Concise tensors of minimal border rank Jelisiejew, Joachim; Landsberg, J. M.; Pal, Arpan
Mathematische Annalen,
01/2024, Letnik:
388, Številka:
3
Journal Article
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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 the classical problem in algebraic complexity theory of classifying minimal border rank tensors in the special case
m
=
5
. Our proofs utilize 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
C
5
⊗
C
5
⊗
C
5
.
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