Arithmetics in number systems with negative base

We study the numeration system with negative basis, introduced by Ito and Sadahiro. We focus on arithmetic operations in the set ${\rm Fin}(-\beta)$ and $\Z_{-\beta}$ of numbers having finite resp. integer $(-\beta)$-expansions. We show that ${\rm Fin}(-\beta)$ is trivial if $\beta$ is smaller than the golden ratio $\frac12(1+\sqrt5)$. For $\beta\geq\frac12(1+\sqrt5)$ we prove that ${\rm Fin}(-\beta)$ is a ring, only if $\beta$ is a Pisot or Salem number with no negative conjugates. We prove the conjecture of Ito and Sadahiro that ${\rm Fin}(-\beta)$ is a ring if $\beta$ is a quadratic Pisot number with positive conjugate. For quadratic Pisot units we determine the number of fractional digits that may appear when adding or multiplying two $(-\beta)$-integers.