Klaus Scheicher: Two new kinds of digit system

Let ${F}$ be a finite field and

$$p(x,y)=\sum_{j=0}^n b_j(x)y^j\in {F}[x,y]$$

with $b_n\in{F}\setminus\{0\}$. We define a new kind of digit system in the quotient ring ${F}[x,y]/p(x,y){F}[x,y]$. There are striking analogies of these digit systems to the well known canonical number systems (CNS) defined in $Z[x]/P(x)Z[x]$ for a polynomial $P(x)\in Z[x]$. Especially, the problem of characterizing the possible bases is solved. In contrast, such a characterization for CNS is still outstanding. Furthermore, we will study $\beta$-expansions in $F((x^{-1}))$, the field of formal Laurent series over $F$ in the variable $x^{-1}$. Once again, we can solve problems corresponding to open problems on classical $\beta$-expansions.

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