We investigate various number system constructions. After summarizing earlier results we prove that for a given lattice Λ
and expansive matrix M: Λ → Λ if ρ(M−1) < 1/2 then there always exists a suitable digit set D for which (Λ, M, D) is a number system. Here ρ means the spectral radius of M−1. We shall prove further that if the polynomial f(x) = c0 + c1x + ··· + ckxk ∈ Z[x], ck = 1 satisfies the condition |c0| > 2 Σ
|ci| then there is a suitable digit set D for which (Zk, M, D) is a number system, where M is the companion matrix of f(x).
Authors:Shigeki Akiyama, Horst Brunotte, and Attila Pethő
The concept of a canonical number system can be regarded as a natural generalization 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.