Automatic β -expansions of formal Laurent series over finite fields

Rational digit systems over finite fields and Christol's Theorem

“…This is true also for digit systems over finite fields: although they often behave similar to their "cousins" defined over Z and R, they also show completely different properties that have no analogue in the integer case. This is highlighted for instance by the p-automaticity results by Rigo [19], Rigo and Waxweiler [20] and, more recently, by Scheicher and Sirvent [23]. The specialty of the digit systems studied in the present paper consists in the fact that they have rational functions as bases and, hence, form the analogues of the rational based digit systems studied by Akiyama et al [3].…”

“…As A is discrete, this is well defined and makes F q ((A)) a topological ring. As mentioned in [23], F q ((A)) is not complete as a topological space. However, the following weaker result holds.…”

“…Formulas of a similar kind have been worked out for β-expansions of formal Laurent series over finite fields in [23]. Even if we use some of the methods developed there, the situation here is more intricate as the expansion graph T (P/Q) has a very difficult structure.…”

“…For this we define the notion of an A-Laurent series (see also [23]). Let x, y ∈ Z 2 be linearly independent.…”

“…Although the anchor was defined in a slightly different way in [23], the following lemma can be proved in the same manner as [23,Lemma 4.10]. , with m := deg P , n := deg Q, and 0 ≤ n < m. We shall now consider the polynomial f = Qb − P .…”