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

Abstract: Let $B(X)=\sum_{n=0}^{\infty}b(n)X^n$ represent a rational function in $\mathbb Q(X)$ and suppose that $b(n)$ is a perfect $d$-th power for all large $n\in\mathbb N$. Pisot's $d$-th root conjecture states that one can choose a $d$-th root $a(n)$ of $b(n)$ such that $A(X):=\sum a(n)X^n$ is again a rational function. In this talk, we propose a function-field analog of Pisot's $d$-th root conjecture and prove it under some ``non-triviality'' assumption. We relate the problem to a result of Pasten-Wang on Buchi's $d$-th power problem and develop a function-field analog of an GCD estimate in a recent work of Levin-Wang. This is a joint work with Ji Guo and Chia-Liang Sun.

number theory

Audience: researchers in the topic

Number Theory Online Conference 2020