Towards signature-based gröbner basis algorithms for computing the nondegenerate locus of a polynomial system

Problem statement. Let K be a field and K be an algebraic closure of K. Consider the polynomial ring R = K[x1,..., xn] over K and a finite sequence of polynomials f1,...,fc in R with c ≤ n. Let V ⊂ Kn be the algebraic set defined by the simultaneous vanishing of the fi's. Recall that V can be decomposed into finitely many irreducible components, whose codimension cannot be greater than c. The set Vc which is the union of all these irreducible components of codimension exactly c is named further the nondegenerate locus of f1,...,fc.