Subword complexity and Laurent series with coefficients in a finite field

الأصل · EN

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 ₂(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.