On Multiplicative Independent Bases for Canonical Number Systems in Cyclotomic Number Fields

  • Manfred G. Madritsch
  • Paul Surer
  • Volker Ziegler
Chapter
First Online:

Abstract

In the present paper we are interested in number systems in the ring of integers of cyclotomic number fields in order to obtain a result equivalent to Cobham’s theorem. For this reason we first search for potential bases. This is done in a very general way in terms of canonical number systems. In a second step we analyse pairs of bases in view of their multiplicative independence. In the last part we state an appropriate variant of Cobham’s theorem.

2010 Mathematics Subject Classification

11R18 11Y40 11A63 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

We want to explicitly give thanks to the anonymous referees. Due to their hints and suggestions we were able to improve a lot the quality and readability of this article. The second author’s research was supported by the Austrian Research Foundation (FWF), Project P23990.

References

  1. 1.
    B. Adamczewski, J. Bell, An analogue of Cobham’s theorem for fractals. Trans. Am. Math. Soc. 363(8), 4421–4442 (2011). MR2792994Google Scholar
  2. 2.
    S. Akiyama, A. Pethő, On canonical number systems. Theor. Comput. Sci. 270(1–2), 921–933 (2002)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    S. Akiyama, H. Rao, New criteria for canonical number systems. Acta Arith. 111(1), 5–25 (2004). MR2038059 (2005d:11007)Google Scholar
  4. 4.
    S. Akiyama, H. Brunotte, A. Pethő, Cubic CNS polynomials, notes on a conjecture of W.J. Gilbert. J. Math. Anal. Appl. 281(1), 402–415 (2003). MR1980100 (2004j:11009)Google Scholar
  5. 5.
    S. Akiyama, T. Borbély, H. Brunotte, A. Pethő, J.M. Thuswaldner, Generalized radix representations and dynamical systems. I. Acta Math. Hungar. 108(3), 207–238 (2005). MR2162561 (2006i:37023)Google Scholar
  6. 6.
    S. Akiyama, H. Brunotte, A. Pethő, J.M. Thuswaldner, Generalized radix representations and dynamical systems. II. Acta Arith. 121(1), 21–61 (2006). MR2216302Google Scholar
  7. 7.
    S. Akiyama, H. Brunotte, A. Pethő, Reducible cubic CNS polynomials. Period. Math. Hungar. 55(2), 177–183 (2007). MR2375040 (2008m:11058)Google Scholar
  8. 8.
    A. Bertrand-Mathis, Comment écrire les nombres entiers dans une base qui n’est pas entière. Acta Math. Hungar. 54(3–4), 237–241 (1989). MR1029085Google Scholar
  9. 9.
    B. Boigelot, J. Brusten, A generalization of Cobham’s theorem to automata over real numbers. Theor. Comput. Sci. 410(18), 1694–1703 (2009). MR2508527Google Scholar
  10. 10.
    A. Bremner, On power bases in cyclotomic number fields. J. Number Theory 28(3), 288–298 (1988). MR932377Google Scholar
  11. 11.
    H. Brunotte, On trinomial bases of radix representations of algebraic integers. Acta Sci. Math. (Szeged) 67(3–4), 521–527 (2001)Google Scholar
  12. 12.
    H. Brunotte, Characterization of CNS trinomials. Acta Sci. Math. (Szeged) 68(3–4), 673–679 (2002). MR1954540 (2003k:11157)Google Scholar
  13. 13.
    H. Brunotte, On cubic CNS polynomials with three real roots. Acta Sci. Math. (Szeged) 70(3–4), 495–504 (2004). MR2107523 (2005h:11055)Google Scholar
  14. 14.
    H. Brunotte, Symmetric CNS trinomials. Integers 9(A19), 201–214 (2009). MR2534909 (2010g:11039)Google Scholar
  15. 15.
    H. Brunotte, A unified proof of two classical theorems on CNS polynomials. Integers 12(4), 709–721 (2012). MR2988542Google Scholar
  16. 16.
    H. Brunotte, Unusual CNS polynomials. Math. Pannon. 24(1), 125–137 (2013). MR3234910Google Scholar
  17. 17.
    H. Brunotte, A. Huszti, A. Pethő, Bases of canonical number systems in quartic algebraic number fields. J. Théor. Nombres Bordeaux 18(3), 537–557 (2006). MR2330426 (2008g:11179)Google Scholar
  18. 18.
    A. Cobham, On the base-dependence of sets of numbers recognizable by finite automata. Math. Syst. Theory 3, 186–192 (1969). MR0250789MathSciNetMATHGoogle Scholar
  19. 19.
    F. Durand, Cobham’s theorem for substitutions. J. Eur. Math. Soc. (JEMS) 13(6), 1799–1814 (2011). MR2835330Google Scholar
  20. 20.
    F. Durand, M. Rigo, On Cobham’s theorem, in Handbook of Automata: From Mathematics to Applications (European Mathematical Society Publishing House, Zurich, 2017)Google Scholar
  21. 21.
    S. Eilenberg, Automata, Languages, and Machines. Vol. A, Pure and Applied Mathematics, vol. 58 (Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York, 1974). MR0530382Google Scholar
  22. 22.
    I. Gaál, L. Robertson, Power integral bases in prime-power cyclotomic fields. J. Number Theory 120(2), 372–384 (2006). MR2257552Google Scholar
  23. 23.
    W.J. Gilbert, Radix representations of quadratic fields. J. Math. Anal. Appl. 83(1), 264–274 (1981). MR632342 (83m:12005)Google Scholar
  24. 24.
    G. Hansel, T. Safer, Vers un théorème de Cobham pour les entiers de Gauss. Bull. Belg. Math. Soc. Simon Stevin 10(Suppl.), 723–735 (2003). MR2073023 (2005c:68236)Google Scholar
  25. 25.
    I. Kátai, B. Kovács, Kanonische Zahlensysteme in der Theorie der quadratischen algebraischen Zahlen. Acta Sci. Math. (Szeged) 42(1–2), 99–107 (1980). MR576942 (81i:12002)Google Scholar
  26. 26.
    I. Kátai, B. Kovács, Canonical number systems in imaginary quadratic fields. Acta Math. Acad. Sci. Hungar. 37(1–3), 159–164 (1981). MR616887 (83a:12005)Google Scholar
  27. 27.
    I. Kátai, J. Szabó, Canonical number systems for complex integers. Acta Sci. Math. (Szeged) 37(3–4), 255–260 (1975). MR0389759 (52 #10590)Google Scholar
  28. 28.
    S.I. Khmelnik, Specialized digital computer for operations with complex numbers. Quest. Radio Electronics XII(2) (1964), 60–82; in Russian.Google Scholar
  29. 29.
    P. Kirschenhofer, J.M. Thuswaldner, Shift radix systems—a survey. RIMS Kôkyûroku Bessatsu B46, 1–59 (2014). MR3330559Google Scholar
  30. 30.
    D.E. Knuth, A imaginary number system. CACM 3(4), 245–247 (1960)MathSciNetCrossRefGoogle Scholar
  31. 31.
    D.E. Knuth, The Art of Computer Programming. Vol. 2: Seminumerical Algorithms (Addison-Wesley, Reading, 1969). MR0286318 (44 #3531)Google Scholar
  32. 32.
    B. Kovács, Canonical number systems in algebraic number fields. Acta Math. Acad. Sci. Hungar. 37(4), 405–407 (1981). MR619892 (82j:12014)Google Scholar
  33. 33.
    B. Kovács, A. Pethő, Number systems in integral domains, especially in orders of algebraic number fields. Acta Sci. Math. (Szeged) 55(3–4), 287–299 (1991). MR1152592 (92m:11116)Google Scholar
  34. 34.
    B. Kovács, A. Pethő, On a representation of algebraic integers. Studia Sci. Math. Hungar. 27(1–2), 169–172 (1992). MR1207568Google Scholar
  35. 35.
    M.G. Madritsch, V. Ziegler, An infinite family of multiplicatively independent bases of number systems in cyclotomic number fields. Acta Sci. Math. (Szeged) 81(1–2), 33–44 (2015). MR3381872Google Scholar
  36. 36.
    M.G. Madritsch, V. Ziegler, On multiplicatively independent bases in cyclotomic number fields. Acta Math. Hungar. 146(1), 224–239 (2015). MR3348190Google Scholar
  37. 37.
    W. Penney, A “binary” system for complex numbers. J. ACM 12(2), 247–248 (April 1965)Google Scholar
  38. 38.
    A. Pethő, On a polynomial transformation and its application to the construction of a public key cryptosystem. Computational Number Theory (Debrecen, 1989) (Walter de Gruyter, Berlin, 1991), pp. 31–43. MR1151853 (93e:94011)Google Scholar
  39. 39.
    A. Pethő, R.F. Tichy, S-unit equations, linear recurrences and digit expansions. Publ. Math. Debr. 42(1–2), 145–154 (1993). MR1208858Google Scholar
  40. 40.
    G. Ranieri, Générateurs de l’anneau des entiers d’une extension cyclotomique. J. Number Theory 128(6), 1576–1586 (2008). MR2419179Google Scholar
  41. 41.
    L. Robertson, Power bases for cyclotomic integer rings. J. Number Theory 69(1), 98–118 (1998). MR1611089Google Scholar
  42. 42.
    L. Robertson, Power bases for 2-power cyclotomic fields. J. Number Theory 88(1), 196–209 (2001). MR1825999Google Scholar
  43. 43.
    L. Robertson, Monogeneity in cyclotomic fields. Int. J. Number Theory 6(7), 1589–1607 (2010). MR2740723Google Scholar
  44. 44.
    L. Robertson, R. Russell, A hybrid Gröbner bases approach to computing power integral bases. Acta Math. Hungar. 147(2), 427–437 (2015). MR3420587Google Scholar
  45. 45.
    J. Sakarovitch, Elements of Automata Theory (Cambridge University Press, Cambridge, 2009); Translated from the 2003 French original by Reuben Thomas. MR2567276Google Scholar
  46. 46.
    H.P. Schlickewei, Linear equations in integers with bounded sum of digits. J. Number Theory 35(3), 335–344 (1990). MR1062338Google Scholar
  47. 47.
    K. Scheicher, J.M. Thuswaldner, On the characterization of canonical number systems. Osaka J. Math. 41(2), 327–351 (2004). MR2069090 (2005c:11013)Google Scholar
  48. 48.
    K. Scheicher, P. Surer, J.M. Thuswaldner, C.E. van de Woestijne, Digit systems over commutative rings. Int. J. Number Theory 10(6), 1459–1483 (2014)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    H.G. Senge, E.G. Straus, PV-numbers and sets of multiplicity. Period. Math. Hungar. 3, 93–100 (1973); Collection of articles dedicated to the memory of Alfréd Rényi, II. MR0340185Google Scholar
  50. 50.
    C.L. Stewart, On the representation of an integer in two different bases. J. Reine Angew. Math. 319, 63–72 (1980). MR586115MathSciNetGoogle Scholar
  51. 51.
    M. Waldschmidt, Diophantine Approximation on Linear Algebraic Groups. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 326 (Springer, Berlin, 2000); Transcendence properties of the exponential function in several variables. MR1756786Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Manfred G. Madritsch
    • 1
    • 2
  • Paul Surer
    • 3
  • Volker Ziegler
    • 4
  1. 1.Institut Elie Cartan de LorraineUniversité de Lorraine, UMR 7502Vandoeuvre-lès-NancyFrance
  2. 2.Institut Elie Cartan de Lorraine, CNRS, UMR 7502Vandoeuvre-lès-NancyFrance
  3. 3.Institut für MathematikUniverstiät für Bodenkultur (BOKU)WienAustria
  4. 4.Institute of MathematicsUniversity of SalzburgSalzburgAustria

Personalised recommendations