LAUSR

Abstract Let P , Q ∈ F q [ X ] ∖ { 0 } be two coprime polynomials over the finite field F q with deg ⁡ P > deg ⁡ Q . We represent each polynomial w over F q by w = ∑ i = 0 k s i Q ( P Q ) i using a rational base P / Q and digits s i ∈ 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 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 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.