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

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 Büchi'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.

Vojta's abc Conjecture for algebraic tori and applications over Function Fields

“…Remarks on the proof of Theorem 5. In [11], we have proved the following gcd theorem. (a) N S,gcd (F (g 1 , .…”

“…, g n ) ∈ (O * S ) n does not belong to Z in the above theorem. Theorem 5 can be derived from the proof of [11,Theorem 8] by replacing the application of [11,Theorem 25] by Theorem 31 in Section 5. We will give more details in Section 5.3.…”

“…Next, we recall some definitions and results from [11]. From now on, we will fix a t satisfying the conditions in Proposition 15 and use the notation η ′ := dη dt for η ∈ K. For the convenience of discussion, we will use the following convention.…”

“…The following is a preliminary version of Theorem 6. We also use this theorem to deduce Theorem 5 from the proof of [11,Theorem 8].…”

“…In the function field setting, the connections between gcd and Vojta's conjectures still exist, and even lead to stronger and unconditional results with techniques that are not available in number fields. Our results in this article are examples of such connections: We first formulate a function field version of Levin's gcd theorem in [17] based on our previous work in [11] and a version of Schmidt's subspace theorem over function fields reproduced in this paper. We then develop a function field version of Vojta's generalized abc conjecture (see [29,Conjecture 23.4 (b)]) for algebraic tori, which is much deeper than Vojta's main conjecture.…”