Pisot’s dth root conjecture, proved by Zannier in 2000, can be stated as follows. Let b be a linear recurrence over a number field k, and d ≥ 2 be an integer. Suppose that b(n) is the dth power of some element in k for all but finitely many n. Then there exists a linear recurrence a over k such that a(n)d = b(n) for all n.
In this talk, we propose a function-field analogue of this result and prove it under some ‘non-triviality’ assumption. We relate the problem to a result of Pasten-Wang on Büchi’s dth power problem and develop a function-field GCD estimate for multivariable polynomials with ‘small coefficients’ evaluating at S-units arguments. We will also discuss its complex analogue in the notion of (generalized Ritt’s) exponential polynomials.
This is a joint work with Ji Guo and Chia-Liang Sun.
This video is part of the Number Theory Web Seminar series.
