The analogue of Büchi's Problem for function fields

Büchi's n Squares Problem asks for an integer M such that any sequence (x0,…,xM−1), whose second difference of squares is the constant sequence (2) (i.e. for all n), satisfies for some integer x. Hensley's Problem for r-th powers (where r is an integer ⩾2) is a generalization of Büchi's Problem asking for an integer M such that, given integers ν and a, the quantity r(ν+n)−a cannot be an r-th power for M or more values of the integer n, unless a=0. The analogues of these problems for rings of functions consider only sequences with at least one non-constant term.Let K be a function field of a curve of genus g. We prove that Hensley's Problem for r-th powers has a positive answer for any r if K has characteristic zero, improving results by Pasten and Vojta. In positive characteristic p we obtain a weaker result, but which is enough to prove that Büchi's Problem has a positive answer if p⩾312g+169 (improving results by Pheidas and the second author).