Abstract
Let f(x, y) be a polynomial in two variables of the form
where D is the degree of f. For \(r > 0\), let
We study the distribution of integer points in \(G^f(r)\). In assuming that f satisfies the so called weekly degenerate condition for main edges of the complete Newton polygon of f, we show that:
-
(i)
There exists a horn-neighborhood neighborhood \(\Omega \) of half-branches at infinity of the curve \(f^{-1}(0) \cup (\frac{\partial f}{\partial y})^{-1}(0)\), which is vertically thin at infinity, such that, if the number of integer points of \(G^f(r)\) is infinitely many, then all of them, except a finite number of points, are contained in the set \(\Omega \), i.e. they are concentrated around the curve \(f^{-1}(0) \cup (\frac{\partial f}{\partial y})^{-1}(0)\).
-
(ii)
The above neighborhood \(\Omega \) can be constructed explicitly via the Newton-Puiseux expansions at infinity of the curve \(f^{-1}(0) \cup (\frac{\partial f}{\partial y})^{-1}(0)\), hence, it is the same for all \(G^f( r )\), \(r > 0\).
-
(iii)
The number of integer points in \(G^f(r) \backslash \Omega \), as r goes to infinity, has the following asymptotics:
$$\begin{aligned} z(G^f(r) \backslash \Omega ) \asymp r^\frac{1}{d}\ln ^{1-k} r, \ \text{ as } r \rightarrow \infty , \end{aligned}$$where d is the Newton distance of f (i.e., the coordinate of the furthest point in the intersection of the complete Newton polygon \({{\tilde{\Gamma }}}(f)\) of f and the diagonal) and \(k \in \{0,1\}\) is the dimension of the smallest face of \({\tilde{\Gamma }}(f)\) containing the point (d, d) in its relative interior.
-
(iv)
If f is non-degenerate in the sense of Kouchnirenko and the Newton distance d of f is greater than 1 then f satisfies the weakly degenerate condition. Hence, the above asymptotic formula holds for all polynomials belonging a Zariski open subset of the space of polynomials having the same Newton polygon.
References
Berndt, B.C., Kim, S., Zaharescu, A.: The circle problem of Gauss and the divisor problem of Dirichlet still unsolved. Am. Math. Mon. 125(2), 99–114 (2018)
Cluckers, R.: Igusa’s conjecture on exponential sums modulo \(p\) and \(p^2\) and motivic oscillation index. Int. Math. Res. Not. IMRN 2008(4), 20 (2008) (article ID rnm118)
Cluckers, R.: Igusa and Denef–Sperber conjecture on non degenerate \(p\)-adic exponential sums. Duke Math. J. 141, 205–216 (2008)
Davenport, H.: On a principle of Lipschitz. J. Lond. Math. Soc. 26, 179–183 (1951)
Gindikin, S., Volevich, L.R.: The method of Newton’s polyhedron in the theory of partial differential equations. Math. Appl. (Sov. Ser.) 86 (1992)
Grafakos, L.: Classical Fourier Analysis. Springer, New York (2008)
Greenblatt, M.: Newton polygons and local integrability of negative powers of smooth functions in the plane. Trans. Am. Math. Soc 352(2), 657–670 (2005)
Greenblatt, M.: Oscillatory integral decay, sublevel set growth, and the Newton polyhedron. Math. Ann. 346, 857–895 (2010)
Ha, H.V., Tran, G.L.: On the volume and the number of lattice points of some semi-algebraic sets. Int. J. Math. 26(10), 13. https://doi.org/10.1142/S0129167X15500780 (2015)
Huxley, M.N.: Integer Points, Exponential Sums and the Exponential Sums. London Mathematical Society Monographs, New Series, vol. 13. Oxford Science Publications, Oxford (1996)
Kouchnirenko, A.G.: Polyhedres de Newton et nombre de Milnor. Invent. Math. 32, 1–31 (1976)
Phong, D.H., Stein, E.M.: The Newton polyhedron and oscillatory integral operators. Acta Math. 179, 107–152 (1997)
Varchenko, A.N.: Newton polyhedra and estimations of oscillatory integrals. Funct. Anal. Appl. 10(3), 175–196 (1976)
Vasiliev, V.A.: Asymptotic exponential integrals, newton’s diagram, and the classification of minimal points. Funct. Anal. Appl. 11(3), 163–172 (1977)
Zung, D.: Number of integral points in a certain set and the approximation of functions of several variables. Mat. Zametki 36(4), 479–491 (1984)
Acknowledgements
The main part of this work was done while the authors were visiting Vietnam Institute for Advanced Study in Mathematics (VIASM), Hanoi, Vietnam. We would like to thank the Institute for financial support and excellent working conditions. We would also like to thank the referee very much for your valuable and constructive suggestion to improve the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Frédéric Pham on the occasion of his eightieth birthday.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Huy Vui Ha and Thi Thao Nguyen were partially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2017.12.
Rights and permissions
About this article
Cite this article
Ha, H.V., Nguyen, T.T. Newton polygon and distribution of integer points in sublevel sets. Math. Z. 295, 1067–1093 (2020). https://doi.org/10.1007/s00209-019-02395-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-019-02395-6
Keywords
- Newton polygon
- Complete Newton polygon
- Kouchnirenko non-degenerate condition
- Weakly degenerate condition
- Distribution of integer points