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Reduction and specialization of polynomials

Volume 172 / 2016

Pierre Dèbes Acta Arithmetica 172 (2016), 175-197 MSC: Primary 12E05, 12E25, 14E20, 12Yxx; Secondary 12E30, 12Fxx, 14Gxx. DOI: 10.4064/aa8176-12-2015 Published online: 3 December 2015

Abstract

We show explicit forms of the Bertini–Noether reduction theorem and of the Hilbert irreducibility theorem. Our approach recasts in a polynomial context the geometric Grothendieck good reduction criterion and the congruence approach to HIT for covers of the line. A notion of ‶bad primes″ of a polynomial $P\in \mathbb Q[T,Y]$ irreducible over $\overline{\mathbb{Q}}$ is introduced, which plays a central and unifying role. For such a polynomial $P$, we deduce a new bound for the least integer $t_0\geq 0$ such that $P(t_0,Y)$ is irreducible in $\mathbb{Q}[Y]$: in the generic case for which the Galois group of $P$ over $\overline{\mathbb{Q}}(T)$ is $S_n$ ($n=\deg_Y(P)$), this bound only depends on the degree of $P$ and the number of bad primes. Similar issues are addressed for algebraic families of polynomials $P(x_1,\ldots,x_s,T,Y)$.

Authors

  • Pierre DèbesLaboratoire Paul Painlevé
    Mathématiques
    Université de Lille
    59655 Villeneuve d'Ascq Cedex, France
    e-mail

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