Generic non-containment of V(S_p) inside V(f_1, …, f_n)

Prove that, for generic vectors p = (p_1, …, p_n) of homogeneous polynomials with deg_x p_i = deg_x f_i, no irreducible component of the zero set V(S_p) is contained in the zero set V(f_1, …, f_n), where S_p := {g_1 p_1 + … + g_n p_n | (g_1, …, g_n) is a syzygy of (f_1, …, f_n)} and Syz(f_1, …, f_n) denotes the syzygy module of (f_1, …, f_n).

Background

The paper studies Canny’s generalized characteristic polynomial and the perturbed resultant obtained via taking the gcd of generalized characteristic polynomials under two generic perturbations. A key technical device in the analysis is the perturbed variety X_p derived from the ε-perturbed system and its relation to the syzygy-generated set S_p.

Lemma 4 (syzygy lemma) shows that X_p ⊆ V(f_1, …, f_n, S_p), with equality provided that no component of V(S_p) is contained in V(f_1, …, f_n). The remark formulates a conjecture asserting that this containment never happens for generic p, which, if true, would yield an exact description of X_p as V(f_1, …, f_n, S_p) generically and consequently sharpen the main results by precisely characterizing the persistent components captured by the perturbed resultant.