[380a17b] | 1 | ////////////////////////////////////////////////////////////////////////////// |
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[f054d7f] | 2 | version="version ainvar.lib 4.3.1.3 Feb_2023 "; // $Id$ |
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[fd3fb7] | 3 | category="Invariant theory"; |
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[5480da] | 4 | info=" |
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[4ac997] | 5 | LIBRARY: ainvar.lib Invariant Rings of the Additive Group |
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[51d95b] | 6 | AUTHORS: Gerhard Pfister (email: pfister@mathematik.uni-kl.de), |
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| 7 | Gert-Martin Greuel (email: greuel@mathematik.uni-kl.de) |
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[4b35a90] | 8 | |
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[f34c37c] | 9 | PROCEDURES: |
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[68e678] | 10 | invariantRing(m..); compute ring of invariants of (K,+)-action given by m |
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[5b9dd1] | 11 | derivate(m,f); derivation of f with respect to the vector field m |
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[68e678] | 12 | actionIsProper(m); tests whether action defined by m is proper |
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| 13 | reduction(p,I); SAGBI reduction of p in the subring generated by I |
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[b9b906] | 14 | completeReduction(); complete SAGBI reduction |
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[68e678] | 15 | localInvar(m,p..); invariant polynomial under m computed from p,... |
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[5b9dd1] | 16 | furtherInvar(m..); compute further invariants of m from the given ones |
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[b9b906] | 17 | sortier(id); sorts generators of id by increasing leading terms |
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[5480da] | 18 | "; |
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[4b35a90] | 19 | |
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| 20 | LIB "inout.lib"; |
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| 21 | LIB "general.lib"; |
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[68e678] | 22 | LIB "algebra.lib"; |
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[468f1c] | 23 | LIB "ring.lib"; |
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[4b35a90] | 24 | /////////////////////////////////////////////////////////////////////////////// |
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| 25 | |
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[4f3359] | 26 | proc sortier(def id) |
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[5b9dd1] | 27 | "USAGE: sortier(id); id ideal/module |
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[49f94f] | 28 | RETURN: the same ideal/module but with generators ordered by their |
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| 29 | leading terms, starting with the smallest |
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[68e678] | 30 | EXAMPLE: example sortier; shows an example |
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| 31 | " |
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[4b35a90] | 32 | { |
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| 33 | if(size(id)==0) |
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[49f94f] | 34 | {return(id); } |
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[4b35a90] | 35 | intvec i=sortvec(id); |
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| 36 | int j; |
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[68e678] | 37 | if( typeof(id)=="ideal") |
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[49f94f] | 38 | { ideal m; } |
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[68e678] | 39 | if( typeof(id)=="module") |
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[49f94f] | 40 | { module m; } |
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| 41 | if( typeof(id)!="ideal" and typeof(id)!="module") |
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| 42 | { ERROR("input must be of type ideal or module"); } |
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| 43 | for (j=1;j<=size(i);j++) |
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[4b35a90] | 44 | { |
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| 45 | m[j] = id[i[j]]; |
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| 46 | } |
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| 47 | return(m); |
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| 48 | } |
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[82716e] | 49 | example |
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[4b35a90] | 50 | { "EXAMPLE:"; echo = 2; |
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| 51 | ring q=0,(x,y,z,u,v,w),dp; |
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| 52 | ideal i=w,x,z,y,v; |
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[68e678] | 53 | sortier(i); |
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[4b35a90] | 54 | } |
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| 55 | /////////////////////////////////////////////////////////////////////////////// |
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| 56 | |
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[4f3359] | 57 | proc derivate (matrix m, def id) |
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[091424] | 58 | "USAGE: derivate(m,id); m matrix, id poly/vector/ideal |
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[b4a463] | 59 | ASSUME: m is an nx1 matrix, where n = number of variables of the basering |
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[b9b906] | 60 | RETURN: poly/vector/ideal (same type as input), result of applying the |
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[5b9dd1] | 61 | vector field by the matrix m componentwise to id; |
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| 62 | NOTE: the vector field is m[1,1]*d/dx(1) +...+ m[1,n]*d/dx(n) |
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[091424] | 63 | EXAMPLE: example derivate; shows an example |
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[d2b2a7] | 64 | " |
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[b9b906] | 65 | { |
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[68e678] | 66 | ideal I = ideal(id); |
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| 67 | matrix mh=matrix(jacob(I))*m; |
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[f054d7f] | 68 | if(typeof(id)=="poly") |
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| 69 | { poly j = mh[1,1]; |
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[68e678] | 70 | } |
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| 71 | else |
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[f054d7f] | 72 | { if (typeof(id)=="vector") { vector j = mh[1]; /*the first column*/ } |
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| 73 | else { ideal j = ideal(mh[1]);} |
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[68e678] | 74 | } |
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| 75 | return(j); |
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[4b35a90] | 76 | } |
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[82716e] | 77 | example |
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[4b35a90] | 78 | { "EXAMPLE:"; echo = 2; |
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| 79 | ring q=0,(x,y,z,u,v,w),dp; |
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| 80 | poly f=2xz-y2; |
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[68e678] | 81 | matrix m[6][1] =x,y,0,u,v; |
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[091424] | 82 | derivate(m,f); |
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[68e678] | 83 | vector v = [2xz-y2,u6-3]; |
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[091424] | 84 | derivate(m,v); |
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| 85 | derivate(m,ideal(2xz-y2,u6-3)); |
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[4b35a90] | 86 | } |
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| 87 | |
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| 88 | /////////////////////////////////////////////////////////////////////////////// |
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| 89 | |
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| 90 | proc actionIsProper(matrix m) |
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[8afd58] | 91 | "USAGE: actionIsProper(m); m matrix |
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[68e678] | 92 | ASSUME: m is a nx1 matrix, where n = number of variables of the basering |
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| 93 | RETURN: int = 1, if the action defined by m is proper, 0 if not |
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| 94 | NOTE: m defines a group action which is the exponential of the vector |
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| 95 | field m[1,1]*d/dx(1) +...+ m[1,n]*d/dx(n) |
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[4b35a90] | 96 | EXAMPLE: example actionIsProper; shows an example |
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[d2b2a7] | 97 | " |
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[4b35a90] | 98 | { |
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| 99 | int i; |
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| 100 | ideal id=maxideal(1); |
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| 101 | def bsr=basering; |
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| 102 | |
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| 103 | //changes the basering bsr to bsr[@t] |
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[62de185] | 104 | ring s = create_ring(ring_list(basering)[1],"("+varstr(basering)+",@t)","dp","no_minpoly"); |
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[4b35a90] | 105 | poly inv,delta,tee,j; |
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| 106 | ideal id=imap(bsr,id); |
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| 107 | matrix @m[size(id)+1][1]; |
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| 108 | @m=imap(bsr,m),0; |
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[7e5454] | 109 | int auxv; |
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[4b35a90] | 110 | |
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| 111 | //computes the exp(@t*m)(var(i)) for all i |
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| 112 | for(i=1;i<=nvars(basering)-1;i++) |
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| 113 | { |
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| 114 | inv=var(i); |
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[091424] | 115 | delta=derivate(@m,inv); |
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[4b35a90] | 116 | j=1; |
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[7e5454] | 117 | auxv=1; |
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[4b35a90] | 118 | tee=@t; |
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| 119 | while(delta!=0) |
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| 120 | { |
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| 121 | inv=inv+1/j*delta*tee; |
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[7e5454] | 122 | auxv=auxv+1; |
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| 123 | j=j*auxv; |
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[4b35a90] | 124 | tee=tee*@t; |
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[091424] | 125 | delta=derivate(@m,delta); |
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[4b35a90] | 126 | } |
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[82716e] | 127 | id=id+ideal(inv); |
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[4b35a90] | 128 | } |
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| 129 | i=inSubring(@t,id)[1]; |
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| 130 | setring(bsr); |
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| 131 | return(i); |
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| 132 | } |
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[82716e] | 133 | example |
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[4b35a90] | 134 | { "EXAMPLE:"; echo = 2; |
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| 135 | |
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[68e678] | 136 | ring rf=0,x(1..7),dp; |
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[4b35a90] | 137 | matrix m[7][1]; |
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| 138 | m[4,1]=x(1)^3; |
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| 139 | m[5,1]=x(2)^3; |
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| 140 | m[6,1]=x(3)^3; |
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| 141 | m[7,1]=(x(1)*x(2)*x(3))^2; |
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| 142 | actionIsProper(m); |
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| 143 | |
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[68e678] | 144 | ring rd=0,x(1..5),dp; |
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[4b35a90] | 145 | matrix m[5][1]; |
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| 146 | m[3,1]=x(1); |
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| 147 | m[4,1]=x(2); |
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[dc3a44] | 148 | m[5,1]=1+x(1)*x(4)^2; |
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[4b35a90] | 149 | actionIsProper(m); |
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| 150 | } |
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| 151 | /////////////////////////////////////////////////////////////////////////////// |
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| 152 | |
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[e801fe] | 153 | proc reduction(poly p, ideal dom, list #) |
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[68e678] | 154 | "USAGE: reduction(p,I[,q,n]); p poly, I ideal, [q monomial, n int (optional)] |
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| 155 | RETURN: a polynomial equal to p-H(f1,...,fr), in case the leading |
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[b9b906] | 156 | term LT(p) of p is of the form H(LT(f1),...,LT(fr)) for some |
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[68e678] | 157 | polynomial H in r variables over the base field, I=f1,...,fr; |
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[5b9dd1] | 158 | if q is given, a maximal power a is computed such that q^a divides |
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[68e678] | 159 | p-H(f1,...,fr), and then (p-H(f1,...,fr))/q^a is returned; |
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| 160 | return p if no H is found |
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[5b9dd1] | 161 | if n=1, a different algorithm is chosen which is sometimes faster |
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[68e678] | 162 | (default: n=0; q and n can be given (or not) in any order) |
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[b9b906] | 163 | NOTE: this is a kind of SAGBI reduction in the subalgebra K[f1,...,fr] of |
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[68e678] | 164 | the basering |
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[4b35a90] | 165 | EXAMPLE: example reduction; shows an example |
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[d2b2a7] | 166 | " |
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[4b35a90] | 167 | { |
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[68e678] | 168 | int i,choose; |
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| 169 | int z=ncols(dom); |
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[4b35a90] | 170 | def bsr=basering; |
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[68e678] | 171 | if( size(#) >0 ) |
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| 172 | { if( typeof(#[1]) == "int") |
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| 173 | { choose = #[1]; |
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| 174 | } |
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| 175 | if( typeof(#[1]) == "poly") |
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| 176 | { poly q = #[1]; |
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| 177 | } |
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| 178 | if( size(#)>1 ) |
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| 179 | { if( typeof(#[2]) == "poly") |
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| 180 | { poly q = #[2]; |
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| 181 | } |
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| 182 | if( typeof(#[2]) == "int") |
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| 183 | { choose = #[2]; |
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| 184 | } |
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| 185 | } |
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[4b35a90] | 186 | } |
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[6a0d85] | 187 | |
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[68e678] | 188 | // -------------------- first algorithm (default) ----------------------- |
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| 189 | if ( choose == 0 ) |
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[4b35a90] | 190 | { |
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[68e678] | 191 | list L = algebra_containment(lead(p),lead(dom),1); |
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| 192 | if( L[1]==1 ) |
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| 193 | { |
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| 194 | // the ring L[2] = char(bsr),(x(1..nvars(bsr)),y(1..z)),(dp(n),dp(m)), |
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[3754ca] | 195 | // contains polynomial check s.t. LT(p) is of the form check(LT(f1),...,LT(fr)) |
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[68e678] | 196 | def s1 = L[2]; |
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| 197 | map psi = s1,maxideal(1),dom; |
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| 198 | poly re = p - psi(check); |
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[5b9dd1] | 199 | // divide by the maximal power of #[1] |
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[5e0891c] | 200 | if ( defined(q) == voice ) |
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[68e678] | 201 | { while ((re!=0) && (re!=#[1]) &&(subst(re,#[1],0)==0)) |
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| 202 | { re=re/#[1]; |
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| 203 | } |
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| 204 | } |
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| 205 | return(re); |
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| 206 | } |
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| 207 | return(p); |
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[b9b906] | 208 | } |
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[68e678] | 209 | // ------------------------- second algorithm --------------------------- |
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| 210 | else |
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| 211 | { |
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| 212 | //----------------- arranges the monomial v for elimination ------------- |
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| 213 | poly v=product(maxideal(1)); |
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[82716e] | 214 | |
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[68e678] | 215 | //------------- changes the basering bsr to bsr[@(0),...,@(z)] ---------- |
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[82e5a3] | 216 | list l2 = ring_list(basering)[2]; |
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[f2aefad] | 217 | for (int ii = 0; ii <= z; ii++) |
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| 218 | { |
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| 219 | l2[size(l2)+1] = "@("+string(ii)+")"; |
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| 220 | } |
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[62de185] | 221 | ring s = create_ring(ring_list(basering)[1], l2, "dp", "no_minpoly"); |
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[68e678] | 222 | //constructs the leading ideal of dom=(p-@(0),dom[1]-@(1),...,dom[z]-@(z)) |
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| 223 | ideal dom=imap(bsr,dom); |
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| 224 | for (i=1;i<=z;i++) |
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[4b35a90] | 225 | { |
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[68e678] | 226 | dom[i]=lead(dom[i])-var(nvars(bsr)+i+1); |
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| 227 | } |
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| 228 | dom=lead(imap(bsr,p))-@(0),dom; |
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[5e0891c] | 229 | |
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[68e678] | 230 | //---------- eliminates the variables of the basering bsr -------------- |
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| 231 | //i.e. computes dom intersected with K[@(0),...,@(z)] (this is hard) |
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| 232 | //### hier Variante analog zu algebra_containment einbauen! |
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| 233 | ideal kern=eliminate(dom,imap(bsr,v)); |
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| 234 | |
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[417a505] | 235 | //--------- test whether @(0)-h(@(1),...,@(z)) is in ker --------------- |
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[3754ca] | 236 | // for some polynomial h and divide by maximal power of q=#[1] |
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[68e678] | 237 | poly h; |
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| 238 | z=size(kern); |
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| 239 | for (i=1;i<=z;i++) |
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| 240 | { |
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| 241 | h=kern[i]/@(0); |
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| 242 | if (deg(h)==0) |
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| 243 | { h=(1/h)*kern[i]; |
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| 244 | // define the map psi : s ---> bsr defined by @(i) ---> p,dom[i] |
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| 245 | setring bsr; |
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| 246 | map psi=s,maxideal(1),p,dom; |
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| 247 | poly re=psi(h); |
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[5b9dd1] | 248 | // divide by the maximal power of #[1] |
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[68e678] | 249 | if (size(#)>0) |
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| 250 | { while ((re!=0) && (re!=#[1]) &&(subst(re,#[1],0)==0)) |
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| 251 | { re=re/#[1]; |
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| 252 | } |
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[4b35a90] | 253 | } |
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[68e678] | 254 | return(re); |
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[4b35a90] | 255 | } |
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| 256 | } |
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[68e678] | 257 | setring bsr; |
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| 258 | return(p); |
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[4b35a90] | 259 | } |
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| 260 | } |
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[091424] | 261 | |
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[82716e] | 262 | example |
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[4b35a90] | 263 | { "EXAMPLE:"; echo = 2; |
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| 264 | ring q=0,(x,y,z,u,v,w),dp; |
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| 265 | poly p=x2yz-x2v; |
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[e801fe] | 266 | ideal dom =x-w,u2w+1,yz-v; |
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| 267 | reduction(p,dom); |
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| 268 | reduction(p,dom,w); |
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[4b35a90] | 269 | } |
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| 270 | |
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| 271 | /////////////////////////////////////////////////////////////////////////////// |
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| 272 | |
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[e801fe] | 273 | proc completeReduction(poly p, ideal dom, list #) |
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[68e678] | 274 | "USAGE: completeReduction(p,I[,q,n]); p poly, I ideal, [q monomial, n int] |
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[b4a463] | 275 | RETURN: a polynomial, the SAGBI reduction of the polynomial p with respect to I |
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[68e678] | 276 | via the procedure 'reduction' as long as possible |
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[5b9dd1] | 277 | if n=1, a different algorithm is chosen which is sometimes faster |
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[68e678] | 278 | (default: n=0; q and n can be given (or not) in any order) |
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| 279 | NOTE: help reduction; shows an explanation of SAGBI reduction |
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[4b35a90] | 280 | EXAMPLE: example completeReduction; shows an example |
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[d2b2a7] | 281 | " |
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[4b35a90] | 282 | { |
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| 283 | poly p1=p; |
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[e801fe] | 284 | poly p2=reduction(p,dom,#); |
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[4b35a90] | 285 | while (p1!=p2) |
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| 286 | { |
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| 287 | p1=p2; |
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[e801fe] | 288 | p2=reduction(p1,dom,#); |
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[4b35a90] | 289 | } |
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| 290 | return(p2); |
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| 291 | } |
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[82716e] | 292 | example |
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[4b35a90] | 293 | { "EXAMPLE:"; echo = 2; |
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| 294 | ring q=0,(x,y,z,u,v,w),dp; |
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| 295 | poly p=x2yz-x2v; |
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[e801fe] | 296 | ideal dom =x-w,u2w+1,yz-v; |
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| 297 | completeReduction(p,dom); |
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| 298 | completeReduction(p,dom,w); |
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[4b35a90] | 299 | } |
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| 300 | |
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| 301 | /////////////////////////////////////////////////////////////////////////////// |
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| 302 | |
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[6a0d85] | 303 | proc completeReductionnew(poly p, ideal dom, list #) |
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| 304 | "USAGE: completeReduction(p,I[,q,n]); p poly, I ideal, [q monomial, n int] |
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| 305 | RETURN: a polynomial, the SAGBI reduction of the polynomial p with I |
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| 306 | via the procedure 'reduction' as long as possible |
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[5b9dd1] | 307 | if n=1, a different algorithm is chosen which is sometimes faster |
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[6a0d85] | 308 | (default: n=0; q and n can be given (or not) in any order) |
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| 309 | NOTE: help reduction; shows an explanation of SAGBI reduction |
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| 310 | EXAMPLE: example completeReduction; shows an example |
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| 311 | " |
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| 312 | { |
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| 313 | if(p==0) |
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| 314 | { |
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| 315 | return(p); |
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| 316 | } |
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| 317 | poly p1=p; |
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| 318 | poly p2=reduction(p,dom,#); |
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| 319 | while (p1!=p2) |
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| 320 | { |
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| 321 | p1=p2; |
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| 322 | p2=reduction(p1,dom,#); |
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| 323 | } |
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| 324 | poly re=lead(p2)+completeReduction(p2-lead(p2),dom,#); |
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| 325 | return(re); |
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| 326 | } |
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| 327 | |
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| 328 | /////////////////////////////////////////////////////////////////////////////// |
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| 329 | |
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[4b35a90] | 330 | proc localInvar(matrix m, poly p, poly q, poly h) |
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[68e678] | 331 | "USAGE: localInvar(m,p,q,h); m matrix, p,q,h polynomials |
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[5b9dd1] | 332 | ASSUME: m(q) and h are invariant under the vector field m, i.e. m(m(q))=m(h)=0 |
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[68e678] | 333 | h must be a ring variable |
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[5b9dd1] | 334 | RETURN: a polynomial, the invariant polynomial of the vector field |
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[51d95b] | 335 | @format |
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| 336 | m = m[1,1]*d/dx(1) +...+ m[n,1]*d/dx(n) |
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| 337 | @end format |
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[b9b906] | 338 | with respect to p,q,h. It is defined as follows: set inv = p if p is |
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[b4a463] | 339 | invariant, and else set |
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[68e678] | 340 | inv = m(q)^N * sum_i=1..N-1{ (-1)^i*(1/i!)*m^i(p)*(q/m(q))^i } |
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[b4a463] | 341 | where m^N(p) = 0, m^(N-1)(p) != 0; the result is inv divided by h |
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| 342 | as often as possible |
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[4b35a90] | 343 | EXAMPLE: example localInvar; shows an example |
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[d2b2a7] | 344 | " |
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[4b35a90] | 345 | { |
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[091424] | 346 | if ((derivate(m,h) !=0) || (derivate(m,derivate(m,q)) !=0)) |
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[4b35a90] | 347 | { |
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[68e678] | 348 | "//the last two polynomials of the input must be invariant functions"; |
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[4b35a90] | 349 | return(q); |
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| 350 | } |
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[68e678] | 351 | int ii,k; |
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| 352 | for ( k=1; k <= nvars(basering); k++ ) |
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| 353 | { if (h == var(k)) |
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| 354 | { ii=1; |
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| 355 | } |
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| 356 | } |
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| 357 | if( ii==0 ) |
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| 358 | { "// the last argument must be a ring variable"; |
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| 359 | return(q); |
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| 360 | } |
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[b9b906] | 361 | |
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[4b35a90] | 362 | poly inv=p; |
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[091424] | 363 | poly dif= derivate(m,inv); |
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| 364 | poly a=derivate(m,q); |
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[4b35a90] | 365 | poly sgn=-1; |
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| 366 | poly coeff=sgn*q; |
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[68e678] | 367 | k=1; |
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[4b35a90] | 368 | if (dif==0) |
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| 369 | { |
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| 370 | return(inv); |
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| 371 | } |
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[82716e] | 372 | while (dif!=0) |
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[4b35a90] | 373 | { |
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| 374 | inv=(a*inv)+(coeff*dif); |
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[091424] | 375 | dif=derivate(m,dif); |
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[4b35a90] | 376 | k=k+1; |
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| 377 | coeff=q*coeff*sgn/k; |
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| 378 | } |
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| 379 | while ((inv!=0) && (inv!=h) &&(subst(inv,h,0)==0)) |
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| 380 | { |
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| 381 | inv=inv/h; |
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| 382 | } |
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| 383 | return(inv); |
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| 384 | } |
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[82716e] | 385 | example |
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[4b35a90] | 386 | { "EXAMPLE:"; echo = 2; |
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| 387 | ring q=0,(x,y,z),dp; |
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| 388 | matrix m[3][1]; |
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| 389 | m[2,1]=x; |
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| 390 | m[3,1]=y; |
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| 391 | poly in=localInvar(m,z,y,x); |
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| 392 | in; |
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| 393 | } |
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| 394 | /////////////////////////////////////////////////////////////////////////////// |
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| 395 | |
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[68e678] | 396 | proc furtherInvar(matrix m, ideal id, ideal karl, poly q, list #) |
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| 397 | "USAGE: furtherInvar(m,id,karl,q); m matrix, id,karl ideals, q poly, n int |
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[5b9dd1] | 398 | ASSUME: karl,id,q are invariant under the vector field m, |
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[68e678] | 399 | moreover, q must be a variable |
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[b9b906] | 400 | RETURN: list of two ideals, the first ideal contains further invariants of |
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[5b9dd1] | 401 | the vector field |
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[51d95b] | 402 | @format |
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| 403 | m = sum m[i,1]*d/dx(i) with respect to id,p,q, |
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| 404 | @end format |
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[b9b906] | 405 | i.e. we compute elements in the (invariant) subring generated by id |
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[b4a463] | 406 | which are divisible by q and divide them by q as often as possible. |
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| 407 | The second ideal contains all invariants given before. |
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| 408 | If n=1, a different algorithm is chosen which is sometimes faster |
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[68e678] | 409 | (default: n=0) |
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[4b35a90] | 410 | EXAMPLE: example furtherInvar; shows an example |
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[d2b2a7] | 411 | " |
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[4b35a90] | 412 | { |
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[68e678] | 413 | list ll = q; |
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| 414 | if ( size(#)>0 ) |
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| 415 | { ll = ll+list(#[1]); |
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[b9b906] | 416 | } |
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[4b35a90] | 417 | int i; |
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[694a1e] | 418 | ideal null,eins; |
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[68e678] | 419 | int z=ncols(id); |
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[4b35a90] | 420 | intvec v; |
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[68e678] | 421 | def br=basering; |
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[4b35a90] | 422 | ideal su; |
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[f054d7f] | 423 | list @l; |
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[68e678] | 424 | for (i=1; i<=z; i++) |
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[4b35a90] | 425 | { |
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[b9b906] | 426 | su[i]=subst(id[i],q,0); |
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[f054d7f] | 427 | @l[i]="y("+string(i)+")"; |
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[4b35a90] | 428 | } |
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[68e678] | 429 | // -- define the map phi : r1 ---> br defined by y(i) ---> id[i](q=0) -- |
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[f054d7f] | 430 | def r1=ring(list(ring_list(br)[1],@l,list(list("dp",1:z)),ideal(0))); |
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[68e678] | 431 | setring br; |
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[4b35a90] | 432 | map phi=r1,su; |
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[82716e] | 433 | setring r1; |
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[68e678] | 434 | // --------------- compute the kernel of phi --------------------------- |
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| 435 | ideal ker=preimage(br,phi,null); |
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[6a0d85] | 436 | ker=mstd(ker)[2]; |
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[68e678] | 437 | // ---- define the map psi : r1 ---> br defined by y(i) ---> id[i] ----- |
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| 438 | setring br; |
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[4b35a90] | 439 | map psi=r1,id; |
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[68e678] | 440 | // ------------------- compute psi(ker(phi)) -------------------------- |
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[4b35a90] | 441 | ideal rel=psi(ker); |
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[417a505] | 442 | // divide by maximal power of q, test whether we really obtain invariants |
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[4b35a90] | 443 | for (i=1;i<=size(rel);i++) |
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| 444 | { |
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| 445 | while ((rel[i]!=0) && (rel[i]!=q) &&(subst(rel[i],q,0)==0)) |
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| 446 | { |
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| 447 | rel[i]=rel[i]/q; |
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[091424] | 448 | if (derivate(m,rel[i])!=0) |
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[4b35a90] | 449 | { |
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[68e678] | 450 | "// error in furtherInvar, function not invariant:"; |
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[4b35a90] | 451 | rel[i]; |
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| 452 | } |
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| 453 | } |
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| 454 | rel[i]=simplify(rel[i],1); |
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| 455 | } |
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[68e678] | 456 | // --------------------------------------------------------------------- |
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[b9b906] | 457 | // test whether some variables occur linearly and then delete the |
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[68e678] | 458 | // corresponding invariant function |
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[4b35a90] | 459 | setring r1; |
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| 460 | int j; |
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| 461 | for (i=1;i<=size(ker);i=i+1) |
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| 462 | { |
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| 463 | for (j=1;j<=z;j++) |
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| 464 | { |
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| 465 | if (deg(ker[i]/y(j))==0) |
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| 466 | { |
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[68e678] | 467 | setring br; |
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| 468 | rel[i]= completeReduction(rel[i],karl,ll); |
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[4b35a90] | 469 | if(rel[i]!=0) |
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| 470 | { |
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| 471 | karl[j+1]=rel[i]; |
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| 472 | rel[i]=0; |
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[694a1e] | 473 | eins=1; |
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[4b35a90] | 474 | } |
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| 475 | setring r1; |
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| 476 | } |
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| 477 | } |
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[82716e] | 478 | |
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[4b35a90] | 479 | } |
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[68e678] | 480 | setring br; |
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[694a1e] | 481 | rel=rel+null; |
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| 482 | if(size(rel)==0){rel=eins;} |
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| 483 | list l=rel,karl; |
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[4b35a90] | 484 | return(l); |
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| 485 | } |
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[82716e] | 486 | example |
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[4b35a90] | 487 | { "EXAMPLE:"; echo = 2; |
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| 488 | ring r=0,(x,y,z,u),dp; |
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| 489 | matrix m[4][1]; |
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| 490 | m[2,1]=x; |
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| 491 | m[3,1]=y; |
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| 492 | m[4,1]=z; |
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| 493 | ideal id=localInvar(m,z,y,x),localInvar(m,u,y,x); |
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| 494 | ideal karl=id,x; |
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| 495 | list in=furtherInvar(m,id,karl,x); |
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| 496 | in; |
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| 497 | } |
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| 498 | /////////////////////////////////////////////////////////////////////////////// |
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| 499 | |
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[091424] | 500 | proc invariantRing(matrix m, poly p, poly q, int b, list #) |
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| 501 | "USAGE: invariantRing(m,p,q,b[,r,pa]); m matrix, p,q poly, b,r int, pa string |
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[68e678] | 502 | ASSUME: p,q variables with m(p)=q and q invariant under m |
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[b9b906] | 503 | i.e. if p=x(i) and q=x(j) then m[j,1]=0 and m[i,1]=x(j) |
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| 504 | RETURN: ideal, containing generators of the ring of invariants of the |
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[5b9dd1] | 505 | additive group (K,+) given by the vector field |
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[51d95b] | 506 | @format |
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| 507 | m = m[1,1]*d/dx(1) +...+ m[n,1]*d/dx(n). |
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| 508 | @end format |
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[b9b906] | 509 | If b>0 the computation stops after all invariants of degree <= b |
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| 510 | (and at least one of higher degree) are found or when all invariants |
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[68e678] | 511 | are computed. |
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[b9b906] | 512 | If b<=0, the computation continues until all generators |
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[68e678] | 513 | of the ring of invariants are computed (should be used only if the |
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[b4a463] | 514 | ring of invariants is known to be finitely generated, otherwise the |
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[091424] | 515 | algorithm might not stop). |
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[68e678] | 516 | If r=1 a different reduction is used which is sometimes faster |
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[b9b906] | 517 | (default r=0). |
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| 518 | DISPLAY: if pa is given (any string as 5th or 6th argument), the computation |
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[68e678] | 519 | pauses whenever new invariants are found and displays them |
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[b4a463] | 520 | THEORY: The algorithm for computing the ring of invariants works in char 0 |
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| 521 | or suffiently large characteristic. |
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| 522 | (K,+) acts as the exponential of the vector field defined by the |
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| 523 | matrix m. |
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| 524 | For background see G.-M. Greuel, G. Pfister, |
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| 525 | Geometric quotients of unipotent group actions, Proc. |
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[5b9dd1] | 526 | London Math. Soc. (3) 67, 75-105 (1993). |
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[b9b906] | 527 | EXAMPLE: example invariantRing; shows an example |
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[d2b2a7] | 528 | " |
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[4b35a90] | 529 | { |
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| 530 | ideal j; |
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| 531 | int i,it; |
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[68e678] | 532 | list ll=q; |
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[091424] | 533 | int bou=b; |
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[68e678] | 534 | if( size(#) >0 ) |
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| 535 | { if( typeof(#[1]) == "int") |
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| 536 | { ll=ll+list(#[1]); |
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| 537 | } |
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| 538 | if( typeof(#[1]) == "string") |
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| 539 | { string pau=#[1]; |
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| 540 | } |
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| 541 | if( size(#)>1 ) |
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[3ed41c] | 542 | { |
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| 543 | if( typeof(#[2]) == "string") |
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| 544 | { string pau=#[2]; |
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| 545 | } |
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| 546 | if( typeof(#[2]) == "int") |
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[68e678] | 547 | { ll=ll+list(#[2]); |
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| 548 | } |
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| 549 | } |
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[4b35a90] | 550 | } |
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[82716e] | 551 | int z; |
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[4b35a90] | 552 | ideal karl; |
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| 553 | ideal k1=1; |
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| 554 | list k2; |
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[68e678] | 555 | //------------------ computation of local invariants ------------------ |
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[4b35a90] | 556 | for (i=1;i<=nvars(basering);i++) |
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| 557 | { |
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| 558 | karl=karl+localInvar(m,var(i),p,q); |
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| 559 | } |
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[091424] | 560 | if( defined(pau) ) |
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[68e678] | 561 | { ""; |
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| 562 | "// local invariants computed:"; |
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| 563 | ""; |
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[4b35a90] | 564 | karl; |
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[68e678] | 565 | ""; |
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[5e0891c] | 566 | pause("// hit return key to continue!"); |
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| 567 | ""; |
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[4b35a90] | 568 | } |
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[68e678] | 569 | //------------------ computation of further invariants ---------------- |
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[4b35a90] | 570 | it=0; |
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| 571 | while (size(k1)!=0) |
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[68e678] | 572 | { |
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[82716e] | 573 | // test if the new invariants are already in the ring generated |
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[68e678] | 574 | // by the invariants we constructed so far |
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[4b35a90] | 575 | it++; |
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| 576 | karl=sortier(karl); |
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| 577 | j=q; |
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| 578 | for (i=1;i<=size(karl);i++) |
---|
| 579 | { |
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[68e678] | 580 | j=j + simplify(completeReduction(karl[i],j,ll),1); |
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[4b35a90] | 581 | } |
---|
| 582 | karl=j; |
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| 583 | j[1]=0; |
---|
| 584 | j=simplify(j,2); |
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| 585 | k2=furtherInvar(m,j,karl,q); |
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| 586 | k1=k2[1]; |
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| 587 | karl=k2[2]; |
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[694a1e] | 588 | if(k1[1]!=1) |
---|
[5e0891c] | 589 | { |
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[694a1e] | 590 | k1=sortier(k1); |
---|
| 591 | z=size(k1); |
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| 592 | for (i=1;i<=z;i++) |
---|
| 593 | { |
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| 594 | k1[i]= completeReduction(k1[i],karl,ll); |
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| 595 | if (k1[i]!=0) |
---|
| 596 | { |
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| 597 | karl=karl+simplify(k1[i],1); |
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| 598 | } |
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| 599 | } |
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| 600 | if( defined(pau) == voice) |
---|
| 601 | { |
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[68e678] | 602 | "// the invariants after",it,"iteration(s):"; ""; |
---|
| 603 | karl;""; |
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[5e0891c] | 604 | pause("// hit return key to continue!"); |
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[68e678] | 605 | ""; |
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[694a1e] | 606 | } |
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| 607 | if( (bou>0) && (size(k1)>0) ) |
---|
| 608 | { |
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| 609 | if( deg(k1[size(k1)])>bou ) |
---|
| 610 | { |
---|
| 611 | return(karl); |
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| 612 | } |
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[4b35a90] | 613 | } |
---|
| 614 | } |
---|
| 615 | } |
---|
| 616 | return(karl); |
---|
| 617 | } |
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[82716e] | 618 | example |
---|
[4b35a90] | 619 | { "EXAMPLE:"; echo = 2; |
---|
| 620 | |
---|
[091424] | 621 | //Winkelmann: free action but Spec(k[x(1),...,x(5)]) --> Spec(invariant ring) |
---|
[68e678] | 622 | //is not surjective |
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[82716e] | 623 | |
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[4b35a90] | 624 | ring rw=0,(x(1..5)),dp; |
---|
| 625 | matrix m[5][1]; |
---|
| 626 | m[3,1]=x(1); |
---|
| 627 | m[4,1]=x(2); |
---|
| 628 | m[5,1]=1+x(1)*x(4)+x(2)*x(3); |
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[091424] | 629 | ideal in=invariantRing(m,x(3),x(1),0); //compute full invarint ring |
---|
[4b35a90] | 630 | in; |
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[82716e] | 631 | |
---|
[4b35a90] | 632 | //Deveney/Finston: The ring of invariants is not finitely generated |
---|
[82716e] | 633 | |
---|
[4b35a90] | 634 | ring rf=0,(x(1..7)),dp; |
---|
| 635 | matrix m[7][1]; |
---|
| 636 | m[4,1]=x(1)^3; |
---|
| 637 | m[5,1]=x(2)^3; |
---|
| 638 | m[6,1]=x(3)^3; |
---|
| 639 | m[7,1]=(x(1)*x(2)*x(3))^2; |
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[68e678] | 640 | ideal in=invariantRing(m,x(4),x(1),6); //all invariants up to degree 6 |
---|
[4b35a90] | 641 | in; |
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[68e678] | 642 | } |
---|
| 643 | /////////////////////////////////////////////////////////////////////////////// |
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| 644 | /* Further examplex |
---|
[b9b906] | 645 | |
---|
[68e678] | 646 | //Deveney/Finston: Proper Ga-action which is not locally trivial |
---|
[4b35a90] | 647 | //r[x(1),...,x(5)] is not flat over the ring of invariants |
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[68e678] | 648 | LIB "invar.lib"; |
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[4b35a90] | 649 | ring rd=0,(x(1..5)),dp; |
---|
| 650 | matrix m[5][1]; |
---|
| 651 | m[3,1]=x(1); |
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| 652 | m[4,1]=x(2); |
---|
| 653 | m[5,1]=1+x(1)*x(4)^2; |
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[68e678] | 654 | ideal in=invariantRing(m,x(3),x(1),0,1); |
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[4b35a90] | 655 | in; |
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[82716e] | 656 | |
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[4b35a90] | 657 | actionIsProper(m); |
---|
[82716e] | 658 | |
---|
[68e678] | 659 | //compute the algebraic relations between the invariants |
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[4b35a90] | 660 | int z=size(in); |
---|
| 661 | ideal null; |
---|
| 662 | ring r1=0,(y(1..z)),dp; |
---|
| 663 | setring rd; |
---|
| 664 | map phi=r1,in; |
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[82716e] | 665 | setring r1; |
---|
| 666 | ideal ker=preimage(rd,phi,null); |
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[4b35a90] | 667 | ker; |
---|
[82716e] | 668 | |
---|
[4b35a90] | 669 | //the discriminant |
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[82716e] | 670 | |
---|
[4b35a90] | 671 | ring r=0,(x(1..2),y(1..2),z,t),dp; |
---|
| 672 | poly p=z+(1+x(1)*y(2)^2)*t+x(1)*y(1)*y(2)*t^2+(1/3)*x(1)*y(1)^2*t^3; |
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[82716e] | 673 | |
---|
[4b35a90] | 674 | matrix m[5][5]; |
---|
| 675 | m[1,1]=z; |
---|
| 676 | m[1,2]=x(1)*y(2)^2+1; |
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| 677 | m[1,3]=x(1)*y(1)*y(2); |
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| 678 | m[1,4]=1/3*x(1)*y(1)^2; |
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| 679 | m[1,5]=0; |
---|
| 680 | m[2,1]=0; |
---|
| 681 | m[2,2]=z; |
---|
| 682 | m[2,3]=x(1)*y(2)^2+1; |
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| 683 | m[2,4]=x(1)*y(1)*y(2); |
---|
| 684 | m[2,5]=1/3*x(1)*y(1)^2; |
---|
| 685 | m[3,1]=x(1)*y(2)^2+1; |
---|
| 686 | m[3,2]=2*x(1)*y(1)*y(2); |
---|
| 687 | m[3,3]=x(1)*y(1)^2; |
---|
| 688 | m[3,4]=0; |
---|
| 689 | m[3,5]=0; |
---|
| 690 | m[4,1]=0; |
---|
| 691 | m[4,2]=x(1)*y(2)^2+1; |
---|
| 692 | m[4,3]=2*x(1)*y(1)*y(2); |
---|
| 693 | m[4,4]=x(1)*y(1)^2; |
---|
| 694 | m[4,5]=0; |
---|
| 695 | m[5,1]=0; |
---|
| 696 | m[5,2]=0; |
---|
| 697 | m[5,3]=x(1)*y(2)^2+1; |
---|
| 698 | m[5,4]=2*x(1)*y(1)*y(2); |
---|
| 699 | m[5,5]=x(1)*y(1)^2; |
---|
[82716e] | 700 | |
---|
[4b35a90] | 701 | poly disc=9*det(m)/(x(1)^2*y(1)^4); |
---|
[82716e] | 702 | |
---|
[68e678] | 703 | LIB "invar.lib"; |
---|
[4b35a90] | 704 | matrix n[6][1]; |
---|
| 705 | n[2,1]=x(1); |
---|
| 706 | n[4,1]=y(1); |
---|
| 707 | n[5,1]=1+x(1)*y(2)^2; |
---|
[82716e] | 708 | |
---|
[091424] | 709 | derivate(n,disc); |
---|
[82716e] | 710 | |
---|
[68e678] | 711 | //x(1)^3*y(2)^6-6*x(1)^2*y(1)*y(2)^3*z+6*x(1)^2*y(2)^4+9*x(1)*y(1)^2*z^2-18*x(1)*y(1)*y(2)*z+9*x(1)*y(2)^2+4 |
---|
[82716e] | 712 | |
---|
[68e678] | 713 | ////////////////////////////////////////////////////////////////////////////// |
---|
| 714 | //constructive approach to Weizenboecks theorem |
---|
[82716e] | 715 | |
---|
[68e678] | 716 | int n=5; |
---|
| 717 | // int n=6; //limit |
---|
| 718 | ring w=32003,(x(1..n)),wp(1..n); |
---|
[82716e] | 719 | |
---|
[5b9dd1] | 720 | // definition of the vector field m=sum m[i]*d/dx(i) |
---|
[4b35a90] | 721 | matrix m[n][1]; |
---|
| 722 | int i; |
---|
| 723 | for (i=1;i<=n-1;i=i+1) |
---|
| 724 | { |
---|
| 725 | m[i+1,1]=x(i); |
---|
| 726 | } |
---|
[68e678] | 727 | ideal in=invariantRing(m,x(2),x(1),0,""); |
---|
[6a0d85] | 728 | |
---|
[4b35a90] | 729 | in; |
---|
[68e678] | 730 | */ |
---|