Sumsets and Veronese varieties

Abstract In this paper, to any subset $$\mathcal {A}\subset \mathbb {Z}^{n}$$ A ? Z n we explicitly associate a unique monomial projection $$Y_{n,d_{\mathcal {A}}}$$ Y , d of Veronese variety, whose Hilbert function coincides with the cardinality t -fold sumsets $$t\mathcal {A}$$ t . This link allows us tackle classical problem determining polynomial $$p_{\mathcal {A}} \in {Q}[t]$$ p ? Q [ ] such that $$|t\mathcal {A}| = p_{\mathcal {A}}(t)$$ | = ( ) for all $$t \ge t_0$$ ? 0 and minimum integer $$n_0(\mathcal {A}) \le ? which condition is satisfied, i.e. so-called phase transition {A}|$$ We use Castelnuovo–Mumford regularity geometry describe derive new bounds {A})$$ under some technical assumptions on convex hull ; vice versa apply theory obtain geometric information varieties