Polynomials and tensors of bounded strength

TL;DR

This paper demonstrates that the concept of strength for tensors and polynomials is universal among Zariski-closed, functorial tensor conditions, extending previous results and unifying various notions of tensor rank.

Contribution

It proves that any non-trivial Zariski-closed functorial tensor condition implies bounded strength, generalizing prior theorems on polynomial and tensor ranks.

Findings

Strength is universal among Zariski-closed functorial tensor conditions.

Generalizes Derksen-Eggermont-Snowden's theorem on cubic polynomials.

Extends Kazhdan-Ziegler's result on derivatives and bounded strength.

Abstract

Notions of rank abound in the literature on tensor decomposition. We prove that strength, recently introduced for homogeneous polynomials by Ananyan-Hochster in their proof of Stillman's conjecture and generalised here to other tensors, is universal among these ranks in the following sense: any non-trivial Zariski-closed condition on tensors that is functorial in the underlying vector space implies bounded strength. This generalises a theorem by Derksen-Eggermont-Snowden on cubic polynomials, as well as a theorem by Kazhdan-Ziegler which says that a polynomial all of whose directional derivatives have bounded strength must itself have bounded strength.