On the volume and the number of lattice of some semialgebraic sets

Abstract

Let $f = (f_1,\ldots,f_m) : \R^n \longrightarrow \R^m$ be a polynomial map; $G^f(r) = \{x\in\R^n : |f_i(x)| \leq r,\ i =1,\ldots, m\}$. We show that if $f$ satisfies the Mikhailov - Gindikin condition then \begin{itemize} \item[(i)] $\text{Volume}\ G^f(r) \asymp r^\theta (\ln r)^k$ \item[(ii)] $\text{Card}\left(G^f(r) \cap \overset{o}{\ \Z^n}\right) \asymp r^{\theta'}(\ln r)^{k'}$, as $r\to \infty$, \end{itemize} where the exponents $\theta,\ k,\ \theta',\ k'$ are determined explicitly in terms of the Newton polyhedra of $f$. \\ \indent Moreover, the polynomial maps satisfy the Mikhailov - Gindikin condition form an open subset of the set of polynomial maps having the same Newton polyhedron.