Specializations of indecomposable polynomials |
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Authors: | Arnaud Bodin Guillaume Chèze Pierre Dèbes |
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Institution: | 1. Laboratoire Paul Painlevé, Mathématiques, Université Lille 1, 59655, Villeneuve d’Ascq Cedex, France 2. Institut de Mathématiques de Toulouse, Université Paul Sabatier Toulouse 3, 31062, Toulouse Cedex 9, France
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Abstract: | 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 ${P(x)\in {\mathbb{Z}}x]}$ to remain indecomposable. We also obtain a Hilbert like result for indecomposability: if f(t 1, . . . , t r , 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 ${f(t_1^\ast+\alpha_1^\ast x,\ldots,t_r^\ast+\alpha_r^\ast x,x)}$ is indecomposable for all ${(t_1^\ast, \ldots, t_r^\ast, \alpha_1^\ast, \ldots,\alpha_r^\ast)\in \overline k^{2r}}$ outside a proper Zariski closed subset. |
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