Sumsets and Veronese varieties

Abstract

In this paper, to any subset $\mathcal{A}\subset {\mathbb{Z}}^n$ we explicitly associate a unique monomial projection $Y_{n,d_{\mathcal{A}}}$ of a Veronese variety, whose Hilbert function coincides with the cardinality of the $t$-fold sumsets $t\mathcal{A}$. This link allows us to tackle the classical problem of determining the polynomial $p_{\mathcal{A}}\in \mathbb{Q}[t]$ such that $|t\mathcal{A}|=p_{\mathcal{A}}(t)$ for all $t\ge t_0$ and the minimum integer $n_0(\mathcal{A})\le t_0$ for which this condition is satisfied, i.e. the so-called {\em phase transition} of $|t\mathcal{A}|$. We use the Castelnuovo--Mumford regularity and the geometry of $Y_{n,d_{\mathcal{A}}}$ to describe the polynomial $p_{\mathcal{A}}(t)$ and to derive new bounds for $n_0(\mathcal{A})$ under some technical assumptions on the convex hull of $\mathcal{A}$; and vice versa we apply the theory of sumsets to obtain geometric information of the varieties $Y_{n,d_{\mathcal{A}}}$.

To appear in Collectanea Mathematica