Small ideals in polynomial rings and applications

TL;DR

This paper establishes effective bounds for ideals generated by polynomials in polynomial rings over finite or algebraically closed fields, with applications to algebraic geometry and additive combinatorics.

Contribution

It provides explicit bounds for generating ideals with R-sequences, extending results on strength, codimension, and Gowers norms in finite fields.

Findings

Bounded the number of generators for ideals in polynomial rings.

Extended results on strength and singular locus from prior theorems.

Applied bounds to Gowers norms and polynomial rank over finite fields.

Abstract

Let $\mathbf{k}$ be a field which is either finite or algebraically closed and let $R = \mathbf{k}[x_1,\ldots,x_n].$ We prove that any $g_1,\ldots,g_s\in R$ homogeneous of positive degrees $\le d$ are contained in an ideal generated by an $R_t$-sequence of $\le A(d)(s+t)^{B(d)}$ homogeneous polynomials of degree $\le d,$ subject to some restrictions on the characteristic of $\mathbf{k}.$ This yields effective bounds for new cases of Ananyan and Hochster's theorem A in arXiv:1610.09268 on strength and the codimension of the singular locus. It also implies effective bounds when $d$ equals the characteristic of $\mathbf{k}$ for Tao and Ziegler's result in arXiv:1101.1469 on rank and $U^d$ Gowers norms of polynomials over finite fields.