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arXiv · 1001.2548

Subword complexity and Laurent series with coefficients in a finite field

Alina Firicel
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Abstract

Decimal expansions of classical constants such as $\sqrt2$, $π$ 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 $π$, $Π_q$, has in general a linear complexity, except in the case $q=2$, when the complexity is quadratic. In particular, this implies the transcendence of $Π_2$ 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.

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Alina Firicel. 2010-01-14. Subword complexity and Laurent series with coefficients in a finite field. https://arxiv.org/abs/1001.2548

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