The Radius of Convergence Function for First Order Differential Equations

We present an algorithm computing, for any first order differential equation L over the affine line and any (Berkovich) point t of this affine line, the p-adic radius of convergence RL(t) of the solutions of L near t. We do explicit computations for the equation (0.1) L(f) def = xf ′ − π(px + ax)f = 0 (πp−1 = −p). where a lies in some valued extension of Qp. For a = −1 and t = 0, a solution of L near t is the Dwork exponential exp(πxp − πx). Among other important properties, it appears that the function RL(t) is entirely determined by its values on a finite subtree of the affine line. The radius of convergence function has been shown to be a basic tool when studying p-adic differential equations. Notably, for first order differential equations, it gives the index of the underlying differential operator acting on various spaces of functions. However explicit computations are far from easy except in the few “trivial” cases where the “small radius theorem” and the logarithmic concavity are sufficient to conclude. In this paper we give an algorithm to compute the radius of convergence function for any first order p-adic differential equation defined on the affine line. It rests crucially on the proposition 2.15 that gives a criterion to decide whether the radius of convergence of a product is the smallest radius of convergence of the factors. As a by-product, we prove a Baldassarri conjecture for first order differential equations without singularities in the affine line (corollary 3.3). Roughly speaking, this conjecture asserts that the radius of convergence function is entirely determined by its values on a finite sub-tree of the whole “quasi-polyhedron” structure made up by Berkovich points. Likely it should mean that the radius of convergence function is “definable” in the sense of [10]. As the computation becomes quickly very tedious, we achieve it only for the differential equation (0.1) which is the simplest but non-trivial case. This example as been already treated in [1] but here we present it following the general algorithm. 2000 Mathematics Subject Classification. Primary 12H25.