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We propose a function-field analog of Pisot's $d$-th root conjecture onlinear 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, ourmain tool, which is also developed in this paper, is a function-field analog ofan GCD estimate in a recent work of Levin and Levin-Wang. As an easy corollaryof such GCD estimate, we also obtain an asymptotic result.

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