Geometric Rank and Linear Determinantal Varieties
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Publication:6387912
DOI10.1007/S40879-023-00615-2arXiv2201.03615MaRDI QIDQ6387912
Publication date: 10 January 2022
Abstract: There are close relations between tripartite tensors with bounded geometric ranks and linear determinantal varieties with bounded codimensions. We study linear determinantal varieties with bounded codimensions, and prove upper bounds of the dimensions of the ambient spaces. Using those results, we classify tensors with geometric rank 3, find upper bounds of multilinear ranks of primitive tensors with geometric rank 4, and prove the existence of such upper bounds in general. We extend results of tripartite tensors to n-part tensors, showing the equivalence between geometric rank 1 and partition rank 1.
Vector and tensor algebra, theory of invariants (15A72) Multilinear algebra, tensor calculus (15A69) Vector spaces, linear dependence, rank, lineability (15A03) Canonical forms, reductions, classification (15A21) Secant varieties, tensor rank, varieties of sums of powers (14N07)
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