G-STABLE RANK OF SYMMETRIC TENSORS AND LOG CANONICAL THRESHOLD

Abstract

Shitov recently gave counterexamples over the real and complex field 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 over perfect fields. We also show that the log-canonical threshold of a complex singularity is bounded by the $G$-stable rank of the defining ideal.