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/10.1007/978-3-030-98767-1
- 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 1-2, October 2001
- Galois Theory of Difference Equations, M. van der Put
and M. F. Singer, V. 1666 in Lecture Notes in Mathematics,
Springer-Verlag, 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
- Telescopers for differential
forms with one variable (with Shashi Chen, Ruyong Feng, Zming
Li, and Stephen Watt, Preprint, 2021
- 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 self-contained 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This is an unpublished manuscript written in 1975.
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. 11-30.
- 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.231-240. Also appears in the Notices
of the American Mathematical Society, June/July 2022, pp.
955-961.
- On the algebraic dependence
of holonomic functions (with J. Roques), Annales Henri
Lebesgue.,Vol. 5, pp.141--177, 2022.
2021
- Separability Problems in
Creative Telescoping (with Shashi Chen, Ruyong Feng, and
Pingchuan Ma), Proceedings of ISSAC 2021, (M.
Mezzarobba, ed.), 83-90, 2021.
- Sparse Interpolation
in Terms of Multivariate Chebyshev Polynomials (with E.
Hubert), Foundations of Computational Mathematics,
Published electronically September 9, 2021,
https://rdcu.be/cxuUX
- On Differentially Algebraic
Generating Series for Walks in the Quarter Plane (with
C.Hardouin), Maple
Worksheets, 2021, Sel. Math. New Ser. 27, 89 (2021).
https://doi.org/10.1007/s00029-021-00703-9
- 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/10.1007/978-3-030-84304-5_3, pp.61-89
2020
- Some Structural Results on
D^n-finite Functions (with V. Pillwein and A.
Jimenez-Pastor), 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, 2751-2792, 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.139-204, 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.12-53.
- Desingularization of Ore
Operators (with S. Chen and M. Kauers), Journal
of Symbolic Computation, Vol. 74, 617-626,
May-June 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, 1733-1793,
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, 671-700.
- 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), 320-343.
2013
- Desingularization Explains
Order-Degree Curves for Ore Operators (with S. Chen, M.
Jaroschek and M. Kauers), Procedings of ISSAC
2013, ( M. Kauers, ed.), 157-164, 2013.
- Projective Isomonodromy and
Galois Groups (with C. Mitschi), Proceedings of the American
Mathematical Society, 141 (2013), no. 2, 605-617.
- Linear Algebraic Groups as
Parameterized Picard-Vessiot Galois Groups, Journal of Algebra, 373, (2013), 151--161, Old version, 2011.
2012
- Monodromy Groups of
Parameterized Linear Differential Equations with Regular
Singularities (with C. Mitschi), Bull. London Math. Soc., 44(5), 2012,
913-930.
- 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.), 130-137, 2012.
2011
- A Jordan-Hoelder Thoerem
for Differentail Algebraic Groups (with P. Cassidy) Journal
of Algebra, 328,
2011, 190–217.
2010
- Liouvillian
solutions of difference-differential equations (with R.
Feng, M. Wu) Journal
of Symbolic Computation, 45, 2010, 287-305.
- An algorithm to
compute liouvillian solutions of prime order
difference-differential equations (with R. Feng, M. Wu)
Journal of Symbolic
Computation, 45,
2010, 306-323.
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, 1-82.
2008
- Differential
Galois Theory of Linear Difference Equations (with C.
Hardouin) Mathematische
Annalen, 342(2)
2008, 333-377 Erratum Some
of the calculations referred to in this paper are contained in
a Maple Worksheet entitled Differential
independence of a class of q-hypergeometric 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), 73-109.
2007
- Model Theory of
Differential Fields: From Commuting to Noncommuting
Derivations Proceedings
of the AMS., 135 (2007), 1929-1934.
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 One-Dimensional Submodules of Laurent-Ore
Modules (with Z. Li, M. Wu, D. Zheng). 2006. Proceedings of ISSAC 2006, pp.
200-208.
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. 3639-3665. An older version is
available here. Related
software is available here.
2002
- Solvable-by-Finite Groups as
Differential Galois Groups (with C. Mitschi). Ann.
Fac. Sci. Toulouse Math. (6) 11/3 (2002),
403-423
- Linear Differential Operators for
Polynomial Equations (with O. Cormier, B.M. Trager and F.
Ulmer) (.pdf file) Journal of Symbolic
Computation, 34 , 2002, pp.355-398.
2000
- Computing the Galois Group
of a Polynomial Using Linear Differential Equations (with O.
Cormier, F. Ulmer) (.ps file) Proceedings of ISSAC
2000., 78-85
1999
- Computing Galois
Groups of Completely Reducible Differential Equations (with
E. Compoint) (.ps file). Journal of Symbolic
Computation, 28/4-5, 1999, 473-494. 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, 139/1-3, 1999, 3-24.
- Solving Difference Equations in
Finite Terms (with P. Hendriks) . Journal of
Symbolic Computation, 27/3, 1999, 239-259.
- 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, 527-554.
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, 659-670. 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, 117-118,
1997, 549-563. 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,
77-104. 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,
185-194.
- Connected Linear Groups as
Differential Galois Groups (with C. Mitschi) (.ps file),
Journal of Algebra, 184, 1996, 333-361.
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, 185-196. 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, 5-56. 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, 343-395.
- 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, 91--104.
- 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(3-4),
1993, 201-211.
1992
- Liouvillian
First Integrals of Differential Equations,
Transactions of the AMS, 333(2), October 1992,
673-688.
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, 329-351, .
- Liouvillian
Solutions of Linear Differential Equations with Liouvillian
Coefficients, J. of Symbolic Computation,
11(3), 1991, 251-274.
- The Interpolation Problem for
k-Sparse Sums of Eigenfunctions of Operators (with D. Yu.
Grigor'ev, M. Karpinski), Advances in Applied
Mathematics, 12, 1991, 76-81
- 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, 133-145 .
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, 59-94.
- 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, 1-20.
- An Outline of Differential
Galois Theory, in Computer Algebra and
Differential Equations, E. Tournier, ed., Academic
Press, 1989, 3-58
1988
- Algebraic Relations Among
Solutions of Linear Differential Equations: Fano's Theorem,
Am. J. of Math., 110, 1988, 115-143.
- 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),
1986,753-763.
- Elementary
and
Liouvillian Solutions of Linear Differential Equations (with
J. Davenport), Journal of Symbolic Computation,
2(3), 1986, 237-260.
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, 966-990 .
- Solving Homogeneous
Linear Differential Equations in Terms of Second Order
Linear Differential Equations, Am. J. of Math.,
107, 1985, 663-696.
- Aplications of
Linear Groups to Differential Equations (with M. Tretkoff),
American Journal of Mathematics, 107, 1985,
1093-1109.
- A Classification of
Differential Equations of Fuchsian Class (with M. Tretkoff),
American Journal of Mathematics, 107, 1985,
1111-1121.
-
A Class of
Vectorfields on S2 that are Topologically Equivalent to
Polynomial Vectorfields (with S. Schecter), Journal
of Differential Equations, 57 (3), 1985,
406-435.
- 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, 635-658.
1983
- Elementary First
Integrals of Differential Equations (with M. Prelle),,
Transactions of the American Mathematical Society ,
279(1), September 1983, 215-229.
1981
- Liouvillian Solutions of
nth Order Homogeneous Linear Differential Equations,
Am. J. Math., 103(4), 1981, 661-682.
1980
- Remarks on Analytic
Continuation (with F. Haimo and M. Tretkoff),
Bulletin of the London Mathematical Society, 12,
1980, 9-12.
- Planar Polynomial
Foliations (with S. Schecter), Proceedings of the
American Mathematical Society, 79 (4), August
1980, 649-656. Addendum .
- Singular Points of Planar Vector Fields (with S. Schecter),
in Globay Theory of Dynamic Systems , Lecture
Notes in Mathematics, 819, Springer-Verlag, 393-410.
- Separatrices at Singular Points of Planar Vector Fields
(with S. Schecter), Acta Mathematica, 145,
1980, 47-78 ; correction in 151, 1983, 297-298.
1979
- A Class of Differential
Fields with Minimal Differential Closures,
Proceedings of the American Mathematical Society, 69(2),
1979, 319-322.
- The Model Theory of
Ordered Differential Fields, The Journal of
Symbolic Logic, 43(1), 1979, 82-91.
- 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.
379-420.
1977
- Functions Satisfying
Elementary Relations,, Transactions of the
American Mathematical Society, 227, 1977,
185-206.
- On
Elementary,
Generalized Elementary, and Liouvillian Extension Fields
(with M. Rosenlicht), in Contributions to Algebra,
(H. Bass et.al., ed.), Academic Press, 1977, 329-342 .
1976
- Solutions of Linear
Differential Equations in Function Fields of One Variable,
Proceedings of the American Mathematical Society, 54,
January 1976, 69-72.
1975
- Elementary Solutions of
Differential Equations, Pacific Journal of
Mathematics, 59(2), 1975, 535-547
Michael Singer's Home Page