Monatshefte für Mathematik

, 155:421 | Cite as

Exhaustive search methods for CNS polynomials



In this paper, we present a method for finding all expansive polynomials with a prescribed degree n and constant term c 0. Our research is motivated by the fact that expansivity is a necessary condition for number system constructions. We use the algorithm for an exhaustive search of CNS polynomials for small values of n and c 0. We also define semi-CNS polynomials and show that producing them the same search method can be used.


Canonical number system Expansive polynomial Generalized binary number system 

Mathematics Subject Classification (2000)



  1. 1.
    Akiyama, S., Borbély, T., Brunotte, H., Pethő, A., Thuswaldner, J.: On a generalization of the radix representation—a survey. In: High Primes and Misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, Fields Inst. Commun., vol. 41, pp. 19–27 (2004)Google Scholar
  2. 2.
    Akiyama S., Brunotte H., Pethő A.: Cubic CNS-polynomials, notes on a conjecture of W.J. Gilbert. J. Math. Anal. Appl. 281, 402–415 (2003)MATHMathSciNetGoogle Scholar
  3. 3.
    Akiyama S., Borbély T., Brunotte H., Pethő A., Thuswaldner J.: Generalized radix representations and dynamical systems I. Acta Math. Hung. 108, 207–238 (2005)MATHCrossRefGoogle Scholar
  4. 4.
    Akiyama S., Pethő A.: On canonical number systems. Theor. Comput. Sci. 270, 921–933 (2002)MATHCrossRefGoogle Scholar
  5. 5.
    Akiyama S., Rao H.: New criteria for canonical number systems. Acta Arith. 111/1, 5–25 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Barbé A., Haeseler F.: Binary number systems for \({\mathbb{Z}^k}\) . J. Number Theory 117/1, 14–30 (2006)CrossRefGoogle Scholar
  7. 7.
    Boyd D.W.: Pisot and salem numbers in intervals of the real line. Math. Comp. 32, 1244–1260 (1978)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Brunotte H.: Characterization of CNS polynomials. Acta Sci. Math. (Szeged) 68, 673–679 (2002)MATHMathSciNetGoogle Scholar
  9. 9.
    Brunotte H.: On trinomial bases of radix representations of algebraic integers. Acta Sci. Math. (Szeged) 67, 407–413 (2001)MathSciNetGoogle Scholar
  10. 10.
    Burcsi P., Kovács A.: algorithm checking a necessary condition of number system constructions. Ann. Univ. Sci. Budapest. Sect. Comput. 25, 143–152 (2005)MATHMathSciNetGoogle Scholar
  11. 11.
    Burcsi, P., Kovács, A.: Zs. Papp-Varga, Decision and classification algorithms for generalized number systems. Ann. Univ. Sci. Budapest. Sect. Comput.Google Scholar
  12. 12.
    Chamfy Ch.: Fonctions méromorphes dans le cercle-unité et leurs séries de Taylor. Ann. Inst. Fourier (Grenoble) 8, 211–262 (1958)MATHMathSciNetGoogle Scholar
  13. 13.
    Dufresnoy, J., Pisot, Ch.: Étude de certaines fonctions méromorphes born’ees sur le cercle unit’e. Application à un ensemble fermé d’entiers alg’ebriques. Ann. Sci. École Norm. Sup. (3) 72(55), 69–92Google Scholar
  14. 14.
    Gilbert W.J.: Radix representation of quadratic fields. J. Math. Anal. Appl. 83, 264–274 (1981)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kátai I., Kovács B.: Canonical number systems in imaginary quadratic fields. Acta Math. Hungar. 37, 159–164 (1981)MATHCrossRefGoogle Scholar
  16. 16.
    Kátai I., Kovács B.: Kanonische Zahlensysteme bei reelen quadratischen Zahlen. Acta Sci. Math. (Szeged) 42, 99–107 (1980)MATHMathSciNetGoogle Scholar
  17. 17.
    Kovács B.: Canonical number systems in algebraic number fields. Acta Math. Hungar. 37, 405–407 (1981)MATHCrossRefGoogle Scholar
  18. 18.
    Kovács A.: Generalized binary number systems. Ann. Univ. Sci. Budapest. Sect. Comput. 20, 195–206 (2001)MATHMathSciNetGoogle Scholar
  19. 19.
    Kovács A.: Number expansion in lattices. Math. Comput. Model. 38, 909–915 (2003)MATHCrossRefGoogle Scholar
  20. 20.
    Kovács, A. Kornafeld, Á., Burcsi, P.: The power of a supercomputer without a supercomputer—project BinSYS (in hungarian). Networkshop 2006, Miskolc, pp. 1–8. (2006)
  21. 21.
    Lehmer D.H.: A machine method for solving polynomial equations. J. ACM 2, 151–162 (1961)CrossRefGoogle Scholar
  22. 22.
    Pethő A.: On a polynomial transformation and its application to the construction of a public key cryptosystem. In: Pethő, A., Pohst, M., Zimmer, H.G., Williams, H.C.(eds) Comput. Number Theory, Proc, pp. 31–43. Walter de Gruyter Publ., Comp., New York (1991)Google Scholar
  23. 23.
    Ralston A.: A First Course in Numerical Analysis. McGraw-Hill Book Co., New York (1965)MATHGoogle Scholar
  24. 24.
    Scheicher K., Thuswaldner J.M.: On the characterization of canonical number systems. Osaka J. Math. 41/2, 327–351 (2004)MathSciNetGoogle Scholar
  25. 25.
    Schur I.: Über Potenzreihen die im Inneren des Einheitskreises beschrankt sind . J. Reine Angew. Math. 147, 205–232 (1917)Google Scholar
  26. 26.
    Schur I.: Über Potenzreihen die im Inneren des Einheitskreises beschrankt sind. J. Reine Angew. Math. 148, 128–145 (1918)Google Scholar
  27. 27.
    SZTAKI Desktop Grid.

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Computer AlgebraEötvös Loránd UniversityBudapestHungary

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