Quantitative problems on the size of G-operators

@article{Lepetit2021QuantitativePO,
title={Quantitative problems on the size of G-operators},
author={Gabriel Lepetit},
journal={manuscripta mathematica},
year={2021}
}
$G$-operators, a class of differential operators containing the differential operators of minimal order annihilating Siegel's $G$-functions, satisfy a condition of moderate growth called Galochkin condition, encoded by a $p$-adic quantity, the size. Previous works of Chudnovsky, Andre and Dwork have provided inequalities between the size of a $G$ -operator and certain computable constants depending among others on its solutions. First, we recall Andre's idea to attach a notion of size to…
2 Citations
On the linear independence of values of G-functions
Abstract We consider a G-function F ( z ) = ∑ k = 0 ∞ A k z k ∈ K 〚 z 〛 , where K is a number field, of radius of convergence R and annihilated by the G-operator L ∈ K ( z ) [ d / d z ] , and a
A note on G-operators of order 2
• Mathematics
• 2021
It is known that G-functions solutions of a linear differential equation of order 1 with coefficients in Q(z) are algebraic (of a very precise form). No general result is known when the order is 2.

References

SHOWING 1-10 OF 19 REFERENCES
Linear independence of values of $G$-functions
• Mathematics
• 2017
Given any non-polynomial $G$-function $F(z)=\sum\_{k=0}^\infty A\_k z^k$ of radius of convergence $R$, we consider the $G$-functions $F\_n^{[s]}(z)=\sum\_{k=0}^\infty \frac{A\_k}{(k+n)^s}z^k$ for any
Séries Gevrey de type arithmétique, I. Théorèmes de pureté et de dualité
Gevrey series are ubiquitous in analysis; any series satisfying some (possibly non-linear) analytic differential equation is Gevrey of some rational order. The present work stems from two
S\\éries Gevrey de type arithm\\étique, II. Transcendance sans transcendance
In this second part, we study the Diophantine properties of values of arithmetic Gevrey series of non-zero order at algebraic points. We rely on the fact, proved in the first part, that the minimal
On the linear independence of values of G-functions
Abstract We consider a G-function F ( z ) = ∑ k = 0 ∞ A k z k ∈ K 〚 z 〛 , where K is a number field, of radius of convergence R and annihilated by the G-operator L ∈ K ( z ) [ d / d z ] , and a
Galois Theory of Linear Differential Equations
• Mathematics
• 2012
Linear differential equations form the central topic of this volume, Galois theory being the unifying theme. A large number of aspects are presented: algebraic theory especially differential Galois
ESTIMATES FROM BELOW OF POLYNOMIALS IN THE VALUES OF ANALYTIC FUNCTIONS OF A CERTAIN CLASS
Estimates from below are obtained for polynomials with integral coefficients in the values of certain Siegel -functions at the algebraic points of a special form. In particular, it is proved that if
MULTILINEAR ALGEBRA
Theorem 1.1. Suppose that V , W are finite dimensional vector spaces over a field F . Then there exists a vector space T over F , and a bilinear map φ : V ×W → T such that T satisfies the following
Finite Dimensional Vector Spaces
In three-dimensional analytic geometry, vectors are defined geometrically. The definition need not be recalled here. The important fact from the algebraic point of view is that a vector ν is
Über einige Anwendungen diophantischer Approximationen
Die bekannte einfache Schlusweise, das bei einer Verteilung von mehr als n Dingen auf n Facher in mindestens einem Fach mindestens zwei Dinge gelegen sind, enthalt eine Verallgemeinerung des