## Section: New Results

### Tensor decomposition and homotopy continuation

Participant : Bernard Mourrain.

A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors of a given rank. By moving away from computing polynomials via elimination theory to computing pseudowitness sets via numerical elimination theory, we develop in [3] computational methods for computing ranks and border ranks of tensors along with decompositions. More generally, we present our approach using joins of any collection of irreducible and nondegenerate projective varieties ${X}_{1},...,{X}_{k}\subset {\mathbb{P}}^{N}$ defined over $\u2102$. After computing ranks over $\u2102$, we also explore computing real ranks. Various examples are included to demonstrate this numerical algebraic geometric approach.

This is a joint work with Alessandra Bernardi, Noah S. Daleo, Jonathan D. Hauenstein.