Vol. 16, No. 5, 2022

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

Harm Derksen

Vol. 16 (2022), No. 5, 1071–1097
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 𝔽3 we discuss an application to the cap set problem.

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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