About the maximal rank of 3-tensors over the real and the complex number field.

Tensor data are becoming important recently in various application fields. In this paper, we consider the maximal rank problem of 3-tensors and extend Atkinson and Stephens’ and Atkinson and Lloyd’s results over the real number field. We also prove the assertion of Atkinson and Stephens: $${{\rm max.rank}_{\mathbb{R}}(m,n,p) \leq m+\lfloor p/2\rfloor n}$$, $${{\rm max.rank}_{\mathbb{R}}(n,n,p) \leq (p+1)n/2}$$ if p is even, $${{\rm max.rank}_{\mathbb{F}}(n,n,3)\leq 2n-1}$$ if $${\mathbb{F}=\mathbb{C}}$$ or n is odd, and $${{\rm max.rank}_{\mathbb{F}}(m,n,3)\leq m+n-1}$$ if m < n where $${\mathbb{F}}$$ stands for $${\mathbb{R}}$$ or $${\mathbb{C}}$$. [ABSTRACT FROM AUTHOR]