The analytic rank of tensors and its applications

TL;DR

This paper introduces the analytic rank of tensors, demonstrating its subadditivity and revealing its applications in understanding tensor roots and replacing other tensor ranks in combinatorial problems.

Contribution

It proves the subadditivity of the analytic rank and shows its applications in tensor analysis and combinatorics, including replacing slice and partition ranks.

Findings

Analytic rank is subadditive for tensor sums.

Common roots of tensors are positively correlated.

Analytic rank can replace slice and partition ranks in certain problems.

Abstract

The analytic rank of a tensor, first defined by Gowers and Wolf in the context of higher-order Fourier analysis, is defined to be the logarithm of the bias of the tensor. We prove that it is a subadditive measure of rank: that is, the analytic rank of the sum of two tensors is at most the sum of their individual analytic ranks. This analytic property turns out to have surprising applications: (i) common roots of tensors are always positively correlated; and (ii) the slice rank and partition rank, which were defined recently in the resolution of the cap-set problem in Ramsey theory, can be replaced by the analytic rank.