{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:36:34Z","timestamp":1776832594922,"version":"3.51.2"},"reference-count":19,"publisher":"American Mathematical Society (AMS)","issue":"256","license":[{"start":{"date-parts":[[2007,6,16]],"date-time":"2007-06-16T00:00:00Z","timestamp":1181952000000},"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 study the problem of finding nonconstant monic integer polynomials, normalized by their degree, with small supremum on an interval\n \n \n \n I<\/mml:mi>\n I<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . The monic integer transfinite diameter\n \n \n \n \n \n t<\/mml:mi>\n \n \n M<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n I<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n t_{\\mathrm {M}}(I)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is defined as the infimum of all such supremums. We show that if\n \n \n \n I<\/mml:mi>\n I<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n has length\n \n \n \n 1<\/mml:mn>\n 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , then\n \n \n \n \n \n t<\/mml:mi>\n \n \n M<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n I<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n \n \n 1<\/mml:mn>\n 2<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:mstyle>\n <\/mml:mrow>\n t_{\\mathrm {M}}(I) = \\tfrac {1}{2}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We make three general conjectures relating to the value of\n \n \n \n \n \n t<\/mml:mi>\n \n \n M<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n I<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n t_{\\mathrm {M}}(I)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for intervals\n \n \n \n I<\/mml:mi>\n I<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of length less than\n \n \n \n 4<\/mml:mn>\n 4<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We also conjecture a value for\n \n \n \n \n \n t<\/mml:mi>\n \n \n M<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n [<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n b<\/mml:mi>\n ]<\/mml:mo>\n )<\/mml:mo>\n <\/mml:mrow>\n t_{\\mathrm {M}}([0,b])<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n where\n \n \n \n \n 0<\/mml:mn>\n ><\/mml:mo>\n b<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n 0>b\\le 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We give some partial results, as well as computational evidence, to support these conjectures. We define functions\n \n \n \n \n \n L<\/mml:mi>\n \n \n \u2212\n \n <\/mml:mo>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n L_{-}(t)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n L<\/mml:mi>\n \n +<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n L_{+}(t)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , which measure properties of the lengths of intervals\n \n \n \n I<\/mml:mi>\n I<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with\n \n \n \n \n \n t<\/mml:mi>\n \n \n M<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n I<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n t_{\\mathrm {M}}(I)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n on either side of\n \n \n \n t<\/mml:mi>\n t<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Upper and lower bounds are given for these functions. We also consider the problem of determining\n \n \n \n \n \n t<\/mml:mi>\n \n \n M<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n I<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n t_{\\mathrm {M}}(I)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n when\n \n \n \n I<\/mml:mi>\n I<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a Farey interval. We prove that a conjecture of Borwein, Pinner and Pritsker concerning this value is true for an infinite family of Farey intervals.\n <\/p>","DOI":"10.1090\/s0025-5718-06-01843-6","type":"journal-article","created":{"date-parts":[[2006,8,16]],"date-time":"2006-08-16T10:28:45Z","timestamp":1155724125000},"page":"1997-2019","source":"Crossref","is-referenced-by-count":4,"title":["The monic integer transfinite diameter"],"prefix":"10.1090","volume":"75","author":[{"given":"K.","family":"Hare","sequence":"first","affiliation":[]},{"given":"C.","family":"Smyth","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,6,16]]},"reference":[{"key":"1","series-title":"CMS Books in Mathematics\/Ouvrages de Math\\'{e}matiques de la SMC","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-21652-2","volume-title":"Computational excursions in analysis and number theory","volume":"10","author":"Borwein, Peter","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/0387954449"},{"issue":"244","key":"2","doi-asserted-by":"publisher","first-page":"1901","DOI":"10.1090\/S0025-5718-03-01477-7","article-title":"Monic integer Chebyshev problem","volume":"72","author":"Borwein, P. 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