Rational digit systems over finite fields and Christol's Theorem

Abstract

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 byDigit 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.