# Reduction and specialization of polynomials

Acta Arithmetica (2016)

- Volume: 172, Issue: 2, page 175-197
- ISSN: 0065-1036

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topPierre 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 -

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