Digit systems over commutative rings

TL;DR

This paper explores digit representations in residue class rings over commutative rings, generalizing number systems and finite field digit systems, and investigates conditions for such representations without assuming zero digit inclusion or monic polynomials.

Contribution

It introduces a generalized framework for digit systems over commutative rings, extending existing concepts to include non-monic polynomials and digit sets without zero.

Findings

Characterization of digit representations over residue class rings

Extension of number system concepts to non-monic polynomials

Analysis of phenomena arising from non-zero digit sets

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.