Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2012; 49(1): 127-133

Printed January 1, 2012

https://doi.org/10.4134/BKMS.2012.49.1.127

Copyright © The Korean Mathematical Society.

Beta-expansions with Pisot bases over $\mathbb{F} _{q}((x^{-1}))$

Mohamed Hbaib

Facult\'e des Sciences de Sfax

Abstract

It is well known that if the $\beta$-expansion of any nonnegative integer is finite, then $\beta$ is a Pisot or Salem number. We prove here that in $\mathbb{F} _{q}((x^{-1}))$, the $\beta$-expansion of the polynomial part of $\beta$ is finite if and only if $\beta$ is a Pisot series. Consequently we give an other proof of Scheicher theorem about finiteness property in $\mathbb{F} _{q}((x^{-1}))$. Finally we show that if the base $\beta$ is a Pisot series, then there is a bound of the length of the fractional part of $\beta$-expansion of any polynomial $P$ in $\mathbb{F} _{q}[x]$.

Keywords: formal power series, $\beta$-expansion, Pisot series

MSC numbers: 11R06, 37B50