URS and bi-URS for Meromorphic Functions in a non-Archimedean Field
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RESEARCH ARTICLES
URS and bi-URS for Meromorphic Functions in a non-Archimedean Field∗ H. H. Khoai1** and V. H. An2, 3*** 1
Thang Long Institute of Mathematics and Applied Sciences, Hanoi, Vietnam 2 Hai Duong Pedagogical College, Hai Duong, Vietnam 3
Thang Long Institute of Mathematics and Applied Sciences, Vietnam
Received September 8, 2020; in final form, October 2, 2020; accepted October 2, 2020
Abstract—Let K be an algebraically closed field of characteristic zero, complete for a nonArchimedean absolute value. In this paper, we give a new class of unique range sets for meromorphic functions on K. We also show the existence of a bi − U RS for M(K) of the form ({a1 , a2 , a3 , a4 , a5 , ∞}), which is different from A. Boutabaa-A. Escassut’s [3]. DOI: 10.1134/S2070046620040020 Key words: Non-Archimedean, meromorphic function, unique range sets.
1. INTRODUCTION Let K be an algebraically closed field of characteristic zero, complete for a non-Archimedean absolute value. We denote by A(K) the ring of entire functions in K, by M(K) the field of meromorphic functions, = K ∪ {∞}. We assume that the reader is familiar with the i.e., the field of fractions of A(K), and K notations in the non-Archimedean Nevanlinna theory (see [2, 9–11, 16]). Let f be a non-constant Denote by Ef (a) the set of a-points of f , meromorphic function on K, and let S be a subset of K. each a-point is counted with its multiplicity, and set Ef (S) = ∪a∈S Ef (a). Two non-constant meromorphic functions f, g are said to share S, counting multiplicities (share S CM) if Ef (S) = Eg (S). If for any two non-constant meromorphic functions f, g, sharing S CM, must be identical, then S is called a unique range set for meromorphic functions or, in brief, a URSM. In the same way, a couple of sets S, T ⊂ K ∪ {∞}, such that S ∩ T = ∅, will be called a bi − U RS for M(K) if the conditions Ef (S) = Eg (S) and Ef (T ) = Eg (T ) imply f = g for any two nonconstant meromorphic functions f and g. Several interesting results on unique range sets for entire and meromorphic functions on K, and bi-URSM have been obtained (see, for example, [3, 5, 12, 14, 15]). So far, the smallest unique range set for meromorphic functions with 10 elements was given by Hu and Yang ([14]). In [3], A. Boutabaa and A. Escassut proved that, there exists a bi-URS for M (K) of the form ({a1 , a2 , a3 , a4 , a5 , ∞}), and in [11] it is established the existence of bi-URS’s for M (K) of the form ({a1 , a2 , a3 , a4 , ∞}). In this paper, we give a new class of unique range sets for meromorphic functions on K. This is a generalization of some results due to P. C. Hu - C. C. Yang ([15]). We also show the existence of a biURS for M(K) of the form ({a1 , a2 , a3 , a4 , a5 , ∞}) which is different from A. Boutabaa-A. Escassut’s [3]. Now let us describe the main results of the paper. ∗
The text was submitted by the authors in English. E-mail: hhkhoai@@math.ac.vn *** E-mail: vuhoaianmai@@yahoo.com **
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URS AND bi-URS
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Let m1 , m2 be positive integers. Consider the following polynomial: m2
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