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 in ℚ[Y]: in the generic case for which the Galois group of P over ℚ̅(T) is Sₙ ($n=deg_Y(P)$), this bound only depends on the degree of P and the number of bad primes. Similar issues are addressed for algebraic families of polynomials $P(x₁,...,x_s,T,Y)$.
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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 of interest: the R-hilbertian property is obtained from the RG-hilbertian property by dropping the condition "Galois", the mordellian property is that every non-trivial extension of K(T) has infinitely many non-trivial specializations, etc. We investigate the connections existing between these properties. In the case of PAC fields we obtain pure Galois-theoretic characterizations. We use them to show that "mordellian" does not imply "hilbertian" and that every PAC R-hilbertian field is hilbertian.
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