On Pisots $d$-th root conjecture for function fields and related GCD estimates

We propose a function-field analog of Pisots $d$-th root conjecture on linear recurrences, and prove it under some non-triviality assumption. Besides a recent result of Pasten-Wang on B{u}chis $d$-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.