  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 $k=\mathbb C$.
