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

Title & Authors
BETA-EXPANSIONS WITH PISOT BASES OVER Fq((x-1))
Hbaib, Mohamed;

Abstract
It is well known that if the $\small{{\beta}}$-expansion of any nonnegative integer is finite, then $\small{{\beta}}$ is a Pisot or Salem number. We prove here that $\small{\mathbb{F}_q((x^{-1}))}$, the $\small{{\beta}}$-expansion of the polynomial part of $\small{{\beta}}$ is finite if and only if $\small{{\beta}}$ is a Pisot series. Consequently we give an other proof of Scheiche theorem about finiteness property in $\small{\mathbb{F}_q((x^{-1}))}$. Finally we show that if the base $\small{{\beta}}$ is a Pisot series, then there is a bound of the length of the fractional part of $\small{{\beta}}$-expansion of any polynomial P in $\small{\mathbb{F}_q[x]}$.
Keywords
formal power series;$\small{{\beta}}$-expansion;Pisot series;
Language
English
Cited by
1.
Continued $$\beta$$ β -fractions with formal power series over finite fields, The Ramanujan Journal, 2016, 39, 1, 95
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