{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T17:39:11Z","timestamp":1781285951715,"version":"3.54.1"},"reference-count":26,"publisher":"American Mathematical Society (AMS)","issue":"276","license":[{"start":{"date-parts":[[2012,2,17]],"date-time":"2012-02-17T00:00:00Z","timestamp":1329436800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We present three algorithms to calculate\n \n \n \n \n \n \n \u03a6\n \n <\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n z<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\Phi _n(z)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , the\n \n \n \n \n n<\/mml:mi>\n \n t<\/mml:mi>\n h<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n n_{th}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n cyclotomic polynomial. The first algorithm calculates\n \n \n \n \n \n \n \u03a6\n \n <\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n z<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\Phi _n(z)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n by a series of polynomial divisions, which we perform using the fast Fourier transform. The second algorithm calculates\n \n \n \n \n \n \n \u03a6\n \n <\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n z<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\Phi _n(z)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n 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\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for which the height of\n \n \n \n \n \n \n \u03a6\n \n <\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n z<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\Phi _n(z)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is greater than\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n n<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n n^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n n<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n n^3<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and\n \n \n \n \n n<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msup>\n n^4<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , respectively. The third algorithm, the big prime algorithm, generates the terms of\n \n \n \n \n \n \n \u03a6\n \n <\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n z<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\Phi _n(z)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n 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\n \n \n \n k<\/mml:mi>\n k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for all cyclotomic polynomials.\n <\/p>","DOI":"10.1090\/s0025-5718-2011-02467-1","type":"journal-article","created":{"date-parts":[[2011,2,17]],"date-time":"2011-02-17T09:41:06Z","timestamp":1297935666000},"page":"2359-2379","source":"Crossref","is-referenced-by-count":17,"title":["Calculating cyclotomic polynomials"],"prefix":"10.1090","volume":"80","author":[{"given":"Andrew","family":"Arnold","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michael","family":"Monagan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2011,2,17]]},"reference":[{"issue":"510","key":"1","doi-asserted-by":"publisher","first-page":"vi+80","DOI":"10.1090\/memo\/0510","article-title":"On the coefficients of cyclotomic polynomials","volume":"106","author":"Bachman, Gennady","year":"1993","journal-title":"Mem. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-9266","issn-type":"print"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1112\/S0024609305018096","article-title":"Flat cyclotomic polynomials of order three","volume":"38","author":"Bachman, Gennady","year":"2006","journal-title":"Bull. London Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6093","issn-type":"print"},{"key":"3","unstructured":"A. S. Bang. Om ligningen \ud835\udf19_{\ud835\udc5b}(\ud835\udc65)=0. Nyt Tidsskrift for Mathematik, (6):6\u201312, 1895."},{"key":"4","isbn-type":"print","first-page":"171","article-title":"On the size of the coefficients of the cyclotomic polynomial","author":"Bateman, P. T.","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/0444865098"},{"key":"5","doi-asserted-by":"publisher","first-page":"372","DOI":"10.2307\/2313417","article-title":"On the coefficients of the cyclotomic polynomials","volume":"75","author":"Bloom, D. M.","year":"1968","journal-title":"Amer. Math. Monthly","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9890","issn-type":"print"},{"key":"6","isbn-type":"print","first-page":"213","article-title":"Computation of cyclotomic polynomials with Magma","author":"Bosma, Wieb","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0792335015"},{"key":"7","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1090\/S0002-9904-1946-08538-9","article-title":"On the coefficients of the cyclotomic polynomial","volume":"52","author":"Erd\u00f6s, Paul","year":"1946","journal-title":"Bull. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9904","issn-type":"print"},{"key":"8","unstructured":"Y. Gallot, P. Moree, and H. Hommerson. Value distribution of cyclotomic polynomial coefficients. Available at http:\/\/arxiv.org\/abs\/0803.2483."},{"key":"9","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/b102438","volume-title":"Algorithms for computer algebra","author":"Geddes, K. O.","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0792392590"},{"key":"10","isbn-type":"print","first-page":"15","article-title":"A numerical method for the determination of the cyclotomic polynomial coefficients","author":"Grytczuk, A.","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/3110123940"},{"key":"11","isbn-type":"print","first-page":"249","article-title":"Vanishing sums of roots of unity","author":"Lenstra, H. W., Jr.","year":"1979","ISBN":"https:\/\/id.crossref.org\/isbn\/9061961696"},{"key":"12","unstructured":"N. Kaplan. Personal correspondence."},{"key":"13","doi-asserted-by":"publisher","first-page":"A30, 357--363","DOI":"10.1515\/INTEG.2010.030","article-title":"Flat cyclotomic polynomials of order four and higher","volume":"10","author":"Kaplan, Nathan","year":"2010","journal-title":"Integers","ISSN":"https:\/\/id.crossref.org\/issn\/1553-1732","issn-type":"print"},{"issue":"1","key":"14","doi-asserted-by":"publisher","first-page":"118","DOI":"10.1016\/j.jnt.2007.01.008","article-title":"Flat cyclotomic polynomials of order three","volume":"127","author":"Kaplan, Nathan","year":"2007","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"31","key":"15","first-page":"31","article-title":"On the calculations of the coefficients of the cyclotomic polynomials","author":"Koshiba, Y\u00f4ichi","year":"1998","journal-title":"Rep. Fac. Sci. 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J.","ISSN":"https:\/\/id.crossref.org\/issn\/0017-0895","issn-type":"print"},{"issue":"3","key":"19","doi-asserted-by":"publisher","first-page":"667","DOI":"10.1016\/j.jnt.2008.10.004","article-title":"Inverse cyclotomic polynomials","volume":"129","author":"Moree, Pieter","year":"2009","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"key":"20","unstructured":"T.D. Noe. Least \ud835\udc58 such that the \ud835\udc65\u207f 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\/\\allowbreak sequences\/A138475."},{"key":"21","unstructured":"T.D. Noe. Maximum possible magnitude of the \ud835\udc65\u207f coefficient of a cyclotomic polynomial. Sequence A138474 in N. J. A. 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Sloane (Ed.), The On-Line Encyclopedia of Integer Sequences (2008), http:\/\/www.research.att.com\/~njas\/sequences\/A146960."},{"key":"25","first-page":"311","article-title":"On the coefficients of cyclotomic polynomials","author":"Thangadurai, R.","year":"2000"},{"key":"26","isbn-type":"print","volume-title":"Modern computer algebra","author":"von zur Gathen, Joachim","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/0521826462","edition":"2"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2011-80-276\/S0025-5718-2011-02467-1\/S0025-5718-2011-02467-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2011-80-276\/S0025-5718-2011-02467-1\/S0025-5718-2011-02467-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:58:26Z","timestamp":1776790706000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2011-80-276\/S0025-5718-2011-02467-1\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,2,17]]},"references-count":26,"journal-issue":{"issue":"276","published-print":{"date-parts":[[2011,10]]}},"alternative-id":["S0025-5718-2011-02467-1"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2011-02467-1","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2011,2,17]]}}}