Abstract
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.
Similar content being viewed by others
References
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)
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)
Akiyama S., Borbély T., Brunotte H., Pethő A., Thuswaldner J.: Generalized radix representations and dynamical systems I. Acta Math. Hung. 108, 207–238 (2005)
Akiyama S., Pethő A.: On canonical number systems. Theor. Comput. Sci. 270, 921–933 (2002)
Akiyama S., Rao H.: New criteria for canonical number systems. Acta Arith. 111/1, 5–25 (2004)
Barbé A., Haeseler F.: Binary number systems for \({\mathbb{Z}^k}\) . J. Number Theory 117/1, 14–30 (2006)
Boyd D.W.: Pisot and salem numbers in intervals of the real line. Math. Comp. 32, 1244–1260 (1978)
Brunotte H.: Characterization of CNS polynomials. Acta Sci. Math. (Szeged) 68, 673–679 (2002)
Brunotte H.: On trinomial bases of radix representations of algebraic integers. Acta Sci. Math. (Szeged) 67, 407–413 (2001)
Burcsi P., Kovács A.: algorithm checking a necessary condition of number system constructions. Ann. Univ. Sci. Budapest. Sect. Comput. 25, 143–152 (2005)
Burcsi, P., Kovács, A.: Zs. Papp-Varga, Decision and classification algorithms for generalized number systems. Ann. Univ. Sci. Budapest. Sect. Comput.
Chamfy Ch.: Fonctions méromorphes dans le cercle-unité et leurs séries de Taylor. Ann. Inst. Fourier (Grenoble) 8, 211–262 (1958)
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), 69–92
Gilbert W.J.: Radix representation of quadratic fields. J. Math. Anal. Appl. 83, 264–274 (1981)
Kátai I., Kovács B.: Canonical number systems in imaginary quadratic fields. Acta Math. Hungar. 37, 159–164 (1981)
Kátai I., Kovács B.: Kanonische Zahlensysteme bei reelen quadratischen Zahlen. Acta Sci. Math. (Szeged) 42, 99–107 (1980)
Kovács B.: Canonical number systems in algebraic number fields. Acta Math. Hungar. 37, 405–407 (1981)
Kovács A.: Generalized binary number systems. Ann. Univ. Sci. Budapest. Sect. Comput. 20, 195–206 (2001)
Kovács A.: Number expansion in lattices. Math. Comput. Model. 38, 909–915 (2003)
Kovács, A. Kornafeld, Á., Burcsi, P.: The power of a supercomputer without a supercomputer—project BinSYS (in hungarian). Networkshop 2006, Miskolc, pp. 1–8. http://nws.iif.hu/ncd2006 (2006)
Lehmer D.H.: A machine method for solving polynomial equations. J. ACM 2, 151–162 (1961)
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)
Ralston A.: A First Course in Numerical Analysis. McGraw-Hill Book Co., New York (1965)
Scheicher K., Thuswaldner J.M.: On the characterization of canonical number systems. Osaka J. Math. 41/2, 327–351 (2004)
Schur I.: Über Potenzreihen die im Inneren des Einheitskreises beschrankt sind . J. Reine Angew. Math. 147, 205–232 (1917)
Schur I.: Über Potenzreihen die im Inneren des Einheitskreises beschrankt sind. J. Reine Angew. Math. 148, 128–145 (1918)
SZTAKI Desktop Grid. http://szdg.lpds.sztaki.hu/szdg/
Author information
Authors and Affiliations
Corresponding author
Additional information
Peter Burcsi was supported by the ELTE-Ericsson Communication Networks Laboratory. The research of Attila Kovács was supported by Bolyai Stipendium, and partly by NKTH RET14/2005 grant under Péter Pázmány program.
Rights and permissions
About this article
Cite this article
Burcsi, P., Kovács, A. Exhaustive search methods for CNS polynomials. Monatsh Math 155, 421–430 (2008). https://doi.org/10.1007/s00605-008-0005-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-008-0005-y