{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T03:41:32Z","timestamp":1776829292821,"version":"3.51.2"},"reference-count":21,"publisher":"American Mathematical Society (AMS)","issue":"214","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We are concerned with the problem of minimizing the supremum norm on an interval of a nonzero polynomial of degree at most\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with integer coefficients. This is an old and hard problem that cannot be exactly solved in any nontrivial cases. We examine the case of the interval\n \n \n \n \n [<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n [0,1]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in most detail. Here we improve the known bounds a small but interesting amount. This allows us to garner further information about the structure of such minimal polynomials and their factors. This is primarily a (substantial) computational exercise. We also examine some of the structure of such minimal \u201cinteger Chebyshev\u201d polynomials, showing for example that on small intevals\n \n \n \n \n [<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n \n \u03b4\n \n <\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n [0, \\delta ]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and for small degrees\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n x<\/mml:mi>\n \n d<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n x^{d}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n achieves the minimal norm. There is a natural conjecture, due to the Chudnovskys and others, as to what the \u201cinteger transfinite diameter\u201d of\n \n \n \n \n [<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n [0,1]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n should be. We show that this conjecture is false. The problem is then related to a trace problem for totally positive algebraic integers due to Schur and Siegel. Several open problems are raised.\n <\/p>","DOI":"10.1090\/s0025-5718-96-00702-8","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"661-681","source":"Crossref","is-referenced-by-count":31,"title":["The integer Chebyshev problem"],"prefix":"10.1090","volume":"65","author":[{"given":"Peter","family":"Borwein","sequence":"first","affiliation":[]},{"given":"Tam\u00e1s","family":"Erd\u00e9lyi","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"issue":"4","key":"1","doi-asserted-by":"publisher","first-page":"885","DOI":"10.5802\/aif.1240","article-title":"Sur le diam\u00e8tre transfini entier d\u2019un intervalle r\u00e9el","volume":"40","author":"Amoroso, Francesco","year":"1990","journal-title":"Ann. Inst. Fourier (Grenoble)","ISSN":"https:\/\/id.crossref.org\/issn\/0373-0956","issn-type":"print"},{"issue":"6","key":"2","first-page":"259","article-title":"Methods for the approximate calculation of the minimum uniform Diophantine deviation from zero on a segment","volume":"38","author":"Aparicio Bernardo, Emiliano","year":"1978","journal-title":"Rev. Mat. Hisp.-Amer. (4)"},{"key":"3","unstructured":"\\bysame, New bounds on the minimal Diophantine deviation from zero on [0,1] and [0,1\/4], Actus Sextas Jour. Mat. Hisp.-Lusitanas (1979), 289\u2013291."},{"key":"4","unstructured":"\\bysame, On some systems of algebraic integers of D.S. Gorshkov and their application in calculus, Rev. Mat. Hisp.-Amer. 41 (1981), 3\u201317 (Spanish)."},{"key":"5","first-page":"6","article-title":"Some results in the problem of Diophantine approximations of functions by polynomials","volume":"163","author":"Aparisio, \u00c8.","year":"1984","journal-title":"Trudy Mat. Inst. Steklov.","ISSN":"https:\/\/id.crossref.org\/issn\/0371-9685","issn-type":"print"},{"key":"6","doi-asserted-by":"crossref","unstructured":"P. Borwein and T. Erd\u00e9lyi, Polynomials and polynomial inequalities, Springer-Verlag, New York, 1995.","DOI":"10.1007\/978-1-4612-0793-1"},{"key":"7","unstructured":"P. Borwein and C. Ingalls, The Prouhet Tarry Escott problem revisited, Math. Enseign. 40 (1994), 3\u201327."},{"key":"8","isbn-type":"print","first-page":"61","article-title":"Number theoretic applications of polynomials with rational coefficients defined by extremality conditions","author":"Chudnovsky, G. V.","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/3764331321"},{"key":"9","doi-asserted-by":"crossref","unstructured":"M. Fekete, \u00dcber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Zeit. 17 (1923), 228\u2013249.","DOI":"10.1007\/BF01504345"},{"key":"10","series-title":"Mathematical Surveys, No. 17","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1090\/surv\/017","volume-title":"Approximation by polynomials with integral coefficients","author":"Ferguson, Le Baron O.","year":"1980","ISBN":"https:\/\/id.crossref.org\/isbn\/0821815172"},{"key":"11","unstructured":"V. Flammang, Sur la longeur des entiers alg\u00e9briques totalement positifs, preprint."},{"key":"12","doi-asserted-by":"crossref","unstructured":"D. Hilbert, Ein Beitrag zur Theorie des Legendreschen Polynoms, Acta Math. 18 (1894), 155\u2013159.","DOI":"10.1007\/BF02418278"},{"key":"13","unstructured":"B. Kashin, On algebraic polynomials with integer coefficients with least deviation from zero on an interval, preprint."},{"issue":"3","key":"14","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1007\/BF03167271","article-title":"Analysis of approximate factorization algorithm. I","volume":"9","author":"Sasaki, Tateaki","year":"1992","journal-title":"Japan J. Indust. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0916-7005","issn-type":"print"},{"key":"15","doi-asserted-by":"crossref","unstructured":"H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, CBMS, Vol. 84, Amer. Math. Soc., Providence, R. I., 1994.","DOI":"10.1090\/cbms\/084"},{"key":"16","doi-asserted-by":"crossref","unstructured":"I. Schur, \u00dcber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganz-zahligen Koeffizienten, Math. Zeit. 1 (1918), 377\u2013402.","DOI":"10.1007\/BF01465096"},{"key":"17","series-title":"Die Grundlehren der mathematischen Wissenschaften, Band 193","volume-title":"Problems and theorems in analysis. Vol. I: Series, integral calculus, theory of functions","author":"P\u00f3lya, G.","year":"1972"},{"issue":"3","key":"18","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1007\/BF01162064","article-title":"On lacunary incomplete polynomials","volume":"177","author":"Saff, Edward B.","year":"1981","journal-title":"Math. Z.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5874","issn-type":"print"},{"key":"19","doi-asserted-by":"publisher","first-page":"425","DOI":"10.2307\/2303037","article-title":"Series for all the roots of the equation (\ud835\udc67-\ud835\udc4e)^{\ud835\udc5a}=\ud835\udc58(\ud835\udc67-\ud835\udc4f)\u207f","volume":"46","author":"Eagle, Albert","year":"1939","journal-title":"Amer. Math. Monthly","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9890","issn-type":"print"},{"issue":"166","key":"20","doi-asserted-by":"publisher","first-page":"663","DOI":"10.2307\/2007609","article-title":"The mean values of totally real algebraic integers","volume":"42","author":"Smyth, C. J.","year":"1984","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"21","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"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1996-65-214\/S0025-5718-96-00702-8\/S0025-5718-96-00702-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-214\/S0025-5718-96-00702-8\/S0025-5718-96-00702-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:07:09Z","timestamp":1776719229000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-214\/S0025-5718-96-00702-8\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996]]},"references-count":21,"journal-issue":{"issue":"214","published-print":{"date-parts":[[1996,4]]}},"alternative-id":["S0025-5718-96-00702-8"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-96-00702-8","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":[[1996]]}}}