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Canonical number systems in imaginary quadratic fields. (English) Zbl 0477.10012

Let \(R=R(\sqrt{-N})\) be an imaginary quadratic field, \(N(\cdot)\) be the norm function, \(\mathcal N_0 = \mathcal N_0(\alpha) =\{0,1,\dots, N(\alpha) -1\}\) for \(\alpha\in R\). We say that \((\alpha, \mathcal N_0)\) is a canonical number system (CNS) if every \(\beta\in R\) can be represented in the form \(\beta = \sum_{i=0}^k a_i \alpha^i\) \((a_i\in \mathcal N_0)\) uniquely. All the CNS are determined (Theorem 1). Furthermore, the following assertion is proved. If \((\alpha, \mathcal N_0)\) is a CNS, then every complex number \(z\) can be written as \(z= \sum_{i=k}^{-\infty}a_i \alpha^i\) \((a_i\in \mathcal N_0\), \(i=k, k-1, \ldots)\) (Theorem 2).

MSC:

11A63 Radix representation; digital problems
11R11 Quadratic extensions
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References:

[1] D. E. Knuth,The art of computer programming. Vol. 2, Addison-Wesley P. C. (London, 1971).
[2] I. Kátai-J. Szabó, Canonical number systems for complex integers,Acta Sci. Math. Szeged,37 (1975), 255–260. · Zbl 0297.12003
[3] I. Kátai-B. Kovács, Kanonische Zahlensysteme in der Theorie der quadratischen algebraischen Zahlen,Acta Sci. Math. Szeged (in print).
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