G-stable rank of symmetric tensors and log canonical threshold

TL;DR

This paper proves that the G-stable rank of symmetric tensors equals the symmetric G-stable rank, and relates the log-canonical threshold of a singularity to the G-stable rank of its defining ideal.

Contribution

It establishes the equality of symmetric G-stable rank and G-stable rank for symmetric tensors, confirming an analog of Comon's conjecture.

Findings

Symmetric G-stable rank equals G-stable rank for symmetric tensors.

Log-canonical threshold is bounded by the G-stable rank of the defining ideal.

Counterexample to Comon's conjecture does not extend to G-stable rank.

Abstract

Shitov recently gave a counterexample to Comon's conjecture that the symmetric tensor rank and tensor rank of a symmetric tensor are the same. In this paper we show that an analog of Comon's conjecture for the G-stable rank introduced by Derksen is true: the symmetric G-stable rank and G-stable rank of a symmetric tensor are the same. We also show that the log-canonical threshold of a singularity is bounded by the G-stable rank of the defining ideal.