 
Summary: NEW CRITERIA FOR CANONICAL NUMBER SYSTEMS
SHIGEKI AKIYAMA AND HUI RAO
Abstract. Let P(x) = xd
+ pd1xd1
+ · · · + p0 be an expanding monic
polynomial with integer coefficients. If each element of Z[x]/P(x)Z[x] has a
polynomial representative with coefficients in [0, p0  1] then P(x) is called
a canonical number system generating polynomial, or a CNS polynomial in
short. A method due to Hollander [6] is employed to study CNS polynomials.
Several new criteria for canonical number system generating polynomials are
given and a conjecture of S.Akiyama & A.Petho [3] is proved. The known
results, especially an algorithm of H. Brunotte's in [4] and a recent work of K.
Scheicher & J.M.Thuswaldner [15], can be derived by this new method in a
simpler way.
1. Introduction
Let P(x) = pdxd
+ pd1xd1
+ · · · + p0 be a polynomial of x with integer
coefficients and pd = 1. Let R be the quotient ring Z[x]/P(x)Z[x]. As a Z
module, R is naturally isomorphic to Zd
