# WEEKLY ACTIVITIES

Newton polygon and the number of integer points in some semialgebraic sets
 Speaker: Ha Huy VuiTime: 9h00, Wednesday, November 22, 2017Location: 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.

### Highlights

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