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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 $f(r,d,|\mathbb{F}|)$, where, forfixed $d$ and $\mathbb{F}$, $f$ is a polynomial in $r$. This is an improvementof a recent result of the author, where he obtained a tower-type bound. Priorto our work, the best known bound was an Ackermann-type function in $r$ and$d$, though it did not depend on $\mathbb{F}$. It follows from our results thata biased polynomial has low rank; there too we obtain a polynomial dependenceimproving the previously known Ackermann-type bound. A similar polynomial boundfor the partition rank was obtained independently and simultaneously byMili\'cevi\'c.

Polynomial bound for the partition rank vs the analytic rank of tensors