Pardue’s Conjecture on generic semi-regular sequences

Prove that, over an infinite field K, a generic sequence of homogeneous polynomials f1, …, fm in the polynomial ring R = K[x1, …, xn] is semi-regular in the sense that, for each i = 1, …, m and every t ≥ deg(fi), the multiplication map (R / ⟨f1, …, f_{i−1}⟩)_{t−deg(fi)} → (R / ⟨f1, …, f_{i−1}⟩)_{t} by fi is injective or surjective.

Background

The paper reviews Pardue’s definition of semi-regular sequences and emphasizes its role in understanding Gröbner basis complexity and Hilbert–Poincaré series. In the context of infinite fields, the authors highlight Pardue’s conjecture that generic polynomial sequences exhibit semi-regularity, a property that ensures controlled behavior of multiplication maps and implies specific forms for Hilbert series upon iteration.

Establishing this conjecture would underpin many complexity estimates for Gröbner basis algorithms and connect to other conjectures such as Fröberg’s, which is known to be equivalent to Pardue’s conjecture. Thus, resolving Pardue’s conjecture would also settle Fröberg’s conjecture in this setting.