Sumsets and Veronese varieties

Abstract

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