2.3.1 Ideal membership

Definition 2..5   A function $NF : K[\underline{x}]^r \times \{G \vert G\ standard basis\} \to K[\underline{x}]^r, (p,G) \mapsto NF(p\vert G)$, is called a normal form if for any $p \in K[\underline{x}]^r$ and any G the following holds: if $NF(p\vert G) \not= 0$ then $L(g) \not\vert L(NF(p\vert G))$ for all $g \in G$. NF(g|G) is called the normal form of $\bf p$ with respect to $\bf G$.

Lemma 2..6   $f \in I$ iff NF(f,std(I)) = 0.

SINGULAR example: 

//f defines a trimodal singularity for generic moduli
ring R = 0,(x,y),ds;
int a1,a2,a3=random(1,100),random(-100,1),random(1,100);
poly f = (x^2-y^3)*(y+a1*x)*(y+a2*x)*(y+a3*x);
ideal J = jacob(f);
ideal I = f;
// J:I, ideal of the closure of V(J) \ V(I)
ideal Q = quotient(J,I);
//the Hessian of f
poly Hess = det(jacob(jacob(f)));
//Hess is contained in Q iff NF is 0
reduce(Hess,std(Q));