Arithmetics of β−expansions in Fq((x −1 ))

Abstract

The aim of this study is to give some arithmetic properties on the set of β-
polynomials i.e. the set of series whose β-expansion has not fractional part, where

|β| > 1 is an algebraic formal power series over the finite field Fq. We will give the
sufficient conditions over β to have the set L

is finite, where L

designates the
maximal finite shift after the comma for the product of two β-polynomials.