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
• Enlaces
  • Texto completo
• 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}}}.

• Referencias bibliográficas
  • Beltrametti, M., Carletti, E., Gallarati, D., Monti Bragadin, G.: Lectures on Curves, Surfaces and Projective Varieties: a classical view...
  • Blancafort, C.: Hilbert functions of graded algebras over Artinian rings. J. Pure Appl. Algebra 125, 55–78 (1998)
  • Bruns, W., Herzog, J.: Cohen-Macaulay rings. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, (1993)
  • Chen, F.J., Chen, Y.G., Wu, J.D.: On the structure of the sumsets. Discrete Math. 311(6), 408–412 (2011)
  • Colarte-Gómez, L., Mezzetti, E., Miró-Roig, R.M.: On the arithmetic Cohen-Macaulayness of varieties parameterized by monomial Togliatti systems....
  • Colarte-Gómez, L., Miró-Roig, R. M.: The canonical module of GT-varieties and the normal bundle of RL-varieties. Mediterranean Journal of...
  • Colarte-Gómez, L.: Gröbner’s problem and the geometry of GT-varieties. PhD Thesis, Universitat de Barcelona, Barcelona, Spain, July of (2021)
  • Curran, M. J., Goldmakher, L.: Khovanskii’s theorem and effective results on sumset structure.arXiv:2009.02140v3
  • Elias, J.: Sumsets and monomial projective curves. Mediterranean Journal of Math., to appear.
  • Freiman, G.A.: Structure theory of set addition Astérisque 258, 1–33 (1999)
  • Geroldinger, A., Ruzsa, I.: Combinatorial number theory and additive group theory. In: Advances courses in Mathematics-CRM Barcelona, Birkhäuser...
  • Gotzmann, G.: Einige einfach-zusammenhängende Hilbertschemata. Math. Z. 180(3), 291–305 (1982)
  • Granville, A., Walker, A.: A tight structure theorem for sumsets.arXiv:2006.01041v2
  • Granville, A., Shakan, G.: The Frobenious postage stamp problem, and beyond. Acta Math. Hungar. 161(2), 700–718 (2020)
  • Granville, A., Shakan, G., Walker, A.: Effective results on the size and structure of sumsets.arXiv:2006.01041v2
  • Green, M.: Restrictions of linear series to hyperplanes, and some results of Macaulay and Gotzmann. Algebraic curves and projective geometry...
  • Harris, J.: Algebraic Geometry: A first course. GTM 133 Springer-Verlag, New York, (1992)
  • Hartshorne, R.: Algebraic Geometry. Springer-Verlag, New York (1977)
  • Herzog, J., Hibi, T.: Castelnuovo-Mumford regularity of simplicial semigroup rings with isolated singularity. Proc. Am. Math. Soc. 131, 2641–2647...
  • Hoa, L.T., Stückrad, J.: Castelnuovo-Mumford regularity of simplicial toric rings. J. Algebra 259, 127–146 (2003)
  • Hochster, M.: Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes. Ann. Math. 96(2), 318–337 (1972)
  • Jelínek, V., Klazar, M.: Generalizations of Khovanskii’s theorems on the growth of sumsets in Abelian semigroups. Adv. Appl. Math. 41, 115–132...
  • Khovanskii, A.G.: Newton polyhedron, Hilbert polynomial, and sums of finite sets. Funct. Anal. Its Appl. 26, 276–281 (1992)
  • Lee, J.: Algebraic proof for the geometric structure of sumsets. Integers 11(4), 477–486 (2011)
  • McCullough, J., Peeva, I.: Counterexamples to the Eisenbud-Goto regularity conjecture. J. Amer. Math. Soc. 31(2), 473–496 (2018) Miller,...
  • Nathanson, M. B.: Additive number theory: Inverse problems and the geometry of sumsets. Graduate Texts in Mathematics, 165, Springer-Verlag,...
  • Nathanson, M.B.: Sums of finite sets of integers. Amer. Math. Monthly 79, 1010–1012 (1972)
  • Nathanson, M.B., Ruzsa, I.Z.: Polynomial growth of sumsets in abelian semigroups. J. de théorie des nombres de Bordeaux 14(2), 553–560 (2002)
  • Nitsche, M.J.: A combinatorial proof of the Eisenbud-Goto conjecture for monomial curves and some simplicial semigroup rings. J. Algebra 397,...
  • O’Carroll, L., Planas-Vilanova, F., Villarreal, R.H.: Degree and algebraic properties of lattice and matrix ideals. SIAM J. Discrete Math....