Polynomials and tensors of 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.