Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Uniqueness of non-Archimedean entire functions sharing sets of values counting multiplicity

Authors: William Cherry and Chung-Chun Yang
Journal: Proc. Amer. Math. Soc. 127 (1999), 967-971
MSC (1991): Primary 11S80, 30D35
MathSciNet review: 1487362
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A set is called a unique range set for a certain class of functions if each inverse image of that set uniquely determines a function from the given class. We show that a finite set is a unique range set, counting multiplicity, for non-Archimedean entire functions if and only if there is no non-trivial affine transformation preserving the set. Our proof uses a theorem of Berkovich to extend, to non-Archimedean entire functions, an argument used by Boutabaa, Escassut, and Haddad to prove this result for polynomials

References [Enhancements On Off] (What's this?)

  • [A-S] W. W. Adams and E. G. Straus, Non-Archimedian analytic functions taking the same values at the same points, Illinois J. Math. 15 (1971), 418-424. MR 43:3504
  • [Ber] V. Berkovich, Spectral Theory and Analytic Geometry over Non-Archimedean Fields, Mathematical Surveys and Monographs 33. Amer. Math. Soc. 1990. MR 91k:32038
  • [B-E-H] A. Boutabaa, A. Escassut, and L. Haddad, On Uniqueness of $p$-Adic Entire Functions, Indag. Math. (N.S.) 8 (1997), 145-155.
  • [Ch] W. Cherry, Non-Archimedean analytic curves in Abelian varieties, Math. Ann. 300 (1994), 393-404. MR 96i:14021
  • [F-R] G. Frank and M. Reinders, A unique range set for meromorphic functions with 11 elements, preprint.
  • [Gr] F. Gross, Factorization of meromorphic functions and some open problems, in Complex Analysis (Proc. Conf., Univ. Kentucky, Lexington, KY, 1976) Lecture Notes in Math. 599, Springer-Verlag, 1977, 51-67. MR 56:8823
  • [G-Y] F. Gross and C.-C. Yang, On preimage and range sets of meromorphic functions, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), 17-20. MR 83d:30027
  • [H-Y] P.-C. Hu and C.-C. Yang, Value distribution theory of $p$-adic meromorphic functions, preprint.
  • [L-Y] P. Li and C.-C. Yang, On the Unique Range Set of Meromorphic Functions, Proc. Amer. Math. Soc. 124 (1996), 177-185. MR 96d:30033
  • [Nev] R. Nevanlinna, Analytic Functions, Springer-Verlag, 1970. MR 43:5003
  • [O-P-Z] I. V. Ostrovskii, F. B. Pakovitch, and M. G. Zaidenberg, A Remark on Complex Polynomials of Least Deviation, Internat. Math. Res. Notices 1996 (1996), 699-703. MR 97i:30007
  • [Y-Y] H. X. Yi and C.-C. Yang On Uniqueness Theorems for Meromorphic Functions (in Chinese), Science Press, China, 1995.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 11S80, 30D35

Retrieve articles in all journals with MSC (1991): 11S80, 30D35

Additional Information

William Cherry
Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
Address at time of publication: Department of Mathematics, University of North Texas, Denton, Texas 76203

Chung-Chun Yang
Affiliation: Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

Received by editor(s): July 18, 1997
Additional Notes: Financial support for the first author was provided by National Science Foundation grants DMS-9505041 and DMS-9304580
The second author’s research was partially supported by a UGC grant of Hong Kong.
Communicated by: David E. Rohrlich
Article copyright: © Copyright 1999 American Mathematical Society