Ticket #213: examples.tst

// paper: Ihsen Yengui, "Dynamical Groebner Bases" (2006), Journal of Algebra
// EX.6 

ring r = (integer, 3,3), (x,y), lp;
ring r = (integer, 27), (x,y), lp;
poly g1 = 9;
poly g2 = x+1;
poly g3 = 3*y^2;
poly g4 = y^3+13*y^2-12;
ideal I = g1,g2,g3,g4;
ideal J = std(I); // is GB, correct

// Ex 7
ring r = (integer, 4), (x,y), lp;
poly f1 = x^4-x;
poly f2 = y^3-1;
poly f3 = 2*x*y;
ideal I = f1,f2,f3;
ideal J = std(I); // correct: 2x instead of 2xy

// Ex. section 2.2 p.456
ring r = (integer), (x,y), lp;
poly f1 = 10*x*y+1;
poly f2 = 6*x^2+3;
ideal I = f1,f2;
ideal J = std(I);
poly F = 62*x^3*y + 11*x^2 + 10*x*y^2 + 56*x*y + y + 8;
NF(F,J); // 0, correct
//cofactors: 
matrix L = lift(I,F);
module S = syz(I);
L = NF(L,S);
(11*x^2+y+8)*f1 + (-8*x*y)*f2 - F; // 0 => even shorter presentation
L[1][1]*f1 + L[1][2]*f2 - F;

// paper 2: A. Hadj Kacem and Ihsen Yengui, "Dynamical Groebner Bases over Dedekind rings" (2006), Journal of Algebra (submitted)

// ex 9, page 7
ring r = (integer, 4), (x,y), lp;
poly f1 = 2*x*y;
poly f2 = 3*y^3+3;
poly f3 = x^2-3*x;
ideal I = f1,f2,f3;
ideal J = std(I); // correct: 2x instead of 2xy, leadcoeffs simplified
module S = syz(I);
module Y =   [-x*y^2-y^2,2*x^2+2*x,2], [-y^5-y^2, 4*x+2*x*y^3,0],[0,x^2-3*x,-3*y^3-3],[1-y^3,2*x*y,0];
print(NF(Y,S)); // 0
print(std(Y)-S); // 0

// section 4, pp. 8-11
ring r = (integer), (x,y,T), lp; // eliminating T
ideal I = 3*x*y+1, (4+2*T)*y + 9, T^2+5;
// a) GB
ideal J = std(I);
// b) syzygies
module S = syz(I); // longer
// todo: TRY with his approach via lifts ?
module Y = [-(4+2*T)*y - 9, 3*x*y+1], [27*x*y - 9 - (4+2*T)*y + 3*(4+2*T)*x*y^2, 1- 9*x^2*y^2];
module Z = [-(4+2*T)*y - 9, 3*x*y+1]; // the only generator; correct
NF(Y,std(Z)); // 0
// c) ideal membership
poly F = (4*T-1)*x^2*y + 6*T*x*y^2 + 9*T*x^2 + 3*x - 4*y -9;
NF(F,J); // 0, ok
list D = division(F,J);
print(transpose(D[1])*transpose(J) + F); // 0 => correct cofactors
print(lift(I,F)); // 
matrix L1 = lift(I,J);
matrix L; ideal K = liftstd(I,L); // fast
matrix ZZ = L*D[1]; // total lift matrix
transpose(ZZ)*transpose(I) + F; // 0