푸크스 미분방정식(Fuchsian differential equation)

개요

선형미분방정식\[\frac{d^n w}{dz^n} + A_1(z)\frac{d^{n-1}w}{dz^{n-1}} + \cdots + A_{n-1}(z)\frac{dw}{dz} + A_n(z)w=0\]
모든 특이점이 정칙특이점(regular singular points)인 경우, 이를 푸크스 미분방정식이라 한다
19세기에 많은 연구가 진행되었음
초기하 미분방정식(Hypergeometric differential equations) 은 대표적인 푸크스 미분방정식의 예이다

Dwork family http://www.math.sunysb.edu/~cschnell/pdf/notes/picardfuchs.pdf
Group Theory and Differential Equations University of Minnesota Lecture Notes 1959-1960 by Lawrence Markus (with permission of the author)
http://www.ima.umn.edu/~miller/Group_Theory_and_Differential_Equations.html

Gray, J. J. 1984. “Fuchs and the Theory of Differential Equations.” Bulletin of the American Mathematical Society 10 (1): 1–26. doi:10.1090/S0273-0979-1984-15186-3.

Liouvillian First Integrals of Differential Equations
- Michael F. Singer, Transactions of the American Mathematical Society, Vol. 333, No. 2 (Oct., 1992), pp. 673-688
Algebraic Relations Among Solutions of Linear Differential Equations: Fano's Theorem
- Michael F. Singer, American Journal of Mathematics, Vol. 110, No. 1 (Feb., 1988), pp. 115-143
Elementary and Liouvillian solutions of linear differential equations
- M. F. Singer and J. H. Davenport, 1985
Some Applications of Linear Groups to Differential Equations
- Michael F. Singer and Marvin D. Tretkoff, American Journal of Mathematics, Vol. 107, No. 5 (Oct., 1985), pp. 1111-1121
Logarithmic singularities of Fuchs equations, and a criterion for the monodromy group to be finite
- N. V. Grigorenko, MATHEMATICAL NOTESVolume 33, Number 6, 453-454
Liouvillian Solutions of n-th Order Homogeneous Linear Differential Equations
- Michael F. SingerAmerican Journal of Mathematics, Vol. 103, No. 4 (Aug., 1981), pp. 661-682
Algebraic Solutions of nth Order Linear Differential Equations
- M. Singer, Proceedings of the Queen's University 1979 Conference on Number Theory, Queens Papers in Pure and Applied Mathematics, (54), pp. 379-420
On Second Order Linear Differential Equations with Algebraic Solutions
- F. Baldassari, B. Dwork, American Journal of Mathematics, Vol. 101, No. 1 (Feb., 1979), pp. 42-76