{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:17:01Z","timestamp":1776795421861,"version":"3.51.2"},"reference-count":31,"publisher":"American Mathematical Society (AMS)","issue":"280","license":[{"start":{"date-parts":[[2013,3,14]],"date-time":"2013-03-14T00:00:00Z","timestamp":1363219200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We present a method for tabulating all cubic function fields over\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper F Subscript q Baseline left-parenthesis t right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">F<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mi>q<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {F}_q(t)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    whose discriminant\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper 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                    has either odd degree or even degree and the leading coefficient of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"minus 3 upper D\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                            <mml:mi>D<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">-3D<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a non-square in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper F Subscript q Superscript asterisk\">\n                        <mml:semantics>\n                          <mml:msubsup>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">F<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>q<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo>\n                              \u2217\n                              \n                            <\/mml:mo>\n                          <\/mml:msubsup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {F}_{q}^*<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , up to a given bound\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper B\">\n                        <mml:semantics>\n                          <mml:mi>B<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">B<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    on\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"degree left-parenthesis upper D right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>deg<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>D<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\deg (D)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Our method is based on a generalization of Belabas\u2019 method for tabulating cubic number fields. The main theoretical ingredient is a generalization of a theorem of Davenport and Heilbronn to cubic function fields, along with a reduction theory for binary cubic forms that provides an efficient way to compute equivalence classes of binary cubic forms. The algorithm requires\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 B Superscript 4 Baseline q Superscript upper B 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>B<\/mml:mi>\n                              <mml:mn>4<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:msup>\n                              <mml:mi>q<\/mml:mi>\n                              <mml:mi>B<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(B^4 q^B)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    field operations as\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper B right-arrow normal infinity\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>B<\/mml:mi>\n                            <mml:mo stretchy=\"false\">\n                              \u2192\n                              \n                            <\/mml:mo>\n                            <mml:mi mathvariant=\"normal\">\n                              \u221e\n                              \n                            <\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">B \\rightarrow \\infty<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The algorithm, examples and numerical data for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"q equals 5 comma 7 comma 11 comma 13\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>q<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>5<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>7<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>11<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>13<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">q=5,7,11,13<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are included.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2012-02591-9","type":"journal-article","created":{"date-parts":[[2012,3,14]],"date-time":"2012-03-14T14:13:10Z","timestamp":1331734390000},"page":"2335-2359","source":"Crossref","is-referenced-by-count":4,"title":["Tabulation of cubic function fields via polynomial binary cubic forms"],"prefix":"10.1090","volume":"81","author":[{"given":"Pieter","family":"Rozenhart","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael Jacobson","family":"Jr.","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Renate","family":"Scheidler","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2012,3,14]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1007\/BF01181074","article-title":"Quadratische K\u00f6rper im Gebiete der h\u00f6heren Kongruenzen. 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