Secant varieties and the complexity of matrix multiplication

2022/08/01 by J. M. Landsberg, Landsberg, J. M. · 1 citation
Computer Science · Mathematics · #14L35 #15A69 #68Q15 #Algebraic Geometry (math.AG) #FOS: Mathematics #Mathematical Approximation and Integration #Polynomial and algebraic computation #Tensor decomposition and applications

paper · pdf · doi:10.48550/arxiv.2208.00857

openalex publication_date 2022/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This is a survey primarily about determining the border rank of tensors, especially those relevant for the study of the complexity of matrix multiplication. This is a subject that on the one hand is of great significance in theoretical computer science, and on the other hand touches on many beautiful topics in algebraic geometry such as classical and recent results on equations for secant varieties (e.g., via vector bundle and representation-theoretic methods) and the geometry and deformation theory of zero dimensional schemes.

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• Waring decompositions of the product of two quadrics: the small rank cases

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