Low analytic rank implies low partition rank for tensors

Abstract: A tensor defined over a finite field $\mathbb{F}$ has low analytic rank if the distribution of its values differs significantly from the uniform distribution. An order $d$ tensor has partition rank 1 if it can be written as a product of two tensors of order less than $d$, and it has partition rank at most $k$ if it can be written as a sum of $k$ tensors of partition rank 1. In this paper, we prove that if the analytic rank of an order $d$ tensor is at most $r$, then its partition rank is at most $f(r,d,|\mathbb{F}|)$. Previously, this was known with $f$ being an Ackermann-type function in $r$ and $d$ but not depending on $\mathbb{F}$. The novelty of our result is that $f$ has only tower-type dependence on its parameters. It follows from our results that a biased polynomial has low rank; there too we obtain a tower-type dependence improving the previously known Ackermann-type bound.