a Noetherian Normalization of the Invariant Ring of |
---|
| 3656 | a Finite Group\" by Decker, Heydtmann, Schreyer (1998)). |
---|
[68e678] | 3657 | EXAMPLE: example primary_charp_no_molien_random; shows an example |
---|
[f34c37c] | 3658 | " |
---|
| 3659 | { degBound=0; |
---|
| 3660 | //----------------- checking input and setting verbose mode ------------------ |
---|
| 3661 | if (char(basering)==0) |
---|
| 3662 | { "ERROR: primary_charp_no_molien_random should only be used with rings of"; |
---|
| 3663 | " characteristic p>0."; |
---|
| 3664 | return(); |
---|
| 3665 | } |
---|
| 3666 | if (size(#)>1) |
---|
| 3667 | { "ERROR: primary_charp_no_molien_random can only have three parameters."; |
---|
| 3668 | return(); |
---|
| 3669 | } |
---|
| 3670 | if (size(#)==1) |
---|
| 3671 | { if (typeof(#[1])<>"int") |
---|
| 3672 | { "ERROR: The third parameter should be of type <int>."; |
---|
| 3673 | return(); |
---|
| 3674 | } |
---|
| 3675 | else |
---|
| 3676 | { int v=#[1]; |
---|
| 3677 | } |
---|
| 3678 | } |
---|
| 3679 | else |
---|
| 3680 | { int v=0; |
---|
| 3681 | } |
---|
| 3682 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 3683 | // as the size of the matrices, as well |
---|
| 3684 | // as the number of primary invariants, |
---|
| 3685 | // we should get |
---|
| 3686 | if (ncols(REY)<>n) |
---|
| 3687 | { "ERROR: First parameter ought to be the Reynolds operator." |
---|
| 3688 | return(); |
---|
| 3689 | } |
---|
| 3690 | //---------------------------------------------------------------------------- |
---|
| 3691 | if (v && voice<>2) |
---|
| 3692 | { " We can start looking for primary invariants..."; |
---|
| 3693 | ""; |
---|
| 3694 | } |
---|
| 3695 | if (v && voice==2) |
---|
| 3696 | { ""; |
---|
| 3697 | } |
---|
| 3698 | //-------------------- initializing variables -------------------------------- |
---|
| 3699 | int dB; |
---|
| 3700 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
| 3701 | // space of invariants of degree d, |
---|
| 3702 | // newdim: dimension the ideal generated |
---|
| 3703 | // the primary invariants plus basis |
---|
| 3704 | // elements, dif=n-i-newdim, i.e. the |
---|
| 3705 | // number of new primary invairants that |
---|
| 3706 | // should be added in this degree - |
---|
| 3707 | ideal P,Pplus,CI,B; // P: will contain primary invariants, |
---|
| 3708 | // Pplus: P+B, CI: a complete |
---|
| 3709 | // intersection with the same Hilbert |
---|
| 3710 | // function as P - |
---|
| 3711 | dB=1; // used as degree bound - |
---|
| 3712 | d=0; // initializing |
---|
| 3713 | int i=0; |
---|
| 3714 | intvec deg_vector; |
---|
| 3715 | //------------------ loop that searches for primary invariants --------------- |
---|
| 3716 | while(1) // repeat until n primary invariants are |
---|
| 3717 | { // found - |
---|
[18bd9c] | 3718 | d++; // degree where we'll search |
---|
[f34c37c] | 3719 | if (v) |
---|
| 3720 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
| 3721 | } |
---|
| 3722 | B=invariant_basis_reynolds(REY,d,intvec(-1,6)); // basis of invariants of |
---|
| 3723 | // degree d |
---|
| 3724 | if (B[1]<>0) |
---|
| 3725 | { Pplus=P+B; |
---|
[f54c83] | 3726 | newdim=dim(groebner(Pplus)); |
---|
[f34c37c] | 3727 | dif=n-i-newdim; |
---|
| 3728 | } |
---|
| 3729 | else |
---|
| 3730 | { dif=0; |
---|
| 3731 | deg_vector=deg_vector,d; |
---|
| 3732 | } |
---|
| 3733 | if (dif<>0) // we have to find dif new primary |
---|
| 3734 | { // invariants |
---|
| 3735 | cd=size(B); |
---|
| 3736 | if (cd<>dif) |
---|
| 3737 | { P,CI,dB=p_search_random(n,d,B,cd,P,i,dif,dB,CI,max); |
---|
| 3738 | } |
---|
| 3739 | else // i.e. we can take all of B |
---|
[18bd9c] | 3740 | { for(j=i+1;j<=i+dif;j++) |
---|
[f34c37c] | 3741 | { CI=CI+ideal(var(j)^d); |
---|
| 3742 | } |
---|
| 3743 | dB=dB+dif*(d-1); |
---|
| 3744 | P=Pplus; |
---|
| 3745 | } |
---|
| 3746 | if (ncols(P)==n+1) |
---|
| 3747 | { "WARNING: The first return value is not a set of primary invariants,"; |
---|
| 3748 | " but polynomials qualifying as the first "+string(i)+" primary invariants."; |
---|
| 3749 | return(matrix(P)); |
---|
| 3750 | } |
---|
| 3751 | if (v) |
---|
[18bd9c] | 3752 | { for (j=1;j<=size(P)-i;j++) |
---|
[f34c37c] | 3753 | { " We find: "+string(P[i+j]); |
---|
| 3754 | } |
---|
| 3755 | } |
---|
| 3756 | i=size(P); |
---|
| 3757 | if (i==n) // found all primary invariants |
---|
| 3758 | { if (v) |
---|
| 3759 | { ""; |
---|
| 3760 | " We found all primary invariants."; |
---|
| 3761 | ""; |
---|
| 3762 | } |
---|
| 3763 | if (deg_vector==0) |
---|
| 3764 | { return(matrix(P)); |
---|
| 3765 | } |
---|
| 3766 | else |
---|
| 3767 | { return(matrix(P),compress(deg_vector)); |
---|
| 3768 | } |
---|
| 3769 | } |
---|
| 3770 | } // done with degree d |
---|
| 3771 | else |
---|
| 3772 | { if (v) |
---|
| 3773 | { " None here..."; |
---|
| 3774 | } |
---|
| 3775 | } |
---|
| 3776 | } |
---|
| 3777 | } |
---|
| 3778 | example |
---|
[9f79aa] | 3779 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7 (changed into char 3)"; echo=2; |
---|
[f34c37c] | 3780 | ring R=3,(x,y,z),dp; |
---|
| 3781 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 3782 | list L=group_reynolds(A); |
---|
| 3783 | list l=primary_charp_no_molien_random(L[1],1); |
---|
| 3784 | print(l[1]); |
---|
| 3785 | } |
---|
[68e678] | 3786 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 3787 | |
---|
| 3788 | proc primary_charp_without_random (list #) |
---|
| 3789 | "USAGE: primary_charp_without_random(G1,G2,...,r[,v]); |
---|
| 3790 | G1,G2,...: <matrices> generating a finite matrix group, r: an <int> |
---|
| 3791 | where -|r| to |r| is the range of coefficients of the random |
---|
| 3792 | combinations of bases elements, v: an optional <int> |
---|
| 3793 | DISPLAY: information about the various stages of the programme if v does not |
---|
| 3794 | equal 0 |
---|
| 3795 | RETURN: primary invariants (type <matrix>) of the invariant ring |
---|
| 3796 | THEORY: Bases of homogeneous invariants are generated successively and random |
---|
| 3797 | linear combinations are chosen as primary invariants that lower the |
---|
| 3798 | dimension of the ideal generated by the previously found invariants |
---|
[9f79aa] | 3799 | (see \"Generating a Noetherian Normalization of the Invariant Ring of |
---|
| 3800 | a Finite Group\" by Decker, Heydtmann, Schreyer (1998)). No Reynolds |
---|
| 3801 | operator or Molien series is used. |
---|
[68e678] | 3802 | EXAMPLE: example primary_charp_without_random; shows an example |
---|
[f34c37c] | 3803 | " |
---|
| 3804 | { degBound=0; |
---|
| 3805 | //--------------------- checking input and setting verbose mode -------------- |
---|
| 3806 | if (char(basering)==0) |
---|
| 3807 | { "ERROR: primary_charp_without_random should only be used with rings of"; |
---|
| 3808 | " characteristic 0."; |
---|
| 3809 | return(); |
---|
| 3810 | } |
---|
| 3811 | if (size(#)<2) |
---|
| 3812 | { "ERROR: There are too few parameters."; |
---|
| 3813 | return(); |
---|
| 3814 | } |
---|
| 3815 | if (typeof(#[size(#)])=="int" && typeof(#[size(#)-1])=="int") |
---|
| 3816 | { int v=#[size(#)]; |
---|
| 3817 | int max=#[size(#)-1]; |
---|
| 3818 | int gen_num=size(#)-2; |
---|
| 3819 | if (gen_num==0) |
---|
| 3820 | { "ERROR: There are no generators of a finite matrix group given."; |
---|
| 3821 | return(); |
---|
| 3822 | } |
---|
| 3823 | } |
---|
| 3824 | else |
---|
| 3825 | { if (typeof(#[size(#)])=="int") |
---|
| 3826 | { int max=#[size(#)]; |
---|
| 3827 | int v=0; |
---|
| 3828 | int gen_num=size(#)-1; |
---|
| 3829 | } |
---|
| 3830 | else |
---|
| 3831 | { "ERROR: The last parameter should be an <int>."; |
---|
| 3832 | return(); |
---|
| 3833 | } |
---|
| 3834 | } |
---|
| 3835 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 3836 | // as the size of the matrices, as well |
---|
| 3837 | // as the number of primary invariants, |
---|
| 3838 | // we should get |
---|
[18bd9c] | 3839 | for (int i=1;i<=gen_num;i++) |
---|
[f34c37c] | 3840 | { if (typeof(#[i])=="matrix") |
---|
| 3841 | { if (nrows(#[i])<>n or ncols(#[i])<>n) |
---|
| 3842 | { "ERROR: The number of variables of the base ring needs to be the same"; |
---|
| 3843 | " as the dimension of the square matrices"; |
---|
| 3844 | return(); |
---|
| 3845 | } |
---|
| 3846 | } |
---|
| 3847 | else |
---|
| 3848 | { "ERROR: The first parameters should be a list of matrices"; |
---|
| 3849 | return(); |
---|
| 3850 | } |
---|
| 3851 | } |
---|
| 3852 | //---------------------------------------------------------------------------- |
---|
| 3853 | if (v && voice==2) |
---|
| 3854 | { ""; |
---|
| 3855 | } |
---|
| 3856 | //---------------------------- initializing variables ------------------------ |
---|
| 3857 | int dB; |
---|
| 3858 | int j,d,cd,newdim,dif; // d: current degree, cd: dimension of |
---|
| 3859 | // space of invariants of degree d, |
---|
| 3860 | // newdim: dimension the ideal generated |
---|
| 3861 | // the primary invariants plus basis |
---|
| 3862 | // elements, dif=n-i-newdim, i.e. the |
---|
| 3863 | // number of new primary invairants that |
---|
| 3864 | // should be added in this degree - |
---|
| 3865 | ideal P,Pplus,CI,B; // P: will contain primary invariants, |
---|
| 3866 | // Pplus: P+B, CI: a complete |
---|
| 3867 | // intersection with the same Hilbert |
---|
| 3868 | // function as P - |
---|
| 3869 | dB=1; // used as degree bound - |
---|
| 3870 | d=0; // initializing |
---|
| 3871 | i=0; |
---|
| 3872 | intvec deg_vector; |
---|
| 3873 | //-------------------- loop that searches for primary invariants ------------- |
---|
| 3874 | while(1) // repeat until n primary invariants are |
---|
| 3875 | { // found - |
---|
[18bd9c] | 3876 | d++; // degree where we'll search |
---|
[f34c37c] | 3877 | if (v) |
---|
| 3878 | { " Computing primary invariants in degree "+string(d)+":"; |
---|
| 3879 | } |
---|
| 3880 | B=invariant_basis(d,#[1..gen_num]); // basis of invariants of degree d |
---|
| 3881 | if (B[1]<>0) |
---|
| 3882 | { Pplus=P+B; |
---|
[f54c83] | 3883 | newdim=dim(groebner(Pplus)); |
---|
[f34c37c] | 3884 | dif=n-i-newdim; |
---|
| 3885 | } |
---|
| 3886 | else |
---|
| 3887 | { dif=0; |
---|
| 3888 | deg_vector=deg_vector,d; |
---|
| 3889 | } |
---|
| 3890 | if (dif<>0) // we have to find dif new primary |
---|
| 3891 | { // invariants |
---|
| 3892 | cd=size(B); |
---|
| 3893 | if (cd<>dif) |
---|
| 3894 | { P,CI,dB=p_search_random(n,d,B,cd,P,i,dif,dB,CI,max); |
---|
| 3895 | } |
---|
| 3896 | else // i.e. we can take all of B |
---|
[18bd9c] | 3897 | { for(j=i+1;j<=i+dif;j++) |
---|
[f34c37c] | 3898 | { CI=CI+ideal(var(j)^d); |
---|
| 3899 | } |
---|
| 3900 | dB=dB+dif*(d-1); |
---|
| 3901 | P=Pplus; |
---|
| 3902 | } |
---|
| 3903 | if (ncols(P)==n+1) |
---|
| 3904 | { "WARNING: The first return value is not a set of primary invariants,"; |
---|
| 3905 | " but polynomials qualifying as the first "+string(i)+" primary invariants."; |
---|
| 3906 | return(matrix(P)); |
---|
| 3907 | } |
---|
| 3908 | if (v) |
---|
[18bd9c] | 3909 | { for (j=1;j<=size(P)-i;j++) |
---|
[f34c37c] | 3910 | { " We find: "+string(P[i+j]); |
---|
| 3911 | } |
---|
| 3912 | } |
---|
| 3913 | i=size(P); |
---|
| 3914 | if (i==n) // found all primary invariants |
---|
| 3915 | { if (v) |
---|
| 3916 | { ""; |
---|
| 3917 | " We found all primary invariants."; |
---|
| 3918 | ""; |
---|
| 3919 | } |
---|
| 3920 | return(matrix(P)); |
---|
| 3921 | } |
---|
| 3922 | } // done with degree d |
---|
| 3923 | else |
---|
| 3924 | { if (v) |
---|
| 3925 | { " None here..."; |
---|
| 3926 | } |
---|
| 3927 | } |
---|
| 3928 | } |
---|
| 3929 | } |
---|
| 3930 | example |
---|
[9f79aa] | 3931 | { "EXAMPLE:"; echo=2; |
---|
[f34c37c] | 3932 | ring R=2,(x,y,z),dp; |
---|
| 3933 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 3934 | matrix P=primary_charp_without_random(A,1); |
---|
| 3935 | print(P); |
---|
| 3936 | } |
---|
[68e678] | 3937 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 3938 | |
---|
| 3939 | proc primary_invariants_random (list #) |
---|
| 3940 | "USAGE: primary_invariants_random(G1,G2,...,r[,flags]); |
---|
| 3941 | G1,G2,...: <matrices> generating a finite matrix group, r: an <int> |
---|
| 3942 | where -|r| to |r| is the range of coefficients of the random |
---|
| 3943 | combinations of bases elements, flags: an optional <intvec> with three |
---|
| 3944 | entries, if the first one equals 0 (also the default), the programme |
---|
| 3945 | attempts to compute the Molien series and Reynolds operator, if it |
---|
| 3946 | equals 1, the programme is told that the Molien series should not be |
---|
| 3947 | computed, if it equals -1 characteristic 0 is simulated, i.e. the |
---|
| 3948 | Molien series is computed as if the base field were characteristic 0 |
---|
| 3949 | (the user must choose a field of large prime characteristic, e.g. |
---|
| 3950 | 32003) and if the first one is anything else, it means that the |
---|
| 3951 | characteristic of the base field divides the group order, the second |
---|
| 3952 | component should give the size of intervals between canceling common |
---|
| 3953 | factors in the expansion of the Molien series, 0 (the default) means |
---|
| 3954 | only once after generating all terms, in prime characteristic also a |
---|
| 3955 | negative number can be given to indicate that common factors should |
---|
| 3956 | always be canceled when the expansion is simple (the root of the |
---|
| 3957 | extension field does not occur among the coefficients) |
---|
| 3958 | DISPLAY: information about the various stages of the programme if the third |
---|
| 3959 | flag does not equal 0 |
---|
| 3960 | RETURN: primary invariants (type <matrix>) of the invariant ring and if |
---|
| 3961 | computable Reynolds operator (type <matrix>) and Molien series (type |
---|
| 3962 | <matrix>), if the first flag is 1 and we are in the non-modular case |
---|
| 3963 | then an <intvec> is returned giving some of the degrees where no |
---|
| 3964 | non-trivial homogeneous invariants can be found |
---|
| 3965 | THEORY: Bases of homogeneous invariants are generated successively and random |
---|
| 3966 | linear combinations are chosen as primary invariants that lower the |
---|
| 3967 | dimension of the ideal generated by the previously found invariants |
---|
[9f79aa] | 3968 | (see \"Generating a Noetherian Normalization of the Invariant Ring of |
---|
| 3969 | a Finite Group\" by Decker, Heydtmann, Schreyer (1998)). |
---|
[68e678] | 3970 | EXAMPLE: example primary_invariants_random; shows an example |
---|
[f34c37c] | 3971 | " |
---|
| 3972 | { |
---|
| 3973 | // ----------------- checking input and setting flags ------------------------ |
---|
| 3974 | if (size(#)<2) |
---|
| 3975 | { "ERROR: There are too few parameters."; |
---|
| 3976 | return(); |
---|
| 3977 | } |
---|
| 3978 | int ch=char(basering); // the algorithms depend very much on the |
---|
| 3979 | // characteristic of the ground field |
---|
| 3980 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 3981 | // as the size of the matrices, as well |
---|
| 3982 | // as the number of primary invariants, |
---|
| 3983 | // we should get |
---|
| 3984 | int gen_num; |
---|
| 3985 | int mol_flag,v; |
---|
| 3986 | if (typeof(#[size(#)])=="intvec" && typeof(#[size(#)-1])=="int") |
---|
| 3987 | { if (size(#[size(#)])<>3) |
---|
| 3988 | { "ERROR: <intvec> should have three entries."; |
---|
| 3989 | return(); |
---|
| 3990 | } |
---|
| 3991 | gen_num=size(#)-2; |
---|
| 3992 | mol_flag=#[size(#)][1]; |
---|
| 3993 | if (#[size(#)][2]<0 && (ch==0 or (ch<>0 && mol_flag<>0))) |
---|
| 3994 | { "ERROR: the second component of <intvec> should be >=0"; |
---|
| 3995 | return(); |
---|
| 3996 | } |
---|
| 3997 | int interval=#[size(#)][2]; |
---|
| 3998 | v=#[size(#)][3]; |
---|
| 3999 | int max=#[size(#)-1]; |
---|
| 4000 | if (gen_num==0) |
---|
| 4001 | { "ERROR: There are no generators of a finite matrix group given."; |
---|
| 4002 | return(); |
---|
| 4003 | } |
---|
| 4004 | } |
---|
| 4005 | else |
---|
| 4006 | { if (typeof(#[size(#)])=="int") |
---|
| 4007 | { gen_num=size(#)-1; |
---|
| 4008 | mol_flag=0; |
---|
| 4009 | int interval=0; |
---|
| 4010 | v=0; |
---|
| 4011 | int max=#[size(#)]; |
---|
| 4012 | } |
---|
| 4013 | else |
---|
| 4014 | { "ERROR: If the two last parameters are not <int> and <intvec>, the last"; |
---|
| 4015 | " parameter should be an <int>."; |
---|
| 4016 | return(); |
---|
| 4017 | } |
---|
| 4018 | } |
---|
[18bd9c] | 4019 | for (int i=1;i<=gen_num;i++) |
---|
[f34c37c] | 4020 | { if (typeof(#[i])=="matrix") |
---|
| 4021 | { if (nrows(#[i])<>n or ncols(#[i])<>n) |
---|
| 4022 | { "ERROR: The number of variables of the base ring needs to be the same"; |
---|
| 4023 | " as the dimension of the square matrices"; |
---|
| 4024 | return(); |
---|
| 4025 | } |
---|
| 4026 | } |
---|
| 4027 | else |
---|
| 4028 | { "ERROR: The first parameters should be a list of matrices"; |
---|
| 4029 | return(); |
---|
| 4030 | } |
---|
| 4031 | } |
---|
| 4032 | //---------------------------------------------------------------------------- |
---|
| 4033 | if (mol_flag==0) |
---|
| 4034 | { if (ch==0) |
---|
| 4035 | { matrix REY,M=reynolds_molien(#[1..gen_num],intvec(0,interval,v)); |
---|
| 4036 | // one will contain Reynolds operator and |
---|
| 4037 | // the other enumerator and denominator |
---|
| 4038 | // of Molien series |
---|
| 4039 | matrix P=primary_char0_random(REY,M,max,v); |
---|
| 4040 | return(P,REY,M); |
---|
| 4041 | } |
---|
| 4042 | else |
---|
| 4043 | { list L=group_reynolds(#[1..gen_num],v); |
---|
| 4044 | if (L[1]<>0) // testing whether we are in the modular |
---|
| 4045 | { string newring="aksldfalkdsflkj"; // case |
---|
| 4046 | if (minpoly==0) |
---|
| 4047 | { if (v) |
---|
| 4048 | { " We are dealing with the non-modular case."; |
---|
| 4049 | } |
---|
[9f79aa] | 4050 | if (typeof(L[2])=="int") |
---|
| 4051 | { molien(L[3..size(L)],newring,L[2],intvec(0,interval,v)); |
---|
| 4052 | } |
---|
| 4053 | else |
---|
| 4054 | { molien(L[2..size(L)],newring,intvec(0,interval,v)); |
---|
| 4055 | } |
---|
[f34c37c] | 4056 | matrix P=primary_charp_random(L[1],newring,max,v); |
---|
| 4057 | return(P,L[1],newring); |
---|
| 4058 | } |
---|
| 4059 | else |
---|
| 4060 | { if (v) |
---|
| 4061 | { " Since it is impossible for this programme to calculate the Molien series for"; |
---|
| 4062 | " invariant rings over extension fields of prime characteristic, we have to"; |
---|
| 4063 | " continue without it."; |
---|
| 4064 | ""; |
---|
| 4065 | |
---|
| 4066 | } |
---|
| 4067 | list l=primary_charp_no_molien_random(L[1],max,v); |
---|
| 4068 | if (size(l)==2) |
---|
| 4069 | { return(l[1],L[1],l[2]); |
---|
| 4070 | } |
---|
| 4071 | else |
---|
| 4072 | { return(l[1],L[1]); |
---|
| 4073 | } |
---|
| 4074 | } |
---|
| 4075 | } |
---|
| 4076 | else // the modular case |
---|
| 4077 | { if (v) |
---|
| 4078 | { " There is also no Molien series, we can make use of..."; |
---|
| 4079 | ""; |
---|
| 4080 | " We can start looking for primary invariants..."; |
---|
| 4081 | ""; |
---|
| 4082 | } |
---|
| 4083 | return(primary_charp_without_random(#[1..gen_num],max,v)); |
---|
| 4084 | } |
---|
| 4085 | } |
---|
| 4086 | } |
---|
| 4087 | if (mol_flag==1) // the user wants no calculation of the |
---|
| 4088 | { list L=group_reynolds(#[1..gen_num],v); // Molien series |
---|
| 4089 | if (ch==0) |
---|
| 4090 | { list l=primary_char0_no_molien_random(L[1],max,v); |
---|
| 4091 | if (size(l)==2) |
---|
| 4092 | { return(l[1],L[1],l[2]); |
---|
| 4093 | } |
---|
| 4094 | else |
---|
| 4095 | { return(l[1],L[1]); |
---|
| 4096 | } |
---|
| 4097 | } |
---|
| 4098 | else |
---|
| 4099 | { if (L[1]<>0) // testing whether we are in the modular |
---|
| 4100 | { list l=primary_charp_no_molien_random(L[1],max,v); // case |
---|
| 4101 | if (size(l)==2) |
---|
| 4102 | { return(l[1],L[1],l[2]); |
---|
| 4103 | } |
---|
| 4104 | else |
---|
| 4105 | { return(l[1],L[1]); |
---|
| 4106 | } |
---|
| 4107 | } |
---|
| 4108 | else // the modular case |
---|
| 4109 | { if (v) |
---|
| 4110 | { " We can start looking for primary invariants..."; |
---|
| 4111 | ""; |
---|
| 4112 | } |
---|
| 4113 | return(primary_charp_without_random(#[1..gen_num],max,v)); |
---|
| 4114 | } |
---|
| 4115 | } |
---|
| 4116 | } |
---|
| 4117 | if (mol_flag==-1) |
---|
| 4118 | { if (ch==0) |
---|
| 4119 | { "ERROR: Characteristic 0 can only be simulated in characteristic p>>0."; |
---|
| 4120 | return(); |
---|
| 4121 | } |
---|
| 4122 | list L=group_reynolds(#[1..gen_num],v); |
---|
| 4123 | string newring="aksldfalkdsflkj"; |
---|
[9f79aa] | 4124 | if (typeof(L[2])=="int") |
---|
| 4125 | { molien(L[3..size(L)],newring,L[2],intvec(0,interval,v)); |
---|
| 4126 | } |
---|
| 4127 | else |
---|
| 4128 | { molien(L[2..size(L)],newring,intvec(0,interval,v)); |
---|
| 4129 | } |
---|
[f34c37c] | 4130 | matrix P=primary_charp_random(L[1],newring,max,v); |
---|
| 4131 | return(P,L[1],newring); |
---|
| 4132 | } |
---|
| 4133 | else // the user specified that the |
---|
| 4134 | { if (ch==0) // characteristic divides the group order |
---|
| 4135 | { "ERROR: The characteristic cannot divide the group order when it is 0."; |
---|
| 4136 | return(); |
---|
| 4137 | } |
---|
| 4138 | if (v) |
---|
| 4139 | { ""; |
---|
| 4140 | } |
---|
| 4141 | return(primary_charp_without_random(#[1..gen_num],max,v)); |
---|
| 4142 | } |
---|
| 4143 | } |
---|
| 4144 | example |
---|
[9f79aa] | 4145 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
[f34c37c] | 4146 | ring R=0,(x,y,z),dp; |
---|
| 4147 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 4148 | list L=primary_invariants_random(A,1); |
---|
| 4149 | print(L[1]); |
---|
| 4150 | } |
---|
[68e678] | 4151 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 4152 | |
---|
| 4153 | proc concat_intmat(intmat A,intmat B) |
---|
| 4154 | { int n=nrows(A); |
---|
| 4155 | int m1=ncols(A); |
---|
| 4156 | int m2=ncols(B); |
---|
| 4157 | intmat C[n][m1+m2]; |
---|
| 4158 | C[1..n,1..m1]=A[1..n,1..m1]; |
---|
| 4159 | C[1..n,m1+1..m1+m2]=B[1..n,1..m2]; |
---|
| 4160 | return(C); |
---|
| 4161 | } |
---|
[68e678] | 4162 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 4163 | |
---|
| 4164 | proc power_products(intvec deg_vec,int d) |
---|
| 4165 | "USAGE: power_products(dv,d); |
---|
| 4166 | dv: an <intvec> giving the degrees of homogeneous polynomials, d: the |
---|
| 4167 | degree of the desired power products |
---|
| 4168 | RETURN: a size(dv)*m <intmat> where each column ought to be interpreted as |
---|
| 4169 | containing the exponents of the corresponding polynomials. The product |
---|
| 4170 | of the powers is then homogeneous of degree d. |
---|
[68e678] | 4171 | EXAMPLE: example power_products; shows an example |
---|
[f34c37c] | 4172 | " |
---|
| 4173 | { ring R=0,x,dp; |
---|
| 4174 | if (d<=0) |
---|
| 4175 | { "ERROR: The <int> may not be <= 0"; |
---|
| 4176 | return(); |
---|
| 4177 | } |
---|
| 4178 | int d_neu,j,nc; |
---|
| 4179 | int s=size(deg_vec); |
---|
| 4180 | intmat PP[s][1]; |
---|
| 4181 | intmat TEST[s][1]; |
---|
[18bd9c] | 4182 | for (int i=1;i<=s;i++) |
---|
[f34c37c] | 4183 | { if (i<0) |
---|
| 4184 | { "ERROR: The entries of <intvec> may not be <= 0"; |
---|
| 4185 | return(); |
---|
| 4186 | } |
---|
| 4187 | d_neu=d-deg_vec[i]; |
---|
| 4188 | if (d_neu>0) |
---|
| 4189 | { intmat PPd_neu=power_products(intvec(deg_vec[i..s]),d_neu); |
---|
| 4190 | if (size(ideal(PPd_neu))<>0) |
---|
| 4191 | { nc=ncols(PPd_neu); |
---|
| 4192 | intmat PPd_neu_gross[s][nc]; |
---|
| 4193 | PPd_neu_gross[i..s,1..nc]=PPd_neu[1..s-i+1,1..nc]; |
---|
[18bd9c] | 4194 | for (j=1;j<=nc;j++) |
---|
[f34c37c] | 4195 | { PPd_neu_gross[i,j]=PPd_neu_gross[i,j]+1; |
---|
| 4196 | } |
---|
| 4197 | PP=concat_intmat(PP,PPd_neu_gross); |
---|
| 4198 | kill PPd_neu_gross; |
---|
| 4199 | } |
---|
| 4200 | kill PPd_neu; |
---|
| 4201 | } |
---|
| 4202 | if (d_neu==0) |
---|
| 4203 | { intmat PPd_neu[s][1]; |
---|
| 4204 | PPd_neu[i,1]=1; |
---|
| 4205 | PP=concat_intmat(PP,PPd_neu); |
---|
| 4206 | kill PPd_neu; |
---|
| 4207 | } |
---|
| 4208 | } |
---|
| 4209 | if (matrix(PP)<>matrix(TEST)) |
---|
| 4210 | { PP=compress(PP); |
---|
| 4211 | } |
---|
| 4212 | return(PP); |
---|
| 4213 | } |
---|
| 4214 | example |
---|
[9f79aa] | 4215 | { "EXAMPLE:"; echo=2; |
---|
[f34c37c] | 4216 | intvec dv=5,5,5,10,10; |
---|
| 4217 | print(power_products(dv,10)); |
---|
| 4218 | print(power_products(dv,7)); |
---|
| 4219 | } |
---|
[68e678] | 4220 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 4221 | |
---|
| 4222 | proc secondary_char0 (matrix P, matrix REY, matrix M, list #) |
---|
| 4223 | "USAGE: secondary_char0(P,REY,M[,v]); |
---|
| 4224 | P: a 1xn <matrix> with primary invariants, REY: a gxn <matrix> |
---|
[68e678] | 4225 | representing the Reynolds operator, M: a 1x2 <matrix> giving numerator |
---|
[f34c37c] | 4226 | and denominator of the Molien series, v: an optional <int> |
---|
| 4227 | ASSUME: n is the number of variables of the basering, g the size of the group, |
---|
| 4228 | REY is the 1st return value of group_reynolds(), reynolds_molien() or |
---|
| 4229 | the second one of primary_invariants(), M the return value of molien() |
---|
| 4230 | or the second one of reynolds_molien() or the third one of |
---|
| 4231 | primary_invariants() |
---|
| 4232 | RETURN: secondary invariants of the invariant ring (type <matrix>) and |
---|
| 4233 | irreducible secondary invariants (type <matrix>) |
---|
| 4234 | DISPLAY: information if v does not equal 0 |
---|
[18bd9c] | 4235 | THEORY: The secondary invariants are calculated by finding a basis (in terms |
---|
| 4236 | of monomials) of the basering modulo the primary invariants, mapping |
---|
[68e678] | 4237 | those to invariants with the Reynolds operator and using these images |
---|
[9f79aa] | 4238 | or their power products such that they are linearly independent modulo |
---|
| 4239 | the primary invariants (see paper \"Some Algorithms in Invariant |
---|
| 4240 | Theory of Finite Groups\" by Kemper and Steel (1997)). |
---|
[68e678] | 4241 | EXAMPLE: example secondary_char0; shows an example |
---|
[f34c37c] | 4242 | " |
---|
| 4243 | { def br=basering; |
---|
| 4244 | degBound=0; |
---|
| 4245 | //----------------- checking input and setting verbose mode ------------------ |
---|
| 4246 | if (char(br)<>0) |
---|
| 4247 | { "ERROR: secondary_char0 should only be used with rings of characteristic 0."; |
---|
| 4248 | return(); |
---|
| 4249 | } |
---|
| 4250 | int i; |
---|
| 4251 | if (size(#)>0) |
---|
| 4252 | { if (typeof(#[size(#)])=="int") |
---|
| 4253 | { int v=#[size(#)]; |
---|
| 4254 | } |
---|
| 4255 | else |
---|
| 4256 | { int v=0; |
---|
| 4257 | } |
---|
| 4258 | } |
---|
| 4259 | else |
---|
| 4260 | { int v=0; |
---|
| 4261 | } |
---|
| 4262 | int n=nvars(br); // n is the number of variables, as well |
---|
| 4263 | // as the size of the matrices, as well |
---|
| 4264 | // as the number of primary invariants, |
---|
| 4265 | // we should get |
---|
| 4266 | if (ncols(P)<>n) |
---|
| 4267 | { "ERROR: The first parameter ought to be the matrix of the primary"; |
---|
| 4268 | " invariants." |
---|
| 4269 | return(); |
---|
| 4270 | } |
---|
| 4271 | if (ncols(REY)<>n) |
---|
| 4272 | { "ERROR: The second parameter ought to be the Reynolds operator." |
---|
| 4273 | return(); |
---|
| 4274 | } |
---|
| 4275 | if (ncols(M)<>2 or nrows(M)<>1) |
---|
| 4276 | { "ERROR: The third parameter ought to be the Molien series." |
---|
| 4277 | return(); |
---|
| 4278 | } |
---|
| 4279 | if (v && voice==2) |
---|
| 4280 | { ""; |
---|
| 4281 | } |
---|
| 4282 | int j, m, counter; |
---|
| 4283 | //- finding the polynomial giving number and degrees of secondary invariants - |
---|
| 4284 | poly p=1; |
---|
[18bd9c] | 4285 | for (j=1;j<=n;j++) // calculating the denominator of the |
---|
[f34c37c] | 4286 | { p=p*(1-var(1)^deg(P[j])); // Hilbert series of the ring generated |
---|
| 4287 | } // by the primary invariants - |
---|
| 4288 | matrix s[1][2]=M[1,1]*p,M[1,2]; // s is used for canceling |
---|
| 4289 | s=matrix(syz(ideal(s))); |
---|
| 4290 | p=s[2,1]; // the polynomial telling us where to |
---|
| 4291 | // search for secondary invariants |
---|
| 4292 | map slead=br,ideal(0); |
---|
| 4293 | p=1/slead(p)*p; // smallest term of p needs to be 1 |
---|
| 4294 | if (v) |
---|
| 4295 | { " Polynomial telling us where to look for secondary invariants:"; |
---|
| 4296 | " "+string(p); |
---|
| 4297 | ""; |
---|
| 4298 | } |
---|
| 4299 | matrix dimmat=coeffs(p,var(1)); // dimmat will contain the number of |
---|
| 4300 | // secondary invariants, we need to find |
---|
| 4301 | // of a certain degree - |
---|
| 4302 | m=nrows(dimmat); // m-1 is the highest degree |
---|
| 4303 | if (v) |
---|
| 4304 | { " In degree 0 we have: 1"; |
---|
| 4305 | ""; |
---|
| 4306 | } |
---|
| 4307 | //-------------------------- initializing variables -------------------------- |
---|
| 4308 | intmat PP; |
---|
| 4309 | poly pp; |
---|
| 4310 | int k; |
---|
| 4311 | intvec deg_vec; |
---|
[f54c83] | 4312 | ideal sP=groebner(ideal(P)); |
---|
[f34c37c] | 4313 | ideal TEST,B,IS; |
---|
| 4314 | ideal S=1; // 1 is the first secondary invariant - |
---|
| 4315 | //--------------------- generating secondary invariants ---------------------- |
---|
[18bd9c] | 4316 | for (i=2;i<=m;i++) // going through dimmat - |
---|
[f34c37c] | 4317 | { if (int(dimmat[i,1])<>0) // when it is == 0 we need to find 0 |
---|
| 4318 | { // elements in the current degree (i-1) |
---|
| 4319 | if (v) |
---|
| 4320 | { " Searching in degree "+string(i-1)+", we need to find "+string(int(dimmat[i,1]))+" invariant(s)..."; |
---|
| 4321 | } |
---|
| 4322 | TEST=sP; |
---|
| 4323 | counter=0; // we'll count up to degvec[i] |
---|
| 4324 | if (IS[1]<>0) |
---|
| 4325 | { PP=power_products(deg_vec,i-1); // finding power products of irreducible |
---|
| 4326 | } // secondary invariants |
---|
| 4327 | if (size(ideal(PP))<>0) |
---|
[18bd9c] | 4328 | { for (j=1;j<=ncols(PP);j++) // going through all the power products |
---|
[f34c37c] | 4329 | { pp=1; |
---|
[18bd9c] | 4330 | for (k=1;k<=nrows(PP);k++) |
---|
[f34c37c] | 4331 | { pp=pp*IS[1,k]^PP[k,j]; |
---|
| 4332 | } |
---|
| 4333 | if (reduce(pp,TEST)<>0) |
---|
| 4334 | { S=S,pp; |
---|
[18bd9c] | 4335 | counter++; |
---|
[f34c37c] | 4336 | if (v) |
---|
| 4337 | { " We find: "+string(pp); |
---|
| 4338 | } |
---|
| 4339 | if (int(dimmat[i,1])<>counter) |
---|
[73054bb] | 4340 | { // TEST=std(TEST+ideal(NF(pp,TEST))); // should be replaced by next |
---|
[f34c37c] | 4341 | // line soon |
---|
[8be7878] | 4342 | TEST=std(TEST,pp); |
---|
[f34c37c] | 4343 | } |
---|
| 4344 | else |
---|
| 4345 | { break; |
---|
| 4346 | } |
---|
| 4347 | } |
---|
| 4348 | } |
---|
| 4349 | } |
---|
| 4350 | if (int(dimmat[i,1])<>counter) |
---|
| 4351 | { B=sort_of_invariant_basis(sP,REY,i-1,int(dimmat[i,1])*6); // B contains |
---|
| 4352 | // images of kbase(sP,i-1) under the |
---|
| 4353 | // Reynolds operator that are linearly |
---|
| 4354 | // independent and that don't reduce to |
---|
| 4355 | // 0 modulo sP - |
---|
| 4356 | if (counter==0 && ncols(B)==int(dimmat[i,1])) // then we can take all of |
---|
| 4357 | { S=S,B; // B |
---|
| 4358 | IS=IS+B; |
---|
| 4359 | if (deg_vec[1]==0) |
---|
| 4360 | { deg_vec=i-1; |
---|
| 4361 | if (v) |
---|
| 4362 | { " We find: "+string(B[1]); |
---|
| 4363 | } |
---|
[18bd9c] | 4364 | for (j=2;j<=int(dimmat[i,1]);j++) |
---|
[f34c37c] | 4365 | { deg_vec=deg_vec,i-1; |
---|
| 4366 | if (v) |
---|
| 4367 | { " We find: "+string(B[j]); |
---|
| 4368 | } |
---|
| 4369 | } |
---|
| 4370 | } |
---|
| 4371 | else |
---|
[18bd9c] | 4372 | { for (j=1;j<=int(dimmat[i,1]);j++) |
---|
[f34c37c] | 4373 | { deg_vec=deg_vec,i-1; |
---|
| 4374 | if (v) |
---|
| 4375 | { " We find: "+string(B[j]); |
---|
| 4376 | } |
---|
| 4377 | } |
---|
| 4378 | } |
---|
| 4379 | } |
---|
| 4380 | else |
---|
| 4381 | { j=0; // j goes through all of B - |
---|
| 4382 | while (int(dimmat[i,1])<>counter) // need to find dimmat[i,1] |
---|
| 4383 | { // invariants that are linearly |
---|
| 4384 | // independent modulo TEST |
---|
[18bd9c] | 4385 | j++; |
---|
[f34c37c] | 4386 | if (reduce(B[j],TEST)<>0) // B[j] should be added |
---|
| 4387 | { S=S,B[j]; |
---|
| 4388 | IS=IS+ideal(B[j]); |
---|
| 4389 | if (deg_vec[1]==0) |
---|
| 4390 | { deg_vec[1]=i-1; |
---|
| 4391 | } |
---|
| 4392 | else |
---|
| 4393 | { deg_vec=deg_vec,i-1; |
---|
| 4394 | } |
---|
[18bd9c] | 4395 | counter++; |
---|
[f34c37c] | 4396 | if (v) |
---|
| 4397 | { " We find: "+string(B[j]); |
---|
| 4398 | } |
---|
| 4399 | if (int(dimmat[i,1])<>counter) |
---|
[73054bb] | 4400 | { //TEST=std(TEST+ideal(NF(B[j],TEST))); // should be replaced by |
---|
[f34c37c] | 4401 | // next line |
---|
[8be7878] | 4402 | TEST=std(TEST,B[j]); |
---|
[f34c37c] | 4403 | } |
---|
| 4404 | } |
---|
| 4405 | } |
---|
| 4406 | } |
---|
| 4407 | } |
---|
| 4408 | if (v) |
---|
| 4409 | { ""; |
---|
| 4410 | } |
---|
| 4411 | } |
---|
| 4412 | } |
---|
| 4413 | if (v) |
---|
| 4414 | { " We're done!"; |
---|
| 4415 | ""; |
---|
| 4416 | } |
---|
| 4417 | return(matrix(S),matrix(IS)); |
---|
| 4418 | } |
---|
| 4419 | example |
---|
[68e678] | 4420 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
[f34c37c] | 4421 | ring R=0,(x,y,z),dp; |
---|
| 4422 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 4423 | list L=primary_invariants(A); |
---|
| 4424 | matrix S,IS=secondary_char0(L[1..3]); |
---|
| 4425 | print(S); |
---|
| 4426 | print(IS); |
---|
| 4427 | } |
---|
[68e678] | 4428 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 4429 | |
---|
| 4430 | proc secondary_charp (matrix P, matrix REY, string ring_name, list #) |
---|
| 4431 | "USAGE: secondary_charp(P,REY,ringname[,v]); |
---|
| 4432 | P: a 1xn <matrix> with primary invariants, REY: a gxn <matrix> |
---|
| 4433 | representing the Reynolds operator, ringname: a <string> giving the |
---|
| 4434 | name of a ring of characteristic 0 where the Molien series is stored, |
---|
| 4435 | v: an optional <int> |
---|
| 4436 | ASSUME: n is the number of variables of the basering, g the size of the group, |
---|
| 4437 | REY is the 1st return value of group_reynolds(), reynolds_molien() or |
---|
[68e678] | 4438 | the second one of primary_invariants(), `ringname` is a ring of |
---|
[9f79aa] | 4439 | char 0 that has been created by molien() or reynolds_molien() or |
---|
| 4440 | primary_invariants() |
---|
[f34c37c] | 4441 | RETURN: secondary invariants of the invariant ring (type <matrix>) and |
---|
| 4442 | irreducible secondary invariants (type <matrix>) |
---|
| 4443 | DISPLAY: information if v does not equal 0 |
---|
[68e678] | 4444 | THEORY: Secondary invariants are calculated by finding a basis (in terms of |
---|
| 4445 | monomials) of the basering modulo primary invariants, mapping those |
---|
[f34c37c] | 4446 | to invariants with the Reynolds operator and using these images or |
---|
[9f79aa] | 4447 | their power products such that they are linearly independent modulo |
---|
| 4448 | the primary invariants (see paper \"Some Algorithms in Invariant |
---|
| 4449 | Theory of Finite Groups\" by Kemper and Steel (1997)). |
---|
[68e678] | 4450 | EXAMPLE: example secondary_charp; shows an example |
---|
[f34c37c] | 4451 | " |
---|
| 4452 | { def br=basering; |
---|
| 4453 | degBound=0; |
---|
| 4454 | //---------------- checking input and setting verbose mode ------------------- |
---|
| 4455 | if (char(br)==0) |
---|
| 4456 | { "ERROR: secondary_charp should only be used with rings of characteristic p>0."; |
---|
| 4457 | return(); |
---|
| 4458 | } |
---|
| 4459 | int i; |
---|
| 4460 | if (size(#)>0) |
---|
| 4461 | { if (typeof(#[size(#)])=="int") |
---|
| 4462 | { int v=#[size(#)]; |
---|
| 4463 | } |
---|
| 4464 | else |
---|
| 4465 | { int v=0; |
---|
| 4466 | } |
---|
| 4467 | } |
---|
| 4468 | else |
---|
| 4469 | { int v=0; |
---|
| 4470 | } |
---|
| 4471 | int n=nvars(br); // n is the number of variables, as well |
---|
| 4472 | // as the size of the matrices, as well |
---|
| 4473 | // as the number of primary invariants, |
---|
| 4474 | // we should get |
---|
| 4475 | if (ncols(P)<>n) |
---|
| 4476 | { "ERROR: The first parameter ought to be the matrix of the primary"; |
---|
| 4477 | " invariants." |
---|
| 4478 | return(); |
---|
| 4479 | } |
---|
| 4480 | if (ncols(REY)<>n) |
---|
| 4481 | { "ERROR: The second parameter ought to be the Reynolds operator." |
---|
| 4482 | return(); |
---|
| 4483 | } |
---|
| 4484 | if (typeof(`ring_name`)<>"ring") |
---|
| 4485 | { "ERROR: The <string> should give the name of the ring where the Molien." |
---|
| 4486 | " series is stored."; |
---|
| 4487 | return(); |
---|
| 4488 | } |
---|
| 4489 | if (v && voice==2) |
---|
| 4490 | { ""; |
---|
| 4491 | } |
---|
| 4492 | int j, m, counter, d; |
---|
| 4493 | intvec deg_dim_vec; |
---|
| 4494 | //- finding the polynomial giving number and degrees of secondary invariants - |
---|
[18bd9c] | 4495 | for (j=1;j<=n;j++) |
---|
[f34c37c] | 4496 | { deg_dim_vec[j]=deg(P[j]); |
---|
| 4497 | } |
---|
| 4498 | setring `ring_name`; |
---|
| 4499 | poly p=1; |
---|
[18bd9c] | 4500 | for (j=1;j<=n;j++) // calculating the denominator of the |
---|
[f34c37c] | 4501 | { p=p*(1-var(1)^deg_dim_vec[j]); // Hilbert series of the ring generated |
---|
| 4502 | } // by the primary invariants - |
---|
| 4503 | matrix s[1][2]=M[1,1]*p,M[1,2]; // s is used for canceling |
---|
| 4504 | s=matrix(syz(ideal(s))); |
---|
| 4505 | p=s[2,1]; // the polynomial telling us where to |
---|
| 4506 | // search for secondary invariants |
---|
| 4507 | map slead=basering,ideal(0); |
---|
| 4508 | p=1/slead(p)*p; // smallest term of p needs to be 1 |
---|
| 4509 | if (v) |
---|
| 4510 | { " Polynomial telling us where to look for secondary invariants:"; |
---|
| 4511 | " "+string(p); |
---|
| 4512 | ""; |
---|
| 4513 | } |
---|
| 4514 | matrix dimmat=coeffs(p,var(1)); // dimmat will contain the number of |
---|
| 4515 | // secondary invariants, we need to find |
---|
| 4516 | // of a certain degree - |
---|
| 4517 | m=nrows(dimmat); // m-1 is the highest degree |
---|
| 4518 | deg_dim_vec=1; |
---|
[18bd9c] | 4519 | for (j=2;j<=m;j++) |
---|
[f34c37c] | 4520 | { deg_dim_vec=deg_dim_vec,int(dimmat[j,1]); |
---|
| 4521 | } |
---|
| 4522 | if (v) |
---|
| 4523 | { " In degree 0 we have: 1"; |
---|
| 4524 | ""; |
---|
| 4525 | } |
---|
| 4526 | //------------------------ initializing variables ---------------------------- |
---|
| 4527 | setring br; |
---|
| 4528 | intmat PP; |
---|
| 4529 | poly pp; |
---|
| 4530 | int k; |
---|
| 4531 | intvec deg_vec; |
---|
[f54c83] | 4532 | ideal sP=groebner(ideal(P)); |
---|
[f34c37c] | 4533 | ideal TEST,B,IS; |
---|
| 4534 | ideal S=1; // 1 is the first secondary invariant |
---|
| 4535 | //------------------- generating secondary invariants ------------------------ |
---|
[18bd9c] | 4536 | for (i=2;i<=m;i++) // going through deg_dim_vec - |
---|
[f34c37c] | 4537 | { if (deg_dim_vec[i]<>0) // when it is == 0 we need to find 0 |
---|
| 4538 | { // elements in the current degree (i-1) |
---|
| 4539 | if (v) |
---|
| 4540 | { " Searching in degree "+string(i-1)+", we need to find "+string(deg_dim_vec[i])+" invariant(s)..."; |
---|
| 4541 | } |
---|
| 4542 | TEST=sP; |
---|
| 4543 | counter=0; // we'll count up to degvec[i] |
---|
| 4544 | if (IS[1]<>0) |
---|
| 4545 | { PP=power_products(deg_vec,i-1); // generating power products of |
---|
| 4546 | } // irreducible secondary invariants |
---|
| 4547 | if (size(ideal(PP))<>0) |
---|
[18bd9c] | 4548 | { for (j=1;j<=ncols(PP);j++) // going through all of those |
---|
[f34c37c] | 4549 | { pp=1; |
---|
[18bd9c] | 4550 | for (k=1;k<=nrows(PP);k++) |
---|
[f34c37c] | 4551 | { pp=pp*IS[1,k]^PP[k,j]; |
---|
| 4552 | } |
---|
| 4553 | if (reduce(pp,TEST)<>0) |
---|
| 4554 | { S=S,pp; |
---|
[18bd9c] | 4555 | counter++; |
---|
[f34c37c] | 4556 | if (v) |
---|
| 4557 | { " We find: "+string(pp); |
---|
| 4558 | } |
---|
| 4559 | if (deg_dim_vec[i]<>counter) |
---|
[73054bb] | 4560 | { //TEST=std(TEST+ideal(NF(pp,TEST))); // should be soon replaced by |
---|
[f34c37c] | 4561 | // next line |
---|
[8be7878] | 4562 | TEST=std(TEST,pp); |
---|
[f34c37c] | 4563 | } |
---|
| 4564 | else |
---|
| 4565 | { break; |
---|
| 4566 | } |
---|
| 4567 | } |
---|
| 4568 | } |
---|
| 4569 | } |
---|
| 4570 | if (deg_dim_vec[i]<>counter) |
---|
| 4571 | { B=sort_of_invariant_basis(sP,REY,i-1,deg_dim_vec[i]*6); // B contains |
---|
| 4572 | // images of kbase(sP,i-1) under the |
---|
| 4573 | // Reynolds operator that are linearly |
---|
| 4574 | // independent and that don't reduce to |
---|
| 4575 | // 0 modulo sP - |
---|
| 4576 | if (counter==0 && ncols(B)==deg_dim_vec[i]) // then we can add all of B |
---|
| 4577 | { S=S,B; |
---|
| 4578 | IS=IS+B; |
---|
| 4579 | if (deg_vec[1]==0) |
---|
| 4580 | { deg_vec=i-1; |
---|
| 4581 | if (v) |
---|
| 4582 | { " We find: "+string(B[1]); |
---|
| 4583 | } |
---|
[18bd9c] | 4584 | for (j=2;j<=deg_dim_vec[i];j++) |
---|
[f34c37c] | 4585 | { deg_vec=deg_vec,i-1; |
---|
| 4586 | if (v) |
---|
| 4587 | { " We find: "+string(B[j]); |
---|
| 4588 | } |
---|
| 4589 | } |
---|
| 4590 | } |
---|
| 4591 | else |
---|
[18bd9c] | 4592 | { for (j=1;j<=deg_dim_vec[i];j++) |
---|
[f34c37c] | 4593 | { deg_vec=deg_vec,i-1; |
---|
| 4594 | if (v) |
---|
| 4595 | { " We find: "+string(B[j]); |
---|
| 4596 | } |
---|
| 4597 | } |
---|
| 4598 | } |
---|
| 4599 | } |
---|
| 4600 | else |
---|
| 4601 | { j=0; // j goes through all of B - |
---|
| 4602 | while (deg_dim_vec[i]<>counter) // need to find deg_dim_vec[i] |
---|
| 4603 | { // invariants that are linearly |
---|
| 4604 | // independent modulo TEST |
---|
[18bd9c] | 4605 | j++; |
---|
[f34c37c] | 4606 | if (reduce(B[j],TEST)<>0) // B[j] should be added |
---|
| 4607 | { S=S,B[j]; |
---|
| 4608 | IS=IS+ideal(B[j]); |
---|
| 4609 | if (deg_vec[1]==0) |
---|
| 4610 | { deg_vec[1]=i-1; |
---|
| 4611 | } |
---|
| 4612 | else |
---|
| 4613 | { deg_vec=deg_vec,i-1; |
---|
| 4614 | } |
---|
[18bd9c] | 4615 | counter++; |
---|
[f34c37c] | 4616 | if (v) |
---|
| 4617 | { " We find: "+string(B[j]); |
---|
| 4618 | } |
---|
| 4619 | if (deg_dim_vec[i]<>counter) |
---|
[73054bb] | 4620 | { //TEST=std(TEST+ideal(NF(B[j],TEST))); // should be soon replaced |
---|
[f34c37c] | 4621 | // by next line |
---|
[8be7878] | 4622 | TEST=std(TEST,B[j]); |
---|
[f34c37c] | 4623 | } |
---|
| 4624 | } |
---|
| 4625 | } |
---|
| 4626 | } |
---|
| 4627 | } |
---|
| 4628 | if (v) |
---|
| 4629 | { ""; |
---|
| 4630 | } |
---|
| 4631 | } |
---|
| 4632 | } |
---|
| 4633 | if (v) |
---|
| 4634 | { " We're done!"; |
---|
| 4635 | ""; |
---|
| 4636 | } |
---|
| 4637 | if (ring_name=="aksldfalkdsflkj") |
---|
| 4638 | { kill `ring_name`; |
---|
| 4639 | } |
---|
| 4640 | return(matrix(S),matrix(IS)); |
---|
| 4641 | } |
---|
| 4642 | example |
---|
[9f79aa] | 4643 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7 (changed into char 3)"; echo=2; |
---|
[f34c37c] | 4644 | ring R=3,(x,y,z),dp; |
---|
| 4645 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 4646 | list L=primary_invariants(A); |
---|
| 4647 | matrix S,IS=secondary_charp(L[1..size(L)]); |
---|
| 4648 | print(S); |
---|
| 4649 | print(IS); |
---|
| 4650 | } |
---|
[68e678] | 4651 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 4652 | |
---|
| 4653 | proc secondary_no_molien (matrix P, matrix REY, list #) |
---|
| 4654 | "USAGE: secondary_no_molien(P,REY[,deg_vec,v]); |
---|
| 4655 | P: a 1xn <matrix> with primary invariants, REY: a gxn <matrix> |
---|
| 4656 | representing the Reynolds operator, deg_vec: an optional <intvec> |
---|
[18bd9c] | 4657 | listing some degrees where no non-trivial homogeneous invariants can |
---|
[68e678] | 4658 | be found, v: an optional <int> |
---|
[f34c37c] | 4659 | ASSUME: n is the number of variables of the basering, g the size of the group, |
---|
| 4660 | REY is the 1st return value of group_reynolds(), reynolds_molien() or |
---|
| 4661 | the second one of primary_invariants(), deg_vec is the second return |
---|
| 4662 | value of primary_char0_no_molien(), primary_charp_no_molien(), |
---|
| 4663 | primary_char0_no_molien_random() or primary_charp_no_molien_random() |
---|
| 4664 | RETURN: secondary invariants of the invariant ring (type <matrix>) |
---|
| 4665 | DISPLAY: information if v does not equal 0 |
---|
[68e678] | 4666 | THEORY: Secondary invariants are calculated by finding a basis (in terms of |
---|
[9f79aa] | 4667 | monomials) of the basering modulo primary invariants, mapping those to |
---|
| 4668 | invariants with the Reynolds operator and using these images as |
---|
[f34c37c] | 4669 | candidates for secondary invariants. |
---|
[68e678] | 4670 | EXAMPLE: example secondary_no_molien; shows an example |
---|
[f34c37c] | 4671 | " |
---|
| 4672 | { int i; |
---|
| 4673 | degBound=0; |
---|
| 4674 | //------------------ checking input and setting verbose ---------------------- |
---|
| 4675 | if (size(#)==1 or size(#)==2) |
---|
| 4676 | { if (typeof(#[size(#)])=="int") |
---|
| 4677 | { if (size(#)==2) |
---|
| 4678 | { if (typeof(#[size(#)-1])=="intvec") |
---|
| 4679 | { intvec deg_vec=#[size(#)-1]; |
---|
| 4680 | } |
---|
| 4681 | else |
---|
| 4682 | { "ERROR: the third parameter should be an <intvec>"; |
---|
| 4683 | return(); |
---|
| 4684 | } |
---|
| 4685 | } |
---|
| 4686 | int v=#[size(#)]; |
---|
| 4687 | } |
---|
| 4688 | else |
---|
| 4689 | { if (size(#)==1) |
---|
| 4690 | { if (typeof(#[size(#)])=="intvec") |
---|
| 4691 | { intvec deg_vec=#[size(#)]; |
---|
| 4692 | int v=0; |
---|
| 4693 | } |
---|
| 4694 | else |
---|
| 4695 | { "ERROR: the third parameter should be an <intvec>"; |
---|
| 4696 | return(); |
---|
| 4697 | } |
---|
| 4698 | } |
---|
| 4699 | else |
---|
| 4700 | { "ERROR: wrong list of parameters"; |
---|
| 4701 | return(); |
---|
| 4702 | } |
---|
| 4703 | } |
---|
| 4704 | } |
---|
| 4705 | else |
---|
| 4706 | { if (size(#)>2) |
---|
| 4707 | { "ERROR: there are too many parameters"; |
---|
| 4708 | return(); |
---|
| 4709 | } |
---|
| 4710 | int v=0; |
---|
| 4711 | } |
---|
| 4712 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 4713 | // as the size of the matrices, as well |
---|
| 4714 | // as the number of primary invariants, |
---|
| 4715 | // we should get |
---|
| 4716 | if (ncols(P)<>n) |
---|
| 4717 | { "ERROR: The first parameter ought to be the matrix of the primary"; |
---|
| 4718 | " invariants." |
---|
| 4719 | return(); |
---|
| 4720 | } |
---|
| 4721 | if (ncols(REY)<>n) |
---|
| 4722 | { "ERROR: The second parameter ought to be the Reynolds operator." |
---|
| 4723 | return(); |
---|
| 4724 | } |
---|
| 4725 | if (v && voice==2) |
---|
| 4726 | { ""; |
---|
| 4727 | } |
---|
| 4728 | int j, m, d; |
---|
| 4729 | int max=1; |
---|
[18bd9c] | 4730 | for (j=1;j<=n;j++) |
---|
[f34c37c] | 4731 | { max=max*deg(P[j]); |
---|
| 4732 | } |
---|
| 4733 | max=max/nrows(REY); |
---|
| 4734 | if (v) |
---|
| 4735 | { " We need to find "+string(max)+" secondary invariants."; |
---|
| 4736 | ""; |
---|
| 4737 | " In degree 0 we have: 1"; |
---|
| 4738 | ""; |
---|
| 4739 | } |
---|
| 4740 | //------------------------- initializing variables --------------------------- |
---|
[f54c83] | 4741 | ideal sP=groebner(ideal(P)); |
---|
[f34c37c] | 4742 | ideal B, TEST; |
---|
| 4743 | ideal S=1; // 1 is the first secondary invariant |
---|
| 4744 | int counter=1; |
---|
| 4745 | i=0; |
---|
| 4746 | if (defined(deg_vec)<>voice) |
---|
| 4747 | { intvec deg_vec; |
---|
| 4748 | } |
---|
| 4749 | int k=1; |
---|
| 4750 | //--------------------- generating secondary invariants ---------------------- |
---|
| 4751 | while (counter<>max) |
---|
[18bd9c] | 4752 | { i++; |
---|
[f34c37c] | 4753 | if (deg_vec[k]<>i) |
---|
| 4754 | { if (v) |
---|
| 4755 | { " Searching in degree "+string(i)+"..."; |
---|
| 4756 | } |
---|
| 4757 | B=sort_of_invariant_basis(sP,REY,i,max); // B contains images of |
---|
| 4758 | // kbase(sP,i) under the Reynolds |
---|
| 4759 | // operator that are linearly independent |
---|
| 4760 | // and that don't reduce to 0 modulo sP |
---|
| 4761 | TEST=sP; |
---|
[18bd9c] | 4762 | for (j=1;j<=ncols(B);j++) |
---|
[f34c37c] | 4763 | { // that are linearly independent modulo |
---|
| 4764 | // TEST |
---|
| 4765 | if (reduce(B[j],TEST)<>0) // B[j] should be added |
---|
| 4766 | { S=S,B[j]; |
---|
[18bd9c] | 4767 | counter++; |
---|
[f34c37c] | 4768 | if (v) |
---|
| 4769 | { " We find: "+string(B[j]); |
---|
| 4770 | } |
---|
| 4771 | if (counter==max) |
---|
| 4772 | { break; |
---|
| 4773 | } |
---|
| 4774 | else |
---|
| 4775 | { if (j<>ncols(B)) |
---|
[73054bb] | 4776 | { //TEST=std(TEST+ideal(NF(B[j],TEST))); // should soon be replaced by |
---|
[f34c37c] | 4777 | // next line |
---|
[8be7878] | 4778 | TEST=std(TEST,B[j]); |
---|
[f34c37c] | 4779 | } |
---|
| 4780 | } |
---|
| 4781 | } |
---|
| 4782 | } |
---|
| 4783 | } |
---|
| 4784 | else |
---|
| 4785 | { if (size(deg_vec)==k) |
---|
[73054bb] | 4786 | { k=1; } |
---|
[f34c37c] | 4787 | else |
---|
[73054bb] | 4788 | { k++; } |
---|
[f34c37c] | 4789 | } |
---|
| 4790 | } |
---|
| 4791 | if (v) |
---|
[73054bb] | 4792 | { ""; } |
---|
[f34c37c] | 4793 | if (v) |
---|
[73054bb] | 4794 | { " We're done!"; ""; } |
---|
[f34c37c] | 4795 | return(matrix(S)); |
---|
| 4796 | } |
---|
| 4797 | example |
---|
[68e678] | 4798 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
[f34c37c] | 4799 | ring R=0,(x,y,z),dp; |
---|
| 4800 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 4801 | list L=primary_invariants(A,intvec(1,1,0)); |
---|
| 4802 | matrix S=secondary_no_molien(L[1..3]); |
---|
| 4803 | print(S); |
---|
| 4804 | } |
---|
[68e678] | 4805 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 4806 | |
---|
| 4807 | proc secondary_and_irreducibles_no_molien (matrix P, matrix REY, list #) |
---|
| 4808 | "USAGE: secondary_and_irreducibles_no_molien(P,REY[,v]); |
---|
| 4809 | P: a 1xn <matrix> with primary invariants, REY: a gxn <matrix> |
---|
| 4810 | representing the Reynolds operator, v: an optional <int> |
---|
| 4811 | ASSUME: n is the number of variables of the basering, g the size of the group, |
---|
| 4812 | REY is the 1st return value of group_reynolds(), reynolds_molien() or |
---|
| 4813 | the second one of primary_invariants() |
---|
| 4814 | RETURN: secondary invariants of the invariant ring (type <matrix>) and |
---|
| 4815 | irreducible secondary invariants (type <matrix>) |
---|
| 4816 | DISPLAY: information if v does not equal 0 |
---|
[68e678] | 4817 | THEORY: Secondary invariants are calculated by finding a basis (in terms of |
---|
[9f79aa] | 4818 | monomials) of the basering modulo primary invariants, mapping those to |
---|
| 4819 | invariants with the Reynolds operator and using these images or their |
---|
| 4820 | power products such that they are linearly independent modulo the |
---|
[f34c37c] | 4821 | primary invariants (see paper \"Some Algorithms in Invariant Theory of |
---|
| 4822 | Finite Groups\" by Kemper and Steel (1997)). |
---|
[68e678] | 4823 | EXAMPLE: example secondary_and_irreducibles_no_molien; shows an example |
---|
[f34c37c] | 4824 | " |
---|
| 4825 | { int i; |
---|
| 4826 | degBound=0; |
---|
| 4827 | //--------------------- checking input and setting verbose mode -------------- |
---|
| 4828 | if (size(#)==1 or size(#)==2) |
---|
| 4829 | { if (typeof(#[size(#)])=="int") |
---|
| 4830 | { if (size(#)==2) |
---|
| 4831 | { if (typeof(#[size(#)-1])=="intvec") |
---|
| 4832 | { intvec deg_vec=#[size(#)-1]; |
---|
| 4833 | } |
---|
| 4834 | else |
---|
| 4835 | { "ERROR: the third parameter should be an <intvec>"; |
---|
| 4836 | return(); |
---|
| 4837 | } |
---|
| 4838 | } |
---|
| 4839 | int v=#[size(#)]; |
---|
| 4840 | } |
---|
| 4841 | else |
---|
| 4842 | { if (size(#)==1) |
---|
| 4843 | { if (typeof(#[size(#)])=="intvec") |
---|
| 4844 | { intvec deg_vec=#[size(#)]; |
---|
| 4845 | int v=0; |
---|
| 4846 | } |
---|
| 4847 | else |
---|
| 4848 | { "ERROR: the third parameter should be an <intvec>"; |
---|
| 4849 | return(); |
---|
| 4850 | } |
---|
| 4851 | } |
---|
| 4852 | else |
---|
| 4853 | { "ERROR: wrong list of parameters"; |
---|
| 4854 | return(); |
---|
| 4855 | } |
---|
| 4856 | } |
---|
| 4857 | } |
---|
| 4858 | else |
---|
| 4859 | { if (size(#)>2) |
---|
| 4860 | { "ERROR: there are too many parameters"; |
---|
| 4861 | return(); |
---|
| 4862 | } |
---|
| 4863 | int v=0; |
---|
| 4864 | } |
---|
| 4865 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 4866 | // as the size of the matrices, as well |
---|
| 4867 | // as the number of primary invariants, |
---|
| 4868 | // we should get |
---|
| 4869 | if (ncols(P)<>n) |
---|
| 4870 | { "ERROR: The first parameter ought to be the matrix of the primary"; |
---|
| 4871 | " invariants." |
---|
| 4872 | return(); |
---|
| 4873 | } |
---|
| 4874 | if (ncols(REY)<>n) |
---|
| 4875 | { "ERROR: The second parameter ought to be the Reynolds operator." |
---|
| 4876 | return(); |
---|
| 4877 | } |
---|
| 4878 | if (v && voice==2) |
---|
| 4879 | { ""; |
---|
| 4880 | } |
---|
| 4881 | int j, m, d; |
---|
| 4882 | int max=1; |
---|
[18bd9c] | 4883 | for (j=1;j<=n;j++) |
---|
[f34c37c] | 4884 | { max=max*deg(P[j]); |
---|
| 4885 | } |
---|
| 4886 | max=max/nrows(REY); |
---|
| 4887 | if (v) |
---|
| 4888 | { " We need to find "+string(max)+" secondary invariants."; |
---|
| 4889 | ""; |
---|
| 4890 | " In degree 0 we have: 1"; |
---|
| 4891 | ""; |
---|
| 4892 | } |
---|
| 4893 | //------------------------ initializing variables ---------------------------- |
---|
| 4894 | intmat PP; |
---|
| 4895 | poly pp; |
---|
| 4896 | int k; |
---|
| 4897 | intvec irreducible_deg_vec; |
---|
[f54c83] | 4898 | ideal sP=groebner(ideal(P)); |
---|
[f34c37c] | 4899 | ideal B,TEST,IS; |
---|
| 4900 | ideal S=1; // 1 is the first secondary invariant |
---|
| 4901 | int counter=1; |
---|
| 4902 | i=0; |
---|
| 4903 | if (defined(deg_vec)<>voice) |
---|
| 4904 | { intvec deg_vec; |
---|
| 4905 | } |
---|
| 4906 | int l=1; |
---|
| 4907 | //------------------- generating secondary invariants ------------------------ |
---|
| 4908 | while (counter<>max) |
---|
[18bd9c] | 4909 | { i++; |
---|
[f34c37c] | 4910 | if (deg_vec[l]<>i) |
---|
| 4911 | { if (v) |
---|
| 4912 | { " Searching in degree "+string(i)+"..."; |
---|
| 4913 | } |
---|
| 4914 | TEST=sP; |
---|
| 4915 | if (IS[1]<>0) |
---|
| 4916 | { PP=power_products(irreducible_deg_vec,i); // generating all power |
---|
| 4917 | } // products of irreducible secondary |
---|
| 4918 | // invariants |
---|
| 4919 | if (size(ideal(PP))<>0) |
---|
[18bd9c] | 4920 | { for (j=1;j<=ncols(PP);j++) // going through all those power products |
---|
[f34c37c] | 4921 | { pp=1; |
---|
[18bd9c] | 4922 | for (k=1;k<=nrows(PP);k++) |
---|
[f34c37c] | 4923 | { pp=pp*IS[1,k]^PP[k,j]; |
---|
| 4924 | } |
---|
| 4925 | if (reduce(pp,TEST)<>0) |
---|
| 4926 | { S=S,pp; |
---|
[18bd9c] | 4927 | counter++; |
---|
[f34c37c] | 4928 | if (v) |
---|
| 4929 | { " We find: "+string(pp); |
---|
| 4930 | } |
---|
| 4931 | if (counter<>max) |
---|
[73054bb] | 4932 | { //TEST=std(TEST+ideal(NF(pp,TEST))); // should soon be replaced by |
---|
[f34c37c] | 4933 | // next line |
---|
[8be7878] | 4934 | TEST=std(TEST,pp); |
---|
[f34c37c] | 4935 | } |
---|
| 4936 | else |
---|
| 4937 | { break; |
---|
| 4938 | } |
---|
| 4939 | } |
---|
| 4940 | } |
---|
| 4941 | } |
---|
| 4942 | if (max<>counter) |
---|
| 4943 | { B=sort_of_invariant_basis(sP,REY,i,max); // B contains images of |
---|
| 4944 | // kbase(sP,i) under the Reynolds |
---|
| 4945 | // operator that are linearly independent |
---|
| 4946 | // and that don't reduce to 0 modulo sP |
---|
[18bd9c] | 4947 | for (j=1;j<=ncols(B);j++) |
---|
[f34c37c] | 4948 | { if (reduce(B[j],TEST)<>0) // B[j] should be added |
---|
| 4949 | { S=S,B[j]; |
---|
| 4950 | IS=IS+ideal(B[j]); |
---|
| 4951 | if (irreducible_deg_vec[1]==0) |
---|
| 4952 | { irreducible_deg_vec[1]=i; |
---|
| 4953 | } |
---|
| 4954 | else |
---|
| 4955 | { irreducible_deg_vec=irreducible_deg_vec,i; |
---|
| 4956 | } |
---|
[18bd9c] | 4957 | counter++; |
---|
[f34c37c] | 4958 | if (v) |
---|
| 4959 | { " We find: "+string(B[j]); |
---|
| 4960 | } |
---|
| 4961 | if (counter==max) |
---|
| 4962 | { break; |
---|
| 4963 | } |
---|
| 4964 | else |
---|
| 4965 | { if (j<>ncols(B)) |
---|
[73054bb] | 4966 | { //TEST=std(TEST+ideal(NF(B[j],TEST))); // should soon be replaced |
---|
[f34c37c] | 4967 | // by next line |
---|
[8be7878] | 4968 | TEST=std(TEST,B[j]); |
---|
[f34c37c] | 4969 | } |
---|
| 4970 | } |
---|
| 4971 | } |
---|
| 4972 | } |
---|
| 4973 | } |
---|
| 4974 | } |
---|
| 4975 | else |
---|
| 4976 | { if (size(deg_vec)==l) |
---|
| 4977 | { l=1; |
---|
| 4978 | } |
---|
| 4979 | else |
---|
[18bd9c] | 4980 | { l++; |
---|
[f34c37c] | 4981 | } |
---|
| 4982 | } |
---|
| 4983 | } |
---|
| 4984 | if (v) |
---|
| 4985 | { ""; |
---|
| 4986 | } |
---|
| 4987 | if (v) |
---|
| 4988 | { " We're done!"; |
---|
| 4989 | ""; |
---|
| 4990 | } |
---|
| 4991 | return(matrix(S),matrix(IS)); |
---|
| 4992 | } |
---|
| 4993 | example |
---|
[68e678] | 4994 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
[f34c37c] | 4995 | ring R=0,(x,y,z),dp; |
---|
| 4996 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 4997 | list L=primary_invariants(A,intvec(1,1,0)); |
---|
| 4998 | matrix S,IS=secondary_and_irreducibles_no_molien(L[1..2]); |
---|
| 4999 | print(S); |
---|
| 5000 | print(IS); |
---|
| 5001 | } |
---|
[68e678] | 5002 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 5003 | |
---|
| 5004 | proc secondary_not_cohen_macaulay (matrix P, list #) |
---|
| 5005 | "USAGE: secondary_not_cohen_macaulay(P,G1,G2,...[,v]); |
---|
| 5006 | P: a 1xn <matrix> with primary invariants, G1,G2,...: nxn <matrices> |
---|
| 5007 | generating a finite matrix group, v: an optional <int> |
---|
| 5008 | ASSUME: n is the number of variables of the basering |
---|
| 5009 | RETURN: secondary invariants of the invariant ring (type <matrix>) |
---|
| 5010 | DISPLAY: information if v does not equal 0 |
---|
[68e678] | 5011 | THEORY: Secondary invariants are generated following \"Generating Invariant |
---|
[9f79aa] | 5012 | Rings of Finite Groups over Arbitrary Fields\" by Kemper (1996). |
---|
[68e678] | 5013 | EXAMPLE: example secondary_not_cohen_macaulay; shows an example |
---|
[f34c37c] | 5014 | " |
---|
| 5015 | { int i, j; |
---|
| 5016 | degBound=0; |
---|
| 5017 | def br=basering; |
---|
| 5018 | int n=nvars(br); // n is the number of variables, as well |
---|
| 5019 | // as the size of the matrices, as well |
---|
| 5020 | // as the number of primary invariants, |
---|
| 5021 | // we should get - |
---|
| 5022 | if (size(#)>0) // checking input and setting verbose |
---|
| 5023 | { if (typeof(#[size(#)])=="int") |
---|
| 5024 | { int gen_num=size(#)-1; |
---|
| 5025 | if (gen_num==0) |
---|
| 5026 | { "ERROR: There are no generators of the finite matrix group given."; |
---|
| 5027 | return(); |
---|
| 5028 | } |
---|
| 5029 | int v=#[size(#)]; |
---|
[18bd9c] | 5030 | for (i=1;i<=gen_num;i++) |
---|
[f34c37c] | 5031 | { if (typeof(#[i])<>"matrix") |
---|
| 5032 | { "ERROR: These parameters should be generators of the finite matrix group."; |
---|
| 5033 | return(); |
---|
| 5034 | } |
---|
| 5035 | if ((n<>nrows(#[i])) or (n<>ncols(#[i]))) |
---|
| 5036 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
| 5037 | return(); |
---|
| 5038 | } |
---|
| 5039 | } |
---|
| 5040 | } |
---|
| 5041 | else |
---|
| 5042 | { int v=0; |
---|
| 5043 | int gen_num=size(#); |
---|
[18bd9c] | 5044 | for (i=1;i<=gen_num;i++) |
---|
[f34c37c] | 5045 | { if (typeof(#[i])<>"matrix") |
---|
| 5046 | { "ERROR: These parameters should be generators of the finite matrix group."; |
---|
| 5047 | return(); |
---|
| 5048 | } |
---|
| 5049 | if ((n<>nrows(#[i])) or (n<>ncols(#[i]))) |
---|
| 5050 | { "ERROR: matrices need to be square and of the same dimensions"; |
---|
| 5051 | return(); |
---|
| 5052 | } |
---|
| 5053 | } |
---|
| 5054 | } |
---|
| 5055 | } |
---|
| 5056 | else |
---|
| 5057 | { "ERROR: There are no generators of the finite matrix group given."; |
---|
| 5058 | return(); |
---|
| 5059 | } |
---|
| 5060 | if (ncols(P)<>n) |
---|
| 5061 | { "ERROR: The first parameter ought to be the matrix of the primary"; |
---|
| 5062 | " invariants." |
---|
| 5063 | return(); |
---|
| 5064 | } |
---|
| 5065 | if (v && voice==2) |
---|
| 5066 | { ""; |
---|
| 5067 | } |
---|
| 5068 | ring alskdfalkdsj=0,x,dp; |
---|
| 5069 | matrix M[1][2]=1,(1-x)^n; // we look at our primary invariants as |
---|
| 5070 | export alskdfalkdsj; |
---|
| 5071 | export M; |
---|
| 5072 | setring br; // such of the subgroup that only |
---|
| 5073 | matrix REY=matrix(maxideal(1)); // contains the identity, this means that |
---|
| 5074 | // ch does not divide the order anymore, |
---|
| 5075 | // this means that we can make use of the |
---|
| 5076 | // Molien series again - M[1,1]/M[1,2] is |
---|
| 5077 | // the Molien series of that group, we |
---|
| 5078 | // now calculate the secondary invariants |
---|
| 5079 | // of this subgroup in the usual fashion |
---|
| 5080 | // where the primary invariants are the |
---|
| 5081 | // ones from the bigger group |
---|
| 5082 | if (v) |
---|
| 5083 | { " The procedure secondary_charp() is called to calculate secondary invariants"; |
---|
| 5084 | " of the invariant ring of the trivial group with respect to the primary"; |
---|
| 5085 | " invariants found previously."; |
---|
| 5086 | ""; |
---|
| 5087 | } |
---|
[9f79aa] | 5088 | matrix trivialS, trivialSI=secondary_charp(P,REY,"alskdfalkdsj",v); |
---|
| 5089 | kill trivialSI; |
---|
[f34c37c] | 5090 | kill alskdfalkdsj; |
---|
| 5091 | // now we have those secondary invariants |
---|
| 5092 | int k=ncols(trivialS); // k is the number of the secondary |
---|
| 5093 | // invariants, we just calculated |
---|
| 5094 | if (v) |
---|
| 5095 | { " We calculate secondary invariants from the ones found for the trivial"; |
---|
| 5096 | " subgroup."; |
---|
| 5097 | ""; |
---|
| 5098 | } |
---|
| 5099 | map f; // used to let generators act on |
---|
| 5100 | // secondary invariants with respect to |
---|
| 5101 | // the trivial group - |
---|
| 5102 | matrix M(1)[gen_num][k]; // M(1) will contain a module |
---|
| 5103 | ideal B; |
---|
[18bd9c] | 5104 | for (i=1;i<=gen_num;i++) |
---|
[9f79aa] | 5105 | { B=ideal(matrix(maxideal(1))*transpose(#[i])); // image of the various |
---|
[f34c37c] | 5106 | // variables under the i-th generator - |
---|
| 5107 | f=br,B; // the corresponding mapping - |
---|
| 5108 | B=f(trivialS)-trivialS; // these relations should be 0 - |
---|
| 5109 | M(1)[i,1..k]=B[1..k]; // we will look for the syzygies of M(1) |
---|
| 5110 | } |
---|
[8632ac] | 5111 | //intvec save_opts=option(get); |
---|
| 5112 | //option(returnSB,redSB); |
---|
| 5113 | //module M(2)=syz(M(1)); // nres(M(1),2)[2]; |
---|
| 5114 | //option(set,save_opts); |
---|
[f34c37c] | 5115 | module M(2)=nres(M(1),2)[2]; |
---|
| 5116 | int m=ncols(M(2)); // number of generators of the module |
---|
| 5117 | // M(2) - |
---|
| 5118 | // the following steps calculates the intersection of the module M(2) with |
---|
| 5119 | // the algebra A^k where A denote the subalgebra of the usual polynomial |
---|
| 5120 | // ring, generated by the primary invariants |
---|
| 5121 | string mp=string(minpoly); // generating a ring where we can do |
---|
| 5122 | // elimination |
---|
[034ce1] | 5123 | execute("ring R=("+charstr(br)+"),(x(1..n),y(1..n),h),dp;"); |
---|
| 5124 | execute("minpoly=number("+mp+");"); |
---|
[f34c37c] | 5125 | map f=br,maxideal(1); // canonical mapping |
---|
| 5126 | matrix M[k][m+k*n]; |
---|
| 5127 | M[1..k,1..m]=matrix(f(M(2))); // will contain a module - |
---|
| 5128 | matrix P=f(P); // primary invariants in the new ring |
---|
[18bd9c] | 5129 | for (i=1;i<=n;i++) |
---|
| 5130 | { for (j=1;j<=k;j++) |
---|
[f34c37c] | 5131 | { M[j,m+(i-1)*k+j]=y(i)-P[1,i]; |
---|
| 5132 | } |
---|
| 5133 | } |
---|
| 5134 | M=elim(module(M),1,n); // eliminating x(1..n), std-calculation |
---|
| 5135 | // is done internally - |
---|
| 5136 | M=homog(module(M),h); // homogenize for 'minbase' |
---|
| 5137 | M=minbase(module(M)); |
---|
| 5138 | setring br; |
---|
| 5139 | ideal substitute=maxideal(1),ideal(P),1; |
---|
| 5140 | f=R,substitute; // replacing y(1..n) by primary |
---|
| 5141 | // invariants - |
---|
| 5142 | M(2)=f(M); // M(2) is the new module |
---|
| 5143 | m=ncols(M(2)); |
---|
| 5144 | matrix S[1][m]; |
---|
| 5145 | S=matrix(trivialS)*matrix(M(2)); // S now contains the secondary |
---|
| 5146 | // invariants |
---|
[18bd9c] | 5147 | for (i=1; i<=m;i++) |
---|
[9f79aa] | 5148 | { S[1,i]=S[1,i]/leadcoef(S[1,i]); // making elements nice |
---|
[f34c37c] | 5149 | } |
---|
| 5150 | S=sort(ideal(S))[1]; |
---|
| 5151 | if (v) |
---|
| 5152 | { " These are the secondary invariants: "; |
---|
[18bd9c] | 5153 | for (i=1;i<=m;i++) |
---|
[f34c37c] | 5154 | { " "+string(S[1,i]); |
---|
| 5155 | } |
---|
| 5156 | ""; |
---|
| 5157 | " We're done!"; |
---|
| 5158 | ""; |
---|
| 5159 | } |
---|
| 5160 | if ((v or (voice==2)) && (m>1)) |
---|
| 5161 | { " WARNING: The invariant ring might not have a Hironaka decomposition"; |
---|
| 5162 | " if the characteristic of the coefficient field divides the"; |
---|
| 5163 | " group order."; |
---|
| 5164 | } |
---|
| 5165 | return(S); |
---|
| 5166 | } |
---|
| 5167 | example |
---|
[9f79aa] | 5168 | { "EXAMPLE:"; echo=2; |
---|
[f34c37c] | 5169 | ring R=2,(x,y,z),dp; |
---|
| 5170 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 5171 | list L=primary_invariants(A); |
---|
| 5172 | matrix S=secondary_not_cohen_macaulay(L[1],A); |
---|
| 5173 | print(S); |
---|
| 5174 | } |
---|
[68e678] | 5175 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 5176 | |
---|
| 5177 | proc invariant_ring (list #) |
---|
| 5178 | "USAGE: invariant_ring(G1,G2,...[,flags]); |
---|
| 5179 | G1,G2,...: <matrices> generating a finite matrix group, flags: an |
---|
| 5180 | optional <intvec> with three entries: if the first one equals 0, the |
---|
| 5181 | program attempts to compute the Molien series and Reynolds operator, |
---|
| 5182 | if it equals 1, the program is told that the Molien series should not |
---|
| 5183 | be computed, if it equals -1 characteristic 0 is simulated, i.e. the |
---|
| 5184 | Molien series is computed as if the base field were characteristic 0 |
---|
| 5185 | (the user must choose a field of large prime characteristic, e.g. |
---|
| 5186 | 32003) and if the first one is anything else, it means that the |
---|
| 5187 | characteristic of the base field divides the group order (i.e. it will |
---|
| 5188 | not even be attempted to compute the Reynolds operator or Molien |
---|
| 5189 | series), the second component should give the size of intervals |
---|
[9f79aa] | 5190 | between canceling common factors in the expansion of Molien series, 0 |
---|
| 5191 | (the default) means only once after generating all terms, in prime |
---|
[f34c37c] | 5192 | characteristic also a negative number can be given to indicate that |
---|
| 5193 | common factors should always be canceled when the expansion is simple |
---|
[68e678] | 5194 | (the root of the extension field occurs not among the coefficients) |
---|
[9f79aa] | 5195 | RETURN: primary and secondary invariants (both of type <matrix>) generating |
---|
| 5196 | the invariant ring with respect to the matrix group generated by the |
---|
[f34c37c] | 5197 | matrices in the input and irreducible secondary invariants (type |
---|
| 5198 | <matrix>) if the Molien series was available |
---|
| 5199 | DISPLAY: information about the various stages of the program if the third flag |
---|
| 5200 | does not equal 0 |
---|
| 5201 | THEORY: Bases of homogeneous invariants are generated successively and those |
---|
| 5202 | are chosen as primary invariants that lower the dimension of the ideal |
---|
[9f79aa] | 5203 | generated by the previously found invariants (see \"Generating a |
---|
[f34c37c] | 5204 | Noetherian Normalization of the Invariant Ring of a Finite Group\" by |
---|
[9f79aa] | 5205 | Decker, Heydtmann, Schreyer (1998)). In the |
---|
[f34c37c] | 5206 | non-modular case secondary invariants are calculated by finding a |
---|
| 5207 | basis (in terms of monomials) of the basering modulo the primary |
---|
| 5208 | invariants, mapping to invariants with the Reynolds operator and using |
---|
| 5209 | those or their power products such that they are linearly independent |
---|
[68e678] | 5210 | modulo the primary invariants (see \"Some Algorithms in Invariant |
---|
[f34c37c] | 5211 | Theory of Finite Groups\" by Kemper and Steel (1997)). In the modular |
---|
| 5212 | case they are generated according to \"Generating Invariant Rings of |
---|
[9f79aa] | 5213 | Finite Groups over Arbitrary Fields\" by Kemper (1996). |
---|
[68e678] | 5214 | EXAMPLE: example invariant_ring; shows an example |
---|
[f34c37c] | 5215 | " |
---|
| 5216 | { if (size(#)==0) |
---|
| 5217 | { "ERROR: There are no generators given."; |
---|
| 5218 | return(); |
---|
| 5219 | } |
---|
| 5220 | int ch=char(basering); // the algorithms depend very much on the |
---|
| 5221 | // characteristic of the ground field - |
---|
| 5222 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 5223 | // as the size of the matrices, as well |
---|
| 5224 | // as the number of primary invariants, |
---|
| 5225 | // we should get |
---|
| 5226 | int gen_num; |
---|
| 5227 | int mol_flag, v; |
---|
| 5228 | //------------------- checking input and setting flags ----------------------- |
---|
| 5229 | if (typeof(#[size(#)])=="intvec") |
---|
| 5230 | { if (size(#[size(#)])<>3) |
---|
| 5231 | { "ERROR: The <intvec> should have three entries."; |
---|
| 5232 | return(); |
---|
| 5233 | } |
---|
| 5234 | gen_num=size(#)-1; |
---|
| 5235 | mol_flag=#[size(#)][1]; |
---|
| 5236 | if (#[size(#)][2]<0 && (ch==0 or (ch<>0 && mol_flag<>0))) |
---|
| 5237 | { "ERROR: the second component of <intvec> should be >=0"; |
---|
| 5238 | return(); |
---|
| 5239 | } |
---|
| 5240 | int interval=#[size(#)][2]; |
---|
| 5241 | v=#[size(#)][3]; |
---|
| 5242 | } |
---|
| 5243 | else |
---|
| 5244 | { gen_num=size(#); |
---|
| 5245 | mol_flag=0; |
---|
| 5246 | int interval=0; |
---|
| 5247 | v=0; |
---|
| 5248 | } |
---|
| 5249 | //---------------------------------------------------------------------------- |
---|
| 5250 | if (mol_flag==0) // calculation Molien series will be |
---|
| 5251 | { if (ch==0) // attempted - |
---|
| 5252 | { matrix REY,M=reynolds_molien(#[1..gen_num],intvec(0,interval,v)); // one |
---|
| 5253 | // will contain Reynolds operator and the |
---|
| 5254 | // other enumerator and denominator of |
---|
| 5255 | // Molien series |
---|
| 5256 | matrix P=primary_char0(REY,M,v); |
---|
| 5257 | matrix S,IS=secondary_char0(P,REY,M,v); |
---|
| 5258 | return(P,S,IS); |
---|
| 5259 | } |
---|
| 5260 | else |
---|
| 5261 | { list L=group_reynolds(#[1..gen_num],v); |
---|
| 5262 | if (L[1]<>0) // testing whether we are in the modular |
---|
| 5263 | { string newring="aksldfalkdsflkj"; // case |
---|
| 5264 | if (minpoly==0) |
---|
| 5265 | { if (v) |
---|
| 5266 | { " We are dealing with the non-modular case."; |
---|
| 5267 | } |
---|
[9f79aa] | 5268 | if (typeof(L[2])=="int") |
---|
| 5269 | { molien(L[3..size(L)],newring,L[2],intvec(0,interval,v)); |
---|
| 5270 | } |
---|
| 5271 | else |
---|
| 5272 | { molien(L[2..size(L)],newring,intvec(0,interval,v)); |
---|
| 5273 | } |
---|
[f34c37c] | 5274 | matrix P=primary_charp(L[1],newring,v); |
---|
| 5275 | matrix S,IS=secondary_charp(P,L[1],newring,v); |
---|
| 5276 | if (defined(aksldfalkdsflkj)==2) |
---|
| 5277 | { kill aksldfalkdsflkj; |
---|
| 5278 | } |
---|
| 5279 | return(P,S,IS); |
---|
| 5280 | } |
---|
| 5281 | else |
---|
| 5282 | { if (v) |
---|
| 5283 | { " Since it is impossible for this programme to calculate the Molien |
---|
| 5284 | series for"; |
---|
| 5285 | " invariant rings over extension fields of prime characteristic, we |
---|
| 5286 | have to"; |
---|
| 5287 | " continue without it."; |
---|
| 5288 | ""; |
---|
| 5289 | |
---|
| 5290 | } |
---|
| 5291 | list l=primary_charp_no_molien(L[1],v); |
---|
| 5292 | if (size(l)==2) |
---|
| 5293 | { matrix S=secondary_no_molien(l[1],L[1],l[2],v); |
---|
| 5294 | } |
---|
| 5295 | else |
---|
[a269e4] | 5296 | { matrix S=secondary_no_molien(l[1],L[1],v); |
---|
[f34c37c] | 5297 | } |
---|
| 5298 | return(l[1],S); |
---|
| 5299 | } |
---|
| 5300 | } |
---|
| 5301 | else // the modular case |
---|
| 5302 | { if (v) |
---|
| 5303 | { " There is also no Molien series or Reynolds operator, we can make use of..."; |
---|
| 5304 | ""; |
---|
| 5305 | " We can start looking for primary invariants..."; |
---|
| 5306 | ""; |
---|
| 5307 | } |
---|
| 5308 | matrix P=primary_charp_without(#[1..gen_num],v); |
---|
| 5309 | matrix S=secondary_not_cohen_macaulay(P,#[1..gen_num],v); |
---|
| 5310 | return(P,S); |
---|
| 5311 | } |
---|
| 5312 | } |
---|
| 5313 | } |
---|
| 5314 | if (mol_flag==1) // the user wants no calculation of the |
---|
| 5315 | { list L=group_reynolds(#[1..gen_num],v); // Molien series |
---|
| 5316 | if (ch==0) |
---|
| 5317 | { list l=primary_char0_no_molien(L[1],v); |
---|
| 5318 | if (size(l)==2) |
---|
[a269e4] | 5319 | { matrix S=secondary_no_molien(l[1],L[1],l[2],v); |
---|
[f34c37c] | 5320 | } |
---|
| 5321 | else |
---|
[a269e4] | 5322 | { matrix S=secondary_no_molien(l[1],L[1],v); |
---|
[f34c37c] | 5323 | } |
---|
| 5324 | return(l[1],S); |
---|
| 5325 | } |
---|
| 5326 | else |
---|
| 5327 | { if (L[1]<>0) // testing whether we are in the modular |
---|
| 5328 | { list l=primary_charp_no_molien(L[1],v); // case |
---|
| 5329 | if (size(l)==2) |
---|
[a269e4] | 5330 | { matrix S=secondary_no_molien(l[1],L[1],l[2],v); |
---|
[f34c37c] | 5331 | } |
---|
| 5332 | else |
---|
[a269e4] | 5333 | { matrix S=secondary_no_molien(l[1],L[1],v); |
---|
[f34c37c] | 5334 | } |
---|
| 5335 | return(l[1],S); |
---|
| 5336 | } |
---|
| 5337 | else // the modular case |
---|
| 5338 | { if (v) |
---|
| 5339 | { " We can start looking for primary invariants..."; |
---|
| 5340 | ""; |
---|
| 5341 | } |
---|
| 5342 | matrix P=primary_charp_without(#[1..gen_num],v); |
---|
| 5343 | matrix S=secondary_not_cohen_macaulay(P,#[1..gen_num],v); |
---|
| 5344 | return(L[1],S); |
---|
| 5345 | } |
---|
| 5346 | } |
---|
| 5347 | } |
---|
| 5348 | if (mol_flag==-1) |
---|
| 5349 | { if (ch==0) |
---|
| 5350 | { "ERROR: Characteristic 0 can only be simulated in characteristic p>>0. |
---|
| 5351 | "; |
---|
| 5352 | return(); |
---|
| 5353 | } |
---|
| 5354 | list L=group_reynolds(#[1..gen_num],v); |
---|
| 5355 | string newring="aksldfalkdsflkj"; |
---|
[9f79aa] | 5356 | if (typeof(L[2])=="int") |
---|
| 5357 | { molien(L[3..size(L)],newring,L[2],intvec(1,interval,v)); |
---|
| 5358 | } |
---|
| 5359 | else |
---|
| 5360 | { molien(L[2..size(L)],newring,intvec(1,interval,v)); |
---|
| 5361 | } |
---|
[f34c37c] | 5362 | matrix P=primary_charp(L[1],newring,v); |
---|
| 5363 | matrix S,IS=secondary_charp(P,L[1],newring,v); |
---|
| 5364 | kill aksldfalkdsflkj; |
---|
| 5365 | return(P,S,IS); |
---|
| 5366 | } |
---|
| 5367 | else // the user specified that the |
---|
| 5368 | { if (ch==0) // characteristic divides the group order |
---|
| 5369 | { "ERROR: The characteristic cannot divide the group order when it is 0. |
---|
| 5370 | "; |
---|
| 5371 | return(); |
---|
| 5372 | } |
---|
| 5373 | if (v) |
---|
| 5374 | { ""; |
---|
| 5375 | } |
---|
| 5376 | matrix P=primary_charp_without(#[1..gen_num],v); |
---|
| 5377 | matrix S=secondary_not_cohen_macaulay(P,#[1..gen_num],v); |
---|
| 5378 | return(L[1],S); |
---|
| 5379 | } |
---|
| 5380 | } |
---|
| 5381 | example |
---|
[9f79aa] | 5382 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
[f34c37c] | 5383 | ring R=0,(x,y,z),dp; |
---|
| 5384 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 5385 | matrix P,S,IS=invariant_ring(A); |
---|
| 5386 | print(P); |
---|
| 5387 | print(S); |
---|
| 5388 | print(IS); |
---|
| 5389 | } |
---|
[68e678] | 5390 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 5391 | |
---|
| 5392 | proc invariant_ring_random (list #) |
---|
| 5393 | "USAGE: invariant_ring_random(G1,G2,...,r[,flags]); |
---|
| 5394 | G1,G2,...: <matrices> generating a finite matrix group, r: an <int> |
---|
| 5395 | where -|r| to |r| is the range of coefficients of random |
---|
| 5396 | combinations of bases elements that serve as primary invariants, |
---|
[9f79aa] | 5397 | flags: an optional <intvec> with three entries: if the first equals 0, |
---|
| 5398 | the program attempts to compute the Molien series and Reynolds |
---|
[f34c37c] | 5399 | operator, if it equals 1, the program is told that the Molien series |
---|
| 5400 | should not be computed, if it equals -1 characteristic 0 is simulated, |
---|
| 5401 | i.e. the Molien series is computed as if the base field were |
---|
| 5402 | characteristic 0 (the user must choose a field of large prime |
---|
[9f79aa] | 5403 | characteristic, e.g. 32003) and if the first one is anything else, |
---|
[68e678] | 5404 | then the characteristic of the base field divides the group order |
---|
| 5405 | (i.e. we will not even attempt to compute the Reynolds operator or |
---|
[f34c37c] | 5406 | Molien series), the second component should give the size of intervals |
---|
[18bd9c] | 5407 | between canceling common factors in the expansion of the Molien |
---|
[9f79aa] | 5408 | series, 0 (the default) means only once after generating all terms, |
---|
| 5409 | in prime characteristic also a negative number can be given to |
---|
| 5410 | indicate that common factors should always be canceled when the |
---|
| 5411 | expansion is simple (the root of the extension field does not occur |
---|
| 5412 | among the coefficients) |
---|
| 5413 | RETURN: primary and secondary invariants (both of type <matrix>) generating |
---|
[f34c37c] | 5414 | invariant ring with respect to the matrix group generated by the |
---|
| 5415 | matrices in the input and irreducible secondary invariants (type |
---|
| 5416 | <matrix>) if the Molien series was available |
---|
| 5417 | DISPLAY: information about the various stages of the program if the third flag |
---|
| 5418 | does not equal 0 |
---|
| 5419 | THEORY: is the same as for invariant_ring except that random combinations of |
---|
| 5420 | basis elements are chosen as candidates for primary invariants and |
---|
| 5421 | hopefully they lower the dimension of the previously found primary |
---|
| 5422 | invariants by the right amount. |
---|
[68e678] | 5423 | EXAMPLE: example invariant_ring_random; shows an example |
---|
[f34c37c] | 5424 | " |
---|
| 5425 | { if (size(#)<2) |
---|
| 5426 | { "ERROR: There are too few parameters."; |
---|
| 5427 | return(); |
---|
| 5428 | } |
---|
| 5429 | int ch=char(basering); // the algorithms depend very much on the |
---|
| 5430 | // characteristic of the ground field |
---|
| 5431 | int n=nvars(basering); // n is the number of variables, as well |
---|
| 5432 | // as the size of the matrices, as well |
---|
| 5433 | // as the number of primary invariants, |
---|
| 5434 | // we should get |
---|
| 5435 | int gen_num; |
---|
| 5436 | int mol_flag, v; |
---|
| 5437 | //------------------- checking input and setting flags ----------------------- |
---|
| 5438 | if (typeof(#[size(#)])=="intvec" && typeof(#[size(#)-1])=="int") |
---|
| 5439 | { if (size(#[size(#)])<>3) |
---|
| 5440 | { "ERROR: <intvec> should have three entries."; |
---|
| 5441 | return(); |
---|
| 5442 | } |
---|
| 5443 | gen_num=size(#)-2; |
---|
| 5444 | mol_flag=#[size(#)][1]; |
---|
| 5445 | if (#[size(#)][2]<0 && (ch==0 or (ch<>0 && mol_flag<>0))) |
---|
| 5446 | { "ERROR: the second component of <intvec> should be >=0"; |
---|
| 5447 | return(); |
---|
| 5448 | } |
---|
| 5449 | int interval=#[size(#)][2]; |
---|
| 5450 | v=#[size(#)][3]; |
---|
| 5451 | int max=#[size(#)-1]; |
---|
| 5452 | if (gen_num==0) |
---|
| 5453 | { "ERROR: There are no generators of a finite matrix group given."; |
---|
| 5454 | return(); |
---|
| 5455 | } |
---|
| 5456 | } |
---|
| 5457 | else |
---|
| 5458 | { if (typeof(#[size(#)])=="int") |
---|
| 5459 | { gen_num=size(#)-1; |
---|
| 5460 | mol_flag=0; |
---|
| 5461 | int interval=0; |
---|
| 5462 | v=0; |
---|
| 5463 | int max=#[size(#)]; |
---|
| 5464 | } |
---|
| 5465 | else |
---|
| 5466 | { "ERROR: If the two last parameters are not <int> and <intvec>, the last"; |
---|
| 5467 | " parameter should be an <int>."; |
---|
| 5468 | return(); |
---|
| 5469 | } |
---|
| 5470 | } |
---|
[18bd9c] | 5471 | for (int i=1;i<=gen_num;i++) |
---|
[f34c37c] | 5472 | { if (typeof(#[i])=="matrix") |
---|
| 5473 | { if (nrows(#[i])<>n or ncols(#[i])<>n) |
---|
| 5474 | { "ERROR: The number of variables of the base ring needs to be the same"; |
---|
| 5475 | " as the dimension of the square matrices"; |
---|
| 5476 | return(); |
---|
| 5477 | } |
---|
| 5478 | } |
---|
| 5479 | else |
---|
| 5480 | { "ERROR: The first parameters should be a list of matrices"; |
---|
| 5481 | return(); |
---|
| 5482 | } |
---|
| 5483 | } |
---|
| 5484 | //---------------------------------------------------------------------------- |
---|
| 5485 | if (mol_flag==0) |
---|
| 5486 | { if (ch==0) |
---|
| 5487 | { matrix REY,M=reynolds_molien(#[1..gen_num],intvec(0,interval,v)); // one |
---|
| 5488 | // will contain Reynolds operator and the |
---|
| 5489 | // other enumerator and denominator of |
---|
| 5490 | // Molien series |
---|
| 5491 | matrix P=primary_char0_random(REY,M,max,v); |
---|
| 5492 | matrix S,IS=secondary_char0(P,REY,M,v); |
---|
| 5493 | return(P,S,IS); |
---|
| 5494 | } |
---|
| 5495 | else |
---|
| 5496 | { list L=group_reynolds(#[1..gen_num],v); |
---|
| 5497 | if (L[1]<>0) // testing whether we are in the modular |
---|
| 5498 | { string newring="aksldfalkdsflkj"; // case |
---|
| 5499 | if (minpoly==0) |
---|
| 5500 | { if (v) |
---|
| 5501 | { " We are dealing with the non-modular case."; |
---|
| 5502 | } |
---|
[9f79aa] | 5503 | if (typeof(L[2])=="int") |
---|
| 5504 | { molien(L[3..size(L)],newring,L[2],intvec(0,interval,v)); |
---|
| 5505 | } |
---|
| 5506 | else |
---|
| 5507 | { molien(L[2..size(L)],newring,intvec(0,interval,v)); |
---|
| 5508 | } |
---|
[f34c37c] | 5509 | matrix P=primary_charp_random(L[1],newring,max,v); |
---|
| 5510 | matrix S,IS=secondary_charp(P,L[1],newring,v); |
---|
| 5511 | if (voice==2) |
---|
| 5512 | { kill aksldfalkdsflkj; |
---|
| 5513 | } |
---|
| 5514 | return(P,S,IS); |
---|
| 5515 | } |
---|
| 5516 | else |
---|
| 5517 | { if (v) |
---|
| 5518 | { " Since it is impossible for this programme to calculate the Molien |
---|
| 5519 | series for"; |
---|
| 5520 | " invariant rings over extension fields of prime characteristic, we |
---|
| 5521 | have to"; |
---|
| 5522 | " continue without it."; |
---|
| 5523 | ""; |
---|
| 5524 | |
---|
| 5525 | } |
---|
| 5526 | list l=primary_charp_no_molien_random(L[1],max,v); |
---|
| 5527 | if (size(l)==2) |
---|
[a269e4] | 5528 | { matrix S=secondary_no_molien(l[1],L[1],l[2],v); |
---|
[f34c37c] | 5529 | } |
---|
| 5530 | else |
---|
[a269e4] | 5531 | { matrix S=secondary_no_molien(l[1],L[1],v); |
---|
[f34c37c] | 5532 | } |
---|
| 5533 | return(l[1],S); |
---|
| 5534 | } |
---|
| 5535 | } |
---|
| 5536 | else // the modular case |
---|
| 5537 | { if (v) |
---|
| 5538 | { " There is also no Molien series, we can make use of..."; |
---|
| 5539 | ""; |
---|
| 5540 | " We can start looking for primary invariants..."; |
---|
| 5541 | ""; |
---|
| 5542 | } |
---|
| 5543 | matrix P=primary_charp_without_random(#[1..gen_num],max,v); |
---|
| 5544 | matrix S=secondary_not_cohen_macaulay(P,#[1..gen_num],v); |
---|
[9f79aa] | 5545 | return(P,S); |
---|
[f34c37c] | 5546 | } |
---|
| 5547 | } |
---|
| 5548 | } |
---|
| 5549 | if (mol_flag==1) // the user wants no calculation of the |
---|
| 5550 | { list L=group_reynolds(#[1..gen_num],v); // Molien series |
---|
| 5551 | if (ch==0) |
---|
| 5552 | { list l=primary_char0_no_molien_random(L[1],max,v); |
---|
| 5553 | if (size(l)==2) |
---|
[a269e4] | 5554 | { matrix S=secondary_no_molien(l[1],L[1],l[2],v); |
---|
[f34c37c] | 5555 | } |
---|
| 5556 | else |
---|
[a269e4] | 5557 | { matrix S=secondary_no_molien(l[1],L[1],v); |
---|
[f34c37c] | 5558 | } |
---|
| 5559 | return(l[1],S); |
---|
| 5560 | } |
---|
| 5561 | else |
---|
| 5562 | { if (L[1]<>0) // testing whether we are in the modular |
---|
| 5563 | { list l=primary_charp_no_molien_random(L[1],max,v); // case |
---|
| 5564 | if (size(l)==2) |
---|
[a269e4] | 5565 | { matrix S=secondary_no_molien(l[1],L[1],l[2],v); |
---|
[f34c37c] | 5566 | } |
---|
| 5567 | else |
---|
[a269e4] | 5568 | { matrix S=secondary_no_molien(l[1],L[1],v); |
---|
[f34c37c] | 5569 | } |
---|
| 5570 | return(l[1],S); |
---|
| 5571 | } |
---|
| 5572 | else // the modular case |
---|
| 5573 | { if (v) |
---|
| 5574 | { " We can start looking for primary invariants..."; |
---|
| 5575 | ""; |
---|
| 5576 | } |
---|
| 5577 | matrix P=primary_charp_without_random(#[1..gen_num],max,v); |
---|
| 5578 | matrix S=secondary_not_cohen_macaulay(P,#[1..gen_num],v); |
---|
| 5579 | return(L[1],S); |
---|
| 5580 | } |
---|
| 5581 | } |
---|
| 5582 | } |
---|
| 5583 | if (mol_flag==-1) |
---|
| 5584 | { if (ch==0) |
---|
| 5585 | { "ERROR: Characteristic 0 can only be simulated in characteristic p>>0. |
---|
| 5586 | "; |
---|
| 5587 | return(); |
---|
| 5588 | } |
---|
| 5589 | list L=group_reynolds(#[1..gen_num],v); |
---|
| 5590 | string newring="aksldfalkdsflkj"; |
---|
[9f79aa] | 5591 | if (typeof(L[2])=="int") |
---|
| 5592 | { molien(L[3..size(L)],newring,L[2],intvec(mol_flag,interval,v)); |
---|
| 5593 | } |
---|
| 5594 | else |
---|
| 5595 | { molien(L[2..size(L)],newring,intvec(mol_flag,interval,v)); |
---|
| 5596 | } |
---|
[f34c37c] | 5597 | matrix P=primary_charp_random(L[1],newring,max,v); |
---|
| 5598 | matrix S,IS=secondary_charp(P,L[1],newring,v); |
---|
| 5599 | kill aksldfalkdsflkj; |
---|
| 5600 | return(P,S,IS); |
---|
| 5601 | } |
---|
| 5602 | else // the user specified that the |
---|
| 5603 | { if (ch==0) // characteristic divides the group order |
---|
| 5604 | { "ERROR: The characteristic cannot divide the group order when it is 0. |
---|
| 5605 | "; |
---|
| 5606 | return(); |
---|
| 5607 | } |
---|
| 5608 | if (v) |
---|
| 5609 | { ""; |
---|
| 5610 | } |
---|
| 5611 | matrix P=primary_charp_without_random(#[1..gen_num],max,v); |
---|
| 5612 | matrix S=secondary_not_cohen_macaulay(P,#[1..gen_num],v); |
---|
| 5613 | return(L[1],S); |
---|
| 5614 | } |
---|
| 5615 | } |
---|
| 5616 | example |
---|
[68e678] | 5617 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
[f34c37c] | 5618 | ring R=0,(x,y,z),dp; |
---|
| 5619 | matrix A[3][3]=0,1,0,-1,0,0,0,0,-1; |
---|
| 5620 | matrix P,S,IS=invariant_ring_random(A,1); |
---|
| 5621 | print(P); |
---|
| 5622 | print(S); |
---|
| 5623 | print(IS); |
---|
| 5624 | } |
---|
[68e678] | 5625 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 5626 | |
---|
| 5627 | proc orbit_variety (matrix F,string newring) |
---|
| 5628 | "USAGE: orbit_variety(F,s); |
---|
| 5629 | F: a 1xm <matrix> defing an invariant ring, s: a <string> giving the |
---|
| 5630 | name for a new ring |
---|
| 5631 | RETURN: a Groebner basis (type <ideal>, named G) for the ideal defining the |
---|
| 5632 | orbit variety (i.e. the syzygy ideal) in the new ring (named `s`) |
---|
| 5633 | THEORY: The ideal of algebraic relations of the invariant ring generators is |
---|
| 5634 | calculated, then the variables of the original ring are eliminated and |
---|
| 5635 | the polynomials that are left over define the orbit variety |
---|
[68e678] | 5636 | EXAMPLE: example orbit_variety; shows an example |
---|
[f34c37c] | 5637 | " |
---|
| 5638 | { if (newring=="") |
---|
| 5639 | { "ERROR: the second parameter may not be an empty <string>"; |
---|
| 5640 | return(); |
---|
| 5641 | } |
---|
| 5642 | if (nrows(F)==1) |
---|
| 5643 | { def br=basering; |
---|
| 5644 | int n=nvars(br); |
---|
| 5645 | int m=ncols(F); |
---|
| 5646 | string mp=string(minpoly); |
---|
[034ce1] | 5647 | execute("ring R=("+charstr(br)+"),("+varstr(br)+",y(1..m)),dp;"); |
---|
| 5648 | execute("minpoly=number("+mp+");"); |
---|
[f34c37c] | 5649 | ideal I=ideal(imap(br,F)); |
---|
[18bd9c] | 5650 | for (int i=1;i<=m;i++) |
---|
[f34c37c] | 5651 | { I[i]=I[i]-y(i); |
---|
| 5652 | } |
---|
| 5653 | I=elim(I,1,n); |
---|
[034ce1] | 5654 | execute("ring "+newring+"=("+charstr(br)+"),(y(1..m)),dp(m);"); |
---|
| 5655 | execute("minpoly=number("+mp+");"); |
---|
[f34c37c] | 5656 | ideal vars; |
---|
[18bd9c] | 5657 | for (i=2;i<=n;i++) |
---|
[f34c37c] | 5658 | { vars[i]=0; |
---|
| 5659 | } |
---|
| 5660 | vars=vars,y(1..m); |
---|
| 5661 | map emb=R,vars; |
---|
| 5662 | ideal G=emb(I); |
---|
| 5663 | kill emb, vars, R; |
---|
| 5664 | keepring `newring`; |
---|
| 5665 | return(); |
---|
| 5666 | } |
---|
| 5667 | else |
---|
| 5668 | { "ERROR: the <matrix> may only have one row"; |
---|
| 5669 | return(); |
---|
| 5670 | } |
---|
| 5671 | } |
---|
| 5672 | example |
---|
[68e678] | 5673 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.3.7:"; echo=2; |
---|
[f34c37c] | 5674 | ring R=0,(x,y,z),dp; |
---|
| 5675 | matrix F[1][7]=x2+y2,z2,x4+y4,1,x2z-1y2z,xyz,x3y-1xy3; |
---|
| 5676 | string newring="E"; |
---|
| 5677 | orbit_variety(F,newring); |
---|
| 5678 | print(G); |
---|
| 5679 | basering; |
---|
| 5680 | } |
---|
[68e678] | 5681 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 5682 | |
---|
| 5683 | proc relative_orbit_variety(ideal I,matrix F,string newring) |
---|
| 5684 | "USAGE: relative_orbit_variety(I,F,s); |
---|
| 5685 | I: an <ideal> invariant under the action of a group, F: a 1xm |
---|
| 5686 | <matrix> defining the invariant ring of this group, s: a <string> |
---|
| 5687 | giving a name for a new ring |
---|
| 5688 | RETURN: a Groebner basis (type <ideal>, named G) for the ideal defining the |
---|
| 5689 | relative orbit variety with respect to I in the new ring (named s) |
---|
| 5690 | THEORY: A Groebner basis of the ideal of algebraic relations of the invariant |
---|
| 5691 | ring generators is calculated, then one of the basis elements plus the |
---|
[18bd9c] | 5692 | ideal generators. The variables of the original ring are eliminated |
---|
[68e678] | 5693 | and the polynomials that are left define the relative orbit variety |
---|
[f34c37c] | 5694 | with respect to I. |
---|
[68e678] | 5695 | EXAMPLE: example relative_orbit_variety; shows an example |
---|
[f34c37c] | 5696 | " |
---|
| 5697 | { if (newring=="") |
---|
| 5698 | { "ERROR: the third parameter may not be empty a <string>"; |
---|
| 5699 | return(); |
---|
| 5700 | } |
---|
| 5701 | degBound=0; |
---|
| 5702 | if (nrows(F)==1) |
---|
| 5703 | { def br=basering; |
---|
| 5704 | int n=nvars(br); |
---|
| 5705 | int m=ncols(F); |
---|
| 5706 | string mp=string(minpoly); |
---|
[034ce1] | 5707 | execute("ring R=("+charstr(br)+"),("+varstr(br)+",y(1..m)),lp;"); |
---|
| 5708 | execute("minpoly=number("+mp+");"); |
---|
[f34c37c] | 5709 | ideal J=ideal(imap(br,F)); |
---|
| 5710 | ideal I=imap(br,I); |
---|
[18bd9c] | 5711 | for (int i=1;i<=m;i++) |
---|
[f34c37c] | 5712 | { J[i]=J[i]-y(i); |
---|
| 5713 | } |
---|
[f54c83] | 5714 | J=groebner(J); |
---|
[f34c37c] | 5715 | J=J,I; |
---|
| 5716 | option(redSB); |
---|
[f54c83] | 5717 | J=groebner(J); |
---|
[f34c37c] | 5718 | ideal vars; |
---|
[9f79aa] | 5719 | //for (i=1;i<=n;i=i+1) |
---|
[f34c37c] | 5720 | //{ vars[i]=0; |
---|
| 5721 | //} |
---|
| 5722 | vars[n]=0; |
---|
| 5723 | vars=vars,y(1..m); |
---|
| 5724 | map emb=R,vars; |
---|
| 5725 | ideal G=emb(J); |
---|
| 5726 | J=J-G; |
---|
[18bd9c] | 5727 | for (i=1;i<=ncols(G);i++) |
---|
[f34c37c] | 5728 | { if (J[i]<>0) |
---|
| 5729 | { G[i]=0; |
---|
| 5730 | } |
---|
| 5731 | } |
---|
| 5732 | G=compress(G); |
---|
[034ce1] | 5733 | execute("ring "+newring+"=("+charstr(br)+"),(y(1..m)),lp;"); |
---|
| 5734 | execute("minpoly=number("+mp+");"); |
---|
[f34c37c] | 5735 | ideal vars; |
---|
[18bd9c] | 5736 | for (i=2;i<=n;i++) |
---|
[f34c37c] | 5737 | { vars[i]=0; |
---|
| 5738 | } |
---|
| 5739 | vars=vars,y(1..m); |
---|
| 5740 | map emb=R,vars; |
---|
| 5741 | ideal G=emb(G); |
---|
| 5742 | kill vars, emb; |
---|
| 5743 | keepring `newring`; |
---|
| 5744 | return(); |
---|
| 5745 | } |
---|
| 5746 | else |
---|
| 5747 | { "ERROR: the <matrix> may only have one row"; |
---|
| 5748 | return(); |
---|
| 5749 | } |
---|
| 5750 | } |
---|
| 5751 | example |
---|
[68e678] | 5752 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.6.3:"; echo=2; |
---|
[f34c37c] | 5753 | ring R=0,(x,y,z),dp; |
---|
| 5754 | matrix F[1][3]=x+y+z,xy+xz+yz,xyz; |
---|
| 5755 | ideal I=x2+y2+z2-1,x2y+y2z+z2x-2x-2y-2z,xy2+yz2+zx2-2x-2y-2z; |
---|
| 5756 | string newring="E"; |
---|
| 5757 | relative_orbit_variety(I,F,newring); |
---|
| 5758 | print(G); |
---|
| 5759 | basering; |
---|
| 5760 | } |
---|
[68e678] | 5761 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f34c37c] | 5762 | |
---|
| 5763 | proc image_of_variety(ideal I,matrix F) |
---|
| 5764 | "USAGE: image_of_variety(I,F); |
---|
| 5765 | I: an arbitray <ideal>, F: a 1xm <matrix> defining an invariant ring |
---|
| 5766 | of a some matrix group |
---|
| 5767 | RETURN: the <ideal> defining the image under that group of the variety defined |
---|
| 5768 | by I |
---|
| 5769 | THEORY: relative_orbit_variety(I,F,s) is called and the newly introduced |
---|
| 5770 | variables in the output are replaced by the generators of the |
---|
| 5771 | invariant ring. This ideal in the original variables defines the image |
---|
| 5772 | of the variety defined by I |
---|
[68e678] | 5773 | EXAMPLE: example image_of_variety; shows an example |
---|
[f34c37c] | 5774 | " |
---|
| 5775 | { if (nrows(F)==1) |
---|
| 5776 | { def br=basering; |
---|
| 5777 | int n=nvars(br); |
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| 5778 | string newring="E"; |
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| 5779 | relative_orbit_variety(I,F,newring); |
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[034ce1] | 5780 | execute("ring R=("+charstr(br)+"),("+varstr(br)+","+varstr(E)+"),lp;"); |
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[f34c37c] | 5781 | ideal F=imap(br,F); |
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[18bd9c] | 5782 | for (int i=1;i<=n;i++) |
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[f34c37c] | 5783 | { F=0,F; |
---|
| 5784 | } |
---|
| 5785 | setring br; |
---|
| 5786 | map emb2=E,F; |
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| 5787 | return(compress(emb2(G))); |
---|
| 5788 | } |
---|
| 5789 | else |
---|
| 5790 | { "ERROR: the <matrix> may only have one row"; |
---|
| 5791 | return(); |
---|
| 5792 | } |
---|
| 5793 | } |
---|
| 5794 | example |
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[68e678] | 5795 | { "EXAMPLE: Sturmfels: Algorithms in Invariant Theory 2.6.8:"; echo=2; |
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[f34c37c] | 5796 | ring R=0,(x,y,z),dp; |
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| 5797 | matrix F[1][3]=x+y+z,xy+xz+yz,xyz; |
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| 5798 | ideal I=xy; |
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| 5799 | print(image_of_variety(I,F)); |
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| 5800 | } |
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[558eb2] | 5801 | /////////////////////////////////////////////////////////////////////////////// |
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