[d2b2a7] | 1 | // $Id: invar.lib,v 1.6 1998-05-05 11:55:30 krueger Exp $ |
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[4b35a90] | 2 | /////////////////////////////////////////////////////// |
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| 3 | // invar.lib |
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| 4 | // algorithm for computing the ring of invariants under |
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[5dc4ea] | 5 | // the action of the additive group (C,+) |
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[4b35a90] | 6 | // written by Gerhard Pfister |
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| 7 | ////////////////////////////////////////////////////// |
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| 8 | |
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[d2b2a7] | 9 | version="$Id: invar.lib,v 1.6 1998-05-05 11:55:30 krueger Exp $"; |
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[5480da] | 10 | info=" |
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[e801fe] | 11 | LIBRARY: invar.lib PROCEDURES FOR COMPUTING INVARIANTS OF (C,+)-ACTIONS |
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[4b35a90] | 12 | |
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| 13 | invariantRing(matrix m,poly p,poly q,int choose) |
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| 14 | // ring of invariants of the action of the additive group |
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| 15 | // defined by the vectorfield corresponding to the matrix m |
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| 16 | // (m[i,1] are the coefficients of d/dx(i)) |
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| 17 | // the polys p and q are assumed to be variables x(i) and x(j) |
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| 18 | // such that m[j,1]=0 and m[i,1]=x(j) |
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| 19 | // if choose=0 the computation stops if generators of the ring |
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| 20 | // of invariants are computed (to be used only if you know that |
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| 21 | // the ring of invariants is finitey generated) |
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| 22 | // if choose<>0 it computes invariants up to degree choose |
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| 23 | |
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| 24 | actionIsProper(matrix m) |
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| 25 | // returns 1 if the action of the additive group defined by the |
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| 26 | // matrix m as above i proper and 0 if not. |
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[5480da] | 27 | "; |
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[4b35a90] | 28 | |
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| 29 | /////////////////////////////////////////////////////////////////////////////// |
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| 30 | |
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| 31 | LIB "inout.lib"; |
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| 32 | LIB "general.lib"; |
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| 33 | |
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| 34 | /////////////////////////////////////////////////////////////////////////////// |
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| 35 | |
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| 36 | |
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| 37 | proc sortier(ideal id) |
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| 38 | { |
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| 39 | if(size(id)==0) |
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| 40 | { |
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| 41 | return(id); |
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| 42 | } |
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| 43 | intvec i=sortvec(id); |
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| 44 | int j; |
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| 45 | ideal m; |
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| 46 | for (j=1;j<=size(i);j++) |
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| 47 | { |
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| 48 | m[j] = id[i[j]]; |
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| 49 | } |
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| 50 | return(m); |
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| 51 | } |
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| 52 | example |
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| 53 | { "EXAMPLE:"; echo = 2; |
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| 54 | ring q=0,(x,y,z,u,v,w),dp; |
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| 55 | ideal i=w,x,z,y,v; |
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| 56 | ideal j=sortier(i); |
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| 57 | j; |
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| 58 | } |
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| 59 | |
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| 60 | |
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| 61 | /////////////////////////////////////////////////////////////////////////////// |
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| 62 | |
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| 63 | |
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| 64 | proc der (matrix m, poly f) |
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[d2b2a7] | 65 | "USAGE: der(m,f); m matrix, f poly |
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[4b35a90] | 66 | RETURN: poly= application of the vectorfield m befined by the matrix m |
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| 67 | (m[i,1] are the coefficients of d/dx(i)) to f |
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| 68 | NOTE: |
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| 69 | EXAMPLE: example der; shows an example |
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[d2b2a7] | 70 | " |
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[4b35a90] | 71 | { |
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| 72 | matrix mh=matrix(jacob(f))*m; |
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| 73 | return(mh[1,1]); |
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| 74 | } |
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| 75 | example |
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| 76 | { "EXAMPLE:"; echo = 2; |
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| 77 | ring q=0,(x,y,z,u,v,w),dp; |
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| 78 | poly f=2xz-y2; |
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| 79 | matrix m[6][1]; |
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| 80 | m[2,1]=x; |
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| 81 | m[3,1]=y; |
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| 82 | m[5,1]=u; |
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| 83 | m[6,1]=v; |
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| 84 | der(m,f); |
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| 85 | } |
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| 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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[d2b2a7] | 91 | "USAGE: actionIsProper(m); m matrix |
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[4b35a90] | 92 | RETURN: int= 1 if is proper, 0 else |
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| 93 | NOTE: |
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| 94 | EXAMPLE: example actionIsProper; shows an example |
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[d2b2a7] | 95 | " |
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[4b35a90] | 96 | { |
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| 97 | int i; |
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| 98 | ideal id=maxideal(1); |
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| 99 | def bsr=basering; |
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| 100 | |
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| 101 | //changes the basering bsr to bsr[@t] |
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| 102 | execute "ring s="+charstr(basering)+",("+varstr(basering)+",@t),dp;"; |
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| 103 | poly inv,delta,tee,j; |
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| 104 | ideal id=imap(bsr,id); |
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| 105 | matrix @m[size(id)+1][1]; |
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| 106 | @m=imap(bsr,m),0; |
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| 107 | |
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| 108 | //computes the exp(@t*m)(var(i)) for all i |
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| 109 | for(i=1;i<=nvars(basering)-1;i++) |
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| 110 | { |
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| 111 | inv=var(i); |
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| 112 | delta=der(@m,inv); |
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| 113 | j=1; |
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| 114 | tee=@t; |
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| 115 | while(delta!=0) |
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| 116 | { |
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| 117 | inv=inv+1/j*delta*tee; |
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| 118 | j=j*(j+1); |
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| 119 | tee=tee*@t; |
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| 120 | delta=der(@m,delta); |
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| 121 | } |
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| 122 | id=id+ideal(inv); |
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| 123 | } |
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| 124 | i=inSubring(@t,id)[1]; |
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| 125 | setring(bsr); |
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| 126 | return(i); |
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| 127 | } |
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| 128 | example |
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| 129 | { "EXAMPLE:"; echo = 2; |
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| 130 | |
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| 131 | ring rf=0,(x(1..7)),dp; |
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| 132 | matrix m[7][1]; |
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| 133 | m[4,1]=x(1)^3; |
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| 134 | m[5,1]=x(2)^3; |
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| 135 | m[6,1]=x(3)^3; |
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| 136 | m[7,1]=(x(1)*x(2)*x(3))^2; |
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| 137 | actionIsProper(m); |
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| 138 | |
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| 139 | ring rd=0,(x(1..5)),dp; |
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| 140 | matrix m[5][1]; |
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| 141 | m[3,1]=x(1); |
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| 142 | m[4,1]=x(2); |
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| 143 | m[5,1]=1+x(1)*x(4)^2 |
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| 144 | actionIsProper(m); |
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| 145 | } |
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| 146 | /////////////////////////////////////////////////////////////////////////////// |
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| 147 | |
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| 148 | |
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[e801fe] | 149 | proc reduction(poly p, ideal dom, list #) |
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[d2b2a7] | 150 | "USAGE: reduction(p,dom,q); p poly, dom ideal, q (optional) monomial |
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[4b35a90] | 151 | RETURN: poly= (p-H(f1,...,fr))/q^a, if Lt(p)=H(Lt(f1),...,Lt(fr)) for |
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| 152 | some polynomial H in r variables over the base field, |
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| 153 | a maximal such that q^a devides p-H(f1,...,fr), |
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[e801fe] | 154 | dom =(f1,...,fr) |
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[4b35a90] | 155 | NOTE: |
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| 156 | EXAMPLE: example reduction; shows an example |
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[d2b2a7] | 157 | " |
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[4b35a90] | 158 | { |
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| 159 | int i; |
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[e801fe] | 160 | int z=size(dom); |
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[4b35a90] | 161 | def bsr=basering; |
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| 162 | |
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| 163 | //arranges the monomial v for elimination |
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| 164 | poly v=var(1); |
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| 165 | for (i=2;i<=nvars(basering);i=i+1) |
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| 166 | { |
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| 167 | v=v*var(i); |
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| 168 | } |
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| 169 | |
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| 170 | //changes the basering bsr to bsr[@(0),...,@(z)] |
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| 171 | execute "ring s="+charstr(basering)+",("+varstr(basering)+",@(0..z)),dp;"; |
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| 172 | |
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[e801fe] | 173 | //costructes the ideal dom=(p-@(0),dom[1]-@(1),...,dom[z]-@(z)) |
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| 174 | ideal dom=imap(bsr,dom); |
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[4b35a90] | 175 | for (i=1;i<=z;i++) |
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| 176 | { |
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[e801fe] | 177 | dom[i]=lead(dom[i])-var(nvars(bsr)+i+1); |
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[4b35a90] | 178 | } |
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[e801fe] | 179 | dom=lead(imap(bsr,p))-@(0),dom; |
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[4b35a90] | 180 | |
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| 181 | //eliminates the variables of the basering bsr |
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[e801fe] | 182 | //i.e. computes dom intersected with K[@(0),...,@(z)] |
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| 183 | ideal kern=eliminate(dom,imap(bsr,v)); |
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[4b35a90] | 184 | |
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| 185 | // test wether @(0)-h(@(1),...,@(z)) is in ker for some poly h |
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| 186 | poly h; |
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| 187 | z=size(kern); |
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| 188 | for (i=1;i<=z;i++) |
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| 189 | { |
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| 190 | h=kern[i]/@(0); |
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| 191 | if (deg(h)==0) |
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| 192 | { |
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| 193 | h=(1/h)*kern[i]; |
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[e801fe] | 194 | // defines the map psi : s ---> bsr defined by @(i) ---> p,dom[i] |
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[4b35a90] | 195 | setring bsr; |
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[e801fe] | 196 | map psi=s,maxideal(1),p,dom; |
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[4b35a90] | 197 | poly re=psi(h); |
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| 198 | |
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| 199 | // devides by the maximal power of #[1] |
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| 200 | if (size(#)>0) |
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| 201 | { |
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| 202 | while ((re!=0) && (re!=#[1]) &&(subst(re,#[1],0)==0)) |
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| 203 | { |
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| 204 | re=re/#[1]; |
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| 205 | } |
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| 206 | } |
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| 207 | |
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| 208 | return(re); |
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| 209 | } |
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| 210 | } |
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| 211 | setring bsr; |
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| 212 | return(p); |
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| 213 | } |
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| 214 | example |
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| 215 | { "EXAMPLE:"; echo = 2; |
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| 216 | ring q=0,(x,y,z,u,v,w),dp; |
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| 217 | poly p=x2yz-x2v; |
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[e801fe] | 218 | ideal dom =x-w,u2w+1,yz-v; |
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| 219 | reduction(p,dom); |
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| 220 | reduction(p,dom,w); |
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[4b35a90] | 221 | } |
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| 222 | |
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| 223 | /////////////////////////////////////////////////////////////////////////////// |
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| 224 | |
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[e801fe] | 225 | proc completeReduction(poly p, ideal dom, list #) |
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[d2b2a7] | 226 | "USAGE: completeReduction(p,dom,q); p poly, dom ideal, |
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[4b35a90] | 227 | q (optional) monomial |
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[e801fe] | 228 | RETURN: poly= the polynomial p reduced with dom via the procedure |
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[4b35a90] | 229 | reduction as long as possible |
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| 230 | NOTE: |
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| 231 | EXAMPLE: example completeReduction; shows an example |
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[d2b2a7] | 232 | " |
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[4b35a90] | 233 | { |
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| 234 | poly p1=p; |
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[e801fe] | 235 | poly p2=reduction(p,dom,#); |
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[4b35a90] | 236 | while (p1!=p2) |
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| 237 | { |
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| 238 | p1=p2; |
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[e801fe] | 239 | p2=reduction(p1,dom,#); |
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[4b35a90] | 240 | } |
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| 241 | return(p2); |
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| 242 | } |
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| 243 | example |
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| 244 | { "EXAMPLE:"; echo = 2; |
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| 245 | ring q=0,(x,y,z,u,v,w),dp; |
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| 246 | poly p=x2yz-x2v; |
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[e801fe] | 247 | ideal dom =x-w,u2w+1,yz-v; |
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| 248 | completeReduction(p,dom); |
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| 249 | completeReduction(p,dom,w); |
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[4b35a90] | 250 | } |
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| 251 | /////////////////////////////////////////////////////////////////////////////// |
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| 252 | |
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[e801fe] | 253 | proc inSubring(poly p, ideal dom) |
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[d2b2a7] | 254 | "USAGE: inSubring(p,dom); p poly, dom ideal |
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[e801fe] | 255 | RETURN: list= 1,string(@(0)-h(@(1),...,@(size(dom)))) :if p = h(dom[1],...,dom[size(dom)]) |
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| 256 | 0,string(h(@(0),...,@(size(dom)))) :if there is only a nonlinear relation |
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| 257 | h(p,dom[1],...,dom[size(dom)])=0. |
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[4b35a90] | 258 | NOTE: |
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| 259 | EXAMPLE: example inSubring; shows an example |
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[d2b2a7] | 260 | " |
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[4b35a90] | 261 | { |
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[e801fe] | 262 | int z=size(dom); |
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[4b35a90] | 263 | int i; |
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| 264 | def gnir=basering; |
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| 265 | list l; |
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| 266 | poly mile=var(1); |
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| 267 | for (i=2;i<=nvars(basering);i++) |
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| 268 | { |
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| 269 | mile=mile*var(i); |
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| 270 | } |
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| 271 | string eli=string(mile); |
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[e801fe] | 272 | // the intersection of ideal nett=(p-@(0),dom[1]-@(1),...) |
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[4b35a90] | 273 | // with the ring k[@(0),...,@(n)] is computed, the result is ker |
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| 274 | execute "ring r1="+charstr(basering)+",("+varstr(basering)+",@(0..z)),dp;"; |
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[e801fe] | 275 | ideal nett=imap(gnir,dom); |
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[4b35a90] | 276 | poly p; |
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| 277 | for (i=1;i<=z;i++) |
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| 278 | { |
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| 279 | execute "p=@("+string(i)+");"; |
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| 280 | nett[i]=nett[i]-p; |
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| 281 | } |
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| 282 | nett=imap(gnir,p)-@(0),nett; |
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| 283 | execute "ideal ker=eliminate(nett,"+eli+");"; |
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| 284 | // test wether @(0)-h(@(1),...,@(z)) is in ker |
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| 285 | l[1]=0; |
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| 286 | l[2]=""; |
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| 287 | for (i=1;i<=size(ker);i++) |
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| 288 | { |
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| 289 | if (deg(ker[i]/@(0))==0) |
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| 290 | { |
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| 291 | string str=string(ker[i]); |
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| 292 | setring gnir; |
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| 293 | l[1]=1; |
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| 294 | l[2]=str; |
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| 295 | return(l); |
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| 296 | } |
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| 297 | if (deg(ker[i]/@(0))>0) |
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| 298 | { |
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| 299 | l[2]=l[2]+string(ker[i]); |
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| 300 | } |
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| 301 | } |
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| 302 | setring gnir; |
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| 303 | return(l); |
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| 304 | } |
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| 305 | example |
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| 306 | { "EXAMPLE:"; echo = 2; |
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| 307 | ring q=0,(x,y,z,u,v,w),dp; |
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| 308 | poly p=xyzu2w-1yzu2w2+u4w2-1xu2vw+u2vw2+xyz-1yzw+2u2w-1xv+vw+2; |
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[e801fe] | 309 | ideal dom =x-w,u2w+1,yz-v; |
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| 310 | inSubring(p,dom); |
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[4b35a90] | 311 | } |
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| 312 | |
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| 313 | /////////////////////////////////////////////////////////////////////////////// |
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| 314 | |
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| 315 | proc localInvar(matrix m, poly p, poly q, poly h) |
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[d2b2a7] | 316 | "USAGE: localInvar(m,p,q,h); m matrix, p,q,h poly |
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[4b35a90] | 317 | RETURN: poly= the invariant of the vectorfield m=Sum m[i,1]*d/dx(i) with respect |
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| 318 | to p,q,h, i.e. |
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| 319 | Sum (-1)^v*(1/v!)*m^v(p)*(q/m(q))^v)*m(q)^N, m^N(q)=0, m^(N-1)(q)<>0 |
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| 320 | it is assumed that m(q) and h are invariant |
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| 321 | the sum above is divided by h as much as possible |
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| 322 | NOTE: |
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| 323 | EXAMPLE: example localInvar; shows an example |
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[d2b2a7] | 324 | " |
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[4b35a90] | 325 | { |
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| 326 | if ((der(m,h) !=0) || (der(m,der(m,q)) !=0)) |
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| 327 | { |
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| 328 | "the last variable defines not an invariant function "; |
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| 329 | return(q); |
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| 330 | } |
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| 331 | poly inv=p; |
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| 332 | poly dif= der(m,inv); |
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| 333 | poly a=der(m,q); |
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| 334 | poly sgn=-1; |
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| 335 | poly coeff=sgn*q; |
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| 336 | int k=1; |
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| 337 | if (dif==0) |
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| 338 | { |
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| 339 | return(inv); |
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| 340 | } |
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| 341 | while (dif!=0) |
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| 342 | { |
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| 343 | inv=(a*inv)+(coeff*dif); |
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| 344 | dif=der(m,dif); |
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| 345 | k=k+1; |
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| 346 | coeff=q*coeff*sgn/k; |
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| 347 | } |
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| 348 | while ((inv!=0) && (inv!=h) &&(subst(inv,h,0)==0)) |
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| 349 | { |
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| 350 | inv=inv/h; |
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| 351 | } |
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| 352 | return(inv); |
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| 353 | } |
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| 354 | example |
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| 355 | { "EXAMPLE:"; echo = 2; |
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| 356 | ring q=0,(x,y,z),dp; |
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| 357 | matrix m[3][1]; |
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| 358 | m[2,1]=x; |
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| 359 | m[3,1]=y; |
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| 360 | poly in=localInvar(m,z,y,x); |
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| 361 | in; |
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| 362 | } |
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| 363 | /////////////////////////////////////////////////////////////////////////////// |
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| 364 | |
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| 365 | |
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| 366 | |
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| 367 | proc furtherInvar(matrix m, ideal id, ideal karl, poly q) |
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[d2b2a7] | 368 | "USAGE: furtherInvar(m,id,karl,q); m matrix, id,karl ideal,q poly |
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[4b35a90] | 369 | RETURN: ideal= further invariants of the vectorfield m=Sum m[i,1]*d/dx(i) with respect |
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| 370 | to id,p,q, i.e. |
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| 371 | the ideal id contains invariants of m and we are looking for elements |
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| 372 | in the subring generated by id which are divisible by q |
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| 373 | it is assumed that m(p) and q are invariant |
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| 374 | the elements mentioned above are computed and divided by q |
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| 375 | as much as possible |
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| 376 | the ideal karl contains all invariants computed yet |
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| 377 | NOTE: |
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| 378 | EXAMPLE: example furtherInvar; shows an example |
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[d2b2a7] | 379 | " |
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[4b35a90] | 380 | { |
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| 381 | int i; |
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| 382 | ideal null; |
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| 383 | int z=size(id); |
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| 384 | intvec v; |
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| 385 | def @r=basering; |
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| 386 | ideal su; |
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| 387 | for (i=1;i<=z;i++) |
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| 388 | { |
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| 389 | su[i]=subst(id[i],q,0); |
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| 390 | } |
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| 391 | // defines the map phi : r1 ---> @r defined by |
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| 392 | // y(i) ---> id[i](q=0) |
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| 393 | execute "ring r1="+charstr(basering)+",(y(1..z)),dp"; |
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| 394 | setring @r; |
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| 395 | map phi=r1,su; |
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| 396 | setring r1; |
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| 397 | // computes the kernel of phi |
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| 398 | execute "ideal ker=preimage(@r,phi,null)"; |
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| 399 | // defines the map psi : r1 ---> @r defined by y(i) ---> id[i] |
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| 400 | setring @r; |
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| 401 | map psi=r1,id; |
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| 402 | // computes psi(ker(phi)) |
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| 403 | ideal rel=psi(ker); |
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| 404 | // devides by the maximal power of q |
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| 405 | // and tests wether we really obtain invariants |
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| 406 | for (i=1;i<=size(rel);i++) |
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| 407 | { |
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| 408 | while ((rel[i]!=0) && (rel[i]!=q) &&(subst(rel[i],q,0)==0)) |
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| 409 | { |
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| 410 | rel[i]=rel[i]/q; |
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| 411 | if (der(m,rel[i])!=0) |
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| 412 | { |
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| 413 | "error in furtherInvar, function not invariant"; |
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| 414 | rel[i]; |
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| 415 | } |
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| 416 | } |
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| 417 | rel[i]=simplify(rel[i],1); |
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| 418 | } |
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| 419 | // test whether some variables occur linearly |
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| 420 | // and deletes the corresponding invariant function |
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| 421 | setring r1; |
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| 422 | int j; |
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| 423 | for (i=1;i<=size(ker);i=i+1) |
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| 424 | { |
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| 425 | for (j=1;j<=z;j++) |
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| 426 | { |
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| 427 | if (deg(ker[i]/y(j))==0) |
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| 428 | { |
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| 429 | setring @r; |
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| 430 | rel[i]= completeReduction(rel[i],karl,q); |
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| 431 | if(rel[i]!=0) |
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| 432 | { |
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| 433 | karl[j+1]=rel[i]; |
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| 434 | rel[i]=0; |
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| 435 | } |
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| 436 | setring r1; |
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| 437 | } |
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| 438 | } |
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| 439 | |
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| 440 | } |
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| 441 | setring @r; |
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| 442 | list l=rel+null,karl; |
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| 443 | return(l); |
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| 444 | } |
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| 445 | example |
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| 446 | { "EXAMPLE:"; echo = 2; |
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| 447 | ring r=0,(x,y,z,u),dp; |
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| 448 | matrix m[4][1]; |
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| 449 | m[2,1]=x; |
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| 450 | m[3,1]=y; |
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| 451 | m[4,1]=z; |
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| 452 | ideal id=localInvar(m,z,y,x),localInvar(m,u,y,x); |
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| 453 | ideal karl=id,x; |
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| 454 | list in=furtherInvar(m,id,karl,x); |
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| 455 | in; |
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| 456 | } |
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| 457 | /////////////////////////////////////////////////////////////////////////////// |
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| 458 | |
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| 459 | |
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| 460 | |
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| 461 | proc invariantRing(matrix m, poly p, poly q,list #) |
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[d2b2a7] | 462 | "USAGE: invariantRing(m,p,q); m matrix, p,q poly |
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[4b35a90] | 463 | RETURN: ideal= the invariants of the vectorfield m=Sum m[i,1]*d/dx(i) |
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| 464 | p,q variables with m(p)=q invariant |
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| 465 | NOTE: |
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| 466 | EXAMPLE: example furtherInvar; shows an example |
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[d2b2a7] | 467 | " |
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[4b35a90] | 468 | { |
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| 469 | ideal j; |
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| 470 | int i,it; |
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| 471 | int bou=-1; |
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| 472 | if(size(#)>0) |
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| 473 | { |
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| 474 | bou=#[1]; |
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| 475 | } |
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| 476 | int z; |
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| 477 | ideal karl; |
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| 478 | ideal k1=1; |
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| 479 | list k2; |
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| 480 | // computation of local invariants |
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| 481 | for (i=1;i<=nvars(basering);i++) |
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| 482 | { |
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| 483 | karl=karl+localInvar(m,var(i),p,q); |
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| 484 | } |
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| 485 | if(bou==0) |
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| 486 | { |
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| 487 | " "; |
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| 488 | "the local invariants:"; |
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| 489 | " "; |
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| 490 | karl; |
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[e801fe] | 491 | // pause; |
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[4b35a90] | 492 | " "; |
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| 493 | } |
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| 494 | // computation of further invariants |
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| 495 | it=0; |
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| 496 | while (size(k1)!=0) |
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| 497 | { |
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| 498 | // test if the new invariants are already in the ring generated |
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| 499 | // by the invariants we constructed already |
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| 500 | it++; |
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| 501 | karl=sortier(karl); |
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| 502 | j=q; |
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| 503 | for (i=1;i<=size(karl);i++) |
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| 504 | { |
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| 505 | j=j+ simplify(completeReduction(karl[i],j,q),1); |
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| 506 | } |
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| 507 | karl=j; |
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| 508 | j[1]=0; |
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| 509 | j=simplify(j,2); |
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| 510 | k2=furtherInvar(m,j,karl,q); |
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| 511 | k1=k2[1]; |
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| 512 | karl=k2[2]; |
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| 513 | k1=sortier(k1); |
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| 514 | z=size(k1); |
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| 515 | for (i=1;i<=z;i++) |
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| 516 | { |
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| 517 | k1[i]= completeReduction(k1[i],karl,q); |
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| 518 | if (k1[i]!=0) |
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| 519 | { |
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| 520 | karl=karl+simplify(k1[i],1); |
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| 521 | } |
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| 522 | } |
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| 523 | if(bou==0) |
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| 524 | { |
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| 525 | " "; |
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| 526 | "the invariants after the iteration"; |
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| 527 | it; |
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| 528 | " "; |
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| 529 | karl; |
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[e801fe] | 530 | // pause; |
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[4b35a90] | 531 | " "; |
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| 532 | } |
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| 533 | if((bou>0) && (size(k1)>0)) |
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| 534 | { |
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| 535 | if(deg(k1[size(k1)])>bou) |
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| 536 | { |
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| 537 | return(karl); |
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| 538 | } |
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| 539 | } |
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| 540 | } |
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| 541 | return(karl); |
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| 542 | } |
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| 543 | example |
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| 544 | { "EXAMPLE:"; echo = 2; |
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| 545 | |
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| 546 | //Winkelmann: free action but Spec k[x(1),...,x(5)]---> Spec In- |
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| 547 | //variantring is not surjective |
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| 548 | |
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| 549 | ring rw=0,(x(1..5)),dp; |
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| 550 | matrix m[5][1]; |
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| 551 | m[3,1]=x(1); |
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| 552 | m[4,1]=x(2); |
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| 553 | m[5,1]=1+x(1)*x(4)+x(2)*x(3); |
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| 554 | ideal in=invariantRing(m,x(3),x(1),0); |
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| 555 | in; |
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| 556 | |
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| 557 | //Deveney/Finston: The ring of invariants is not finitely generated |
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| 558 | |
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| 559 | ring rf=0,(x(1..7)),dp; |
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| 560 | matrix m[7][1]; |
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| 561 | m[4,1]=x(1)^3; |
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| 562 | m[5,1]=x(2)^3; |
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| 563 | m[6,1]=x(3)^3; |
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| 564 | m[7,1]=(x(1)*x(2)*x(3))^2; |
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| 565 | ideal in=invariantRing(m,x(4),x(1),6); |
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| 566 | in; |
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| 567 | |
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| 568 | |
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| 569 | //Deveney/Finston:Proper Ga-action which is not locally trivial |
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| 570 | //r[x(1),...,x(5)] is not flat over the ring of invariants |
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| 571 | |
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| 572 | ring rd=0,(x(1..5)),dp; |
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| 573 | matrix m[5][1]; |
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| 574 | m[3,1]=x(1); |
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| 575 | m[4,1]=x(2); |
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| 576 | m[5,1]=1+x(1)*x(4)^2; |
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| 577 | ideal in=invariantRing(m,x(3),x(1)); |
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| 578 | in; |
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| 579 | |
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| 580 | actionIsProper(m); |
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| 581 | |
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| 582 | //computes the relations between the invariants |
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| 583 | int z=size(in); |
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| 584 | ideal null; |
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| 585 | ring r1=0,(y(1..z)),dp; |
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| 586 | setring rd; |
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| 587 | map phi=r1,in; |
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| 588 | setring r1; |
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| 589 | ideal ker=preimage(rd,phi,null); |
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| 590 | ker; |
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| 591 | |
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| 592 | //the discriminant |
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| 593 | |
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| 594 | ring r=0,(x(1..2),y(1..2),z,t),dp; |
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| 595 | 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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| 596 | |
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| 597 | matrix m[5][5]; |
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| 598 | m[1,1]=z; |
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| 599 | m[1,2]=x(1)*y(2)^2+1; |
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| 600 | m[1,3]=x(1)*y(1)*y(2); |
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| 601 | m[1,4]=1/3*x(1)*y(1)^2; |
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| 602 | m[1,5]=0; |
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| 603 | m[2,1]=0; |
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| 604 | m[2,2]=z; |
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| 605 | m[2,3]=x(1)*y(2)^2+1; |
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| 606 | m[2,4]=x(1)*y(1)*y(2); |
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| 607 | m[2,5]=1/3*x(1)*y(1)^2; |
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| 608 | m[3,1]=x(1)*y(2)^2+1; |
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| 609 | m[3,2]=2*x(1)*y(1)*y(2); |
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| 610 | m[3,3]=x(1)*y(1)^2; |
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| 611 | m[3,4]=0; |
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| 612 | m[3,5]=0; |
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| 613 | m[4,1]=0; |
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| 614 | m[4,2]=x(1)*y(2)^2+1; |
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| 615 | m[4,3]=2*x(1)*y(1)*y(2); |
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| 616 | m[4,4]=x(1)*y(1)^2; |
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| 617 | m[4,5]=0; |
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| 618 | m[5,1]=0; |
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| 619 | m[5,2]=0; |
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| 620 | m[5,3]=x(1)*y(2)^2+1; |
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| 621 | m[5,4]=2*x(1)*y(1)*y(2); |
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| 622 | m[5,5]=x(1)*y(1)^2; |
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| 623 | |
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| 624 | poly disc=9*det(m)/(x(1)^2*y(1)^4); |
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| 625 | |
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| 626 | LIB "invar.lib"; |
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| 627 | matrix n[6][1]; |
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| 628 | n[2,1]=x(1); |
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| 629 | n[4,1]=y(1); |
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| 630 | n[5,1]=1+x(1)*y(2)^2; |
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| 631 | |
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| 632 | der(n,disc); |
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| 633 | |
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| 634 | //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 |
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| 635 | |
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| 636 | |
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| 637 | //constructive approach to Weizenbcks theorem |
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| 638 | |
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| 639 | int n=5; |
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| 640 | |
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| 641 | ring w=0,(x(1..n)),wp(1..n); |
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| 642 | |
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| 643 | // definition of the vectorfield m=sum m[i]*d/dx(i) |
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| 644 | matrix m[n][1]; |
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| 645 | int i; |
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| 646 | for (i=1;i<=n-1;i=i+1) |
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| 647 | { |
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| 648 | m[i+1,1]=x(i); |
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| 649 | } |
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| 650 | ideal in=invariantRing(m,x(2),x(1),0); |
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| 651 | in; |
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| 652 | |
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| 653 | |
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| 654 | |
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| 655 | } |
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