Bases of canonical number systems in quartic algebraic number fields

 Access to full text

 Full (PDF)

 Access to full text

1. S. Akiyama, T. Borbély, H. Brunotte, A. Pethő and J. M. Thuswaldner, 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 Institute Commucations, vol. 41 (2004), 19–27. Zbl1135.11007
2. S. Akiyama, T. Borbély, H. Brunotte, A. Pethő and J. M. Thuswaldner, Generalized radix representations and dynamical systems I, Acta Math. Hung., 108 (2005), 207–238. Zbl1110.11003MR2162561
3. S. Akiyama, H. Brunotte and A. Pethő, Cubic CNS polynomials, notes on a conjecture of W.J. Gilbert, J. Math. Anal. and Appl., 281 (2003), 402–415. Zbl1021.11005MR1980100
4. S. Akiyama and H. Rao, New criteria for canonical number systems, Acta Arith., 111 (2004), 5–25. Zbl1049.11008MR2038059
5. S. Akiyama and J. M. Thuswaldner, On the topological structure of fractal tilings generated by quadratic number systems, Comput. Math. Appl. 49 (2005), no. 9-10, 1439–1485. Zbl1123.11004MR2149493
6. T. Borbély, Általánosított számrendszerek, Master Thesis, University of Debrecen, 2003. 
7. H. Brunotte, On trinomial bases of radix representations of algebraic integers, Acta Sci. Math. (Szeged), 67 (2001), 521–527. Zbl0996.11067MR1876451
8. H. Brunotte, On cubic CNS polynomials with three real roots, Acta Sci. Math. (Szeged), 70 (2004), 495 – 504. Zbl1064.11005MR2107523
9. I. Gaál, Diophantine equations and power integral bases, Birkhäuser (Berlin), (2002). Zbl1016.11059MR1896601
10. W. J. Gilbert, Radix representations of quadratic fields, J. Math. Anal. Appl., 83 (1981), 264–274. Zbl0472.10011MR632342
11. E. H. Grossman, Number bases in quadratic fields, Studia Sci. Math. Hungar., 20 (1985), 55–58. Zbl0562.12004MR886005
12. V. Grünwald, Intorno all’aritmetica dei sistemi numerici a base negativa con particolare riguardo al sistema numerico a base negativo-decimale per lo studio delle sue analogie coll’aritmetica ordinaria (decimale), Giornale di matematiche di Battaglini, 23 (1885), 203–221, 367. 
13. K. Győry, Sur les polynômes à coefficients entiers et de discriminant donné III, Publ. Math. (Debrecen), 23 (1976), 141–165. Zbl0354.10041MR437491
14. I. Kátai and B. Kovács, Kanonische Zahlensysteme in der Theorie der quadratischen algebraischen Zahlen, Acta Sci. Math. (Szeged), 42 (1980), 99–107. Zbl0386.10007MR576942
15. I. Kátai and B. Kovács, Canonical number systems in imaginary quadratic fields, Acta Math. Acad. Sci. Hungar., 37 (1981), 159–164. Zbl0477.10012MR616887
16. I. Kátai and J. Szabó, Canonical number systems for complex integers, Acta Sci. Math. (Szeged), 37 (1975), 255–260. Zbl0309.12001MR389759
17. D. E. Knuth, An imaginary number system, Comm. ACM, 3 (1960), 245 – 247. MR127508
18. D. E. Knuth, The Art of Computer Programming, Vol. 2 Semi-numerical Algorithms, Addison Wesley (1998), London 3rd edition. Zbl0895.65001MR633878
19. B. Kovács, Canonical number systems in algebraic number fields, Acta Math. Acad. Sci. Hungar., 37 (1981), 405–407. Zbl0505.12001MR619892
20. B. Kovács and A. Pethő, Number systems in integral domains, especially in orders of algebraic number fields, Acta Sci. Math. (Szeged), 55 (1991), 287–299. Zbl0760.11002MR1152592
21. S. Körmendi, Canonical number systems in $\mathbb{Q}\left(\sqrt[3]{2}\right)$, Acta Sci. Math. (Szeged), 50 (1986), 351–357. Zbl0616.10007MR882046
22. G. Lettl and A. Pethő, Complete solution of a family of quartic Thue equations, Abh. Math. Sem. Univ. Hamburg 65 (1995), 365–383. Zbl0853.11021MR1359142
23. M. Mignotte, A. Pethő and R. Roth, Complete solutions of quartic Thue and index form equations, Math. Comp. 65 (1996), 341–354. Zbl0853.11022MR1316596
24. P. Olajos, Power integral bases in the family of simplest quartic fields, Experiment. Math. 14 (2005), 129–132. Zbl1092.11042MR2169516
25. A. Pethő, On a polynomial transformation and its application to the construction of a public key cryptosystem,Computational Number Theory, Proc., Walter de Gruyter Publ. Comp. Eds.: A. Pethő, M. Pohst, H. G. Zimmer and H. C. Williams (1991), 31–43. Zbl0733.94014MR1151853
26. A. Pethő, Notes on CNS polynomials and integral interpolation, More sets, graphs and numbers, 301–315, Bolyai Soc. Math. Stud., 15, Springer, Berlin, 2006. Zbl1136.11021MR2223397
27. A. Pethő, Connections between power integral bases and radix representations in algebraic number fields, Proc. of the 2003 Nagoya Conf. “Yokoi-Chowla Conjecture and Related Problems”, Furukawa Total Pr. Co. (2004), 115–125. 
28. R. Robertson, Power bases for cyclotomic integer rings, J. Number Theory, 69 (1998), 98–118. Zbl0923.11150MR1611089
29. R. Robertson, Power bases for 2-power cyclotomic integer rings, J. Number Theory, 88 (2001), 196–209. Zbl0973.11091MR1825999
30. K. Scheicher,Kanonische Ziffernsysteme und Automaten, Grazer Math. Ber., 333 (1997), 1–17. Zbl0905.11009MR1640469
31. D. Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137–1152. Zbl0307.12005MR352049
32. J. M. Thuswaldner, Elementary properties of canonical number systems in quadratic fields, in: Applications of Fibonacci Numbers, Volume 7, G. E. Bergum et al. (eds.), Kluwer Academic Publishers, Dordrecht (1998), 405–414. Zbl0917.11054MR1638467