Newton polygon and the number of integer points in some semialgebraic sets
Speaker: Ha Huy Vui

Time: 9h00, Wednesday, November 22, 2017
Room semina, Floor 6th, Building A6,, Institute of Mathematics, 18 Hoang Quoc Viet, Cau Giay, Hanoi

Abstract: Let $f(x; y)$ be a polynomial in two variables of the form
$$f(x; y) = a_0y^D + a_1(x)y^{D-1} + ldots + a_D(x),$$
where $D$ is the degree of $f$. For $r > 0$, let
$$G^f(r) := {(x; y) in mathbb R^2: |f(x; y)| leq r}.$$
For $K subseteq mathbb R^2$, let $z(G^f(r) cap K)$ denote the number of integer points in the set $G^f(r) cap K$. We show that if $f$ satisfies the so called {it weakly degenerate condition} w.r.t. its Newton polygon $Gamma(f)$ then there exists a neighborhood $Omega_A$ of the set
$$({f=0}cup {frac{delta f}{delta y}=0})cap {|x|>A},$$
vertically thin at infinity, such that
$$z(G^f(r)setminus Omega_A) = r^{frac{1}{d}} ln^{1-k}r; text{ as } r rightarrow infty,$$
where $d$ is the coordinate of the furthest point in the intersection of the so called complete Newton polygon $tilde Gamma(f)$ of $f$ and the diagonal, and $k in{0; 1}$ is the dimension of the face of $tilde Gamma(f)$ containing the point $(d; d)$ in its relative interior.
This is a joint work with Nguyen Thi Thao.


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