Reduction and specialization of polynomials

Pierre Dèbes. "Reduction and specialization of polynomials." Acta Arithmetica 172.2 (2016): 175-197. <http://eudml.org/doc/278906>.

@article{PierreDèbes2016,
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 ∈ 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 $P(x₁,...,x_s,T,Y)$.},
author = {Pierre Dèbes},
journal = {Acta Arithmetica},
keywords = {polynomials; reduction; specialization; Bertini-Noether theorem; Hilbert irreducibility theorem; Grothendieck good reduction criterion},
language = {eng},
number = {2},
pages = {175-197},
title = {Reduction and specialization of polynomials},
url = {http://eudml.org/doc/278906},
volume = {172},
year = {2016},
}

TY - JOUR
AU - Pierre Dèbes
TI - Reduction and specialization of polynomials
JO - Acta Arithmetica
PY - 2016
VL - 172
IS - 2
SP - 175
EP - 197
AB - 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 $P(x₁,...,x_s,T,Y)$.
LA - eng
KW - polynomials; reduction; specialization; Bertini-Noether theorem; Hilbert irreducibility theorem; Grothendieck good reduction criterion
UR - http://eudml.org/doc/278906
ER -