The analogue of B\"uchi's problem for function fields

TL;DR

This paper extends classical problems about squares and powers to function fields, proving new results that confirm the existence of bounds and solutions in characteristic zero and positive characteristic cases.

Contribution

It proves that Hensley's problem for r-th powers has a positive answer over function fields of characteristic zero, and improves bounds for B"uchi's problem in positive characteristic.

Findings

Hensley's problem has a positive answer in characteristic zero for any r.

B"uchi's problem has a positive answer if the characteristic p is at least 312g+169.

Improves previous results by Pasten, Vojta, and Pheidas.

Abstract

B\"uchi's $n$ Squares Problem asks for an integer $M$ such that any sequence $(x_0,...,x_{M-1})$, whose second difference of squares is the constant sequence $(2)$ (i.e. $x^2_n-2x^2_{n-1}+x_{n-2}^2=2$ for all $n$), satisfies $x_n^2=(x+n)^2$ for some integer $x$. Hensley's problem for $r$-th powers (where $r$ is an integer $\geq2$) is a generalization of B\"{u}chi's problem asking for an integer $M$ such that, given integers $\nu$ and $a$, the quantity $(\nu+n)^r-a$ 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 B\"uchi's problem has a positive answer if $p\geq 312g+169$ (improving results by Pheidas and the second author).