Alessandro Oneto: Power ideals, Fröberg conjecture and Waring problems

This thesis is divided into two chapters. First, we want to study particular classes of power ideals, with particular attention to their relation with the Fröberg conjecture on the Hilbert series of generic ideals. In the second part, we study a generalization (introduced in Ralf Fröberg, Giorgio Ottaviani, and Boris Shapiro) of the classical Waring problem for polynomials about writing homogeneous polynomials as sums of powers. We see also how the theories of fat points and secant varieties of Veronese varieties play a crucial role in the relation between those chapters and in providing tools to find an answer to our questions.

The main results in this thesis are: the proof of the Fröberg conjecture in the case of ideals with eight generators in four variables with the same degree; the study, in terms of Hilbert series and Betti numbers, of a class of power ideals (from a joint paper with Jörgen Backelin); the study of a generalized version of Waring problem with focus on the cases of sum of squares and monomial (from a joint paper with Enrico Carlini).