Arithmetics of β- expansions in F q (( x-1 ))

The aim of this study is to give some arithmetic properties on the set of beta-polynomials in Fq((x(-1))) i.e. the set of series whose beta-expansion has not fractional part, where |beta| > 1 is an algebraic formal power series over the finite field Fq. We will give sufficient conditions over beta to have the quantity L-circle dot is finite, where L-circle dot designates the maximal finite shift after the comma for the product of two beta-polynomials.