On ranks of polynomials

TL;DR

This paper establishes an upper bound on the rank of polynomials over fields with characteristic greater than their degree, based on the ranks of their derivatives, using Gowers norms in finite fields.

Contribution

It introduces a bound on polynomial rank depending on derivative ranks, leveraging Gowers norms, and highlights the lack of a direct proof over complex fields.

Findings

Existence of a function C(r,d) bounding polynomial rank.

Bound depends on the rank of derivatives of the polynomial.

Proof utilizes Gowers norms for finite fields.

Abstract

Let $V$ be a vector space over a field $k, P:V\to k, d\geq 3$. We show the existence of a function $C(r,d)$ such that $rank (P)\leq C(r,d)$ for any field $k,char (k)>d$, a finite-dimensional $k$-vector space $V$ and a polynomial $P:V\to k$ of degree $d$ such that $rank(\partial P/\partial t)\leq r$ for all $t\in V-0$. Our proof of this theorem is based on the application of results on Gowers norms for finite fields $k$. We don't know a direct proof in the case when $k=\mathbb C$.