Towards Signature-Based Grobner 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[ x 1 ,..., x n ] over K and a finite sequence of polynomials f 1 ,..., f c in R with c ≤ n. Let V ⊂ K n be the algebraic set defined by the simultaneous vanishing of the f i 's. Recall that V can be decomposed into finitely many irreducible components, whose codimension cannot be greater than c. The set V c which is the union of all these irreducible components of codimension exactly c is named further the nondegenerate locus of f 1 ,..., f c .