On Ranks of Polynomials

Let V be a vector space over a field k, P : V → k, d ≥ 3. We show the existence of a function C(r, d) such that rank(P) ≤ C(r, d) for any field k, char(k) > d, a finite-dimensional k-vector space V and a polynomial P : V → k of degree d such that rank(∂P/∂t) ≤ r for all t ∈ 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 even in the case when k = ℂ.