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Reduction and specialization of polynomials

Pierre Dèbes — 2016

Acta Arithmetica

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 ∈ ℚ[T,Y] irreducible over ℚ̅ is introduced, which plays a central and unifying role. For such a polynomial P, we deduce a new bound for the least integer t₀ ≥ 0 such that P(t₀,Y) is irreducible...

Almost hilbertian fields

Pierre DèbesDan Haran — 1999

Acta Arithmetica

This paper is devoted to some variants of the Hilbert specialization property. For example, the RG-hilbertian property (for a field K), which arose in connection with the Inverse Galois Problem, requires that the specialization property holds solely for extensions of K(T) that are Galois and regular over K. We show that fields inductively obtained from a real hilbertian field by adjoining real pth roots (p odd prime) are RG-hilbertian; some of these fields are not hilbertian. There are other variants...

Galois Covers and the Hilbert-Grunwald Property

Pierre DèbesNour Ghazi — 2012

Annales de l’institut Fourier

Our main result combines three topics: it contains a Grunwald-Wang type conclusion, a version of Hilbert’s irreducibility theorem and a p -adic form à la Harbater, but with good reduction, of the Regular Inverse Galois Problem. As a consequence we obtain a statement that questions the RIGP over . The general strategy is to study and exploit the good reduction of certain twisted models of the covers and of the associated moduli spaces.

Familles de Hurwitz et cohomologie non abélienne

Pierre DèbesJean-Claude DouaiMichel Emsalem — 2000

Annales de l'institut Fourier

Nous nous intéressons à la question de l’existence de familles de Hurwitz au-dessus d’un espace de modules de revêtements de la droite. On sait que de telles familles existent dans le cas où les revêtements n’ont pas d’automorphismes. Dans le cas général, il y a une obstruction cohomologique, de nature non-abélienne. Nous donnons une double description de cette obstruction : la première en termes de gerbe, l’outil le mieux adapté à des situations cohomologiques non-abéliennes et la deuxièmes en...

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