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. I contributed the appendix to “The conjectures of Kumbhakar, Roy, and Srinivasan’’ by Freitag, León Sánchez, Li, and Nagloo. In this appendix I give a proof of the following theorem:

   Let K be a nonconstant ordinary differential field of characteristic zero with algebraically closed constants C that is finitely generated over C in the differential sense and let G be a connected linear algebraic group defined over C. Then G is the differential Galois group of a Picard-Vessiot extension of K.

   This generalizes a result of Mitschi and myself where K is only assumed to be finitely generated in the algebraic sense over C and the proof follows in a similar manner. The complete paper can be found here . The appendix can be read independently of the rest of the paper.


2. On Deciding Transcendence of Power Series (with A. Bostan, B. Salvy). Submitted 2025
   


3. 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
   


4. 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.
   


5. 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".


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 This is an unpublished manuscript written in 1975.