Towards a Geometric Approach to Strassen's Asymptotic Rank Conjecture

2018/11/30 by Austin Conner, Fulvio Gesmundo, Joseph M. Landsberg +2 · 1 citation
Mathematics · Computer Science · #math.AG #cs.CC #msc:15A69 #msc:14L35 #msc:68Q15

paper · pdf · doi:10.1007/s13348-020-00280-8

published as Collectanea Mathematica, 2020 · Final version. Revisions in Section 1 and Section 3

arxiv created 2020/02/11 · arxiv updated 2020/02/12

Abstract

We make a first geometric study of three varieties in ℂ^m ⊗ ℂ^m ⊗ ℂ^m (for each m), including the Zariski closure of the set of tight tensors, the tensors with continuous regular symmetry. Our motivation is to develop a geometric framework for Strassen's Asymptotic Rank Conjecture that the asymptotic rank of any tight tensor is minimal. In particular, we determine the dimension of the set of tight tensors. We prove that this dimension equals the dimension of the set of oblique tensors, a less restrictive class introduced by Strassen.

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