On Pisot's $d$-th root conjecture for function fields and related GCD estimates

TL;DR

This paper introduces a function-field analog of Pisot's $d$-th root conjecture for linear recurrences, proves it under certain conditions, and develops related GCD estimates with asymptotic applications.

Contribution

It formulates and proves a function-field version of Pisot's conjecture, extending previous work with new GCD estimates and asymptotic results.

Findings

Proved the function-field analog of Pisot's $d$-th root conjecture under non-triviality assumptions.

Developed a new GCD estimate in the function-field setting based on Levin and Levin-Wang's work.

Derived an asymptotic result as a corollary of the GCD estimate.

Abstract

We propose a function-field analog of Pisot's $d$-th root conjecture on linear recurrences, and prove it under some "non-triviality" assumption. Besides a recent result of Pasten-Wang on B{\"u}chi's $d$-th power problem, our main tool, which is also developed in this paper, is a function-field analog of an GCD estimate in a recent work of Levin and Levin-Wang. As an easy corollary of such GCD estimate, we also obtain an asymptotic result.