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On The Factorization of Partial Differential Equations

Published online by Cambridge University Press:  20 November 2018

W. Dale Brownawell
W. Dale Brownawell*
Affiliation:
Pennsylvania State University, University Park, Pennsylvania
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In [4] N. Steinmetz used Nevanlinna theory to establish remarkably versatile theorems on the factorization of ordinary differential equations which implied numerous previous results of various authors. (Here factorization is taken in the sense of function composition as introduced by F. Gross in [2].) The thrust of Steinmetz’ central results on factorization is that if g(z) is entire and f(z) is meromorphic in C such that the composite fog satisfies an algebraic differential equation, then so do f(z) and, degenerate cases aside, g(z). In addition, the more one knows about the equation for fog (e.g. degree, weight, autonomy), the more one can conclude about the equations for f and g.

In this note we generalize Steinmetz’ work to show the following:

  • a) Steinmetz’ two basic results, Satz 1 and Korollar 1 of [4] can be seen as one-variable specializations of a single two variable result, and

  • b) the function g(z) can itself be allowed to be a function of several variables.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 4 , 01 August 1987 , pp. 825 - 834
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Chuang, C.-T., On the inversion of a linear differential polynomial, in Analytic functions of one complex variable, Contemp. Math. 48 (Amer. Math. Soc, Providence, R.I., 1986).Google Scholar
2. Gross, F., Factorization of meromorphic functions (Math. Research Center, Naval Research Lab., Washington, D. C, 1972).Google Scholar
3. Shiffman, B., An introduction to the Carlson-Griffiths equidistrihution theory, in Value Distribution Theory, Joensuu (1981), Lecture Notes in Mathematics 981 (Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1983).Google Scholar
4. Steinmetz, N., Über die faktorisierharen Lôsungen gewöhnlicher Differentialgleichungen, Math. Z. 170 (1980), 169–180.Google Scholar
5. Vitter, A., The lemma of the logarithmic derivative in several complex variables, Duke Math. J. 44 (1977), 89–104.Google Scholar