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Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/193020
Sumsets and Veronese varieties
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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 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 $Y_{n, d_{\mathcal{A}}}$.
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COLARTE GÓMEZ, Liena, ELÍAS GARCÍA, Joan and MIRÓ-ROIG, Rosa M. (Rosa Maria). Sumsets and Veronese varieties. Collectanea Mathematica. 2022. ISSN 0010-0757. [consulted: 15 of October of 2026]. Available at: https://hdl.handle.net/2445/193020