Polynomial bound for partition rank in terms of analytic rank

Abstract

Let G1, … , Gk be vector spaces over a finite field F= Fq with a non-trivial additive character χ. The analytic rank of a multilinear form α: G1× ⋯ × Gk→ F is defined as arank(α)=-logqEx1∈G1,…,xk∈Gkχ(α(x1,…,xk)). The partition rank prank (α) of α is the smallest number of maps of partition rank 1 that add up to α, where a map is of partition rank 1 if it can be written as a product of two multilinear forms, depending on different coordinates. It is easy to see that arank (α) ≤ O(prank (α)) and it has been known that prank (α) can be bounded from above in terms of arank (α). In this paper, we improve the latter bound to polynomial, i.e. we show that there are quantities C, D depending on k only such that prank (α) ≤ C(arank (α) D+ 1). As a consequence, we prove a conjecture of Kazhdan and Ziegler. The same result was obtained independently and simultaneously by Janzer.