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arXiv · 1805.01816

Polynomials and tensors of bounded strength

Arthur Bik·Jan Draisma·Rob H. Eggermont
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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.

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Arthur Bik, Jan Draisma, Rob H. Eggermont. 2018-05-04. Polynomials and tensors of bounded strength. https://doi.org/10.1142/s0219199718500621

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