Polynomials and tensors of bounded strength

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 generalized 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 generalizes 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.