Number systems and the Chinese Remainder Theorem

TL;DR

This paper explores polynomial-based number systems and their properties under the Chinese Remainder Theorem, providing conditions for their combination and applications to simultaneous number systems.

Contribution

It establishes precise conditions for combining polynomial residue systems and applies the Chinese Remainder Theorem to develop simultaneous number systems.

Findings

Conditions for polynomial number system combinations

Generalized Chinese Remainder Theorem form

Applications to simultaneous number systems

Abstract

A well-known generalisation of positional numeration systems is the case where the base is the residue class of $x$ modulo a given polynomial $f(x)$ with coefficients in (for example) the integers, and where we try to construct finite expansions for all residue classes modulo $f(x)$, using a suitably chosen digit set. We give precise conditions under which direct or fibred products of two such polynomial number systems are again of the same form. The main tool is a general form of the Chinese Remainder Theorem. We give applications to simultaneous number systems in the integers.