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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
DOI: https://doi.org/10.1090/S0002-9939-08-09393-3
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
MR Author ID: 325057
Email: dcoman@syr.edu

Evgeny A. Poletsky
Affiliation: Department of Mathematics, 215 Carnegie Hall, Syracuse University, Syracuse, New York 13244-1150
MR Author ID: 197859
Email: eapolets@syr.edu

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.