## Section: New Results

### Waring loci and the Strassen conjecture

Participant : Alessandro Oneto.

In [8], we introduce the notion of the *Waring locus* of a homogeneous polynomial. A *Waring decomposition* is an expression of a polynomial as sum of powers of linear forms. The smallest length of such a decomposition is called the *Waring rank* of the polynomial. A very difficult challenge is to compute the rank and a minimal decomposition of a given form. The Waring locus of a polynomial is the locus of linear forms that appear in a minimal decomposition of it. The idea behind this construction is to find an iterative approach to construct Waring decompositions *step-by-step*, by adding one power at the time. Moreover, we give a version of the famous *Strassen conjecture* on the additivity of rank for sums of polynomials in independent sets of variables. We compute the Waring loci in several cases as binary forms, quadrics, monomials and plane cubics and for some other particular families of polynomials.

This is a joint work with E. Carlini, and M. V. Catalisano.