Let $(K,\nu )$ be a valued field, where $\nu$ is a rank one discrete valuation. Let $R$ be its ring of valuation, $𝔪$ its maximal ideal, and $L$ an extension of $K$, defined by a monic irreducible polynomial $F\left(X\right)\in R\left[X\right]$. Assume that $\overline{F}\left(X\right)$ factors as a product of $r$ distinct powers of monic irreducible polynomials. In this paper a condition which guarantees the existence of exactly $r$ distinct valuations of $K$ extending $\nu$ is given, in such a way that it generalizes the results given in the paper “Prolongations of valuations to finite...

We consider function field analogues of the conjecture of Győry, Sárközy and Stewart (1996) on the greatest prime divisor of the product $(ab+1)(ac+1)(bc+1)$ for distinct positive integers $a$, $b$ and $c$. In particular, we show that, under some natural conditions on rational functions $F,G,H\in ℂ\left(X\right)$, the number of distinct zeros and poles of the shifted products $FH+1$ and $GH+1$ grows linearly with $degH$ if $degH\ge max\{degF,degG\}$. We also obtain a version of this result for rational functions over a finite field.

1. Let R be a domain and f ∈ R[X] a polynomial. A k-tuple $x₀,x₁,...,x_{k-1}$ of distinct elements of R is called a cycle of f if $f\left(x_i\right)=x_{i+1}$ for i=0,1,...,k-2 and $f\left(x_{k-1}\right)=x₀$. 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...

Let $K$ be a number field defined by an irreducible polynomial $F\left(X\right)\in \mathbb{Z}\left[X\right]$ and ${\mathbb{Z}}_K$ its ring of integers. For every prime integer $p$, we give sufficient and necessary conditions on $F\left(X\right)$ that guarantee the existence of exactly $r$ prime ideals of ${\mathbb{Z}}_K$ lying above $p$, where $\overline{F}\left(X\right)$ factors into powers of $r$ monic irreducible polynomials in ${𝔽}_p\left[X\right]$. The given result presents a weaker condition than that given by S. K. Khanduja and M. Kumar (2010), which guarantees the existence of exactly $r$ prime ideals of ${\mathbb{Z}}_K$ lying above $p$. We further specify...

A subfield K ⊆ ℚ̅ has the Bogomolov property if there exists a positive ε such that no non-torsion point of $K^{\times }$ has absolute logarithmic height below ε. We define a relative extension L/K to be Bogomolov if this holds for points of $L^{\times }\setminus K^{\times }$. 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.

On cherche à donner une méthode effective de calcul de la constante d’Eisenstein [3] d’une fonction algébrique. On commence en précisant les liens entre cette constante et les rayons de convergence $p$-adiques de la fonction pour les différents nombres premiers $p$. Puis on donne une démonstration entièrement effective du résultat bien connu liant fonctions algébriques et diagonales de fractions rationnelles. Enfin on explique comment en déduire une méthode de calcul générale. On illustre la méthode...