Books written or edited by Michael F. Singer
 Integration in Finite Terms: Fundamental Sources, C.G.
Raab, M.F. Singer (editors), Texts & Monographs in
Symbolic Computation, Springer Cham, 2022,
https://link.springer.com/book/10.1007/9783030987671
 Galois Theories of Linear Difference Equations: An
Introduction, C. Hardouin, J. Sauloy, M.F. Singer
(editors), Mathematical Surveys and Monographs, Volume
211,
American
Mathematical Society, 2016
 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
 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),
Errors in proof of Theorem 3.1 of Chapter 3
 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)
Papers by Michael F. Singer
Preprints
 Comments on
Ueber algebraisch integrirbare lineare
Differntialgleichungen by F.G. Frobenius In this
1875 paper, Frobenius proves (among other things) that if all
solution of an irreducble homogneous linear differential
equation of order greater than 2 can be expressed as a rational
function of one of them, then all solutions of the equation are
algebraic.I present the proof in English and replace Frobenius's
use of the monodromy group with the differential Galois group.
2023.
 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".
 Asymptotic Behavior of
Solutions of Differential Equations and Hardy Fields:
Preliminary Report. This is an unpublished manuscript
written in 1975.
2024
 Telescopers for
differential forms with one parameter (with S. Chen,
R. Feng, Z. Li, S.M. Watt) Selecta Mathematica
30:34,2024. https://doi.org/10.1007/s00029024009266
2022
 Comments on
Rosenlicht's Integration in Finite Terms
In: Integration in Finite Terms:
Fundamental Sources, C.G.Raab, M.F. Singer
(editors), Texts & Monographs in Symbolic
Computation, Springer Cham, 2022,pp. 1130.
 MSRI Addresses the
Challenge (with H. Barcelo) In: Count Me
In: Community and Belonging in Mathematics, D.
Dumbaugh, D. Haunsperger (eds.), Classroom Resource
Materials, Vol. 68, American Mathematical Society,
2022, pp.231240. Also appears in the Notices of
the American Mathematical Society, June/July
2022, pp. 955961.
 On the
algebraic dependence of holonomic functions (with
J. Roques), Annales Henri Lebesgue.,Vol.
5, pp.141177, 2022.
 Sparse
Interpolation in Terms of Multivariate Chebyshev
Polynomials (with E. Hubert), Foundations
of Computational Mathematics, vol.22,
pp.18011862,2022.
2021
 Separability
Problems in Creative Telescoping (with Shashi
Chen, Ruyong Feng, and Pingchuan Ma), Proceedings
of ISSAC 2021, (M. Mezzarobba, ed.), 8390,
2021.
 On Differentially
Algebraic Generating Series for Walks in the
Quarter Plane (with C.Hardouin), Maple Worksheets,
2021, Selecta Mathematica New Ser. 27, 89
(2021). https://doi.org/10.1007/s00029021007039
 On the kernel
curves associated with walks in the quarter
plane (with T. Dreyfus, C. Hardouin, J. Roques),
Discriminant calculations: maple code,
pdf file,
In: Bostan A., Raschel K. (eds) Transcendence in
Algebra, Combinatorics, Geometry and Number
Theory. TRANS 2019. Springer Proceedings in
Mathematics & Statistics, vol 373. Springer,
Cham. https://doi.org/10.1007/9783030843045_3,
pp.6189
2020
 Some Structural
Results on D^nfinite Functions (with V.
Pillwein and A. JimenezPastor), Advances
in Applied Mathematics, Vol. 117, June 2020
 Walks on
the Quarter Plane, Genus Zero Case (with T.
Dreyfus, C. Hardouin, J. Roques), Journal
of Combinatorial Theory A, Vol. 174, August
2020.
2019
 Consistent
systems of linear differential and difference
equations (with R. Schaefke), Journal of
the European Mathematical Society, Vol.21,
Issue 9, 27512792, 2019
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, pp.139204, 2018.
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 , Applicable Algebra in
Engineering, Communication and Computing, 7(2),
1996, 77104.
 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) ,
Journal of Symbolic Computation, 16,
July 1993, 9  36.
 Liouvillian
and Algebraic Solutions of Second and Third
Order Linear Differential Equations, (with F.
Ulmer), 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
Michael Singer's Home Page
Comments on
Frobenius's article "Uber algebraisch integrirbare lineare
Differentialgleichungen'' In this paper, Frobenius
considers homogeneous linear differential equations $L(y) =
0$ with coefficients in $\CX(x)$ such that every solution
can be expressed as a rational function in one of the
solutions. Among other things, he shows that if the equation
is irreducible and of order greater than or equal to 3, then
all solutions are algebraic. After reading Frobenius’s
paper, I realized that some of his arguments could be
replaced with results from "On the algebraic dependence of
holonomic functions", #89 below. In this note, I use those
results and some of Frobenius’s arguments to prove his
result. I then discuss Frobenius’s arguments.