Section: New Results

Waring-like decompositions of polynomials

Participant : Alessandro Oneto.

In [9], we consider particular types of additive decompositions of homogeneous polynomials. The classical decomposition is the Waring decomposition, where we decompose polynomials as sums of powers of linear forms. Another well studied decomposition is the sometimes-called Chow decomposition, where we decompose polynomials as sums of products of linear forms. These are the extremal cases of the additive decompositions considered in this work. For a fixed partition $(d_1,...,d_s)⊢d$ of the degree of the polynomial, we consider decompositions as sums of degree forms of the form ${\ell }_1^{d_1}\cdots {\ell }_s^{d_s}$, where the $\ell$'s are linear forms. The homogeneous polynomials of the form ${\ell }_1^{d_1}\cdots {\ell }_s^{d_s}$ are parametrized by particular linear projections of certain Segre-Veronese varieties. The main results of this work concerns the dimension of the secant varieties to these projections of Segre-Veronese varieties. In particular, we compute their dimensions in the binary case (forms in two variables) and the case of secant lines varieties for any partition and any number of variables. From these results, we deduce the dimension of higher secant varieties in some particular cases.

This is a joint work with M. V. Catalisano, Luca Chiantini, and A. V. Geramita.