On the generic and typical ranks of 3-tensors

We study the generic and typical ranks of 3-tensors of dimension l×m×n using results from matrices and algebraic geometry. We state a conjecture about the exact values of the generic rank of 3-tensors over the complex numbers, which is verified numerically for l,m,n≤14. We also discuss the typical ranks over the real numbers, and give an example of an infinite family of 3-tensors of the form l=m,n=(m-1)2+1,m=3,4,…, which have at least two typical ranks.