{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:47:50Z","timestamp":1776836870382,"version":"3.51.2"},"reference-count":22,"publisher":"American Mathematical Society (AMS)","issue":"342","license":[{"start":{"date-parts":[[2024,3,7]],"date-time":"2024-03-07T00:00:00Z","timestamp":1709769600000},"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":"<p>\n                    In this paper, a new method to compute lower and upper bounds for Salem numbers with a given trace and a given degree is given. With this method, it is proven that the smallest trace of Salem numbers of degree\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"22\">\n                        <mml:semantics>\n                          <mml:mn>22<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">22<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"negative 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">-1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Further, new lower bounds for degree of Salem numbers with minimal trace\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"negative 5\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>5<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">-5<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"negative 6\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>6<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">-6<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are given. All Salem numbers of trace\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"negative 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">-2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and degree\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"24\">\n                        <mml:semantics>\n                          <mml:mn>24<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">24<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"26\">\n                        <mml:semantics>\n                          <mml:mn>26<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">26<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are given. This includes\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"7\">\n                        <mml:semantics>\n                          <mml:mn>7<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">7<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    additional Salem numbers of degree\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"26\">\n                        <mml:semantics>\n                          <mml:mn>26<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">26<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    beyond what was previously known. The auxiliary functions related to Chebyshev polynomials, which are adapted to Salem number, are used in this work.\n                  <\/p>","DOI":"10.1090\/mcom\/3833","type":"journal-article","created":{"date-parts":[[2023,2,9]],"date-time":"2023-02-09T08:16:26Z","timestamp":1675930586000},"page":"1779-1790","source":"Crossref","is-referenced-by-count":1,"title":["Salem numbers with minimal trace"],"prefix":"10.1090","volume":"92","author":[{"given":"Qiong","family":"Chen","sequence":"first","affiliation":[]},{"given":"Qiang","family":"Wu","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2023,3,7]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/CBO9780511721274.003","article-title":"The trace problem for totally positive algebraic integers","author":"Aguirre, Juli\u00e1n","year":"2008","ISBN":"https:\/\/id.crossref.org\/isbn\/9780521714679"},{"key":"2","series-title":"Wiley-Interscience Series in Discrete Mathematics and Optimization","isbn-type":"print","volume-title":"Linear programming in infinite-dimensional spaces","author":"Anderson, Edward J.","year":"1987","ISBN":"https:\/\/id.crossref.org\/isbn\/0471912506"},{"key":"3","unstructured":"C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier, GP-Pari version 2.3.2, 2007."},{"key":"4","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":"214","key":"5","doi-asserted-by":"publisher","first-page":"661","DOI":"10.1090\/S0025-5718-96-00702-8","article-title":"The integer Chebyshev problem","volume":"65","author":"Borwein, Peter","year":"1996","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"6","doi-asserted-by":"publisher","first-page":"23","DOI":"10.11650\/tjm\/8208","article-title":"Finding all Salem numbers of trace -1 and degree up to 20","volume":"22","author":"Chen, Youyan","year":"2018","journal-title":"Taiwanese J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1027-5487","issn-type":"print"},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"179","DOI":"10.5802\/jtnb.1116","article-title":"Linear relations with conjugates of a Salem number","volume":"32","author":"Dubickas, Art\u016bras","year":"2020","journal-title":"J. Th\\'{e}or. Nombres Bordeaux","ISSN":"https:\/\/id.crossref.org\/issn\/1246-7405","issn-type":"print"},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1017\/S0305004107000692","article-title":"On the lines passing through two conjugates of a Salem number","volume":"144","author":"Dubickas, Art\u016bras","year":"2008","journal-title":"Math. Proc. Cambridge Philos. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0305-0041","issn-type":"print"},{"issue":"266","key":"9","doi-asserted-by":"publisher","first-page":"1119","DOI":"10.1090\/S0025-5718-08-02120-0","article-title":"Trace of totally positive algebraic integers and integer transfinite diameter","volume":"78","author":"Flammang, V.","year":"2009","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"10","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1142\/S1793042119500064","article-title":"The absolute trace of totally positive algebraic integers","volume":"15","author":"Flammang, V.","year":"2019","journal-title":"Int. J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/1793-0421","issn-type":"print"},{"issue":"4","key":"11","doi-asserted-by":"publisher","first-page":"515","DOI":"10.1007\/BF01457454","article-title":"Factoring polynomials with rational coefficients","volume":"261","author":"Lenstra, A. K.","year":"1982","journal-title":"Math. Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"issue":"3","key":"12","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1017\/S1446788711001030","article-title":"The trace problem for totally positive algebraic integers","volume":"90","author":"Liang, Yanhua","year":"2011","journal-title":"J. Aust. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/1446-7887","issn-type":"print"},{"issue":"274","key":"13","doi-asserted-by":"publisher","first-page":"1041","DOI":"10.1090\/S0025-5718-2010-02424-X","article-title":"Computing totally positive algebraic integers of small trace","volume":"80","author":"McKee, James","year":"2011","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"14","doi-asserted-by":"publisher","first-page":"409","DOI":"10.1016\/j.jnt.2015.09.019","article-title":"Salem numbers of trace -2, and a conjecture of Estes and Guralnick","volume":"160","author":"McKee, James","year":"2016","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"key":"15","isbn-type":"print","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1007\/978-3-540-24847-7_25","article-title":"Salem numbers of trace -2 and traces of totally positive algebraic integers","author":"McKee, James","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/3540221565"},{"issue":"1","key":"16","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1112\/S0024609304003790","article-title":"There are Salem numbers of every trace","volume":"37","author":"McKee, James","year":"2005","journal-title":"Bull. London Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6093","issn-type":"print"},{"issue":"1","key":"17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1080\/10586458.2013.849213","article-title":"Finding degree-16 monic irreducible integer polynomials of minimal trace by optimization methods","volume":"23","author":"El Otmani, S.","year":"2014","journal-title":"Exp. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1058-6458","issn-type":"print"},{"key":"18","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1016\/j.jnt.2014.11.013","article-title":"A Salem number with degree 34 and trace -3","volume":"150","author":"El Otmani, S.","year":"2015","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"230","key":"19","doi-asserted-by":"publisher","first-page":"827","DOI":"10.1090\/S0025-5718-99-01099-6","article-title":"Salem numbers of negative trace","volume":"69","author":"Smyth, C. J.","year":"2000","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"20","doi-asserted-by":"publisher","first-page":"1","DOI":"10.5802\/aif.985","article-title":"Totally positive algebraic integers of small trace","volume":"34","author":"Smyth, Christopher","year":"1984","journal-title":"Ann. Inst. Fourier (Grenoble)","ISSN":"https:\/\/id.crossref.org\/issn\/0373-0956","issn-type":"print"},{"issue":"331","key":"21","doi-asserted-by":"publisher","first-page":"2317","DOI":"10.1090\/mcom\/3636","article-title":"Totally positive algebraic integers with small trace","volume":"90","author":"Wang, Cong","year":"2021","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"242","key":"22","doi-asserted-by":"publisher","first-page":"901","DOI":"10.1090\/S0025-5718-02-01442-4","article-title":"On the linear independence measure of logarithms of rational numbers","volume":"72","author":"Wu, Qiang","year":"2003","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-342\/S0025-5718-2023-03833-9\/S0025-5718-2023-03833-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:58:36Z","timestamp":1776833916000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-342\/S0025-5718-2023-03833-9\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,7]]},"references-count":22,"journal-issue":{"issue":"342","published-print":{"date-parts":[[2023,7]]}},"alternative-id":["S0025-5718-2023-03833-9"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3833","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":[[2023,3,7]]}}}