BETA-EXPANSIONS WITH PISOT BASES OVER Fq((x-1))
Mohamed Hbaib · Bulletin of the Korean Mathematical Society · 2012
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 $\mathbb{F}_q((x^{-1}))$ , 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 Scheiche 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 $\mathbb{F}_q[x]$ .