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A tensor defined over a finite field $\mathbb{F}$ has low analytic rank ifthe distribution of its values differs significantly from the uniformdistribution. An order $d$ tensor has partition rank 1 if it can be written asa product of two tensors of order less than $d$, and it has partition rank atmost $k$ if it can be written as a sum of $k$ tensors of partition rank 1. Inthis paper, we prove that if the analytic rank of an order $d$ tensor is atmost $r$, 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 notdepending on $\mathbb{F}$. The novelty of our result is that $f$ has onlytower-type dependence on its parameters. It follows from our results that abiased polynomial has low rank; there too we obtain a tower-type dependenceimproving the previously known Ackermann-type bound.

Low analytic rank implies low partition rank for tensors