Books written or edited by Michael F. Singer

1. 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)
   
   
2. 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)
   
   
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 
   
   

Papers by Michael F. Singer

Preprints


1. Introduction to the Galois Theory of Linear Differential Equations (82 pages). 2007. This is an expanded version of the 10 lectures I gave as the 2006 London Mathematical Society Invited Lecture Series at the Heriot-Watt University 31 July - 4 August 2006.
   
   
2008


2. Differential Galois Theory of Linear Difference Equations (with C. Hardouin) Mathematische Annalen, 342(2) 2008, 333-377. 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).
   
   
3. 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


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

2006


5. 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).


6. 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


7. 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
   
   
8. Solvable-by-Finite Groups as Differential Galois Groups (with C. Mitschi) (.ps file). Also available as a .dvi file, Ann. Fac. Sci. Toulouse Math. (6) 11/3 (2002), 403-423
   
   
9. 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
   
   
10. 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
    
    
11. 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
    
    
12. 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.
    
    
13. Solving Difference Equations in Finite Terms (with P. Hendriks) . Journal of Symbolic Computation, 27/3, 1999, 239-259.
    
    
14. 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
    
    
15. 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
    
    
16. 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
    
    
17. 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
    
    
18. 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. Also available as a .dvi file
    
    
19. 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
    
    
20. 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
    
    
21. 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
    
    
22. 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.
    
    
23. 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
    
    
24. 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
    
    
25. 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
    
    
26. Moduli of Linear Differential Equations on the Rieman Sphere with Fixed Galois Groups , Pacific Journal of Mathematics, 106(2), 1993, 343-395.
    
    
27. 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.
    
    
28. 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
    
    
29. 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
    
    
30. 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.
    
    
31. 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
    
    
32. Liouvillian First Integrals of Differential Equations, Transactions of the AMS, 333(2), October 1992, 673-688.
    
    1991
    
    
33. 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, .
    
    
34. Liouvillian Solutions of Linear Differential Equations with Liouvillian Coefficients, J. of Symbolic Computation, 11(3), 1991, 251-274.
    
    
35. 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
    
    
36. 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
    
    
37. 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.
    
    
38. Formal Solutions of Differential Equations, J. of Symbolic Computation, 10, 1990, 59-94.
    
    
39. 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
    
    
40. Algebraic Properties of the Ring of General Exponential Polynomials (with C. W. Henson, L. Rubel), Complex Variables Theory and Applications, 13, 1989, 1-20.
    
    
41. An Outline of Differential Galois Theory, in Computer Algebra and Differential Equations, E. Tournier, ed., Academic Press, 1989, 3-58
    
    1988
    
    
42. Algebraic Relations Among Solutions of Linear Differential Equations: Fano's Theorem, Am. J. of Math., 110, 1988, 115-143.
    
    
43. Autonomous Functions (with L. Rubel), Journal of Differential Equations, 75(2), 1988
    
    1986
    
    
44. Algebraic Relations Among Solutions of Linear Differential Equations, Transactions of the American Mathematics Society , 295(2), 1986,753-763.
    
    
45. Elementary and Liouvillian Solutions of Linear Differential Equations (with J. Davenport), Journal of Symbolic Computation, 2(3), 1986, 237-260.
    
    1985
    
    
46. 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 .
    
    
47. Solving Homogeneous Linear Differential Equations in Terms of Second Order Linear Differential Equations, Am. J. of Math., 107, 1985, 663-696.
    
    
48. Aplications of Linear Groups to Differential Equations (with M. Tretkoff), American Journal of Mathematics, 107, 1985, 1093-1109.
    
    
49. A Classification of Differential Equations of Fuchsian Class (with M. Tretkoff), American Journal of Mathematics, 107, 1985, 1111-1121.
    
    
50. 
    
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.


51. 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
    
    
52. Elementary First Integrals of Differential Equations (with M. Prelle),, Transactions of the American Mathematical Society , 279(1), September 1983, 215-229.
    
    1981
    
    
53. Liouvillian Solutions of nth Order Homogeneous Linear Differential Equations, Am. J. Math., 103(4), 1981, 661-682.
    
    1980
    
    
54. Remarks on Analytic Continuation (with F. Haimo and M. Tretkoff), Bulletin of the London Mathematical Society, 12, 1980, 9-12.
    
    
55. Planar Polynomial Foliations (with S. Schecter), Proceedings of the American Mathematical Society, 79 (4), August 1980, 649-656. Addendum .
    
    
56. Singular Points of Planar Vector Fields (with S. Schecter), in Globay Theory of Dynamic Systems , Lecture Notes in Mathematics, 819, Springer-Verlag, 393-410.
    
    
57. Separatrices at Singular Points of Planar Vector Fields (with S. Schecter), Acta Mathematica, 145, 1980, 47-78 ; correction in 151, 1983, 297-298.
    
    1979
    
    
58. A Class of Differential Fields with Minimal Differential Closures, Proceedings of the American Mathematical Society, 69(2), 1979, 319-322.
    
    
59. The Model Theory of Ordered Differential Fields, The Journal of Symbolic Logic, 43(1), 1979, 82-91.
    
    
60. 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
    
    
61. Functions Satisfying Elementary Relations,, Transactions of the American Mathematical Society, 227, 1977, 185-206.
    
    
62. 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
    
    
63. Solutions of Linear Differential Equations in Function Fields of One Variable, Proceedings of the American Mathematical Society, 54, January 1976, 69-72.
    
    1975
    
    
64. Elementary Solutions of Differential Equations, Pacific Journal of Mathematics, 59(2), 1975, 535-547
    
    
    
65. Asymptotic Behavior of Solutions of Differential Equations and Hardy Fields: Preliminary Report. This is an unpublished manuscript written in 1975.