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

Let $f=(f_1,\dots ,f_m):{\mathbb{R}}^n⟶{\mathbb{R}}^m$ be a polynomial map; $G^f(r)=\{x\in {\mathbb{R}}^n:\mathrm{\mid }{f}_i(x)\mathrm{\mid }\le r,i=1,\dots ,m\}$. 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}$\left(G^f® \cap \overset{o}{\ \Z^n}\right) \asymp r^{\theta’}(\ln r)^{k’}$,as$r\to \infty, \end{itemize} where the exponents\theta,\ k,
\theta’,\ k’$aredeterminedexplicitlyintermsoftheNewtonpolyhedraof$f$. \ \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.