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Reduction and specialization of polynomials

Pierre Dèbes (2016)

Acta Arithmetica

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...

Reduction of differential equations

Krystyna Skórnik, Joseph Wloka (2000)

Banach Center Publications

Let (F,D) be a differential field with the subfield of constants C (c ∈ C iff Dc=0). We consider linear differential equations (1) L y = D n y + a n - 1 D n - 1 y + . . . + a 0 y = 0 , where a 0 , . . . , a n ∈ F , and the solution y is in F or in some extension E of F (E ⊇ F). There always exists a (minimal, unique) extension E of F, where Ly=0 has a full system y 1 , . . . , y n of linearly independent (over C) solutions; it is called the Picard-Vessiot extension of F E = PV(F,Ly=0). The Galois group G(E|F) of an extension field E ⊇ F consists of all differential automorphisms of...

Reduction of semialgebraic constructible functions

Ludwig Bröcker (2005)

Annales Polonici Mathematici

Let R be a real closed field with a real valuation v. A ℤ-valued semialgebraic function on Rⁿ is called algebraic if it can be written as the sign of a symmetric bilinear form over R[X₁,. .., Xₙ]. We show that the reduction of such a function with respect to v is again algebraic on the residue field. This implies a corresponding result for limits of algebraic functions in definable families.

Relations de Fuchs pour les systèmes différentiels réguliers

Eduardo Corel (2001)

Bulletin de la Société Mathématique de France

Dans cet article, nous montrons que la notion analytique d’exposants développée par Levelt pour les systèmes différentiels linéaires en une singularité régulière s’interprète algébriquement en termes d’invariants de réseaux, relatifs à un réseau stable maximal que nous appelons « réseau de Levelt ». Nous obtenons en particulier un encadrement pour la somme des exposants des systèmes n’ayant que des singularités régulières sur ℙ 1 ( ℂ ).

Relative Bogomolov extensions

Robert Grizzard (2015)

Acta Arithmetica

A subfield K ⊆ ℚ̅ has the Bogomolov property if there exists a positive ε such that no non-torsion point of K × has absolute logarithmic height below ε. We define a relative extension L/K to be Bogomolov if this holds for points of L × ∖ K × . We construct various examples of extensions which are and are not Bogomolov. We prove a ramification criterion for this property, and use it to show that such extensions can always be constructed if some rational prime has bounded ramification index in K.

Relatively complete ordered fields without integer parts

Mojtaba Moniri, Jafar S. Eivazloo (2003)

Fundamenta Mathematicae

We prove a convenient equivalent criterion for monotone completeness of ordered fields of generalized power series [ [ F G ] ] with exponents in a totally ordered Abelian group G and coefficients in an ordered field F. This enables us to provide examples of such fields (monotone complete or otherwise) with or without integer parts, i.e. discrete subrings approximating each element within 1. We include a new and more straightforward proof that [ [ F G ] ] is always Scott complete. In contrast, the Puiseux series field...

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