1 November 2021 Apolarity, border rank, and multigraded Hilbert scheme
Weronika Buczyńska, Jarosław Buczyński
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Duke Math. J. 170(16): 3659-3702 (1 November 2021). DOI: 10.1215/00127094-2021-0048


We introduce an elementary method to study the border rank of polynomials and tensors, analogous to the apolarity lemma. This can be used to describe the border rank of all cases uniformly, including those very special ones that resisted a systematic approach. We also define a border rank version of the variety of sums of powers and analyze its usefulness in studying tensors and polynomials with large symmetries. In particular, it can be applied to provide lower bounds for the border rank of some interesting tensors, such as the matrix multiplication tensor. We work in a general setting, where the base variety is not necessarily a Segre or Veronese variety, but an arbitrary smooth toric projective variety. A critical ingredient of our work is an irreducible component of a multigraded Hilbert scheme related to the toric variety in question.


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Weronika Buczyńska. Jarosław Buczyński. "Apolarity, border rank, and multigraded Hilbert scheme." Duke Math. J. 170 (16) 3659 - 3702, 1 November 2021. https://doi.org/10.1215/00127094-2021-0048


Received: 17 December 2019; Revised: 17 October 2020; Published: 1 November 2021
First available in Project Euclid: 14 October 2021

MathSciNet: MR4332674
zbMATH: 1481.14006
Digital Object Identifier: 10.1215/00127094-2021-0048

Primary: 14C05
Secondary: 14M25 , 15A69 , 68Q17

Keywords: apolarity , border rank , Cox ring , invariant ideals , multigraded Hilbert scheme , variety of sums of powers

Rights: Copyright © 2021 Duke University Press


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Vol.170 • No. 16 • 1 November 2021
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