#### 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 ${\mathbb{𝔽}}_{3}$ 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