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Zbl 1010.12007
Nagata, Makoto
A generalization of the sizes of differential equations and its applications to $G$-function theory. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 30, No.2, 465-497 (2001). ISSN 0391-173X

Let $K$ be an algebraic number field. If we consider a differential system $$ \frac{dy}{dx}=Ay,\quad A\in M_n(K(x)),\tag 1 $$ the size $\sigma (A)$ measures properties (heights) of $A$, together with those of coefficients of the systems obtained from (1) by differentiation, with respect to all finite places of $K$ [see {\it B. Dwork, G. Gerotto} and {\it F. J. Sullivan}, An introduction to $G$-functions. Annals of Mathematics Studies 133. Princeton (1994; Zbl 0830.12004)]. A related notion of the size $\sigma (y)$ is used also for a matrix solution of (1). \par The author studies a more general equation $$ \frac{dz}{dx}=A_1z-zB\tag 2 $$ (a solution $z$ is a matrix function). A notion of the size $\sigma (A_1,B)$ coinciding with $\sigma (A_1)$ for $B=0$ is introduced, and an estimate for $\sigma (A_1,B)$ via $\sigma (z)$ is obtained. As an application, the author considers a transform [introduced by {\it Y. André}, $G$-functions and geometry. Aspects of Mathematics 13. Vieweg (1989; Zbl 0688.10032)] reducing (1) to a system (2), in which $A_1$ has a simple pole at $x=0$, and $B=x^{-1}\operatorname{Res}(A_1)$. This leads to some information about solutions of the initial equation (1).
[A.N.Kochubei (Ky\" iv)]

MSC 2000:: *12H25 p-adic differential equations
11R47 Other analytic theory

Keywords: size of a matrix; size of a solution; $G$-function; height

Citations: Zbl 0830.12004; Zbl 0688.10032

Cited in: Zbl 1016.12500

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