Books written or edited by Michael F. Singer



  1. 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

  2. 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

  3. 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.

  4. 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 

  5. 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

  6. 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




  1. Integrals of Solutions of Linear Differential Equations. This is a manuscript containing an explication of some results from Daniel Bertrand's paper ``Extensions de D-modules et groupes de Galois différentielles"; in particular, a result giving necessary and sufficient condtions for the integral of a solution of a homogeneous linear differential equation to be algebraically dependent on the solutions of this equation. 2025

  2. 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.

  3. 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".

  4. Asymptotic Behavior of Solutions of Differential Equations and Hardy Fields: Preliminary Report. This is an unpublished manuscript written in 1975.
    5.

    2024

  1. 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/s00029-024-00926-6


    2. 2022

  2. 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.

  3. 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.

  4. On the algebraic dependence of holonomic functions (with J. Roques), Annales Henri Lebesgue.,Vol. 5, pp.141--177, 2022.

  5. Sparse Interpolation in Terms of Multivariate Chebyshev Polynomials (with E. Hubert), Foundations of Computational Mathematics, vol.22, pp.1801-1862,2022. 


    6. 2021

  6. Separability Problems in Creative Telescoping (with Shashi Chen, Ruyong Feng, and Pingchuan Ma), Proceedings of ISSAC 2021, (M. Mezzarobba, ed.), 83-90, 2021.

  7. 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/s00029-021-00703-9

  8. 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


    9. 2020
  9. Some Structural Results on D^n-finite Functions (with V. Pillwein and A. Jimenez-Pastor), Advances in Applied Mathematics, Vol. 117, June 2020

  10. Walks on the Quarter Plane, Genus Zero Case (with T. Dreyfus, C. Hardouin, J. Roques), Journal of Combinatorial Theory A, Vol. 174, August 2020.


    11. 2019

  11. Consistent systems of linear differential and difference equations (with R. Schaefke), Journal of the European Mathematical Society, Vol.21, Issue 9, 2751-2792, 2019


    12. 2018

  12. 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.


    13. 2017

  13. 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


    14. 2016

  14. 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.

  15. Desingularization of Ore Operators (with S. Chen and M. Kauers),  Journal of Symbolic Computation, Vol. 74, 617-626, May-June 2016.


    16. 2015

  16. 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.

    17. 2014

  17. 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.

  18. 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.

  19. On the Summability of Bivariate Rational Functions (with S. Chen), Journal of Algebra, 409, (2014), 320-343.

    20. 2013

  20. 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.

  21. Projective Isomonodromy and Galois Groups (with C. Mitschi), Proceedings of the American Mathematical Society, 141 (2013), no. 2, 605-617.

    22.
  22. Linear Algebraic Groups as Parameterized Picard-Vessiot Galois Groups, Journal of Algebra, 373, (2013), 151--161, Old version, 2011.

    23. 2012

  23. Monodromy Groups of Parameterized Linear Differential Equations with Regular Singularities (with C. Mitschi), Bull. London Math. Soc., 44(5), 2012, 913-930. 

    24.
  24. Residues and Telescopers for Bivariate Rational Functions (with S. Chen),  Advances in Applied Mathematics, 49 (2012) 111–133.  

  25. 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.

    26. 2011

  26. A Jordan-Hoelder Thoerem for Differentail Algebraic Groups (with P. Cassidy) Journal of Algebra, 328, 2011, 190–217.

    27. 2010

  27. Liouvillian solutions of difference-differential equations (with R. Feng, M. Wu)  Journal of Symbolic Computation, 45, 2010, 287-305.

  28. 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.

    29. 2009

  29. 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.

    30. 2008

  30. 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).

  31. 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.


    32. 2007

  32. Model Theory of Differential Fields: From Commuting to Noncommuting Derivations Proceedings of the AMS., 135 (2007), 1929-1934.


    33. 2006

  33. 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).

  34. 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.


    35. 2005

  35. 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

  36. Solvable-by-Finite Groups as Differential Galois Groups (with C. Mitschi). Ann. Fac. Sci. Toulouse Math. (6) 11/3 (2002), 403-423

  37. 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

  38. 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

  39. 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

  40. 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.

  41. Solving Difference Equations in Finite Terms (with P. Hendriks) . Journal of Symbolic Computation, 27/3, 1999, 239-259.

  42. 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

  43. 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

  44. 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

  45. Testing Reducibility of Linear Differential Operators: A Group Theoretic Perspective , Applicable Algebra in Engineering, Communication and Computing, 7(2), 1996, 77-104.

  46. 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.

  47. 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

  48. 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

  49. 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

  50. 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.

  51. 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

  52. 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

  53. 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

  54. Moduli of Linear Differential Equations on the Rieman Sphere with Fixed Galois Groups , Pacific Journal of Mathematics, 106(2), 1993, 343-395.

  55. 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.

  56. Galois Groups of Second and Third Order Linear Differential Equations (with F. Ulmer) , Journal of Symbolic Computation, 16, July 1993, 9 - 36.

  57. 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

  58. 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.

  59. 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

  60. Liouvillian First Integrals of Differential Equations, Transactions of the AMS, 333(2), October 1992, 673-688.

    1991

  61. 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, .

  62. Liouvillian Solutions of Linear Differential Equations with Liouvillian Coefficients, J. of Symbolic Computation, 11(3), 1991, 251-274.

  63. 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

  64. 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

  65. 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.

  66. Formal Solutions of Differential Equations, J. of Symbolic Computation, 10, 1990, 59-94.

  67. 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

  68. Algebraic Properties of the Ring of General Exponential Polynomials (with C. W. Henson, L. Rubel), Complex Variables Theory and Applications, 13, 1989, 1-20.

  69. An Outline of Differential Galois Theory, in Computer Algebra and Differential Equations, E. Tournier, ed., Academic Press, 1989, 3-58

    1988

  70. Algebraic Relations Among Solutions of Linear Differential Equations: Fano's Theorem, Am. J. of Math., 110, 1988, 115-143.

  71. Autonomous Functions (with L. Rubel), Journal of Differential Equations, 75(2), 1988

    1986

  72. Algebraic Relations Among Solutions of Linear Differential Equations, Transactions of the American Mathematics Society , 295(2), 1986,753-763.

  73. Elementary and Liouvillian Solutions of Linear Differential Equations (with J. Davenport), Journal of Symbolic Computation, 2(3), 1986, 237-260.

    1985

  74. 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 .

  75. Solving Homogeneous Linear Differential Equations in Terms of Second Order Linear Differential Equations, Am. J. of Math., 107, 1985, 663-696.

  76. Aplications of Linear Groups to Differential Equations (with M. Tretkoff), American Journal of Mathematics, 107, 1985, 1093-1109.

  77. A Classification of Differential Equations of Fuchsian Class (with M. Tretkoff), American Journal of Mathematics, 107, 1985, 1111-1121.


    79. 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.

  79. 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

  80. Elementary First Integrals of Differential Equations (with M. Prelle),, Transactions of the American Mathematical Society , 279(1), September 1983, 215-229.

    1981

  81. Liouvillian Solutions of nth Order Homogeneous Linear Differential Equations, Am. J. Math., 103(4), 1981, 661-682.

    1980

  82. Remarks on Analytic Continuation (with F. Haimo and M. Tretkoff), Bulletin of the London Mathematical Society, 12, 1980, 9-12.

  83. Planar Polynomial Foliations (with S. Schecter), Proceedings of the American Mathematical Society, 79 (4), August 1980, 649-656. Addendum .

  84. Singular Points of Planar Vector Fields (with S. Schecter), in Globay Theory of Dynamic Systems , Lecture Notes in Mathematics, 819, Springer-Verlag, 393-410.

  85. Separatrices at Singular Points of Planar Vector Fields (with S. Schecter), Acta Mathematica, 145, 1980, 47-78 ; correction in 151, 1983, 297-298.

    1979

  86. A Class of Differential Fields with Minimal Differential Closures, Proceedings of the American Mathematical Society, 69(2), 1979, 319-322.

  87. The Model Theory of Ordered Differential Fields, The Journal of Symbolic Logic, 43(1), 1979, 82-91.

  88. 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

  89. Functions Satisfying Elementary Relations,, Transactions of the American Mathematical Society, 227, 1977, 185-206.

  90. 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

  91. Solutions of Linear Differential Equations in Function Fields of One Variable, Proceedings of the American Mathematical Society, 54, January 1976, 69-72.

    1975

  92. Elementary Solutions of Differential Equations, Pacific Journal of Mathematics, 59(2), 1975, 535-547





    93.

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.
    •