On the geometry of geometric rank

Geng, Runshi; Landsberg, Joseph M.

doi:10.2140/ant.2022.16.1141

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On the geometry of geometric rank

Runshi Geng and Joseph M. Landsberg

Vol. 16 (2022), No. 5, 1141–1160

DOI: 10.2140/ant.2022.16.1141

Abstract

We make a geometric study of the geometric rank of tensors recently introduced by Kopparty et al. Results include classification of tensors with degenerate geometric rank in $ℂ^{3} \otimes ℂ^{3} \otimes ℂ^{3}$ , classification of tensors with geometric rank two, and showing that upper bounds on geometric rank imply lower bounds on tensor rank.

Keywords

geometric rank, matrix multiplication complexity, tensor rank, asymptotic rank, laser method

Mathematical Subject Classification

Primary: 14L30, 15A69, 68Q17

Milestones

Received: 8 December 2020

Revised: 5 August 2021

Accepted: 14 August 2021

Published: 16 August 2022

Authors

Runshi Geng
Department of Mathematics Texas A&M University College Station, TX United States

Joseph M. Landsberg
Department of Mathematics Texas A&M University College Station, TX United States

Vol. 16, No. 5, 2022