Concise tensors of minimal border rank

2022/05/11 by Jelisiejew, Joachim, Landsberg, J. M., Pal, Arpan · 1 citation
#14C05 #15A69 #68Q15 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #Computational Complexity (cs.CC) #FOS: Computer and information sciences #FOS: Mathematics

paper · doi:10.48550/arxiv.2205.05713

Abstract

We determine defining equations for the set of concise tensors of minimal border rank in C^m⊗ C^m⊗ C^m when m=5 and the set of concise minimal border rank 1_*-generic tensors when m=5,6. We solve this classical problem in algebraic complexity theory with the aid of two recent developments: the 111-equations defined by Buczyńska-Buczyński and results of Jelisiejew-Šivic on the variety of commuting matrices. We introduce a new algebraic invariant of a concise tensor, its 111-algebra, and exploit it to give a strengthening of Friedland's normal form for 1-degenerate tensors satisfying Strassen's equations. We use the 111-algebra to characterize wild minimal border rank tensors and classify them in C⁵⊗ C⁵⊗ C⁵.

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