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Summary: NEW CRITERIA FOR CANONICAL NUMBER SYSTEMS
SHIGEKI AKIYAMA AND HUI RAO
Abstract. Let P (x) = x d + p d 1 x d 1 + + p 0 be an expanding monic
polynomial with integer coeÆcients. If each element of Z[x]=P (x)Z[x] has a
polynomial representative with coeÆcients in [0; jp 0 j 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.Peth}o [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) = p d x d + p d 1 x d 1 + + p 0 be a polynomial of x with integer
coeÆcients and p d = 1. Let R be the quotient ring Z[x]=P (x)Z[x]. As a Z-
module, R is naturally isomorphic to Z d and each element of R is represented
uniquely in the form
(1)
d 1
X
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