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 (), this bound only depends on the degree of P and the number of bad primes. Similar issues are addressed for algebraic families of polynomials .
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa8176-12-2015,
author = {Pierre D\`ebes},
title = {Reduction and specialization of polynomials},
journal = {Acta Arithmetica},
volume = {172},
year = {2016},
pages = {175-197},
zbl = {06545346},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8176-12-2015}
}
Pierre Dèbes. Reduction and specialization of polynomials. Acta Arithmetica, Tome 172 (2016) pp. 175-197. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8176-12-2015/