Resumen de Sumsets and Veronese varieties

Liena Colarte Gómez, Joan Elías García , Rosa María Miró-Roig

• In this paper, to any subset A⊂Zn we explicitly associate a unique monomial projection Yn,dA of a Veronese variety, whose Hilbert function coincides with the cardinality of the t-fold sumsets tA. This link allows us to tackle the classical problem of determining the polynomial pA∈Q[t] such that |tA|=pA(t) for all t≥t0 and the minimum integer n0(A)≤t0 for which this condition is satisfied, i.e. the so-called phase transition of |tA|. We use the Castelnuovo–Mumford regularity and the geometry of Yn,dA to describe the polynomial pA(t) and to derive new bounds for n0(A) under some technical assumptions on the convex hull of A; and vice versa we apply the theory of sumsets to obtain geometric information of the varieties Yn,dA.