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The uniqueness problem of meromorphic maps into the complex projective space

Published online by Cambridge University Press:  22 January 2016

Hirotaka Fujimoto
Hirotaka Fujimoto*
Affiliation:
Nagoya University
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In 1921, G. Pólya showed that non-constant meromorphic functions ϕ and ψ of finite genera on the complex plane C are necessarily equal if there are distinct five values ai(1 ≦ i ≦ 5) such that ϕ(z) — ai and ψ(z) — ai have the same zeros of the same multiplicities for each i ([8]). Afterwards, R. Nevanlinna obtained the same conclusion for arbitrary ϕp and ψ satisfying ϕ— 1(ai) = ψ— 1(1 ≦ i ≦ 5) regardless of multiplicities. And, some other results relating to this were given by H. Cartan ([2], [3]), E. M. Schmid ([9]) and others. The purpose of this paper is to give some types of generalizations of these results to the case of meromorphic maps into the N-dimensional complex projective space PN(C).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 58 , September 1975 , pp. 1 - 23
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1975

References

[1] Borel, E., Sur les zeros des fonctions entières, Acta Math., 20 (1897), 357–396.CrossRefGoogle Scholar
[2] Cartan, H., Sur les systemes de fonctions holomorphes a varietes lineaire lacunaires et leurs applications, Ann. E. N. S., 45 (1928), 255–346.Google Scholar
[3] Cartan, H., Sur les zeros des combinaisons lineaires de p fonctions holomorphes donnees, Mathematica 7 (1933), 5–31.Google Scholar
[4] Fujimoto, H., Extension of the big Picard’s theorem, Tohoku Math. J., 24 (1972), 415–422.CrossRefGoogle Scholar
[5] Fujimoto, H., On meromorphic maps into the complex projective space, J. Math. Soc. Japan, 26 (1974), 272–288.Google Scholar
[6] Green, M. L., Holomorphic maps into complex projective space omitting hyperplanes, Trans. A. M. S., 160 (1972), 89–103.CrossRefGoogle Scholar
[7] Nevanlinna, R., Einige Eindeutigkeitssatze in der Theorie der meromorphen Funktionen, Acta Math., 48 (1926), 367–391.CrossRefGoogle Scholar
[8] Pólya, G., Bestimmung einer ganzen Funktion endlichen Geschlechts durch viererlei Stellen, Math. Tidsskrift B, KÕbenhavn 1921, 16–21.Google Scholar
[9] Schmid, E. M., Some theorems on value distributions of meromorphic functions, Math. Z., 120 (1971), 61–92.Google Scholar