Sumsets and Veronese varieties

• Colarte-Gómez, Liena ^[1] ; Elias, Joan ^[1] ; Miró-Roig, Rosa M. ^[1]
  1. [1] Universitat de Barcelona

     Universitat de Barcelona

     Barcelona, España

     
• 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 A⊂Zn we explicitly associate a unique monomial projection Yn,dA of a Veronese variety, whose Hilbert function coincides with the cardinality of the t-fold sumsets tA. This link allows us to tackle the classical problem of determining the polynomial pA∈Q[t] such that |tA|=pA(t) for all t≥t0 and the minimum integer n0(A)≤t0 for which this condition is satisfied, i.e. the so-called phase transition of |tA|. We use the Castelnuovo–Mumford regularity and the geometry of Yn,dA to describe the polynomial pA(t) and to derive new bounds for n0(A) under some technical assumptions on the convex hull of A; and vice versa we apply the theory of sumsets to obtain geometric information of the varieties Yn,dA.