{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:52:19Z","timestamp":1776797539116,"version":"3.51.2"},"reference-count":12,"publisher":"American Mathematical Society (AMS)","issue":"296","license":[{"start":{"date-parts":[[2016,4,22]],"date-time":"2016-04-22T00:00:00Z","timestamp":1461283200000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Let\n \n \n \n \n \u03b1\n \n <\/mml:mi>\n \\alpha<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be an algebraic integer of degree\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , not\n \n \n \n 0<\/mml:mn>\n 0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n or a root of unity, all of whose conjugates\n \n \n \n \n \n \u03b1\n \n <\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n \\alpha _i<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n lie in a sector\n \n \n \n \n |<\/mml:mo>\n arg<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n z<\/mml:mi>\n |<\/mml:mo>\n \n \u2264\n \n <\/mml:mo>\n \n \u03b8\n \n <\/mml:mi>\n <\/mml:mrow>\n \\vert \\arg z \\vert \\leq \\theta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . In 1995, G.\u00a0Rhin and C.\u00a0Smyth computed the greatest lower bound\n \n \n \n \n c<\/mml:mi>\n (<\/mml:mo>\n \n \u03b8\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n c(\\theta )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of the absolute Mahler measure (\n \n \n \n \n \n \n \u220f\n \n <\/mml:mo>\n \n i<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:munderover>\n max<\/mml:mo>\n (<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \n \u03b1\n \n <\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n \n |<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n \n )<\/mml:mo>\n \n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n \\prod _{i=1}^d \\max (1, | \\alpha _i |))^{1\/d}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of\n \n \n \n \n \u03b1\n \n <\/mml:mi>\n \\alpha<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , for\n \n \n \n \n \u03b8\n \n <\/mml:mi>\n \\theta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n belonging to nine subintervals of\n \n \n \n \n [<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 2<\/mml:mn>\n \n \u03c0\n \n <\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 3<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n [0, 2 \\pi \/3]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . More recently, in 2004, G. Rhin and Q. Wu improved the result to thirteen subintervals of\n \n \n \n \n [<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n \n \u03c0\n \n <\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n [0, \\pi ]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and extended some existing subintervals. In this paper, for the first time we find a complete subinterval where\n \n \n \n \n c<\/mml:mi>\n (<\/mml:mo>\n \n \u03b8\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n c(\\theta )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is known exactly, as well as a fourteenth subinterval. Moreover, we slightly extend further all the existing subintervals.\n <\/p>","DOI":"10.1090\/mcom\/2959","type":"journal-article","created":{"date-parts":[[2015,4,22]],"date-time":"2015-04-22T12:10:19Z","timestamp":1429704619000},"page":"2927-2938","source":"Crossref","is-referenced-by-count":7,"title":["On the absolute Mahler measure of polynomials having all zeros in a sector. 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(2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"key":"8","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1112\/jlms\/s1-37.1.341","article-title":"On some inequalities for polynomials in several variables","volume":"37","author":"Mahler, K.","year":"1962","journal-title":"J. London Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6107","issn-type":"print"},{"issue":"209","key":"9","doi-asserted-by":"publisher","first-page":"295","DOI":"10.2307\/2153335","article-title":"On the absolute Mahler measure of polynomials having all zeros in a sector","volume":"64","author":"Rhin, Georges","year":"1995","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"249","key":"10","doi-asserted-by":"publisher","first-page":"383","DOI":"10.1090\/S0025-5718-04-01676-X","article-title":"On the absolute Mahler measure of polynomials having all zeros in a sector. 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