Algebraic complexities and algebraic curves over finite fields

Maximal Rank of ${m}\times{n}\times 2$ Tensors over Arbitrary Fields

. Waring’s problem, of expressing an integer as the sum of powers, has a very long history going back to the 17th century, and the problem has been studied in many diﬀerent contexts. In this paper we introduce the notion of a Waring subspace and a Waring identiﬁable subspace with respect to a projective algebraic variety X . When X is the Veronese variety, these subspaces play a fundamental role in the theory of symmetric tensors and are related to the Waring decomposition and Waring identiﬁability of symmetric tensors (homo-geneous polynomials). We give several constructions and classiﬁcation results of Waring identiﬁable subspaces with respect to the Veronese variety in P 5 ( F q ) and in P 9 ( F q ), and include some applications to the theory of linear systems of quadrics in P 3 ( F q ).

Sharon J. Laskowski

Papers

Computing lower bounds on tensor rank over finite fields