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.

中文翻译：

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张量的 G-stable 秩和上限集问题

我们介绍$G$- 高阶张量在完美场上的稳定排名。这$G$-stable rank 与几何不变理论中的 Hilbert-Mumford 稳定性标准有关。我们将把$G$-张量等级和切片等级的稳定等级。对于数值应用，我们表示$G$-作为优化问题的解决方案的稳定排名。在场上${\mathbb{𝔽}}_3$我们讨论上限集问题的应用。