Arithmetics on beta-expansions with Pisot bases over $F_q((x^{-1}))$

Abstract

In this paper we consider finite $\beta$-expansions in the field of formal series with Pisot basis $\beta$. We are studying the arithmetic operations on $\beta$-expansions and provide bounds on the number of fractional digits arising in multiplication for arbitrary $\beta$-polynomials noted $L_\odot$. This value is given explicitly for families of Pisot basis. The last part of this paper is devoted to quadratic Pisot series where we will give the exact value for $L_\odot$.