The G-stable rank for tensors and the cap set problem

Derksen, Harm

doi:10.2140/ant.2022.16.1071

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The $G$-stable rank for tensors and the cap set problem

Harm Derksen

Vol. 16 (2022), No. 5, 1071–1097

DOI: 10.2140/ant.2022.16.1071

Abstract

We introduce the $G$ -stable rank of a higher order tensors over perfect fields. The $G$ -stable rank is related to the Hilbert–Mumford criterion for stability in geometric invariant theory. We will relate the $G$ -stable rank to the tensor rank and slice rank. For numerical applications, we express the $G$ -stable rank as a solution to an optimization problem. Over the field $𝔽_{3}$ we discuss an application to the cap set problem.

Keywords

tensor rank, geometric invariant theory, $G$-stable rank, cap set problem

Mathematical Subject Classification

Primary: 11B25, 15A69

Secondary: 13A50, 14L24, 14N07

Milestones

Received: 10 August 2020

Revised: 8 July 2021

Accepted: 30 August 2021

Published: 16 August 2022

Authors

Harm Derksen
Department of Mathematics Northeastern University Boston, MA United States

Vol. 16, No. 5, 2022