The next gap in the subrank of 3-tensors

Author:

Fulvio Gesmundo, Jeroen Zuiddam

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), Computational Complexity (cs.CC), Combinatorics (math.CO), Quantum Physics (quant-ph)

journal:

date:

2023-07-11 16:00:00

Abstract

Recent works of Costa-Dalai, Christandl-Gesmundo-Zuiddam, Blatter-Draisma-Rupniewski, and Bri\"et-Christandl-Leigh-Shpilka-Zuiddam have investigated notions of discreteness and gaps in the possible values that asymptotic tensor ranks can take. In particular, it was shown that the asymptotic subrank and asymptotic slice rank of any nonzero 3-tensor is equal to 1, equal to 1.88, or at least 2 (over any field), and that the set of possible values of these parameters is discrete (in several regimes). We determine exactly the next gap, showing that the asymptotic subrank and asymptotic slice rank of any nonzero 3-tensor is equal to 1, equal to 1.88, equal to 2, or at least 2.68.

PDF: The next gap in the subrank of 3-tensors.pdf