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

DOI:
https://doi.org/10.1090/S0002-9939-99-04789-9

MathSciNet review:
1487362

Full-text PDF Free Access

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

**[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**0277771****[Ber]**Vladimir G. Berkovich,*Spectral theory and analytic geometry over non-Archimedean fields*, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, 1990. MR**1070709****[B-E-H]**A. Boutabaa, A. Escassut, and L. Haddad,*On Uniqueness of -Adic Entire Functions*, Indag. Math. (N.S.)**8**(1997), 145-155.**[Ch]**William Cherry,*Non-Archimedean analytic curves in abelian varieties*, Math. Ann.**300**(1994), no. 3, 393–404. MR**1304429**, https://doi.org/10.1007/BF01450493**[F-R]**G. Frank and M. Reinders,*A unique range set for meromorphic functions with 11 elements*, preprint.**[Gr]**Fred Gross,*Factorization of meromorphic functions and some open problems*, Complex analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976), Springer, Berlin, 1977, pp. 51–67. Lecture Notes in Math., Vol. 599. MR**0450529****[G-Y]**Fred Gross and Chung Chun Yang,*On preimage and range sets of meromorphic functions*, Proc. Japan Acad. Ser. A Math. Sci.**58**(1982), no. 1, 17–20. MR**649056****[H-Y]**P.-C. Hu and C.-C. Yang,*Value distribution theory of -adic meromorphic functions*, preprint.**[L-Y]**Ping Li and Chung-Chun Yang,*On the unique range set of meromorphic functions*, Proc. Amer. Math. Soc.**124**(1996), no. 1, 177–185. MR**1291784**, https://doi.org/10.1090/S0002-9939-96-03045-6**[Nev]**Rolf Nevanlinna,*Analytic functions*, Translated from the second German edition by Phillip Emig. Die Grundlehren der mathematischen Wissenschaften, Band 162, Springer-Verlag, New York-Berlin, 1970. MR**0279280****[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**14**(1996), 699–703. MR**1411590**, https://doi.org/10.1155/S1073792896000438**[Y-Y]**H. X. Yi and C.-C. Yang*On Uniqueness Theorems for Meromorphic Functions*(in Chinese), Science Press, China, 1995.

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

Email:
mayang@uxmail.ust.hk

DOI:
https://doi.org/10.1090/S0002-9939-99-04789-9

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