Generic radius of convergence of differential equations with polynomial coefficients over a number field
Let \( k \) be a number field, let \( S \) be the set of finite places of \( k \) and let \( L \) be a ``differential operator'', namely some element of \( k(x)[{d\over dx}]\). For each \( \nu \) in \( S \) and \( r \) in \( ]0,\infty[\), one can define the ``generic radius of convergence'' \( R_{\nu}(L,r) \) of (the solutions of) \( L \) in the ``generic disk'' of radius \( r\). When \( \nu \) is fixed, the function \( r\mapsto R_{\nu}(L,r) \) is known to be piecewise logarithmically affine. The main technical result proved in this paper is that, for a given \( L\), the set of all (logarithmic) slopes of the functions \( R_{\nu}(L,r) \) for all places \( \nu \) is finite. As an application the authors show equivalence between ``Bombieri'' and ``local Bombieri'' characterizations of G-functions and that both imply the ``local Galochkin'' one. These equivalences were already known [see \textit{Y. André}, G-functions and geometry. Aspects of Mathematics, 13. Wiesbaden: Vieweg (1989; Zbl 0688.10032)]. Actually they make up a part of G-operator theory (namely the simplest one). However the proof given here is original, even if it could likely be reduced to André's one when deeply examined.
scientific article; zbMATH DE number 1805815
scientific article