 
Summary: REDUCIBLE CUBIC CNS POLYNOMIALS
SHIGEKI AKIYAMA, HORST BRUNOTTE, AND ATTILA PETHO
Abstract. The concept of a canonical number system can be regarded as a natural gener
alization of decimal representations of rational integers to elements of residue class rings of
polynomial rings. Generators of canonical number systems are CNS polynomials which are
known in the linear and quadratic cases, but whose complete description is still open. In the
present note reducible CNS polynomials are treated, and the main result is the characterization
of reducible cubic CNS polynomials.
1. Introduction
Canonical number systems have been introduced as natural generalizations of the classical
decimal representation of the rational integers to algebraic integers. We refer the reader to [7] for
a detailed account on the historical development and the connections of the concept of canonical
number systems to other theories, e.g. shift radix systems, finite automata or fractal tilings.
Let us briefly recall the main definitions for our purposes here. Consider a monic integral
polynomial P = Xd
+ pd1Xd1
+ · · · + p0 with p0 = 0. P is called a CNS polynomial (see [18])
if for every A Z[X] there exist a0, ..., a {0, 1, . . . , p0  1} such that
A a0 + a1X + · · · + a X (mod P).
In this case, the pair (, {0, 1, . . . , P(0)  1}) is called a canonical number system (CNS) where
