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arXiv · 1004.3729

Digit systems over commutative rings

Klaus Scheicher·Paul Surer·Jörg M. Thuswaldner·Christiaan E. van de Woestijne
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Abstract

Let $\E$ be a commutative ring with identity and $P\in\E[x]$ be a polynomial. In the present paper we consider digit representations in the residue class ring $\E[x]/(P)$. In particular, we are interested in the question whether each $A\in\E[x]/(P)$ can be represented modulo $P$ in the form $e_0+e_1 X + \cdots + e_h X^h$, where the $e_i\in\E[x]/(P)$ are taken from a fixed finite set of digits. This general concept generalises both canonical number systems and digit systems over finite fields. Due to the fact that we do not assume that $0$ is an element of the digit set and that $P$ need not be monic, several new phenomena occur in this context.

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Klaus Scheicher, Paul Surer, Jörg M. Thuswaldner, Christiaan E. van de Woestijne. 2010-04-21. Digit systems over commutative rings. https://arxiv.org/abs/1004.3729

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