{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:38:26Z","timestamp":1776724706788,"version":"3.51.2"},"reference-count":16,"publisher":"American Mathematical Society (AMS)","issue":"224","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Given a monic real polynomial with all its roots on the unit circle, we ask to what extent one can perturb its middle coefficient and still have a polynomial with all its roots on the unit circle. We show that the set of possible perturbations forms a closed interval of length at most\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"4\">\n                        <mml:semantics>\n                          <mml:mn>4<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">4<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"4\">\n                        <mml:semantics>\n                          <mml:mn>4<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">4<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    achieved only for polynomials of the form\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x Superscript 2 n Baseline plus c x Superscript n plus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                                <mml:mi>n<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:msup>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">x^{2n}+cx^n+1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"c\">\n                        <mml:semantics>\n                          <mml:mi>c<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">c<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-bracket negative 2 comma 2 right-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">[-2,2]<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The problem can also be formulated in terms of perturbing the constant coefficient of a polynomial having all its roots in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-bracket negative 1 comma 1 right-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">[-1,1]<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . If we restrict to integer coefficients, then the polynomials in question are products of cyclotomics. We show that in this case there are no perturbations of length\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"3\">\n                        <mml:semantics>\n                          <mml:mn>3<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    that do not arise from a perturbation of length\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"4\">\n                        <mml:semantics>\n                          <mml:mn>4<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">4<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We also investigate the connection between slightly perturbed products of cyclotomic polynomials and polynomials with small Mahler measure. We describe an algorithm for searching for polynomials with small Mahler measure by perturbing the middle coefficients of products of cyclotomic polynomials. We show that the complexity of this algorithm is\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis upper C Superscript StartRoot d EndRoot Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>C<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:msqrt>\n                                  <mml:mi>d<\/mml:mi>\n                                <\/mml:msqrt>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(C^{\\sqrt {d}})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d\">\n                        <mml:semantics>\n                          <mml:mi>d<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">d<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is the degree, and we report on the polynomials found by this algorithm through degree 64.\n                  <\/p>","DOI":"10.1090\/s0025-5718-98-01007-2","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"1707-1726","source":"Crossref","is-referenced-by-count":17,"title":["Perturbing polynomials with all their roots on the unit circle"],"prefix":"10.1090","volume":"67","author":[{"given":"Michael","family":"Mossinghoff","sequence":"first","affiliation":[]},{"given":"Christopher","family":"Pinner","sequence":"additional","affiliation":[]},{"given":"Jeffrey","family":"Vaaler","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1998]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"80","DOI":"10.1006\/jnth.1996.0114","article-title":"Algebraic numbers close to 1 and variants of Mahler\u2019s measure","volume":"60","author":"Amoroso, Francesco","year":"1996","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"152","key":"2","doi-asserted-by":"publisher","first-page":"1361","DOI":"10.2307\/2006402","article-title":"Reciprocal polynomials having small measure","volume":"35","author":"Boyd, David W.","year":"1980","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"4","key":"3","doi-asserted-by":"publisher","first-page":"453","DOI":"10.4153\/CMB-1981-069-5","article-title":"Speculations concerning the range of Mahler\u2019s measure","volume":"24","author":"Boyd, David W.","year":"1981","journal-title":"Canad. Math. Bull.","ISSN":"https:\/\/id.crossref.org\/issn\/0008-4395","issn-type":"print"},{"key":"4","isbn-type":"print","first-page":"7","article-title":"Cyclotomic partitions","author":"Boyd, David W.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/3110117231"},{"issue":"4","key":"5","doi-asserted-by":"publisher","first-page":"391","DOI":"10.4064\/aa-34-4-391-401","article-title":"On a question of Lehmer and the number of irreducible factors of a polynomial","volume":"34","author":"Dobrowolski, E.","year":"1979","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"6","unstructured":"E. Hille, Analytic Function Theory, Vol. II, Chelsea, 1987."},{"key":"7","doi-asserted-by":"crossref","unstructured":"D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. (2) 34 (1933), 461\u2013479.","DOI":"10.2307\/1968172"},{"key":"8","doi-asserted-by":"crossref","unstructured":"E. T. Lehmer, A numerical function applied to cyclotomy, Bull. Amer. Math. Soc. 36 (1930), 291\u2013298.","DOI":"10.1090\/S0002-9904-1930-04939-3"},{"key":"9","unstructured":"M. J. Mossinghoff, Algorithms for the determination of polynomials with small Mahler measure, Ph.D. Thesis, The University of Texas at Austin, 1995."},{"key":"10","doi-asserted-by":"crossref","unstructured":"M. J. Mossinghoff, Polynomials with small Mahler measure, Math. Comp. 67 (1998), 1697\u20131705.","DOI":"10.1090\/S0025-5718-98-01006-0"},{"key":"11","series-title":"Springer Study Edition","isbn-type":"print","volume-title":"Problems and theorems in analysis. Vol. II","author":"P\u00f3lya, G.","year":"1976","ISBN":"https:\/\/id.crossref.org\/isbn\/0387902910"},{"key":"12","series-title":"Translations of Mathematical Monographs","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/mmono\/068","volume-title":"Introduction to analytic number theory","volume":"68","author":"Postnikov, A. 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