The Symmetric Rank of 2 × 2 × ⋯ × 2 Symmetric Tensors over an Arbitrary Field

Abstract

The symmetric rank problem of 2 × 2 × ⋯ × 2 symmetric tensors are related to Waring’s problem of binary forms. In this paper, we characterize the symmetric rank of 2 × 2 × ⋯ × 2 symmetric tensors when the symmetric rank is 1 and 2 in arbitrary characteristic (either zero or strictly larger than the order of a tensor). Moreover, we characterize the symmetric rank for 2 × 2 × 2 symmetric tensors over the fields where the characteristic is zero or larger than three or $\mathbb{F}_{2}$ or $\mathbb{F}_{3}$.

Keywords

Sylvester Theorem / Waring’s problem / binary forms / symmetric tensor rank / arbitrary field