Rational digit systems over finite fields and Christol's Theorem

Abstract

Let P,Q∈Fq[X]∖{0} be two coprime polynomials over the finite field Fq with deg⁡P>deg⁡Q. We represent each polynomial w over Fq byw=∑i=0ksiQ(PQ)i using a rational baseP/Q and digitssi∈Fq[X] satisfying deg⁡si<deg⁡P. Digit expansions of this type are also defined for formal Laurent series over Fq. We prove uniqueness and automatic properties of these expansions. Although the ω-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 Fq[X]. Finally, we study relations between digit expansions of formal Laurent series and a finite fields version of Mahler's 3/2-problem.