On the generic and typical ranks of 3-tensors

We study the generic and typical ranks of 3-tensors of dimension l x m x 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. (C) 2011 Elsevier Inc. All rights reserved.