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We consider a finite set of power series in one variable with coefficients in a field of characteristic zero having a chosen nonarchimedean valuation. We study the growth of these series near the boundary of their common "open" disk of convergence. Our results are definitive when the wronskian is bounded. The main application involves local solutions of ordinary linear differential equations with analytic coefficients. The effective determination of the common radius of convergence remains open (and is not treated here). Let K be an algebraically closed field of characteristic zero complete under a nonarchimedean valuation with residue class field of characteristic p. Let D = d/dx L = D n + CnID n+** +Co (1) be a linear differential operator with coefficients meromorphic in some neighborhood of the origin. Let u = ao + ax +... (2) be a power series solution of L which converges in an open (p-adic) disk of radius r. Our object is to describe the asymptotic behavior of lairs as s o-* . In a series of articles we have shown that subject to certain restrictions we may conclude that rsi asi = 0(s n-I 1).3 The object of the present article is to make this estimate effective. At the same time we greatly simplify, and generalize, our best previous results [12] for the noneffective form. Our previous work was based on the notion of a generic disk together with a condition for reducibility of differential operators with unbounded solutions [4, Theorem 4]. Thus we transferred our given differential equation to a generic disk, reduced the problem there to the case of first order differential equations and then transferred the information back. The present method eliminates the notion of generic disks and avoids the question of reducibility. We also avoid direct hypotheses on the coefficients of the differential equation. We impose conditions only on the solutions and on the wronskian. The main tools are the Frobenius formulation of the basis of the solution space (equation (3.7) below) and the boundary seminorm [10]. Received by the editors March 1, 1979 and, in revised form, April 27, 1979. AMS (MOS) subject classifications (1970). Primary 12B40. ( 1980 American Mathematical Society 0002-9947/80/0000-0264/$05.75

Effective 𝑝-adic bounds for solutions of homogeneous linear differential equations