Hayman, W. K.; Rubel, Lee A. Unavoidable systems of functions. (English) Zbl 0873.30016 Math. Proc. Camb. Philos. Soc. 117, No. 2, 345-351 (1995). Let \(D\) denote a domain in the complex plane. A set of functions \(g_1(z),\dots,g_n(z)\) meromorphic in \(D\) is said to form an unavoidable system in \(D\), if whenever \(f\) is meromorphic (\(\not\equiv\) constant) in \(D\), at least one of the equations \(f(z)=g_i(z)\) \((i=1,2,\dots,n)\) has a root in \(D\). It was shown by L. A. Rubel and the reviewer [Mich. Math. J. 20, 289-296 (1974; Zbl 0263.30022)] that the minimum cardinality of an avoidable system in the complex plane is 3 and three distinct polynomials \(a_1\), \(a_2\), \(a_3\) such that \(a_1-a_2\) and \(a_2-a_3\) are not both constant form an unavoidable system.In this paper, the authors show that the above result holds for an arbitrary plane domain or a Riemann surface. The authors illustrate how an unavoidable triplet can be constructed for an arbitrary non-compact Riemann surface. They also discuss the related problem about unavoidable sets of functions which avoid each other. Reviewer: C.-C.Yang (Kowloon) Cited in 4 Documents MSC: 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory Citations:Zbl 0263.30022 PDFBibTeX XMLCite \textit{W. K. Hayman} and \textit{L. A. Rubel}, Math. Proc. Camb. Philos. Soc. 117, No. 2, 345--351 (1995; Zbl 0873.30016) Full Text: DOI References: [1] DOI: 10.1007/BF02392595 · Zbl 0619.30027 · doi:10.1007/BF02392595 [2] Behnke, Entwicklungen analytischer Funktionen auf Riemannschen Fl?chen. Math. Ann 120 pp 430– (1949) · Zbl 0038.23502 [3] DOI: 10.2307/2048323 · Zbl 0741.32009 · doi:10.2307/2048323 [4] Rubel, Michigan Math J. 20 pp 289– (1973) [5] Forster, Lectures on Riemann Surfaces (1981) · doi:10.1007/978-1-4612-5961-9 [6] DOI: 10.1007/BF01360812 · Zbl 0179.11402 · doi:10.1007/BF01360812 [7] Luecking, Complex analysis: a functional analysis approach (1984) · Zbl 0546.30002 [8] Gauthier, Canadian J. Math 28 pp 885– (1976) · Zbl 0321.46032 · doi:10.4153/CJM-1976-085-x [9] Gauthier, Canadian J. Math 27 pp 1110– (1975) · Zbl 0322.46006 · doi:10.4153/CJM-1975-116-x [10] Hayman., Meromorphic functions (1964) · Zbl 0115.06203 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.