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Pólya fields and Pólya numbers

Amandine Leriche (2010)

Actes des rencontres du CIRM

A number field K , with ring of integers 𝒪 K , is said to be a Pólya field if the 𝒪 K -algebra formed by the integer-valued polynomials on 𝒪 K admits a regular basis. In a first part, we focus on fields with degree less than six which are Pólya fields. It is known that a field K is a Pólya field if certain characteristic ideals are principal. Analogously to the classical embedding problem, we consider the embedding of K in a Pólya field. We give a positive answer to this embedding problem by showing that...

Pólya fields, Pólya groups and Pólya extensions: a question of capitulation

Amandine Leriche (2011)

Journal de Théorie des Nombres de Bordeaux

A number field K , with ring of integers 𝒪 K , is said to be a Pólya field when the 𝒪 K -algebra formed by the integer-valued polynomials on 𝒪 K admits a regular basis. It is known that such fields are characterized by the fact that some characteristic ideals are principal. Analogously to the classical embedding problem in a number field with class number one, when K is not a Pólya field, we are interested in the embedding of K in a Pólya field. We study here two notions which can be considered as measures...

Poly-Bernoulli numbers

Masanobu Kaneko (1997)

Journal de théorie des nombres de Bordeaux

By using polylogarithm series, we define “poly-Bernoulli numbers” which generalize classical Bernoulli numbers. We derive an explicit formula and a duality theorem for these numbers, together with a von Staudt-type theorem for di-Bernoulli numbers and another proof of a theorem of Vandiver.

Polynômes à groupe de Galois diédral

Dominique Martinais, Leila Schneps (1992)

Journal de théorie des nombres de Bordeaux

Soit K un corps et K 1 une extension quadratique de K . Étant donné un polynôme P de K 1 [ X ] à groupe de Galois cyclique, nous donnons une méthode pour construire un polynôme Q de K [ X ] à groupe de Galois diédral, à partir des racines de P . Cette méthode est tout à fait explicite : nous donnons de nombreux exemples de polynômes à groupe de Galois diédral sur le corps ℚ .

Polynômes de Lagrange sur les entiers d'un corps quadratique imaginaire

M. Ably, M. M'Zari (1998)

Journal de théorie des nombres de Bordeaux

L'objet de ce texte est de donner une estimation arithmétique des valeurs prises par les polynômes de Lagrange sur les entiers d'un corps quadratique imaginaire en des points de ce corps. Ces polynômes interviennent dans l'étude des fonctions entières arithmétiques et dans les minorations de formes linéaires de Logarithmes.

Polynomial Automorphisms Over Finite Fields

Maubach, Stefan (2001)

Serdica Mathematical Journal

It is shown that the invertible polynomial maps over a finite field Fq , if looked at as bijections Fn,q −→ Fn,q , give all possible bijections in the case q = 2, or q = p^r where p > 2. In the case q = 2^r where r > 1 it is shown that the tame subgroup of the invertible polynomial maps gives only the even bijections, i.e. only half the bijections. As a consequence it is shown that a set S ⊂ Fn,q can be a zero set of a coordinate if and only if #S = q^(n−1).

Polynomial cycles in certain local domains

T. Pezda (1994)

Acta Arithmetica

1. Let R be a domain and f ∈ R[X] a polynomial. A k-tuple x ₀ , x ₁ , . . . , x k - 1 of distinct elements of R is called a cycle of f if f ( x i ) = x i + 1 for i=0,1,...,k-2 and f ( x k - 1 ) = x ₀ . The number k is called the length of the cycle. A tuple is a cycle in R if it is a cycle for some f ∈ R[X]. It has been shown in [1] that if R is the ring of all algebraic integers in a finite extension K of the rationals, then the possible lengths of cycles of R-polynomials are bounded by the number 7 7 · 2 N , depending only on the degree N of K. In this note we consider...

Polynomial cycles in certain rings of rationals

Władysław Narkiewicz (2002)

Journal de théorie des nombres de Bordeaux

It is shown that the methods established in [HKN3] can be effectively used to study polynomial cycles in certain rings. We shall consider the rings 𝐙 [ 1 N ] and shall describe polynomial cycles in the case when N is either odd or twice a prime.

Currently displaying 301 – 320 of 694