ON PERIODIC CONTINUED FRACTIONS OVER $\mathbb{F}_q((X^{−1}))$

Abstract

Let $\mathbb{F}_q$ be a field with $q$ elements of characteristic $p$ and $\mathbb{F}_q((X^{−1}))$ be the field of formal power series over $\mathbb{F}_q$. Let $f$ be a quadratic formal power series of continued fraction expansion $[b_0; b_1, \ldots, b_s, \overline{a_1, \ldots, a_t}]$, we denote by $t = \operatorname{Per}(f)$ the period length of the partial quotients of $f$. The aim of this paper is to study the continued fraction expansion of $Af$ where $A$ is a polynomial $\in \mathbb{F}_q[X]$. In particular we study the asymptotic behavior of the functions \[ S(N,n) = \sup_{\operatorname{deg} A = N} \sup_{f \in \Lambda_{n}} \operatorname{Per}(Af) \quad \textrm{and} \quad R(N) = \sup_{n \geq 1} \frac{S(N,n)}{n}, \] where $\Lambda_{n}$ is the set of quadratic formal power series of period $n$ in $\mathbb{F}_q((X^{−1}))$.