Reducible cubic CNS polynomials
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REDUCIBLE CUBIC CNS POLYNOMIALS ˝3 Shigeki Akiyama1 , Horst Brunotte2 and Attila Petho [Communicated by Andr´ as S´ ark¨ ozy] 1
Department of Mathematics, Faculty of Science, Niigata University Ikarashi 2-8050, Niigata 950-2181, Japan E-mail: [email protected] 2
3
Haus-Endt-Strasse 88, D-40593 D¨ usseldorf, Germany E-mail: [email protected]
Department of Computer Science, University of Debrecen P.O. Box 12, H-4010 Debrecen, Hungary E-mail: [email protected] (Received April 29, 2007; Accepted August 7, 2007)
Abstract
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.
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 = X d + pd−1 X d−1 + · · · + p0 with p0 = 0. P is Mathematics subject classification number : 11A63, 12D99. Key words and phrases: CNS polynomial, canonical number system, radix representation. 1 Supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology, Grand-in Aid for fundamental research 18540022, 2006–2008. 3 Supported partially by the Hungarian NFSR Grant No. K67580. 0031-5303/2007/$20.00
c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
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˝ S. AKIYAMA, H. BRUNOTTE and A. PETHO
called a CNS polynomial (see [18]) if for every A ∈ Z[X] there exist a0 , . . . , a ∈ {0, 1, . . . , |p0 | − 1} such that A ≡ a0 + a1 X + · · · + a X
(modP ).
In this case, the pair (α, {0, 1, . . . , |P (0)| − 1}) is called a canonical number system (CNS) where α is a root of P . As the main ingredient of a canonical number system is the CNS polynomial P we restrict our attention to CNS polynomials. The characterization of linear and quadratic CNS polynomials is well-known (see e.g. [14], [13], [10], [11]), however, for higher degrees only partial results have been achieved (see e.g. [15], [14], [13], [5], [6], [21], [4], [20], [9]). An important class of reducible CNS polynomials of arbitrary degrees has systematically been studied by Peth˝ o [19] in connection with integral interpolation. Similar investigations have been performed by Kane [12]. In particular, the complete description of cubic CNS polynomials is still an open problem. Therefore, the characteri
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