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16
Linear meromorphic differential equations: a modern point of view
 Bulletin (New Series) of the American Mathematical Society, 33(1):1 – 42
, 1996
"... Abstract. A large part of the modern theory of differential equations in the complex domain is concerned with regular singularities and holonomic systems. However the theory of differential equations with irregular singularities has a long history and has become very active in recent years. Substant ..."
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Cited by 16 (1 self)
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Abstract. A large part of the modern theory of differential equations in the complex domain is concerned with regular singularities and holonomic systems. However the theory of differential equations with irregular singularities has a long history and has become very active in recent years. Substantial links of this theory to the theory of algebraic groups, commutative algebra, resurgent functions, and Galois differential methods have been discovered. This survey attempts a general introduction to some of these aspects, with emphasis on reduction theory, asymptotic analysis, Stokes phenomena, and certain moduli problems. 1.
Moduli of linear differential equations on the Riemann sphere with fixed Galois groups
 Pac. J. of Math
, 1993
"... For fixed m and n, we consider the vector space of linear differential equations of order n whose coefficients are polynomials of degree at most m. We show that for G in a large class of linear algebraic groups, if we fix the exponents and determining factors at the singular points (but not the sing ..."
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Cited by 12 (8 self)
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For fixed m and n, we consider the vector space of linear differential equations of order n whose coefficients are polynomials of degree at most m. We show that for G in a large class of linear algebraic groups, if we fix the exponents and determining factors at the singular points (but not the singular points themselves) then the set of such differential equations with this fixed data, fixed Galois group G and fixed (/module for the solution space forms a constructible set (i.e., an element of the Boolean algebra generated by the Zariski closed sets). Our class of groups includes finite groups, connected groups, and groups whose connected component of the identity is semisimple or unipotent. We give an example of a group for which this result is false and also apply this result to the inverse problem in differential Galois theory. 1. Introduction. In this paper we consider the set J?(n, m) of homogeneous linear differential equations
Connected Linear Groups as Differential Galois Groups
 Journal of Algebra
, 1995
"... this paper we give a proof of the following ..."
The Inverse Problem in Differential Galois Theory
, 1996
"... this paper will be assumed to be of characteristic zero. the subscript k and refer to C). A linear differential equation is an equation ..."
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Cited by 9 (1 self)
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this paper will be assumed to be of characteristic zero. the subscript k and refer to C). A linear differential equation is an equation
Galois groups of basic hypergeometric equations
, 2007
"... Abstract. In this paper we compute the Galois groups of basic hypergeometric equations. In this paper q is a complex number such that 0 < q  < 1. 1 Basic hypergeometric series and equations The theory of hypergeometric functions and equations dates back at least as far as Gauss. It has long ..."
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Cited by 5 (3 self)
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Abstract. In this paper we compute the Galois groups of basic hypergeometric equations. In this paper q is a complex number such that 0 < q  < 1. 1 Basic hypergeometric series and equations The theory of hypergeometric functions and equations dates back at least as far as Gauss. It has long been and is still an integral part of the mathematical literature. In particular, the Galois theory of (generalized) hypergeometric equations attracted the attention of many authors. For this issue, we refer the reader to [2, 3, 13] and to the references therein. We also single out the papers [8, 14], devoted to the calculation of some Galois groups by means of a density theorem (Ramis theorem). In this paper we focus our attention on the Galois theory of the basic hypergeometric equations, the later being natural qanalogues of the hypergeometric equations. The basic hypergeometric series φ(z) = 2φ1 (a,b;c;z) with parameters (a,b,c) ∈ (C ∗ ) 3 defined by:
On Ramis's solution of the local inverse problem of differential Galois theory
 J. Pure Appl. Alg. 110 Ž 1996
, 1994
"... : Recently, J.P. Ramis gave necessary and sufficient conditions for a linear algebraic group to be the Galois group of a PicardVessiot extension of the field Cfxg[x \Gamma1 ] of germs of meromorphic functions at zero. The conditions of Ramis are stated in terms of the Lie algebra of the group. In ..."
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Cited by 2 (2 self)
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: Recently, J.P. Ramis gave necessary and sufficient conditions for a linear algebraic group to be the Galois group of a PicardVessiot extension of the field Cfxg[x \Gamma1 ] of germs of meromorphic functions at zero. The conditions of Ramis are stated in terms of the Lie algebra of the group. In this paper, we give equivalent simple group theoretic conditions, and show how these generalize previous conditions of Kovacic in the solvable case. 1 Introduction The general inverse problem in differential Galois theory can be stated as follows: Let k denote a differential field of characteristic 0 and C the subfield of constants of k, which we assume to be algebraically closed. Characterize those linear algebraic groups G that are Galois groups of PicardVessiot extensions of k. An early contribution to this problem is due to BialynickiBirula [1] who showed that if the transcendence degree of k over C is finite and nonzero then any connected nilpotent group is a Galois group over k. ...
Inverse Problem for Semisimple Frobenius Manifolds Monodromy Data and the Painlevé VI Equation
, 2000
"... This work is a part the Ph.D. thesis of Davide Guzzetti, with the supervision of professor B. Dubrovin. We study the inverse problem for semisimple Frobenius manifolds of dimension three. We explicitly compute a parametric form of the solutions of the WDVV equations of associativity in terms of solu ..."
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This work is a part the Ph.D. thesis of Davide Guzzetti, with the supervision of professor B. Dubrovin. We study the inverse problem for semisimple Frobenius manifolds of dimension three. We explicitly compute a parametric form of the solutions of the WDVV equations of associativity in terms of solutions of a special Painlevé VI equation and we show that the solutions are labelled by a set of monodromy data. The procedure is a relevant application of the theory of isomonodromic deformations. We use the parametric form to construct polynomial and algebraic solutions of the WDVV equations. We also apply the parametric form to construct the generating function of GromovWitten invariants corresponding to the Frobenius manifold called quantum cohomology of the two dimensional projective space. As a necessary step, we give a contribution to the analysis of the Painlevé VI equation. We find a class of solutions that covers almost all the values of the monodromy data associated to the equation, except one point in the space of the data. We describe the asymptotic behavior close to the critical points in terms of two parameters and we find the relation among the parameters at the different critical points (connection problem). In this way we unify and extend preexisting results. In particular, we This work is devoted to the construction of solutions of the WDVV equations of associativity in 2D topological field theory as an application of the theory of Frobenius manifolds, isomonodromic deformations and Painlevé equations. It is a part of my Ph.D. thesis in mathematical physics submitted at
ON A GENERIC INVERSE DIFFERENTIAL GALOIS PROBLEM FOR GLn
, 2001
"... Abstract. In this paper we construct a generic PicardVessiot extension for the general linear groups. In the case when the differential base field has finite transcendence degree over its field of constants we provide necessary and sufficient conditions for solving the inverse differential Galois p ..."
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Abstract. In this paper we construct a generic PicardVessiot extension for the general linear groups. In the case when the differential base field has finite transcendence degree over its field of constants we provide necessary and sufficient conditions for solving the inverse differential Galois problem for this groups via specialization from our generic extension.
On Ramis’s solution of the local inverse problem of differential Galois theory
, 1994
"... Recently, Ramis gave necessary and sufficient conditions for a linear algebraic group to be the Galois group of a PicardVessiot extension of the field @{x}[x‘1 of germs of meromorphic functions at zero. In this paper, we give equivalent simple group theoretic conditions, and show how these general ..."
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Recently, Ramis gave necessary and sufficient conditions for a linear algebraic group to be the Galois group of a PicardVessiot extension of the field @{x}[x‘1 of germs of meromorphic functions at zero. In this paper, we give equivalent simple group theoretic conditions, and show how these generalize previous conditions of Kovacic in the solvable case. 1.