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.

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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