Combinatorical and Algebraic Structures Seminar

Session details

Date: 6.10.2015
Speaker: Klaus Scheicher, IMA, Universitat für Bodenkultur Wien
Title: On automaticity of certain digit expansions of Laurent series
Abstract: (joint work with Jörg Thuswaldner)
Let $P, Q\in\mathbb{F}_q[X]\setminus\{0\}$ be two coprime polynomials over the finite field $\mathbb{F}_q$ with $\deg P > \deg Q$. We represent each polynomial $w$ over $\mathbb{F}_q$ by \[ w=\sum_{i=0}^k\frac{s_i}{Q}{\Big(\frac{P}{Q}\Big)}^i \] using a rational \emph{base} $P/Q$ and \emph{digits} $s_i\in\mathbb{F}_q[X]$ satisfying $\deg s_i < \deg P$. \emph{Digit expansions} of this kind can be extended to the field of Laurent series over $\mathbb{F}_q$. We prove uniqueness properties of these expansions and show that their digits can be described by finite automata. Although the language of the possible digit strings is not regular we are able to characterize the digit expansions of algebraic elements. In particular, we give a version of Christol's Theorem by showing that the digit string of a digit expansion is automatic if and only if the expanded element is algebraic over $\mathbb{F}_q[X]$. Finally, we study relations between digit expansions and a finite fields version of Mahler's $3/2$-problem.

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