Arithmetics in the set of beta-polynomials

Abstract

Let be a formal series with deg( ) 2, the aim of this paper is to prove that the maximal length of the nite -fractional parts in the -expansion of product of two beta-polynomials (a formal series that have not -fractional part), denoted L ( ) is nite when is Pisot or Salem series. Especially, we give its exact value if have one conjugate with absolute value smaller than 1 j j and if is a Pisot series verifying d +Ad 1 d 1 + +A0 = 0 such that deg( ) = m 2 and deg(A0) = s deg(Ai) 80 i d 2.