On ranks of polynomials

2018/02/14 by Kazhdan, David, Ziegler, Tamar
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1802.04984

Abstract

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 in the case when k=\mathbb C.

Related

• Extending weakly polynomial functions from high rank varieties
• The inverse conjecture for the Gowers norm over finite fields in low characteristic
• Polynomial bound for partition rank in terms of analytic rank
• Properties of high rank subvarieties of affine spaces
• Ranges of polynomials control degree ranks of Green and Tao over finite prime fields