On the generic and typical ranks of 3-tensors
This work addresses fundamental questions in tensor rank theory, providing conjectures and examples that advance understanding of generic and typical ranks for 3-tensors.
The authors study generic and typical ranks of 3-tensors, proposing a conjecture for exact generic rank values over complex numbers (verified numerically up to dimensions 14) and identifying an infinite family of tensors with at least two typical ranks over real numbers.
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 not greater than 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.