HENSLEY'S PROBLEM FOR FUNCTION FIELDS

Abstract

Büchi's square problem asks if there exists a positive integer M such that all x1, …, xM ∈ ℤ satisfying the equations [Formula: see text] for all 3 ≤ r ≤ M must also satisfy [Formula: see text] for some integer x and for all 1 ≤ r ≤ M. Hensley's 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 all 1 ≤ r ≤ M, then a = 0. It is not difficult to see that a positive answer to Hensley's problem implies a positive answer to Büchi's square problem. One can ask a more general version of Hensley's problem by replacing the square power by an nth power for any integer n ≥ 2 which is called Hensley's problem for nth powers. In this paper, we will study Hensley's problem for nth powers over function fields of any characteristic.