Ce texte donne une nouvelle présentation, et une généralisation, des groupes qui apparaissent dans les travaux de Rhin-Viola ([8], [9]) sur les mesures d’irrationalité de $\zeta$(2) et $\zeta$(3). D’une part, on interprète ces groupes comme des groupes d’automorphismes, ce qui permet de déduire chacune des relations entre intégrales utilisées par Rhin-Viola d’un changement de variables. D’autre part, on considère plusieurs familles d’intégrales $n$-uples, et on montre que chacune d’elles est munie d’une action...

A result on the orders of the reductions of an element of the group of S-units of a number field is obtained by investigating three height functions for groups of S-units of number fields which are analogous to local, global and canonical height functions for elliptic curves.

We present bounds for the degree and the height of the polynomials arising in some problems in effective algebraic geometry including the implicitization of rational maps and the effective Nullstellensatz over a variety. Our treatment is based on arithmetic intersection theory in products of projective spaces and extends to the arithmetic setting constructions and results due to Jelonek. A key role is played by the notion of canonical mixed height of a multiprojective variety. We study this notion...

We prove inequalities that compare the size of an S-regulator with a product of heights of multiplicatively independent S-units. Our upper bound for the S-regulator follows from a general upper bound for the determinant of a real matrix proved by Schinzel. The lower bound for the S-regulator follows from Minkowski's theorem on successive minima and a volume formula proved by Meyer and Pajor. We establish similar upper bounds for the relative regulator of an extension l/k of number fields.

Let K be a number field, X be a smooth projective curve over it and D be a reduced divisor on X. Let (E,∇) be a vector bundle with connection having meromorphic singularities on D. Let $p_1,...,p_s\in X\left(K\right)$ and $X^o:=X̅\setminus D,p_1,...,p_s$ (the $p_j$’s may be in the support of D). Using tools from Nevanlinna theory and formal geometry, we give the definition of E-section of arithmetic type of the vector bundle E with respect to the points $p_j$; this is the natural generalization of the notion of E-function defined in Siegel-Shidlovskiĭ theory. We prove...

Many new types of Hurwitz continued fractions have been studied by the author. In this paper we show that all of these closed forms can be expressed by using confluent hypergeometric functions ${}_0{F}_1(;c;z)$. In the application we study some new Hurwitz continued fractions whose closed form can be expressed by using confluent hypergeometric functions.

The purpose of this article is twofold. The first is to find the dimension of the set of integral points off divisors in subgeneral position in a projective algebraic variety $V\subset {\mathbb{P}}_{k̅}^m$, where k is a number field. As consequences, the results of Ru-Wong (1991), Ru (1993), Noguchi-Winkelmann (2003) and Levin (2008) are recovered. The second is to show the complete hyperbolicity of the complement of divisors in subgeneral position in a projective algebraic variety $V\subset {\mathbb{P}}_{ℂ}^m.$

Soient $A,B,C=A+B$ trois éléments de l’ensemble ${\mathbb{N}}^{*}$ des entiers > $0$ (resp. $ℂ\left[X\right]$) des polynômes complexes) premiers entre eux ; on note $r\left(ABC\right)$ le produit des facteurs premiers (resp. le nombre des facteurs premiers dans $ℂ\left[X\right]$) du produit $ABC$. La conjecture $\left(abc\right)$ énonce que, pour tout $ϵ\>0$, il existe $C_{ϵ}\>0$ pour lequel l’inégalité : $r\left(ABC\right)\ge C_{ϵ}{S}^{1-ϵ}$ avec $S=$ max$(A,B,C)$) est toujours vérifiée. Le théorème de Mason établit l’inégalité, $D$ (supposé > $0$) désignant le plus grand des degrés des polynômes $A,B,C:r\left(ABC\right)\ge D+1$. Les cas de triplets de polynômes où l’égalité...