Subword Complexity and Laurent Series with Coefficients in a Finite Field

Decimal expansions of classical constants such as √ 2, π and ζ(3) have long been a source of difficult questions. In the case of Laurent series with coefficients in a finite field, where no carry-over difficulties appear, the situation seems to be simplified and drastically different. On the other hand, Carlitz introduced analogs of real numbers such as π, e or ζ(3). Hence, it became reasonable to enquire how “complex” the Laurent representation of these “numbers” is. In this paper we prove that the inverse of Carlitz’s analog of π, Πq, has in general a linear complexity, except in the case q = 2, when the complexity is quadratic. In particular, this implies the transcendence of Π2 over F2(T ). In the second part, we consider the classes of Laurent series of at most polynomial complexity and of zero entropy. We show that these satisfy some nice closure properties.