Relative Rank and Regularization

TL;DR

This paper introduces the concept of relative rank for polynomials, establishing a relation with relative bias, which improves regularization efficiency and broadens applications in algebra and combinatorics.

Contribution

It defines relative rank, relates it to bias, and demonstrates its advantages over Schmidt rank for regularization and applications in algebraic geometry and combinatorics.

Findings

Established a relation between relative rank and bias over finite fields.

Provided an efficient polynomial regularization procedure.

Proved that polynomials can be contained in an ideal generated by a regular sequence of bounded size.

Abstract

We introduce a new concept of rank - relative rank associated to a filtered collection of polynomials. When the filtration is trivial our relative rank coincides with Schmidt rank (also called strength). We also introduce the notion of relative bias. The main result of the paper is a relation between these two quantities over finite fields (as a special case we obtain a new proof of the results in arXiv:1902.09830). This relation allows us to get an accurate estimate for the number of points on an affine variety given by a collection of polynomials which is high relative rank (Lemma 3.2). The key advantage of relative rank is that it allows one to perform an efficient regularization procedure which is polynomial in the initial number of polynomials (the regularization process with Schmidt rank is far worse than tower exponential). The main result allows us to replace Schmidt rank with relative rank in many key applications in combinatorics, algebraic geometry and algebra. For example, we prove that any collection of polynomials $\mathcal{P}=(P_i)_{i=1}^c$ of degrees $\le d$ in a polynomial ring over an algebraically closed field of characteristic $>d$ is contained in an ideal $\mathcal{I}(\mathcal{Q})$, generated by a collection $\mathcal{Q}$ of polynomials of degrees $\le d$ which form a regular sequence, and $\mathcal{Q}$ is of size $\le A c^{A}$, where $A=A(d)$ is independent of the number of variables.