Explicit tensors of border rank at least 2d−2 in Kd ⊗ Kd ⊗ Kd in arbitrary characteristic. Issue 10 (3rd October 2019)

• Record Type: Journal Article
• Authors: Derksen, Harm; Makam, Visu
• Is Part Of: Linear & multilinear algebra. Volume 67:Issue 10(2019)
• Page Start: 2104
• Page End: 2116
• Publication Date: 2019-10-03
• Subjects: Border rank -- lower bounds -- Koszul flattenings -- linear subspaces of matrices -- tensor blow-ups; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2018.1484419
• Languages: English
• ISSNs: 0308-1087

Abstract

ABSTRACT: For tensors in C d ⊗ C d ⊗ C d, Landsberg provides non-trivial equations for tensors of border rank 2 d −3 for d even and 2 d −5 for d odd in Landsberg [Non-triviality of equations and explicit tensors in C m ⊗ C m ⊗ C m of border rank at least 2 m −2. J Pure Appl Algebra. 2015;219(8):3677–3684]. In Derksen and Makam [On non-commutative rank and tensor rank, Linear Multilinear Algebra, published online 2017], we observe that Landsberg's method can be interpreted in the language of tensor blow-ups of matrix spaces, and using concavity of blow-ups, we improve the case for odd d from 2 d −5 to 2 d −4. The purpose of this paper is to show that the aforementioned results extend to tensors in K d ⊗ K d ⊗ K d for any field K .