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

Abstract

Let $f=(f_1,\dots ,f_m):{\mathbb{R}}^n⟶{\mathbb{R}}^m$ be a polynomial map; $G^f(r)=\{x\in {\mathbb{R}}^n:|f_i(x)|\le r,i=1,\dots ,m\}$. 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)] $\text{Card}\left(G^f(r)\cap \stackrel{o}{{\mathbb{Z}}^n}\right)\asymp r^{{\theta }^{\prime }}(\ln r{)}^{k^{\prime }}$, as $r\to \infty$, \end{itemize} where the exponents $\theta ,k,{\theta }^{\prime },k^{\prime }$ 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.