Dilation of Newton Polytope and p-adic Estimate

Let f(X) be a polynomial in n variables over the finite field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{F}_{q}$\end{document}. Its Newton polytope Δ(f) is the convex closure in ℝn of the origin and the exponent vectors (viewed as points in ℝn) of monomials in f(X). The minimal dilation of Δ(f) such that it contains at least one lattice point of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{Z}_{>0}^{n}$\end{document} plays a vital pole in the p-adic estimate of the number of zeros of f(X) in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{F}_{q}$\end{document}. Using this fact, we obtain several tight and computational bounds for the dilation which unify and improve a number of previous results in this direction.