Explicit tensors of border rank at least 2d−2 in Kd ⊗ Kd ⊗ Kd in arbitrary characteristic

Abstract

ABSTRACTFor tensors in , Landsberg provides non-trivial equations for tensors of border rank 2d−3 for d even and 2d−5 for d odd in Landsberg [Non-triviality of equations and explicit tensors in of border rank at least 2m−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 2d−5 to 2d−4. The purpose of this paper is to show that the aforementioned results extend to tensors in for any field K.