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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Stable algebras of entire functions


Authors: Dan Coman and Evgeny A. Poletsky
Journal: Proc. Amer. Math. Soc. 136 (2008), 3993-4002
MSC (2000): Primary 32A38; Secondary 30H05
Published electronically: June 11, 2008
MathSciNet review: 2425740
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Abstract: Suppose that $ h$ and $ g$ belong to the algebra $ \mathcal{B}$ generated by the rational functions and an entire function $ f$ of finite order on $ \mathbb{C}^n$ and that $ h/g$ has algebraic polar variety. We show that either $ h/g\in\mathcal{B}$ or $ f=q_1e^p+q_2$, where $ p$ is a polynomial and $ q_1,q_2$ are rational functions. In the latter case, $ h/g$ belongs to the algebra generated by the rational functions, $ e^p$ and $ e^{-p}$.

The stability property is related to the problem of algebraic dependence of entire functions over the ring of polynomials. The case of algebraic dependence over $ \mathbb{C}$ of two entire or meromorphic functions on $ \mathbb{C}^n$ is completely resolved in this paper.


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Additional Information

Dan Coman
Affiliation: Department of Mathematics, 215 Carnegie Hall, Syracuse University, Syracuse, New York 13244-1150
Email: dcoman@syr.edu

Evgeny A. Poletsky
Affiliation: Department of Mathematics, 215 Carnegie Hall, Syracuse University, Syracuse, New York 13244-1150
Email: eapolets@syr.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09393-3
PII: S 0002-9939(08)09393-3
Received by editor(s): April 11, 2007
Received by editor(s) in revised form: October 18, 2007
Published electronically: June 11, 2008
Additional Notes: Both authors are supported by NSF Grants.
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.