Hensleyʼs problem for complex and non-Archimedean meromorphic functions

Büchiʼs problem asks if there exists a positive integer M such that all x 1 , … , x M ∈ Z satisfying the equations x r 2 − 2 x r − 1 2 + x r − 2 2 = 2 for all 3 ⩽ r ⩽ M must also satisfy x r 2 = ( x + r ) 2 for some integer x. Hensleyʼs problem asks if there exists a positive integer M such that, fo...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications Vol. 381; no. 2; pp. 661 - 677
Main Authors: An, Ta Thi Hoai, Wang, Julie Tzu-Yueh
Format: Journal Article
Language:English
Published: Amsterdam Elsevier Inc 15.09.2011
Elsevier
Subjects:
Buchiʼs problem
Exact sciences and technology
Functions of a complex variable
Hensleyʼs problem
Hilbertʼs tenth problem
Mathematical analysis
Mathematics
Meromorphic functions
Nevanlinna theory
Non-Archimedean meromorphic functions
Partial differential equations
Sciences and techniques of general use
Several complex variables and analytic spaces
Value distribution theory
Hilbertʼs tenth problem
Buchiʼs problem
Value distribution theory
Non-Archimedean meromorphic functions
Hensleyʼs problem
Meromorphic functions
Nevanlinna theory
Complex function
Integer
Mathematical analysis
Distribution function
Hensley's problem
Meromorphic function
Buchi's problem
Hilbert's tenth problem
ISSN:0022-247X, 1096-0813
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Summary:Büchiʼs problem asks if there exists a positive integer M such that all x 1 , … , x M ∈ Z satisfying the equations x r 2 − 2 x r − 1 2 + x r − 2 2 = 2 for all 3 ⩽ r ⩽ M must also satisfy x r 2 = ( x + r ) 2 for some integer x. Hensleyʼs problem asks if there exists a positive integer M such that, for any integers ν and a, if ( ν + r ) 2 − a is a square for 1 ⩽ r ⩽ M , then a = 0 . It is not difficult to see that a positive answer to Hensleyʼs problem implies a positive answer to Büchiʼs problem. One can ask a more general version of the Hensleyʼs problem by replacing the square by n-th power for any integer n ⩾ 2 which is called the Hensleyʼs n-th power problem. In this paper we will solve Hensleyʼs n-th power problem for complex meromorphic functions and non-Archimedean meromorphic functions.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2011.03.025