On ranks of polynomials

Abstract

Let $V$ be a vector space over a field $k,P:V\to k,d\ge 3$. We show the existence of a function $C(r,d)$ such that $rank(P)\le C(r,d)$ 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 $rank(\partial P/\partial t)\le r$ 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 $k=\mathbb{C}$.