부산대학교 도서관

Let $\alpha\in{\bf Z}$ and let $N=\{\beta_0,\beta_1,\cdots,\beta_s\}$ be a subset of ${\bf Z}$. The couple $\{\alpha,N\}$ is called a canonical system (CS) in ${\bf Z}$ if every $\gamma\in{\bf Z}$ has a unique representation of the form $\gamma=a_0+a_1\alpha+\cdots +a_m\alpha^m$ with $a_i\in N$ and some $m>0$. [For instance if $\alpha=-k<-2$, and $N=\{0,1,\cdots,k-1\}$, then $\{\alpha,N\}$ is a CS for ${\bf Z}$.] \par The authors show: To any $\alpha<-2$ there exist infinitely many subsets $N\subset{\bf Z}$ with ${\rm card}\,N=|\alpha|$ such that $\{\alpha,N\}$ is a CS in ${\bf Z}$. There is an algorithm to decide whether a given $\{\alpha,N\}$ is a CS or not. The problem to find all CS in ${\bf Z}$ remains open. \par The notion introduced above can be generalized. Instead of ${\bf Z}$ we consider a ring $R$, $\alpha\in R$, $N\subset R$, and define in an obvious way a CS in $R$. In particular if $R$ is a ring, $\alpha\in R$, and $N$ is restricted to be the set $N=\{0,1,2,\cdots,n\}$, it is called a canonical number system (CNS) in $R$. A ring may have a CS but not a CNS. A characterisation of rings having a CNS will appear elsewhere.