Beta-expansions with Pisot bases over $\mathbb{F} _{q}((x^{-1}))$
Bull. Korean Math. Soc. 2012 Vol. 49, No. 1, 127-133
https://doi.org/10.4134/BKMS.2012.49.1.127
Published online January 1, 2012
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
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