Rational digit systems over finite fields and Christol's Theorem

Let P, Q is an element of F-q[X] \ {0} be two coprime polynomials over the finite field Fq with deg P > deg Q. We represent each polynomial w over F-q by w = Sigma(k)(i=0)s(i/)Q (P/Q)(i) using a rational base P/Q and digits s(i) is an element of F-q[X] satisfying deg s(i) < deg P. Digit expansions of this type are also defined for formal Laurent series over 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 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. (C) 2016 Elsevier Inc. All rights reserved.