restart;
with(Groebner):
DoExist := proc(tau, n)
if rtable_num_elems(tau) >= n then
return tau[n];
else
return 0;
end if;
end proc;
IsZero := proc(a, b)
if a=0 then
return 0;
else
return a/b;
end if
end proc;
g1 := x^2-w*y;
g2 := x*y - w*z;
g3 := y^2 - x*z;
gro := Basis([g1,g2,g3],plex(x,y,z,w));
X1 := `*`(LeadingTerm(g1, tdeg(x, y, z, w)));
X2 := `*`(LeadingTerm(g2, tdeg(x, y, z, w)));
X3 := `*`(LeadingTerm(g3, tdeg(x, y, z, w)));
X12 := lcm(X1,X2);
X13 := lcm(X1,X3);
X23 := lcm(X2,X3);
S12 := SPolynomial(g1, g2, lexdeg([x, y, z, w]));
S23 := SPolynomial(g2, g3, lexdeg([x, y, z, w]));
S13 := SPolynomial(g1, g3, lexdeg([x, y, z, w]));
e1 := Vector([1,0,0]);
e2 := Vector([0,1,0]);
e3 := Vector([0,0,1]);
eq1:= S12 = h121*g1 + h122*g2 + h123*g3;
eq1 := S12 - expand(h121*g1 + h122*g2 + h123*g3):
NormalForm(eq1, Basis([g1, g2, g3], tdeg(x, y, z, w, h121, h122, h123)), tdeg(x, y, z, w, h121, h122, h123), 'Q');
h121 := 0;
h122 := 0;
h123 := 0;
s12 := IsZero(X12,X1)*e1-IsZero(X12,X2)*e2-Vector([DoExist(<Q>,3), DoExist(<Q>,2), DoExist(<Q>,1)]);
eq1 := S13-expand(h131*g1+h132*g2+h133*g3):
NormalForm(eq1, Basis([g1, g2, g3], tdeg(x, y, z, w, h131, h132, h133)), tdeg(x, y, z, w, h131, h132, h133), 'Q');
h131 := 0;
h132 := 0;
h133 := 0;
s13 := IsZero(X13,X1)*e1-IsZero(X13,X3)*e3-Vector([DoExist(<Q>,3), DoExist(<Q>,2), DoExist(<Q>,1)]);
eq1:= S23 - expand(h231*g1 + h232*g2 + h233*g3);
NormalForm(eq1, Basis([g1, g2, g3], tdeg(x, y, z, w, h231, h232, h233)), tdeg(x, y, z, w, h231, h232, h233), 'Q');
h231 := 0;
h232 := 0;
h233 := 0;
s23 := IsZero(X23,X2)*e2-IsZero(X23,X3)*e3-Vector([DoExist(<Q>,3), DoExist(<Q>,2), DoExist(<Q>,1)]);
with(LinearAlgebra):
#F = Syz*GrobnerBasis
F := simplify(MatrixMatrixMultiply(Matrix([[s12[1],s13[1],s23[1]],[s12[2],s13[2],s23[2]],[s12[3],s13[3],s23[3]]]), Matrix([[gro[1]],[gro[2]],[gro[3]]])));
F[1][1] - g1 = 0;
F[2][1] - g2 = 0;
F[3][1] - g3 = 0;
F := simplify(MatrixMatrixMultiply(Matrix([[s12[1],0,s23[1]],[s12[2],0,s23[2]],[s12[3],0,s23[3]]]), Matrix([[gro[1]],[gro[2]],[gro[3]]])));
F[1][1] - g1 = 0;
F[2][1] - g2 = 0;
F[3][1] - g3 = 0;
#F = GrobnerBasis*Syz
F := simplify(MatrixMatrixMultiply(Matrix([[gro[1],gro[2],gro[3]]]), Matrix([[s12[1],s13[1],s23[1]],[s12[2],s13[2],s23[2]],[s12[3],s13[3],s23[3]]])));
F[1][1] - g1 = 0;
F[1][2] - g2 = 0;
F[1][3] - g3 = 0;
F := simplify(MatrixMatrixMultiply(Matrix([[gro[1],gro[2],gro[3]]]), Matrix([[s12[1],0,s23[1]],[s12[2],0,s23[2]],[s12[3],0,s23[3]]])));
F[1][1] - g1 = 0;
F[1][2] - g2 = 0;
F[1][3] - g3 = 0;
#F = GrobnerBasis*Syz
F := simplify(MatrixMatrixMultiply(Matrix([[gro[1],gro[2],gro[3]]]), Matrix([[s12[1],s12[1],s12[1]],[s13[2],s13[2],s13[2]],[s23[3],s23[3],s23[3]]])));
F[1][1] - g1 = 0;
F[1][2] - g2 = 0;
F[1][3] - g3 = 0;
F := simplify(MatrixMatrixMultiply(Matrix([[gro[1],gro[2],gro[3]]]), Matrix([[s12[1],0,s12[1]],[s13[2],0,s13[2]],[s23[3],0,s23[3]]])));
F[1][1] - g1 = 0;
F[1][2] - g2 = 0;
F[1][3] - g3 = 0;

syz result is s12, s23

but after verify, F is not equal to GrobnerBasis*Syz