论文标题
汇总和Veronese品种
Sumsets and Veronese varieties
论文作者
论文摘要
在本文中，到任何子集$ \ Mathcal {a} \ subset \ Mathbb {z}^{n} $我们明确地将唯一的单次投影$ y_ { $ t \ mathcal {a} $。此链接允许我们解决确定多项式$ p _ {\ mathcal {a}}} \ in \ mathbb {q} [t] $的经典问题。 = P _ {\ MathCal {a}}（T）$ for $ t \ geq T_0 $和最小整数$ n_0（\ nathcal {a}）\ leq t_0 $，为此条件满足，即所谓的{\ em em em ex em e em eaptrion}的$ | t \ nmathcal |我们使用castelnuovo-蒙福福德的规律性和$ y_ {n，d _ {\ Mathcal {a}}} $的几何形状来描述多项式$ p _ {\ nathcal {a}}}（t）（t）$，并在$ n_0（\ new_0（\ new_0）$ new bunds of Mathcal of new bunds of new bunds in} a} a} a} a} a} $ { $ \ Mathcal {a} $;反之亦然，我们应用总和的理论来获取品种的几何信息$ y_ {n，d _ {\ Mathcal {a}}} $。
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 $t \geq t_0$ and the minimum integer $n_0(\mathcal{A}) \leq t_0$ 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 $Y_{n,d_{\mathcal{A}}}$.
