Reduction and specialization of polynomials

• Autorzy: Pierre Dèbes
• Języki publikacji: EN

Abstrakty

EN

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 $P(x₁,...,x_s,T,Y)$.

Kategorie tematyczne

• 12Fxx: Field extensions
• 12Yxx: Computational aspects of field theory and polynomials
• 12E25: Hilbertian fields; Hilbert's irreducibility theorem
• 14Gxx: Arithmetic problems. Diophantine geometry
• 12E05: Polynomials (irreducibility, etc.)
• 12E30: Field arithmetic
• 14E20: Coverings

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2016
• Tom: 172
• Numer: 2
• Strony: 175-197

Daty

wydano

2016

Twórcy

autor

Pierre Dèbes

• Laboratoire Paul Painlevé, Mathématiques, Université de Lille, 59655 Villeneuve d'Ascq Cedex, France

Identyfikatory

DOI

10.4064/aa8176-12-2015