Books written or edited by Michael F. Singer
 Differential Equations and Computer Algebra, M.F.
Singer, editor, Academic Press, 1991. (This is a collection of
papers from the Computer Algebra and Differential Equations
Conference held in Ithaca in May 1990)
 Galois Theory of Difference Equations M. van der Put
and M. F. Singer, V. 1666 in Lecture Notes in Mathematics,
SpringerVerlag, 1997.
Errata (.ps file)
 Differential Galois Theory M. van der Put and M.
F. Singer, (.ps file)
(manuscript, 433 pages), 2001.
Revised edition July 2002 Published as Galois Theory
of Linear Differential Equations Grundlehren der
mathematischen Wissenschaften, Volume 328, Springer,
2003.
 Effective Methods in Algebraic Geometry, M.F.
Singer (editor). Selected papers from the conference
Methodes Effectives en Geometrie Algebriques (MEGA2000),
Journal of Pure and Applied Algebra, Volume 164,
Issues 12, October 2001
Papers by Michael F. Singer
Preprints
 Walks on the Quarter Plane,
Genus Zero Case (with T. Dreyfus, C. Hardouin, J. Roques),
Preprint, 2017
 Consistent
systems of linear differential and difference equations (with
R. Schaefke), Journal of the European Mathematical
Society, to appear, 2016
 Mahler equations and
rationality (with R. Schaefke), Preprint, 2016. This
contains another proof of a result of Adamczewski and Bell
concerning Mahler equations: A formal power series satisfying a
p− and a q−Mahler equation over C(x) with multiplicatively
independent positive integers p and q is a rational function.
The proof presented here is selfcontained and is essentially a
compilation of proofs contained in the preprint
"Consistent systems of linear differential and difference
equations".
2018
 On the Nature of the
Generating Series of Walks in the Quarter Plane (with T.
Dreyfus, C. Hardouin, J. Roques), Inventiones
mathematicae, Vol. 213, Issue 1, 2018, pp.205236.
2017
 Galois groups for integrable
and projectively integrable linear differential equations
(with C. Arreche), Journal of Algebra, Vol.
480, 423–449, 15 June 2017,
dx.doi.org/10.1016/j.jalgebra.2017.02.032
2016
 Algebraic and Algorithmic Aspects
of Linear Difference Equations, Galois Theories of
Linear Difference Equations: An Introduction (C. Hardouin,
J. Sauloy, M. F. Singer), Mathematical Surveys and Monographs,
Volume 211, American Mathematical Society, 2016, pp.1253.
 Desingularization of Ore Operators
(with S. Chen and M. Kauers), Journal of
Symbolic Computation, Vol. 74, 617626, MayJune
2016.
2015
 Reductive linear differential
algebraic groups and the Galois groups of parameterized linear
differential equations (with A. Minchenko and A. Ovchinnikov),
International Mathematics Research Notices, Vol.
215, Issue 7, 17331793, 2015.
2014
 Unipotent differential
algebraic groups as parameterized differential Galois groups
(with A. Minchenko and A. Ovchinnikov), Journal of the
Institute of Mathematics of Jussieu, volume 13, issue 04,
671700.
 Parallel Telescoping and
Parameterized Picard–Vessiot Theory (with S. Chen, R. Feng and
Z. Li), Proceedings of ISSAC 2014, pp. 99–106,
ACM Press, 2014.
 On the Summability of
Bivariate Rational Functions (with S. Chen), Journal
of Algebra, 409, (2014), 320343.
2013
 Desingularization Explains
OrderDegree Curves for Ore Operators (with S. Chen, M.
Jaroschek and M. Kauers), Procedings of ISSAC 2013,
( M. Kauers, ed.), 157164, 2013.
 Projective Isomonodromy and
Galois Groups (with C. Mitschi), Proceedings of the American Mathematical Society,
141 (2013), no. 2,
605617.
 Linear Algebraic Groups as
Parameterized PicardVessiot Galois Groups, Journal of Algebra, 373, (2013), 151161, Old version, 2011.
2012
 Monodromy Groups of
Parameterized Linear Differential Equations with Regular
Singularities (with C. Mitschi), Bull. London Math. Soc., 44(5), 2012, 913930.
 Residues and Telescopers for
Bivariate Rational Functions (with S. Chen), Advances in Applied Mathematics,
49 (2012)
111–133.
 Telescopers for Raltional and
Algebraic Functions via Residues (with S. Chen and M. Kauers),
Procedings of ISSAC 2012,
(J. van der Hoeven and M. van Hoeij, eds.), 130137, 2012.
2011
 A JordanHoelder Thoerem
for Differentail Algebraic Groups (with P. Cassidy) Journal
of Algebra, 328,
2011, 190–217.
2010
 Liouvillian solutions
of differencedifferential equations (with R. Feng, M. Wu)
Journal of Symbolic
Computation, 45,
2010, 287305.
 An algorithm to
compute liouvillian solutions of prime order
differencedifferential equations (with R. Feng, M. Wu) Journal of Symbolic Computation, 45, 2010, 306323.
2009
 Introduction to the Galois
Theory of Linear Differential Equations Algebraic Theory of Differential
Equations, M.A.H. MacCallum and A.V. Mikhalov, eds.,
London Mathematical Society Lecture Note Series (no. 357),
Cambridge University Press, 2009, 182.
2008
 Differential
Galois Theory of Linear Difference Equations (with C.
Hardouin) Mathematische
Annalen, 342(2)
2008, 333377 Erratum Some of
the calculations referred to in this paper are contained in a
Maple Worksheet entitled Differential
independence of a class of qhypergeometric difference
equations (a pdf version of this may be found here).
 On the
Definitions of Difference Galois Groups (with Z. Chatzidakis,
C. Hardouin) Model
Theory with applications to algebra and analysis, I and II,
(Z. Chatzidakis, H.D. Macpherson, A. Pillay, A.J. Wilkie
editors), Cambridge University Press, Cambridge.(2008), 73109.
2007
 Model Theory of
Differential Fields: From Commuting to Noncommuting
Derivations Proceedings
of the AMS., 135 (2007), 19291934.
2006
 Galois Theory of
Parameterized Differential Equations and Linear Differential
Algebraic Groups (with P.J. Cassidy) Differential Equations and Quantum
Groups (IRMA Lectures in Mathematics and Theoretical
Physics Vol. 9), ed. D. Bertrand, B. Enriquez, C. Mitschi, C.
Sabbah, R. Schaefke, EMS Publishing house pp. 113 157
(2006).
 A Recursive Method for
Determining the OneDimensional Submodules of LaurentOre
Modules (with Z. Li, M. Wu, D. Zheng). 2006. Proceedings of ISSAC 2006, pp.
200208.
2005
 On the Constructive
Inverse Problem in Differential Galois Theory, (with W. Cook
and C. Mitschi) (.pdf file) , Comm. in Algebra ,
33/10, 2005, pp. 36393665. An older version is
available here. Related software
is available here.
2002
 SolvablebyFinite Groups as
Differential Galois Groups (with C. Mitschi). Ann.
Fac. Sci. Toulouse Math. (6) 11/3 (2002),
403423
 Linear Differential Operators for
Polynomial Equations (with O. Cormier, B.M. Trager and F.
Ulmer) (.pdf file) Journal of Symbolic Computation,
34 , 2002, pp.355398.
2000
 Computing the Galois Group of
a Polynomial Using Linear Differential Equations (with O.
Cormier, F. Ulmer) (.ps file) Proceedings of ISSAC
2000., 7885
1999
 Computing Galois
Groups of Completely Reducible Differential Equations (with E.
Compoint) (.ps file). Journal of Symbolic Computation,
28/45, 1999, 473494. Also available as a .dvi file
 Calculating the Galois
group of L_1(L_2(y))=0, L_1, L_2 Completely Reducible
Operators (with P. Berman) (.pdf file) and Abstract. Journal of
Pure and Applied Algebra, 139/13, 1999, 324.
 Solving Difference Equations in
Finite Terms (with P. Hendriks) . Journal of
Symbolic Computation, 27/3, 1999, 239259.
 Direct and Inverse Problems in
Differential Galois Theory (.ps file). Selected
Works of Ellis Kolchin with Commentary , Bass, Buium,
Cassidy, eds., American Mathematical Society, 1999, 527554.
Also available as a .dvi file
1998
 Relations Lineaires entre
Solutions d'une Equation Differentielle (with E. Compoint)
(.ps file). Annales des Fac. des Science de Toulouse,
Vol. VII, No. 4, 1998, 659670. Also available as a .dvi file
1997
 Linear Differential Equations and
Products of Linear Forms (with F. Ulmer) (.ps file).
Journal of Pure and Applied Algebra, 117118,
1997, 549563. Also available as a
.dvi file
1996
 Testing Reducibility of Linear
Differential Operators: A Group Theoretic Perspective (.ps
file), Applicable Algebra in Engineering,
Communication and Computing, 7(2), 1996, 77104.
Also available as a .dvi file
 On Ramis's Solution of the Local
Inverse Problem of Differential Galois Theory (with C.
Mitschi), Journal of
Pure and Applied Algebra, 110, 1996, 185194.
 Connected Linear Groups as
Differential Galois Groups (with C. Mitschi) (.ps file),
Journal of Algebra, 184, 1996, 333361. Also
available as a .dvi file
 The Inverse Problem in
Differential Galois Theory (with C. Mitschi) (.ps file) ,
in The Stokes Phenomenon and Hilbert's 16th Problem,
B.l.J. Braaksma, et. al., eds., World Scientific, Singapore,
1996, 185196. Also available as a
.dvi file
 On the Infinitesimal Geometry
of Integrable Systems (with A. Baider, R. Churchill, D. Rod)
(.ps file) , in Mechanics Day, Shadwich et.
al., eds, Fields Institute Communications, 7 ,
American Mathematical Society, 1996, 556. Also available as a .dvi file
1995
 Necessary Conditions for
Liouvillian Solutions of (Third Order) Linear Differential
Equations (with F. Ulmer) (.ps file), Applied
Algebra in Engineering, Communication and Computing,
6(1), 1995, 1  22. Also available as a .dvi file ; an extended
abstract of this paper appeared in the Proceedings of the
International Symposium on Symbolic and Algebraic Computation
(ISSAC `92), ACM Press.
 Group Theoretic Obstructions to
Integrability (with R. Churchill, D. Rod) (.ps file),
Ergodic Theory and Dynamical Systems} , 15,
1995, 15 48 . Also available as a .dvi file
 On Computing Algebraic Functions
using Logarithms and Exponentials (with D. Grigoriev, A. Yao)
(.ps file) , SIAM J. Comp., 24(2),
1995, 242  246. Also available as a
.dvi file
1994
 Computational
Complexity of Sparse Rational Function Interpolation (with D.
Yu. Grigor'ev, M. Karpinski) (.ps file), SIAM J. of
Computing, 23, 1994, 1 11. Also available as a .dvi file
1993
 Moduli of Linear Differential
Equations on the Rieman Sphere with Fixed Galois Groups ,
Pacific Journal of Mathematics, 106(2), 1993,
343395.
 Computational Complexity of Sparse Real Algebraic Function
Interpolation (with D. Yu. Grigor'ev, M. Karpinski), in the
Proceedings of the Conference on Effective Methods in
Algebraic Geometry (MEGA '92), April 1992, Progress in
Math., Birkhaeuser, 109 1993, 91104.
 Galois Groups of Second
and Third Order Linear Differential Equations (with F. Ulmer)
(.ps file), Journal of Symbolic Computation,
16, July 1993, 9  36. Also available as a .dvi file
 Liouvillian and Algebraic
Solutions of Second and Third Order Linear Differential
Equations, (with F. Ulmer)(.ps file), Journal of
Symbolic Computation, 16, July 1993, 37  74.
Also available as a .dvi
file
 On a Third Order Differential
Equation Whose Differential Galois Group is a Simple Group
with 168 Elements (with F. Ulmer), Proceedings of
the 10th International Symposium on Applied Algebra, Algebraic
Algorithms and Error Correcting Codes, Puerto Rico, May 1993,
in Lecture Notes in Computer Science, 519.
 On Integer Zeros of Exponential Polynomials (with C. W.
Henson, L. A. Rubel, L. van den Dries), Complex Variables
Theory and Applications, 23(34), 1993, 201211.
1992
 Liouvillian First
Integrals of Differential Equations, Transactions of
the AMS, 333(2), October 1992, 673688.
1991
 Solving Ordinary
Differential Equations in Terms of Series with Real Exponents
(with D. Yu. Grigor'ev), Transactions of the A.M.S.,
327(1), 1991, 329351, .
 Liouvillian Solutions
of Linear Differential Equations with Liouvillian Coefficients,
J. of Symbolic Computation, 11(3), 1991,
251274.
 The Interpolation Problem for
kSparse Sums of Eigenfunctions of Operators (with D. Yu.
Grigor'ev, M. Karpinski), Advances in Applied
Mathematics, 12, 1991, 7681
 Size efficient parallel algebraic circuits for partial
derivatives (with E. Kaltofen), in IV International
Conference on Computer Algebra in Physical Research, D.
V. Shirkov, V. A. Rostovtsev, and V. P. Gerdt, ed., World
Scientific Publ., Singapore, 1991, 133145 .
1990
 Fast
Parallel Algorithms for Sparse Multivariate Polynomial
Interpolation over Finite Fields (with D. Yu. Grigor'ev and M.
Karpinski), SIAM J. of Computation, 19(6),
December
1990,
1059  1063.
 Formal Solutions of Differential
Equations, J. of Symbolic Computation, 10,
1990, 5994.
 Interpolation of Sparse
Rational Functions without Knowing the Bounds on Exponents
(with D. Yu. Grigor'ev, M. Karpinski), Proceedings
of the 1990 IEEE Foundations of Computer Science Conference,
IEEE Computer Society Press, 1990, 840  847.
1989
 Algebraic
Properties of the Ring of General Exponential Polynomials
(with C. W. Henson, L. Rubel), Complex Variables Theory and
Applications, 13, 1989, 120.
 An Outline of Differential
Galois Theory, in Computer Algebra and Differential
Equations, E. Tournier, ed., Academic Press, 1989, 358
1988
 Algebraic Relations Among
Solutions of Linear Differential Equations: Fano's Theorem,
Am. J. of Math., 110, 1988, 115143.
 Autonomous Functions (with L.
Rubel), Journal of Differential Equations,
75(2), 1988
1986
 Algebraic Relations Among
Solutions of Linear Differential Equations,
Transactions of the American Mathematics Society , 295(2),
1986,753763.
 Elementary
and
Liouvillian Solutions of Linear Differential Equations (with
J. Davenport), Journal of Symbolic Computation,
2(3), 1986, 237260.
1985
 An Extension
of Liouville's Theorem on Integration in Finite Terms (with B.
D. Saunders and B. F. Caviness), SIAM Journal of
Computing, 14, 1985, 966990 .
 Solving Homogeneous Linear
Differential Equations in Terms of Second Order Linear
Differential Equations, Am. J. of Math.,
107, 1985, 663696.
 Aplications of Linear
Groups to Differential Equations (with M. Tretkoff),
American Journal of Mathematics, 107, 1985,
10931109.
 A Classification of
Differential Equations of Fuchsian Class (with M. Tretkoff),
American Journal of Mathematics, 107, 1985,
11111121.

A Class of Vectorfields
on S2 that are Topologically Equivalent to Polynomial
Vectorfields (with S. Schecter), Journal of
Differential Equations, 57 (3), 1985, 406435.
 A Differentially Algebraic Elimination Theorem with
Applications to Analog Computatbility in the Calculus of
Variations (with L. Rubel), Proceedings of the American
Mathematical Society, 94(4), 1985, 635658.
1983
 Elementary First
Integrals of Differential Equations (with M. Prelle),,
Transactions of the American Mathematical Society ,
279(1), September 1983, 215229.
1981
 Liouvillian Solutions of nth
Order Homogeneous Linear Differential Equations, Am.
J. Math., 103(4), 1981, 661682.
1980
 Remarks on Analytic Continuation
(with F. Haimo and M. Tretkoff), Bulletin of the London
Mathematical Society, 12, 1980, 912.
 Planar Polynomial Foliations
(with S. Schecter), Proceedings of the American
Mathematical Society, 79 (4), August 1980,
649656. Addendum .
 Singular Points of Planar Vector Fields (with S. Schecter),
in Globay Theory of Dynamic Systems , Lecture Notes
in Mathematics, 819, SpringerVerlag, 393410.
 Separatrices at Singular Points of Planar Vector Fields (with
S. Schecter), Acta Mathematica, 145, 1980,
4778 ; correction in 151, 1983, 297298.
1979
 A Class of Differential
Fields with Minimal Differential Closures,
Proceedings of the American Mathematical Society, 69(2),
1979, 319322.
 The Model Theory of
Ordered Differential Fields, The Journal of Symbolic
Logic, 43(1), 1979, 8291.
 Algebraic Solutions of nth Order
Linear Differential Equations, Proceedings of the
Queen's University 1979 Conference on Number Theory,
Queens Papers in Pure and Applied Mathematics, (54), pp.
379420.
1977
 Functions Satisfying Elementary
Relations,, Transactions of the American
Mathematical Society, 227, 1977, 185206.
 On
Elementary,
Generalized Elementary, and Liouvillian Extension Fields
(with M. Rosenlicht), in Contributions to Algebra,
(H. Bass et.al., ed.), Academic Press, 1977, 329342 .
1976
 Solutions of Linear
Differential Equations in Function Fields of One Variable,
Proceedings of the American Mathematical Society, 54,
January 1976, 6972.
1975
 Elementary Solutions of
Differential Equations, Pacific Journal of
Mathematics, 59(2), 1975, 535547
 Asymptotic Behavior of Solutions of
Differential Equations and Hardy Fields: Preliminary Report.
This is an unpublished manuscript written in 1975.
Michael Singer's Home Page