| 1 | LIB "tst.lib"; tst_init(); |
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| 2 | LIB "solve.lib"; |
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| 3 | ring r=0,x(1..5),dp; |
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| 4 | poly f0= x(1)^3+x(2)^2+x(3)^2+x(4)^2-x(5)^2; |
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| 5 | poly f1= x(2)^3+x(1)^2+x(3)^2+x(4)^2-x(5)^2; |
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| 6 | poly f2=x(3)^3+x(1)^2+x(2)^2+x(4)^2-x(5)^2; |
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| 7 | poly f3=x(4)^2+x(1)^2+x(2)^2+x(3)^2-x(5)^2; |
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| 8 | poly f4=x(5)^2+x(1)^2+x(2)^2+x(3)^2; |
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| 9 | ideal i=f0,f1,f2,f3,f4; |
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| 10 | ideal si=std(i); |
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| 11 | // |
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| 12 | // dimension of a solution set (here: 0) can be read from a Groebner bases |
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| 13 | // (with respect to any global monomial ordering) |
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| 14 | dim(si); |
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| 15 | // |
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| 16 | // the number of complex solutions (counted with multiplicities) is: |
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| 17 | vdim(si); |
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| 18 | // |
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| 19 | // The given system has a multiple solution at the origin. We use facstd |
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| 20 | // to compute equations for the non-zero solutions: |
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| 21 | option(redSB); |
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| 22 | ideal maxI=maxideal(1); |
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| 23 | ideal j=sat(si,maxI)[1]; // output is Groebner basis |
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| 24 | vdim(j); // number of non-zero solutions (with mult's) |
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| 25 | // |
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| 26 | // We compute a triangular decomposition for the ideal I. This requires first |
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| 27 | // the computation of a lexicographic Groebner basis (we use the FGLM |
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| 28 | // conversion algorithm): |
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| 29 | ring R=0,x(1..5),lp; |
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| 30 | ideal j=fglm(r,j); |
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| 31 | list L=triangMH(j); |
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| 32 | size(L); // number of triangular components |
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| 33 | L[1]; // the first component |
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| 34 | // |
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| 35 | // We compute floating point approximations for the solutions (with 30 digits) |
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| 36 | def S=triang_solve(L,30); |
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| 37 | setring S; |
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| 38 | size(rlist); // number of different non-zero solutions |
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| 39 | rlist[1]; // the first solution |
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| 40 | // |
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| 41 | // Alternatively, we could have applied directly the solve command: |
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| 42 | setring r; |
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| 43 | def T=solve(i,30,1,"nodisplay"); // compute all solutions with mult's |
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| 44 | setring T; |
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| 45 | size(SOL); // number of different solutions |
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| 46 | SOL[1][1]; SOL[1][2]; // first solution and its multiplicity |
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| 47 | SOL[size(SOL)]; // solutions of highest multiplicity |
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| 48 | // |
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| 49 | // Or, we could remove the multiplicities first, by computing the |
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| 50 | // radical: |
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| 51 | setring r; |
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| 52 | ideal k=std(radical(i)); |
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| 53 | vdim(k); // number of different complex solutions |
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| 54 | def T1=solve(k,30,"nodisplay"); // compute all solutions with mult's |
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| 55 | setring T1; |
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| 56 | size(SOL); // number of different solutions |
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| 57 | SOL[1]; |
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| 58 | tst_status(1);$ |
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