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Polynomials and tensors of bounded strength

Arthur Bik

Universität Bern, Mathematisches Institut, Alpeneggstrasse 22, 3012 Bern, Switzerland

E-mail Address: [email protected]

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Jan Draisma

Universität Bern, Mathematisches Institut, Sidlerstrasse 5, 3012 Bern, Switzerland

Eindhoven University of Technology, 5612 AZ Eindhoven, The Netherlands

E-mail Address: [email protected]

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Rob H. Eggermont

Department of Mathematics and Computer Science, Eindhoven University of Technology, P. O. Box 513, 5600 MB Eindhoven, The Netherlands

E-mail Address: [email protected]

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https://doi.org/10.1142/S0219199718500621Cited by:10 (Source: Crossref)

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 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.

Keywords:

AMSC: 15A69, 15A72, 13A50, 14M12

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