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

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 \(\mathbb{F}_{2}\) $${\mathbb{F}}_2$$ or \(\mathbb{F}_{3}\) $${\mathbb{F}}_3$$ .