Hensleyʼs problem for complex and non-Archimedean meromorphic functions

Büchiʼs problem asks if there exists a positive integer M such that all x1,…,xM∈Z satisfying the equations for all 3⩽r⩽M must also satisfy for some integer x. Hensleyʼs problem asks if there exists a positive integer M such that, for any integers ν and a, if 2(ν+r)−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.