• Publications
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Spectral asymmetry and Riemannian geometry. II
In Part I of this paper (6) we proved various index theorems for manifolds with boundary including an extension of the Hirzebruch signature theorem. We now propose to investigate the geometric andExpand
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Galois Theory of Linear Differential Equations
Linear differential equations form the central topic of this volume, Galois theory being the unifying theme. A large number of aspects are presented: algebraic theory especially differential GaloisExpand
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Galois Theory of Difference Equations
Picard-Vessiot rings.- Algorithms for difference equations.- The inverse problem for difference equations.- The ring S of sequences.- An excursion in positive characteristic.- Difference modules overExpand
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Liouvillian First Integrals of Differential Equations
  • M. Singer
  • Mathematics, Computer Science
  • ISSAC
  • 4 July 1988
TLDR
We show that if a system of differential equations has a generic solution that satisfies a liouvillian relation, that is, there is a liOUvillian function of several variables vanishing on the curve defined by this solution, then the system has a nonconstant first integral that is constant on solution curves in some nonempty open set. Expand
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Galois Theory of Parameterized Differential Equations and Linear Differential Algebraic Groups
We present a Galois theory of parameterized linear differential equations where the Galois groups are linear differential algebraic groups, that is, groups of matrices whose entries are functions ofExpand
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Liouvillian Solutions of Linear Differential Equations with Liouvillian Coefficients
  • M. Singer
  • Computer Science, Mathematics
  • J. Symb. Comput.
  • 1 April 1991
TLDR
Let L(y) = b be a linear differential equation with coefficients in a differential field K. We discuss the problem of deciding if such an equation has a non-zero solution in K and give a procedure to find a basis for the space of solutions, liouvillian over K. Expand
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Liouvillian and Algebraic Solutions of Second and Third Order Linear Differential Equations
TLDR
We show that the index of a 1-reducible subgroup of the differential Galois group of an ordinary homogeneous linear differential equation L(y) =0 yields the best possible bound for the degree of the minimal polynomial of an algebraic solution of the Riccati equation associated to L( y) = 0. Expand
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Elementary first integrals of differential equations
TLDR
It is not always possible and sometimes not even advantageous to write the solutions of a system of differential equations explicitly in terms of elementary functions. Expand
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Extremal metrics on blowups
In this paper we provide conditions that are sufficient to guarantee the existence of extremal metrics on blowups at finitely many points of Kahler manifolds which already carry an extremal metric.Expand
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