{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:13:29Z","timestamp":1776788009418,"version":"3.51.2"},"reference-count":18,"publisher":"American Mathematical Society (AMS)","issue":"263","license":[{"start":{"date-parts":[[2009,2,11]],"date-time":"2009-02-11T00:00:00Z","timestamp":1234310400000},"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                    We study families of integer circulant matrices and methods for determining which are unimodular. This problem arises in the study of cyclically presented groups, and leads to the following problem concerning polynomials with integer coefficients: given a polynomial\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f left-parenthesis x right-parenthesis element-of double-struck upper Z left-bracket x right-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">f(x)\\in \\mathbb {Z}[x]<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , determine all those\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n element-of double-struck upper N\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">N<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n\\in \\mathbb {N}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    such that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper R e s left-parenthesis f left-parenthesis x right-parenthesis comma x Superscript n Baseline minus 1 right-parenthesis equals plus-or-minus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>Res<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo>\n                              \u00b1\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\operatorname {Res}(f(x),x^n-1)=\\pm 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In this paper we describe methods for resolving this problem, including a method based on the use of Strassman\u2019s Theorem on\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -adic power series, which are effective in many cases. The methods are illustrated with examples arising in the study of cyclically presented groups and further examples which illustrate the strengths and weaknesses of the methods for polynomials of higher degree.\n                  <\/p>","DOI":"10.1090\/s0025-5718-08-02089-9","type":"journal-article","created":{"date-parts":[[2008,4,22]],"date-time":"2008-04-22T17:24:54Z","timestamp":1208885094000},"page":"1639-1652","source":"Crossref","is-referenced-by-count":9,"title":["Unimodular integer circulants"],"prefix":"10.1090","volume":"77","author":[{"given":"J.","family":"Cremona","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2008,2,11]]},"reference":[{"key":"1","isbn-type":"print","first-page":"67","article-title":"Cyclotomic points on curves","author":"Beukers, F.","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/1568811268"},{"issue":"3-4","key":"2","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1006\/jsco.1996.0125","article-title":"The Magma algebra system. 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Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1058-6458","issn-type":"print"},{"issue":"2","key":"6","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1515\/jgth.2003.019","article-title":"On irreducible cyclic presentations","volume":"6","author":"Edjvet, Martin","year":"2003","journal-title":"J. Group Theory","ISSN":"https:\/\/id.crossref.org\/issn\/1433-5883","issn-type":"print"},{"issue":"2","key":"7","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1142\/S0218196704001724","article-title":"On a class of cyclically presented groups","volume":"14","author":"Edjvet, Martin","year":"2004","journal-title":"Internat. J. Algebra Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0218-1967","issn-type":"print"},{"issue":"2","key":"8","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1080\/10586458.2001.10504450","article-title":"Cyclic presentations of the trivial group","volume":"10","author":"Edjvet, Martin","year":"2001","journal-title":"Experiment. 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PARI\/GP, version 2.4.1, 2006. available from \\url{http:\/\/pari.math.u-bordeaux.fr\/}."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2008-77-263\/S0025-5718-08-02089-9\/S0025-5718-08-02089-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2008-77-263\/S0025-5718-08-02089-9\/S0025-5718-08-02089-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:22:32Z","timestamp":1776784952000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2008-77-263\/S0025-5718-08-02089-9\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,2,11]]},"references-count":18,"journal-issue":{"issue":"263","published-print":{"date-parts":[[2008,7]]}},"alternative-id":["S0025-5718-08-02089-9"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-08-02089-9","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":[[2008,2,11]]}}}