BETA-EXPANSIONS WITH PISOT BASES OVER Fq((x-1))

Abstract

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