Subword complexity and Laurent series with coefficients in a finite field

TL;DR

This paper investigates the subword complexity of Laurent series over finite fields, especially Carlitz's analogs of constants like pi, revealing their complexity patterns and implications for transcendence.

Contribution

It establishes the complexity behavior of Carlitz's analog of pi, showing linear complexity generally and quadratic at q=2, and explores properties of series with low complexity.

Findings

Inverse of Carlitz's pi analog has linear complexity for most q

At q=2, the complexity of the series is quadratic

Series with polynomial complexity or zero entropy have closure properties

Abstract

Decimal expansions of classical constants such as $\sqrt2$, $\pi$ and $\zeta(3)$ 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 $\pi$, $e$ or $\zeta(3)$. 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 $\pi$, $\Pi_q$, has in general a linear complexity, except in the case $q=2$, when the complexity is quadratic. In particular, this implies the transcendence of $\Pi_2$ over $\F_2(T)$. 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.