Several remarks on tensor rank computation

Yaroslav Shitov

Vol. 334 (2025), No. 1, 143–151

DOI: 10.2140/pjm.2025.334.143

Abstract

The Waring rank of a homogeneous polynomial $f$ is the smallest number $\mathrm{wr}<!--FUNCTION\; APPLICATION-->f$ for which $f$ is the sum of $\mathrm{wr}<!--FUNCTION\; APPLICATION-->f$ powers of linear forms. We show

and answer a question of Teitler. We also discuss further questions of Koiran and Schaefer regarding the algorithmic computation of tensor ranks.

Keywords

tensor rank, Waring decomposition, algorithms

Mathematical Subject Classification

Primary: 14N07, 15A27, 15A69

Milestones

Received: 3 October 2024

Revised: 5 January 2025

Accepted: 7 January 2025

Published: 23 January 2025