We consider an n x n linear system of ODEs with an irregular singularity of Poincare rank 1 at z = infinity, holomorphically depending on parameter t within a polydisk in C-n centered at t = 0, such that the eigenvalu...
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We consider an n x n linear system of ODEs with an irregular singularity of Poincare rank 1 at z = infinity, holomorphically depending on parameter t within a polydisk in C-n centered at t = 0, such that the eigenvalues of the leading matrix at z = infinity coalesce along a locus Delta contained in the polydisk, passing through t = 0. Namely, z = infinity is a resonant irregular singularity for t is an element of Delta. We analyze the case when the leading matrix remains diagonalizable at Delta. We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon, and monodromy data as t varies in the polydisk, and their limits for t tending to points of Delta. When the system also has a Fuchsian singularity at z = 0, we show, under minimal vanishing conditions on the residue matrix at z = 0, that isomonodromic deformations can be extended to the whole polydisk (including Delta) in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisk. These data can be computed just by considering the system at the fixed coalescence point t = 0. Conversely, when the system is isomonodromic in a small domain not intersecting Delta inside the polydisk, we give certain vanishing conditions on some entries of the Stokes matrices, ensuring that Delta is not a branching locus for the t-continuation of fundamental matrix solutions. The importance of these results for the analytic theory of Frobenius manifolds is explained. An application to Painleve equations is discussed.
The notion of strict equivalence for order one differential equations of the form f (y', y, z) = 0 with coefficients in a finite extension K of C(z) is introduced. The equation gives rise to a curve X over K and a...
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The notion of strict equivalence for order one differential equations of the form f (y', y, z) = 0 with coefficients in a finite extension K of C(z) is introduced. The equation gives rise to a curve X over K and a derivation D on its function field K (X). Procedures are described for testing strict equivalence, strict equivalence to an autonomous equation, computing algebraic solutions and verifying the Painleve property. These procedures use known algorithms for isomorphisms of curves over an algebraically closed field of characteristic zero, the Risch algorithm and computation of algebraic solutions. The most involved cases concern curves X of genus 0 or 1. This paper complements work of M. Matsuda and of G. Muntingh & M. van der Put. (C) 2014 Elsevier Ltd. All rights reserved.
We study the Pade approximations of first and second kinds of families of so-called Lerch functions. They can be efficiently determined by using ideas of Riemann and Chudnovsky on the monodromy of rigid systems of dif...
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We study the Pade approximations of first and second kinds of families of so-called Lerch functions. They can be efficiently determined by using ideas of Riemann and Chudnovsky on the monodromy of rigid systems of differential equations. In addition, we give some applications to diophantine approximations. (C) 2012 Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG).
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