Calculating cyclotomic polynomials

Authors:
Andrew Arnold and Michael Monagan

Journal:
Math. Comp. **80** (2011), 2359-2379

MSC (2010):
Primary 11Y16; Secondary 12-04

DOI:
https://doi.org/10.1090/S0025-5718-2011-02467-1

Published electronically:
February 17, 2011

MathSciNet review:
2813365

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We present three algorithms to calculate , the cyclotomic polynomial. The first algorithm calculates by a series of polynomial divisions, which we perform using the fast Fourier transform. The second algorithm calculates as a quotient of products of sparse power series. These two algorithms, described in detail in the paper, were used to calculate cyclotomic polynomials of large height and length. In particular, we have found the least for which the height of is greater than , , , and , respectively. The third algorithm, the big prime algorithm, generates the terms of sequentially, in a manner which reduces the memory cost. We use the big prime algorithm to find the minimal known height of cyclotomic polynomials of order five. We include these results as well as other examples of cyclotomic polynomials of unusually large height, and bounds on the coefficient of the term of degree for all cyclotomic polynomials.

**1.**Gennady Bachman,*On the coefficients of cyclotomic polynomials*, Mem. Amer. Math. Soc.**106**(1993), no. 510, vi+80. MR**1172916**, https://doi.org/10.1090/memo/0510**2.**Gennady Bachman,*Flat cyclotomic polynomials of order three*, Bull. London Math. Soc.**38**(2006), no. 1, 53–60. MR**2201603**, https://doi.org/10.1112/S0024609305018096**3.**A. S. Bang.

Om ligningen .*Nyt Tidsskrift for Mathematik*, (6):6-12, 1895.**4.**P. T. Bateman, C. Pomerance, and R. C. Vaughan,*On the size of the coefficients of the cyclotomic polynomial*, Topics in classical number theory, Vol. I, II (Budapest, 1981) Colloq. Math. Soc. János Bolyai, vol. 34, North-Holland, Amsterdam, 1984, pp. 171–202. MR**781138****5.**D. M. Bloom,*On the coefficients of the cyclotomic polynomials*, Amer. Math. Monthly**75**(1968), 372–377. MR**0227086**, https://doi.org/10.2307/2313417**6.**Wieb Bosma,*Computation of cyclotomic polynomials with Magma*, Computational algebra and number theory (Sydney, 1992) Math. Appl., vol. 325, Kluwer Acad. Publ., Dordrecht, 1995, pp. 213–225. MR**1344932****7.**Paul Erdös,*On the coefficients of the cyclotomic polynomial*, Bull. Amer. Math. Soc.**52**(1946), 179–184. MR**0014110**, https://doi.org/10.1090/S0002-9904-1946-08538-9**8.**Y. Gallot, P. Moree, and H. Hommerson.

Value distribution of cyclotomic polynomial coefficients.

Available at`http://arxiv.org/abs/0803.2483`.**9.**K. O. Geddes, S. R. Czapor, and G. Labahn,*Algorithms for computer algebra*, Kluwer Academic Publishers, Boston, MA, 1992. MR**1256483****10.**A. Grytczuk and B. Tropak,*A numerical method for the determination of the cyclotomic polynomial coefficients*, Computational number theory (Debrecen, 1989) de Gruyter, Berlin, 1991, pp. 15–19. MR**1151850****11.**H. W. Lenstra Jr.,*Vanishing sums of roots of unity*, Proceedings, Bicentennial Congress Wiskundig Genootschap (Vrije Univ., Amsterdam, 1978) Math. Centre Tracts, vol. 101, Math. Centrum, Amsterdam, 1979, pp. 249–268. MR**541398****12.**N. Kaplan.

Personal correspondence.**13.**Nathan Kaplan,*Flat cyclotomic polynomials of order four and higher*, Integers**10**(2010), A30, 357–363. MR**2684126**, https://doi.org/10.1515/INTEG.2010.030**14.**Nathan Kaplan,*Flat cyclotomic polynomials of order three*, J. Number Theory**127**(2007), no. 1, 118–126. MR**2351667**, https://doi.org/10.1016/j.jnt.2007.01.008**15.**Yôichi Koshiba,*On the calculations of the coefficients of the cyclotomic polynomials*, Rep. Fac. Sci. Kagoshima Univ.**31**(1998), 31–44. MR**1718402****16.**Yôichi Koshiba,*On the calculations of the coefficients of the cyclotomic polynomials. II*, Rep. Fac. Sci. Kagoshima Univ.**33**(2000), 55–59. MR**1833785****17.**Helmut Maier,*The size of the coefficients of cyclotomic polynomials*, Analytic number theory, Vol. 2 (Allerton Park, IL, 1995) Progr. Math., vol. 139, Birkhäuser Boston, Boston, MA, 1996, pp. 633–639. MR**1409383****18.**H. L. Montgomery and R. C. Vaughan,*The order of magnitude of the 𝑚th coefficients of cyclotomic polynomials*, Glasgow Math. J.**27**(1985), 143–159. MR**819835**, https://doi.org/10.1017/S0017089500006145**19.**Pieter Moree,*Inverse cyclotomic polynomials*, J. Number Theory**129**(2009), no. 3, 667–680. MR**2488596**, https://doi.org/10.1016/j.jnt.2008.10.004**20.**T.D. Noe.

Least such that the coefficient of cyclotomic polynomial phi(k,x) has the largest possible magnitude.

Sequence A138475 in N. J. A. Sloane (Ed.), The On-Line Encyclopedia of Integer Sequences (2008),`http://www.research.att.com/~njas/sequences/A138475`.**21.**T.D. Noe.

Maximum possible magnitude of the coefficient of a cyclotomic polynomial.

Sequence A138474 in N. J. A. Sloane (Ed.), The On-Line Encyclopedia of Integer Sequences (2008), published electronically at`http://www.research.att.com/~njas/sequences/A138474`.**22.**T.D. Noe.

Numbers n such that phi(n,x) is a flat cyclotomic polynomial of order four.

Sequence A117318 in N. J. A. Sloane (Ed.), The On-Line Encyclopedia of Integer Sequences (2008),`http://www.research.att.com/~njas/sequences/A117318`.**23.**T.D. Noe.

Numbers n such that phi(n,x) is a flat cyclotomic polynomial of order three.

Sequence A117223 in N. J. A. Sloane (Ed.), The On-Line Encyclopedia of Integer Sequences (2008),`http://www.research.att.com/~njas/sequences/A117223`.**24.**T.D. Noe.

Odd squarefree numbers n such that the cyclotomic polynomial phi(n,x) is not coefficient convex.

Sequence A146960 in N. J. A. Sloane (Ed.), The On-Line Encyclopedia of Integer Sequences (2008),`http://www.research.att.com/~njas/sequences/A146960`.**25.**R. Thangadurai,*On the coefficients of cyclotomic polynomials*, Cyclotomic fields and related topics (Pune, 1999) Bhaskaracharya Pratishthana, Pune, 2000, pp. 311–322. MR**1802391****26.**Joachim von zur Gathen and Jürgen Gerhard,*Modern computer algebra*, 2nd ed., Cambridge University Press, Cambridge, 2003. MR**2001757**

Retrieve articles in *Mathematics of Computation*
with MSC (2010):
11Y16,
12-04

Retrieve articles in all journals with MSC (2010): 11Y16, 12-04

Additional Information

**Andrew Arnold**

Affiliation:
Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada

Email:
ada26@sfu.ca

**Michael Monagan**

Affiliation:
Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada

Email:
mmonagan@cecm.sfu.ca

DOI:
https://doi.org/10.1090/S0025-5718-2011-02467-1

Received by editor(s):
October 10, 2008

Received by editor(s) in revised form:
July 17, 2010

Published electronically:
February 17, 2011

Additional Notes:
This work was supported by NSERC of Canada and the MITACS NCE of Canada

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.