Alessandro Oneto (SU): Ideals of points and Waring problems for polynomials
Alessandro Oneto (Stockholm) gives the Seminar in Algebra and Algebraic Geometry:
Ideals of points and Waring problems for polynomials
Abstract: A Waring decomposition of a homogeneous polynomials F (with complex coefficients) is an expression of the polynomial F as a sum of powers of linear forms. The smallest possible length of such a decomposition is called Waring rank of F. Via Apolarity Theory, finding such decompositions is equivalent to look for ideals of reduced points inside the so-called perp-ideal of F, i.e., the ideal of polynomials annihilating F by acting as partial differentials. Such a set of points in said to be apolar to F.
The aim of this talk is to describe old and recent results about Waring decompositions of polynomials obtained by studying ideals of reduced points. In particular, we focus on minimal sets of apolar points, i.e., sets of points apolar to F and of cardinality equal to the rank of F. We see that, for several familes of homogeneous polynomials (binary forms, quadrics, monomials, plane cubics) there are always points that cannot appear in a minimal set of apolar points and we describe their loci. We call them forbidden points. This is part of a recent joint work with E.Carlini and M.V.Catalisano.
In the case of monomials, we also present a recent result classifying the monomials that can have a minimal set of apolar points with only real coefficients. This is a joint work with E.Carlini, M.Kummer and E.Ventura.