Specializations of indecomposable polynomials

We address some questions concerning indecomposable polynomials and their behaviour under specialization. For instance we give a bound on a prime p for the reduction modulo p of an indecomposable polynomial \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${P(x)\in {\mathbb{Z}}[x]}$$\end{document} to remain indecomposable. We also obtain a Hilbert like result for indecomposability: if f(t1, . . . , tr, x) is an indecomposable polynomial in several variables with coefficients in a field of characteristic p = 0 or p > deg(f), then the one variable specialized polynomial \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f(t_1^\ast+\alpha_1^\ast x,\ldots,t_r^\ast+\alpha_r^\ast x,x)}$$\end{document} is indecomposable for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(t_1^\ast, \ldots, t_r^\ast, \alpha_1^\ast, \ldots,\alpha_r^\ast)\in \overline k^{2r}}$$\end{document} outside a proper Zariski closed subset.