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For 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. [ABSTRACT FROM AUTHOR]

We study the category of wheeled PROPs using tools from Invariant Theory. A typical example of a wheeled PROP is the mixed tensor algebra V = T (V) ⊗ T (V ⋆) , where T (V) is the tensor algebra on an n -dimensional vector space over a field K of characteristic 0. First we classify all the ideals of the initial object Z in the category of wheeled PROPs. We show that non-degenerate sub-wheeled PROPs of V are exactly subalgebras of the form V G where G is a closed, reductive subgroup of the general linear group GL (V). When V is a finite dimensional Hilbert space, a similar description of invariant tensors for an action of a compact group was given by Schrijver. We also generalize the theorem of Procesi that says that trace rings satisfying the n -th Cayley-Hamilton identity can be embedded in an n × n matrix ring over a commutative algebra R. Namely, we prove that a wheeled PROP can be embedded in R ⊗ V for a commutative K -algebra R if and only if it satisfies certain relations. [ABSTRACT FROM AUTHOR]

We study the relationship between the commutative and the non-commutative rank of a linear matrix. We give examples that show that the ratio of the two ranks comes arbitrarily close to 2. Such examples can be used for giving lower bounds for the border rank of a given tensor. Landsberg used such techniques to give non-trivial equations for the tensors of border rank at most in if m is even. He also gave such equations for tensors of border rank at most in if m is odd. Using concavity of tensor blow-ups we show non-trivial equations for tensors of border rank in for odd m for any field of characteristic 0. We also give another proof of the regularity lemma by Ivanyos, Qiao and Subrahmanyam. [ABSTRACT FROM AUTHOR]