| 1 | LIB "tst.lib"; |
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| 2 | tst_init(); |
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| 3 | |
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| 4 | |
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| 5 | //====================== Exercise 2.1 ============================= |
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| 6 | ring R = 0, (x,y), dp; |
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| 7 | poly f1, f2, f3 = x2+y2, x2y2, x3y-xy3; |
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| 8 | if (not(defined(algDependent))) { LIB "algebra.lib"; } |
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| 9 | def L = algDependent(ideal(f1,f2,f3)); |
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| 10 | L[1]; |
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| 11 | def S = L[2]; |
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| 12 | setring S; |
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| 13 | ker; |
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| 14 | |
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| 15 | setring R; |
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| 16 | poly g, g1, g2 = x4+y4, x+y, xy; |
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| 17 | L = algebra_containment(g,ideal(g1,g2),1); |
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| 18 | def S2 = L[2]; |
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| 19 | setring S2; |
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| 20 | check; |
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| 21 | |
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| 22 | ring T = 0, x(1..3), dp; |
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| 23 | qring Q = groebner(x(1)*x(2)*x(3)-1); |
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| 24 | map phi = Q, x(2)*x(3), x(1)*x(3), x(1)*x(2); |
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| 25 | is_bijective(phi,Q); |
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| 26 | //-> 1 |
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| 27 | |
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| 28 | |
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| 29 | kill R,Q,S,S2,T,L; |
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| 30 | //====================== Exercise 2.2 ============================= |
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| 31 | ring R = 0, (w,x,y,z), dp; |
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| 32 | ideal I = y-x2, z-x3; |
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| 33 | ideal SI = groebner(I); |
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| 34 | ideal Ih = homog(SI,w); // generators are homogenized |
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| 35 | Ih; |
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| 36 | |
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| 37 | ideal J = homog(I,w); // generators are homogenized |
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| 38 | size(reduce(J,groebner(Ih),1)); // J is contained in Ih |
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| 39 | size(reduce(Ih,groebner(J),1)); // Ih is not contained in J |
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| 40 | if (not(defined(sat))){ LIB "elim.lib"; } |
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| 41 | ideal Iinf = sat(J,Ih)[1]; |
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| 42 | Iinf; |
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| 43 | |
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| 44 | |
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| 45 | kill R; |
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| 46 | //====================== Exercise 2.3 ============================= |
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| 47 | ring S = 0, (y,x), dp; |
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| 48 | ideal I = maxideal(3),maxideal(2),x,y,1; |
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| 49 | ring R = 0, (x,y,a(1..10)), (dp(2),dp(10)); |
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| 50 | ideal I = imap(S,I); |
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| 51 | matrix A[10][1] = a(1..10); |
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| 52 | poly f = (matrix(I)*A)[1,1]; |
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| 53 | ideal J = f, diff(f,x), diff(f,y); |
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| 54 | J = groebner(J); |
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| 55 | // check for generators for J that do not depend on x,y |
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| 56 | ideal JJ; |
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| 57 | for (int i=1; i<=size(J); i++) |
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| 58 | { |
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| 59 | if (J[i][1]<y) { JJ = JJ,J[i]; } |
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| 60 | } |
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| 61 | JJ = simplify(JJ,2); // erase zero generators |
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| 62 | size(JJ); |
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| 63 | //-> 1 |
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| 64 | homog(JJ); |
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| 65 | //-> 1 |
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| 66 | poly D = JJ[1]; |
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| 67 | deg(D); size(D); |
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| 68 | //-> 12 |
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| 69 | //-> 2040 |
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| 70 | |
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| 71 | |
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| 72 | kill i,R,S; |
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| 73 | //====================== Exercise 2.4 ============================= |
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| 74 | //======== Procedures are stored in the library file sol.lib ======= |
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| 75 | //================================================================== |
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| 76 | LIB "sol.lib"; |
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| 77 | example ideal_intersect; |
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| 78 | example ideal_quotient; |
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| 79 | example saturate; |
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| 80 | |
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| 81 | tst_status(1);$ |
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| 82 | |
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