Please use this identifier to cite or link to this item:
http://hdl.handle.net/2445/193020
Title: | Sumsets and Veronese varieties |
Author: | Colarte Gómez, Liena Elías García, Joan Miró-Roig, Rosa M. (Rosa Maria) |
Keywords: | Àlgebra commutativa Teoria de nombres Commutative algebra Number theory |
Issue Date: | 28-Mar-2022 |
Publisher: | Springer |
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 $t \geq t_0$ 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 $Y_{n, d_{\mathcal{A}}}$. |
Note: | Reproducció del document publicat a: https://doi.org/10.1007/s13348-022-00352-x |
It is part of: | Collectanea Mathematica, 2022 |
URI: | http://hdl.handle.net/2445/193020 |
Related resource: | https://doi.org/10.1007/s13348-022-00352-x |
ISSN: | 0010-0757 |
Appears in Collections: | Articles publicats en revistes (Matemàtiques i Informàtica) |
Files in This Item:
File | Description | Size | Format | |
---|---|---|---|---|
719114.pdf | 466.29 kB | Adobe PDF | View/Open |
This item is licensed under a Creative Commons License