Rational digit systems over finite fields and Christol's Theorem

Abstract

Let $P,Q\in {\mathbb{F}}_q[X]\setminus \{0\}$ be two coprime polynomials over the finite field ${\mathbb{F}}_q$ with $\mathrm{deg}P>\mathrm{deg}Q$. We represent each polynomial $w$ over ${\mathbb{F}}_q$ by \[w=\sum_{i=0}^k\frac{s_i}{Q}{\left(\frac{P}{Q}\right)}^i\] using a rational base $P/Q$ and digits $s_i\in {\mathbb{F}}_q[X]$ satisfying $\mathrm{deg}{s}_i<\mathrm{deg}P$. Digit expansions of this type are also defined for formal Laurent series over ${\mathbb{F}}_q$. We prove uniqueness and automatic properties of these expansions. Although the $\omega$-language of the possible digit strings is not regular, we are able to characterize the digit expansions of algebraic elements. In particular, we give a version of Christol's Theorem by showing that the digit string of the digit expansion of a formal Laurent series is automatic if and only if the series is algebraic over ${\mathbb{F}}_q[X]$. Finally, we study relations between digit expansions of formal Laurent series and a finite fields version of Mahler's $3/2$-problem.

26 pages, 3 figures