Nagoya Mathematical Journal

On uniqueness polynomials for meromorphic functions

Hirotaka Fujimoto

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A polynomial $P(w)$ is called a uniqueness polynomial (or a uiqueness polynomial in a broad sense) if $P(f) = cP(g)$ (or $P(f) = P(g)$) implies $f = g$ for any nonzero constant $c$ and nonconstant meromorphic functions $f$ and $g$ on $\mathbf C$. We consider a monic polynomial $P(w)$ without multiple zero whose derivative has mutually distinct $k$ zeros $e_j$ with multiplicities $q_j$. Under the assumption that $P(e_\ell) \not= P(e_m)$ for all distinct $\ell$ and $m$, we prove that $P(w)$ is a uniqueness polynomial in a broad sense if and only if $\sum_{\ell>m}q_\ell q_m > \sum_{\ell} q_\ell$. We also give some sufficient conditions for uniqueness polynomials.

Article information

Nagoya Math. J., Volume 170 (2003), 33-46.

First available in Project Euclid: 27 April 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30D35: Distribution of values, Nevanlinna theory


Fujimoto, Hirotaka. On uniqueness polynomials for meromorphic functions. Nagoya Math. J. 170 (2003), 33--46.

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  • H. Fujimoto, On uniqueness of meromorphic functions sharing finite sets , Amer. J. Math., 122 (2000), 1175–1203.
  • W. Fulton, Algebraic curves, W. A. Benjamin, New York (1969).
  • G. Frank and M. Reinders, A unique range set for meromorphic functions with 11 elements , Complex Variables, 37 (1998), 185–193.
  • F. Gross and C. C. Yang, On preimage and range sets of meromorphic functions , Proc. Japan Acad., 58 (1982), 17–20.
  • P. Li and C. C. Yang, Some further results on the unique range sets of meromorphic functions , Kodai Math. J., 18 (1995), 437–450.
  • R. Miranda, Algebraic curves and Riemann surfaces, Amer. Math. Soc. (1995).
  • B. Shiffman, Uniqueness of entire and meromorphic functions sharing finite sets , Complex Variables, 43 (2001), 433–449.
  • C. C. Yang and X. Hua, Unique polynomials of entire and meromorphic functions , Matematicheskaya fizika, analiz, geometriya, 3 (1997), 391–398.
  • H. Yi, The unique range sets of entire or meromorphic functions , Complex Variables, 28 (1995), 13–21.
  • H. Yi, Some further results on uniqueness of meromorphic functions , Complex Varialbles, 38 (1999), 375–385.