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Newton polygon and distribution of integer points in sublevel sets

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Abstract

Let f(xy) be a polynomial in two variables of the form

$$\begin{aligned} f(x,y) = a_0y^D + a_1(x)y^{D-1} + \cdots + a_D(x), \end{aligned}$$

where D is the degree of f. For \(r > 0\), let

$$\begin{aligned} G^f(r) = \{(x,y) \in \mathbb {R}^2 : |f(x,y)| \le r\}. \end{aligned}$$

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:

  1. (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 \(f^{-1}(0) \cup (\frac{\partial f}{\partial y})^{-1}(0)\).

  2. (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 \(G^f( r )\), \(r > 0\).

  3. (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 (dd) in its relative interior.

  4. (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.

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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.

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Correspondence to Thi Thao Nguyen.

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Dedicated to Professor Frédéric Pham on the occasion of his eightieth birthday.

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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.

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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

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