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- Title
Newton polygon and distribution of integer points in sublevel sets.
- Authors
Ha, Huy Vui; Nguyen, Thi Thao
- Abstract
Let f(x, y) be a polynomial in two variables of the form f (x , y) = a 0 y D + a 1 (x) y D - 1 + ⋯ + a D (x) , where D is the degree of f. For r > 0 , let G f (r) = { (x , y) ∈ R 2 : | f (x , y) | ≤ r }. 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: There exists a horn-neighborhood neighborhood Ω of half-branches at infinity of the curve f - 1 (0) ∪ (∂ f ∂ 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 Ω , i.e. they are concentrated around the curve f - 1 (0) ∪ (∂ f ∂ y) - 1 (0) . The above neighborhood Ω can be constructed explicitly via the Newton-Puiseux expansions at infinity of the curve f - 1 (0) ∪ (∂ f ∂ y) - 1 (0) , hence, it is the same for all G f (r) , r > 0 . The number of integer points in G f (r) \ Ω , as r goes to infinity, has the following asymptotics: z (G f (r) \ Ω) ≍ r 1 d ln 1 - k r , as r → ∞ , where d is the Newton distance of f (i.e., the coordinate of the furthest point in the intersection of the complete Newton polygon Γ ~ (f) of f and the diagonal) and k ∈ { 0 , 1 } is the dimension of the smallest face of Γ ~ (f) containing the point (d, d) in its relative interior. 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.
- Publication
Mathematische Zeitschrift, 2020, Vol 295, Issue 3/4, p1067
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/s00209-019-02395-6