{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,6]],"date-time":"2026-05-06T16:03:46Z","timestamp":1778083426595,"version":"3.51.4"},"reference-count":4,"publisher":"American Mathematical Society (AMS)","issue":"222","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We consider bounds on the smallest possible root with a specified argument\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03d5\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\phi<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of a power series\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f left-parenthesis z right-parenthesis equals 1 plus sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts a Subscript i Baseline z Superscript i\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:munderover>\n                                <mml:mo>\n                                  \u2211\n                                  \n                                <\/mml:mo>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi>n<\/mml:mi>\n                                  <mml:mo>=<\/mml:mo>\n                                  <mml:mn>1<\/mml:mn>\n                                <\/mml:mrow>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"normal\">\n                                    \u221e\n                                    \n                                  <\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:munderover>\n                            <\/mml:mrow>\n                            <mml:msub>\n                              <mml:mi>a<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>i<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:msup>\n                              <mml:mi>z<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>i<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">f(z)=1+{ \\sum _{n=1}^{\\infty }} a_{i}z^{i}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with coefficients\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a Subscript i\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">a_{i}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in the interval\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-bracket negative g comma g 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:mi>g<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>g<\/mml:mi>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">[-g,g]<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We describe the form that the extremal power series must take and hence give an algorithm for computing the optimal root when\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi slash 2 pi\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03d5\n                              \n                            <\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>\n                              \u03c0\n                              \n                            <\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\phi \/2\\pi<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is rational. When\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"g greater-than-or-equal-to 2 StartRoot 2 EndRoot plus 3\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>g<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:msqrt>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msqrt>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">g\\geq 2\\sqrt {2}+3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    we show that the smallest disc containing two roots has radius\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis StartRoot g EndRoot plus 1 right-parenthesis Superscript negative 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msqrt>\n                              <mml:mi>g<\/mml:mi>\n                            <\/mml:msqrt>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(\\sqrt {g}+1)^{-1}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    coinciding with the smallest double real root possible for such a series. It is clear from our computations that the behaviour is more complicated for smaller\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"g\">\n                        <mml:semantics>\n                          <mml:mi>g<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">g<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We give a similar procedure for computing the smallest circle with a real root and a pair of conjugate roots of a given argument. We conclude by briefly discussing variants of the beta-numbers (where the defining integer sequence is generated by taking the nearest integer rather than the integer part). We show that the conjugates,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"lamda\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03bb\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\lambda<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , of these pseudo-beta-numbers either lie inside the unit circle or their reciprocals must be roots of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-bracket negative 1 slash 2 comma 1 slash 2 right-parenthesis\">\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:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">[-1\/2,1\/2)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    power series; in particular we obtain the sharp inequality\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartAbsoluteValue lamda EndAbsoluteValue less-than-or-equal-to 3 slash 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>\n                              \u03bb\n                              \n                            <\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">|\\lambda |\\leq 3\/2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-98-00960-0","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:13:45Z","timestamp":1027707225000},"page":"715-736","source":"Crossref","is-referenced-by-count":9,"title":["Power series with restricted coefficients and a root on a given ray"],"prefix":"10.1090","volume":"67","author":[{"given":"Franck","family":"Beaucoup","sequence":"first","affiliation":[]},{"given":"Peter","family":"Borwein","sequence":"additional","affiliation":[]},{"given":"David","family":"Boyd","sequence":"additional","affiliation":[]},{"given":"Christopher","family":"Pinner","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1998]]},"reference":[{"key":"1","unstructured":"F. Beaucoup, P. Borwein, D. W. Boyd and C. Pinner, Multiple roots of [-1,1], J. London Math. Soc. to appear."},{"issue":"3-4","key":"2","first-page":"317","article-title":"Zeros of polynomials with 0,1 coefficients","volume":"39","author":"Odlyzko, A. M.","year":"1993","journal-title":"Enseign. Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0013-8584","issn-type":"print"},{"issue":"3","key":"3","doi-asserted-by":"publisher","first-page":"477","DOI":"10.1112\/plms\/s3-68.3.477","article-title":"Conjugates of beta-numbers and the zero-free domain for a class of analytic functions","volume":"68","author":"Solomyak, Boris","year":"1994","journal-title":"Proc. London Math. Soc. (3)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6115","issn-type":"print"},{"issue":"5","key":"4","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1006\/jsco.1994.1056","article-title":"On some bounds for zeros of norm-bounded polynomials","volume":"18","author":"Yamamoto, Osami","year":"1994","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1998-67-222\/S0025-5718-98-00960-0\/S0025-5718-98-00960-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-222\/S0025-5718-98-00960-0\/S0025-5718-98-00960-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:44:55Z","timestamp":1776721495000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-222\/S0025-5718-98-00960-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998]]},"references-count":4,"journal-issue":{"issue":"222","published-print":{"date-parts":[[1998,4]]}},"alternative-id":["S0025-5718-98-00960-0"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-98-00960-0","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":[[1998]]}}}