Specializations of indecomposable polynomials

Arnaud Bodin (1) , Guillaume Chèze (2) , Pierre Dèbes (1)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 IMT - Institut de Mathématiques de Toulouse UMR5219

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 $P(x)\in \Zz[x]$ to remain indecomposable. We also obtain a Hilbert like result for indecomposability: if $f(t_1,\ldots,t_r,x)$ 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 $(t_1^\ast, \ldots, t_r^\ast, \alpha_1^\ast, \ldots,\alpha_r^\ast)\in \overline k^{2r}$ off a proper Zariski closed subset.

Mots clés: indecomposable polynomials; irreducible polynomials

Domaines: Algèbre commutative [math.AC]; Géométrie algébrique [math.AG]; Théorie des nombres [math.NT]; Anneaux et algèbres [math.RA]

Arnaud Bodin, Guillaume Chèze, Pierre Dèbes. Specializations of indecomposable polynomials. Manuscripta mathematica, 2012, 139, pp.391-403. ⟨hal-00574896⟩