Research Article
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Year 2018, Volume: 23 Issue: 23, 167 - 175, 11.01.2018
https://doi.org/10.24330/ieja.373660

Abstract

References

  • S. Akiyama, Private communication, 2012.
  • S. Akiyama, T. Borbely, H. Brunotte, A. Peth}o and J. M. Thuswaldner, Gen- eralized radix representations and dynamical systems I, Acta Math. Hungar., 108(3) (2005), 207-238.
  • S. Akiyama, H. Brunotte and A. Peth}o, Cubic CNS polynomials, notes on a conjecture of W. J. Gilbert, J. Math. Anal. Appl., 281(1) (2003), 402-415.
  • S. Akiyama and A. Peth}o, On canonical number systems, Theoret. Comput. Sci., 270(1-2) (2002), 921-933.
  • S. Akiyama and H. Rao, New criteria for canonical number systems, Acta Arith., 111(1) (2004), 5-25.
  • S. Akiyama and K. Scheicher, Symmetric shift radix systems and nite expan- sions, Math. Pannon., 18(1) (2007), 101-124.
  • G. Barat, V. Berthe, P. Liardet and J. Thuswaldner, Dynamical directions in numeration, Numeration, pavages, substitutions, Ann. Inst. Fourier (Grenoble), 56(7) (2006), 1987-2092.
  • V. Berthe, Numeration and discrete dynamical systems, Computing, 94(2-4) (2012), 369-387.
  • T. Borbely,  Altalanostott szamrendszerek, Master Thesis, University of Debrecen, 2003.
  • J. Borcea and P. Branden, The Lee-Yang and Polya-Schur programs II, The- ory of stable polynomials and applications, Comm. Pure Appl. Math., 62(12) (2009), 1595-1631.
  • H. Brunotte, Characterization of CNS trinomials, Acta Sci. Math. (Szeged), 68(3-4) (2002), 673-679.
  • H. Brunotte, On the roots of expanding integer polynomials, Acta Math. Acad. Paedagog. Nyhazi. (N.S.), 27(2) (2011), 161-171.
  • H. Brunotte, A uni ed proof of two classical theorems on CNS polynomials, Integers, 12(4) (2012), 709-721.
  • H. Brunotte, Unusual CNS polynomials, Math. Pannon., 24(1) (2013), 125-137. [15] H. Brunotte, Small degree CNS polynomials with dominant condition, Math. Pannon., 25(1) (2014/15), 113-133.
  • P. Burcsi and A. Kovacs, Exhaustive search methods for CNS polynomials, Monatsh. Math., 155(3-4) (2008), 421-430.
  • A. Chen, On the reducible quintic complete base polynomials, J. Number Theory, 129(1) (2009), 220-230.
  • A. Dubickas, Roots of polynomials with dominant term, Int. J. Number Theory, 7(5) (2011), 1217-1228.
  • L. German and A. Kovacs, On number system constructions, Acta Math. Hungar., 115(1-2) (2007), 155-167.
  • W. J. Gilbert, Radix representations of quadratic elds, J. Math. Anal. Appl., 83(1) (1981), 264-274.
  • V. Grunwald, 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.
  • D. M. Kane, Generalized base representations, J. Number Theory, 120(1) (2006), 92-100.
  • I. Katai and B. Kovacs, Kanonische Zahlensysteme in der Theorie der quadratischen algebraischen Zahlen, Acta Sci. Math. (Szeged), 42 (1980), 99- 107.
  • I. Katai and B. Kovacs, Canonical number systems in imaginary quadratic elds, Acta Math. Acad. Sci. Hungar., 37(1-3) (1981), 159-164.
  • I. Katai and J. Szabo, Canonical number systems for complex integers, Acta Sci. Math. (Szeged), 37(3-4) (1975), 255-260.
  • D. E. Knuth, An imaginary number system, Comm. ACM, 3 (1960), 245-247.
  • B. Kovacs, Canonical number systems in algebraic number elds, Acta Math. Acad. Sci. Hungar., 37(4) (1981), 405-407.
  • A. Kovacs, Generalized binary number systems, Ann. Univ. Sci. Budapest. Sect. Comput., 20 (2001), 195-206.
  • B. Kovacs and A. Peth}o, Number systems in integral domains, especially in orders of algebraic number elds, Acta Sci. Math. (Szeged), 55(3-4) (1991), 287-299.
  • A. Peth}o, On a polynomial transformation and its application to the construc- tion of a public key cryptosystem, in Computational number theory (Debrecen, 1989), de Gruyter, Berlin, (1991), 31-43.
  • A. Peth}o, Private communication, 2000.
  • A. Peth}o, Connections between power integral bases and radix representations in algebraic number elds, in Proceedings of the 2003 Nagoya Conference \Yokoi-Chowla Conjecture and Related Problems", S. Katayama, C. Levesque, and T. Nakahara, eds., Saga Univ., Saga, (2004), 115-125.
  • K. Scheicher and J. M. Thuswaldner, On the characterization of canonical number systems, Osaka J. Math., 41(2) (2004), 327-351.
  • P. Surer, "-shift radix systems and radix representations with shifted digit sets, Publ. Math. Debrecen, 74(1-2) (2009), 19-43.
  • A. Tatrai, Parallel implementations of Brunotte's algorithm, J. Parallel Distrib. Comput., 71(4) (2011), 565-572.

Some comments on Akiyama's conjecture on CNS polynomials

Year 2018, Volume: 23 Issue: 23, 167 - 175, 11.01.2018
https://doi.org/10.24330/ieja.373660

Abstract

It is well-known that in general polynomials lose their CNS property
by addition of small positive integers. We comment on a conjecture of S.
Akiyama on addition of suciently large positive constants to CNS polynomials.

References

  • S. Akiyama, Private communication, 2012.
  • S. Akiyama, T. Borbely, H. Brunotte, A. Peth}o and J. M. Thuswaldner, Gen- eralized radix representations and dynamical systems I, Acta Math. Hungar., 108(3) (2005), 207-238.
  • S. Akiyama, H. Brunotte and A. Peth}o, Cubic CNS polynomials, notes on a conjecture of W. J. Gilbert, J. Math. Anal. Appl., 281(1) (2003), 402-415.
  • S. Akiyama and A. Peth}o, On canonical number systems, Theoret. Comput. Sci., 270(1-2) (2002), 921-933.
  • S. Akiyama and H. Rao, New criteria for canonical number systems, Acta Arith., 111(1) (2004), 5-25.
  • S. Akiyama and K. Scheicher, Symmetric shift radix systems and nite expan- sions, Math. Pannon., 18(1) (2007), 101-124.
  • G. Barat, V. Berthe, P. Liardet and J. Thuswaldner, Dynamical directions in numeration, Numeration, pavages, substitutions, Ann. Inst. Fourier (Grenoble), 56(7) (2006), 1987-2092.
  • V. Berthe, Numeration and discrete dynamical systems, Computing, 94(2-4) (2012), 369-387.
  • T. Borbely,  Altalanostott szamrendszerek, Master Thesis, University of Debrecen, 2003.
  • J. Borcea and P. Branden, The Lee-Yang and Polya-Schur programs II, The- ory of stable polynomials and applications, Comm. Pure Appl. Math., 62(12) (2009), 1595-1631.
  • H. Brunotte, Characterization of CNS trinomials, Acta Sci. Math. (Szeged), 68(3-4) (2002), 673-679.
  • H. Brunotte, On the roots of expanding integer polynomials, Acta Math. Acad. Paedagog. Nyhazi. (N.S.), 27(2) (2011), 161-171.
  • H. Brunotte, A uni ed proof of two classical theorems on CNS polynomials, Integers, 12(4) (2012), 709-721.
  • H. Brunotte, Unusual CNS polynomials, Math. Pannon., 24(1) (2013), 125-137. [15] H. Brunotte, Small degree CNS polynomials with dominant condition, Math. Pannon., 25(1) (2014/15), 113-133.
  • P. Burcsi and A. Kovacs, Exhaustive search methods for CNS polynomials, Monatsh. Math., 155(3-4) (2008), 421-430.
  • A. Chen, On the reducible quintic complete base polynomials, J. Number Theory, 129(1) (2009), 220-230.
  • A. Dubickas, Roots of polynomials with dominant term, Int. J. Number Theory, 7(5) (2011), 1217-1228.
  • L. German and A. Kovacs, On number system constructions, Acta Math. Hungar., 115(1-2) (2007), 155-167.
  • W. J. Gilbert, Radix representations of quadratic elds, J. Math. Anal. Appl., 83(1) (1981), 264-274.
  • V. Grunwald, 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.
  • D. M. Kane, Generalized base representations, J. Number Theory, 120(1) (2006), 92-100.
  • I. Katai and B. Kovacs, Kanonische Zahlensysteme in der Theorie der quadratischen algebraischen Zahlen, Acta Sci. Math. (Szeged), 42 (1980), 99- 107.
  • I. Katai and B. Kovacs, Canonical number systems in imaginary quadratic elds, Acta Math. Acad. Sci. Hungar., 37(1-3) (1981), 159-164.
  • I. Katai and J. Szabo, Canonical number systems for complex integers, Acta Sci. Math. (Szeged), 37(3-4) (1975), 255-260.
  • D. E. Knuth, An imaginary number system, Comm. ACM, 3 (1960), 245-247.
  • B. Kovacs, Canonical number systems in algebraic number elds, Acta Math. Acad. Sci. Hungar., 37(4) (1981), 405-407.
  • A. Kovacs, Generalized binary number systems, Ann. Univ. Sci. Budapest. Sect. Comput., 20 (2001), 195-206.
  • B. Kovacs and A. Peth}o, Number systems in integral domains, especially in orders of algebraic number elds, Acta Sci. Math. (Szeged), 55(3-4) (1991), 287-299.
  • A. Peth}o, On a polynomial transformation and its application to the construc- tion of a public key cryptosystem, in Computational number theory (Debrecen, 1989), de Gruyter, Berlin, (1991), 31-43.
  • A. Peth}o, Private communication, 2000.
  • A. Peth}o, Connections between power integral bases and radix representations in algebraic number elds, in Proceedings of the 2003 Nagoya Conference \Yokoi-Chowla Conjecture and Related Problems", S. Katayama, C. Levesque, and T. Nakahara, eds., Saga Univ., Saga, (2004), 115-125.
  • K. Scheicher and J. M. Thuswaldner, On the characterization of canonical number systems, Osaka J. Math., 41(2) (2004), 327-351.
  • P. Surer, "-shift radix systems and radix representations with shifted digit sets, Publ. Math. Debrecen, 74(1-2) (2009), 19-43.
  • A. Tatrai, Parallel implementations of Brunotte's algorithm, J. Parallel Distrib. Comput., 71(4) (2011), 565-572.
There are 34 citations in total.

Details

Journal Section Articles
Authors

Horst Brunotte This is me

Publication Date January 11, 2018
Published in Issue Year 2018 Volume: 23 Issue: 23

Cite

APA Brunotte, H. (2018). Some comments on Akiyama’s conjecture on CNS polynomials. International Electronic Journal of Algebra, 23(23), 167-175. https://doi.org/10.24330/ieja.373660
AMA Brunotte H. Some comments on Akiyama’s conjecture on CNS polynomials. IEJA. January 2018;23(23):167-175. doi:10.24330/ieja.373660
Chicago Brunotte, Horst. “Some Comments on Akiyama’s Conjecture on CNS Polynomials”. International Electronic Journal of Algebra 23, no. 23 (January 2018): 167-75. https://doi.org/10.24330/ieja.373660.
EndNote Brunotte H (January 1, 2018) Some comments on Akiyama’s conjecture on CNS polynomials. International Electronic Journal of Algebra 23 23 167–175.
IEEE H. Brunotte, “Some comments on Akiyama’s conjecture on CNS polynomials”, IEJA, vol. 23, no. 23, pp. 167–175, 2018, doi: 10.24330/ieja.373660.
ISNAD Brunotte, Horst. “Some Comments on Akiyama’s Conjecture on CNS Polynomials”. International Electronic Journal of Algebra 23/23 (January 2018), 167-175. https://doi.org/10.24330/ieja.373660.
JAMA Brunotte H. Some comments on Akiyama’s conjecture on CNS polynomials. IEJA. 2018;23:167–175.
MLA Brunotte, Horst. “Some Comments on Akiyama’s Conjecture on CNS Polynomials”. International Electronic Journal of Algebra, vol. 23, no. 23, 2018, pp. 167-75, doi:10.24330/ieja.373660.
Vancouver Brunotte H. Some comments on Akiyama’s conjecture on CNS polynomials. IEJA. 2018;23(23):167-75.