Sumsets and Veronese varieties
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[1] Universitat de Barcelona
Universitat de Barcelona
Barcelona, España
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- Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 74, Fasc. 2, 2023, págs. 353-374
- Idioma: inglés
- DOI: 10.1007/s13348-022-00352-x
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Resumen
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 \ge t_0 and the minimum integer n_0(\mathcal {A}) \le 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}}}.
