{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:46:23Z","timestamp":1776843983993,"version":"3.51.2"},"reference-count":14,"publisher":"American Mathematical Society (AMS)","issue":"274","license":[{"start":{"date-parts":[[2011,9,9]],"date-time":"2011-09-09T00:00:00Z","timestamp":1315526400000},"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                    Exploiting symmetry in Gr\u00f6bner basis computations is difficult when the symmetry takes the form of a group acting by automorphisms on monomials in finitely many variables. This is largely due to the fact that the group elements, being invertible, cannot preserve a term order. By contrast, inspired by work of Aschenbrenner and Hillar, we introduce the concept of\n                    <italic>equivariant Gr\u00f6bner basis<\/italic>\n                    in a setting where a\n                    <italic>monoid<\/italic>\n                    acts by\n                    <italic>homomorphisms<\/italic>\n                    on monomials in potentially\n                    <italic>infinitely<\/italic>\n                    many variables. We require that the action be compatible with a term order, and under some further assumptions derive a Buchberger-type algorithm for computing equivariant Gr\u00f6bner bases.\n                  <\/p>\n                  <p>\n                    Using this algorithm and the monoid of strictly increasing functions\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper N right-arrow double-struck upper N\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">N<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">\n                              \u2192\n                              \n                            <\/mml:mo>\n                           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comma y Subscript i j Baseline right-arrow from bar s Subscript i Baseline s Subscript j Baseline plus t Subscript i Baseline t Subscript j Baseline\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>y<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>i<\/mml:mi>\n                                <mml:mi>j<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2223\n                              \n                            <\/mml:mo>\n                            <mml:mi>i<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>j<\/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:mo>,<\/mml:mo>\n                            <mml:mi>i<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>j<\/mml:mi>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                            <mml:mo stretchy=\"false\">\n                              \u2192\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>s<\/mml:mi>\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2223\n                              \n                            <\/mml:mo>\n                            <mml:mi>i<\/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:mo stretchy=\"false\">]<\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mtext>\u00a0<\/mml:mtext>\n                            <mml:msub>\n                              <mml:mi>y<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>i<\/mml:mi>\n                                <mml:mi>j<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">\n                              \u21a6\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>s<\/mml:mi>\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:msub>\n                              <mml:mi>s<\/mml:mi>\n                              <mml:mi>j<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:msub>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mi>j<\/mml:mi>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {R}[y_{ij} \\mid i,j \\in \\mathbb {N}, i &gt; j] \\to \\mathbb {R}[s_i,t_i \\mid i \\in \\mathbb {N}],\\ y_{ij} \\mapsto s_is_j + t_it_j<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    is generated by two types of polynomials:\n                    <italic>\n                      off-diagonal\n                      <inline-formula content-type=\"math\/mathml\">\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"3 times 3\">\n                          <mml:semantics>\n                            <mml:mrow>\n                              <mml:mn>3<\/mml:mn>\n                              <mml:mo>\n                                \u00d7\n                                \n                              <\/mml:mo>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:mrow>\n                            <mml:annotation encoding=\"application\/x-tex\">3 \\times 3<\/mml:annotation>\n                          <\/mml:semantics>\n                        <\/mml:math>\n                      <\/inline-formula>\n                      -minors\n                    <\/italic>\n                    and\n                    <italic>pentads<\/italic>\n                    . This confirms a conjecture by Drton, Sturmfels, and Sullivant on the Gaussian two-factor model from algebraic statistics.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2010-02415-9","type":"journal-article","created":{"date-parts":[[2010,12,29]],"date-time":"2010-12-29T14:39:28Z","timestamp":1293633568000},"page":"1123-1133","source":"Crossref","is-referenced-by-count":25,"title":["Equivariant Gr\u00f6bner bases and the Gaussian two-factor model"],"prefix":"10.1090","volume":"80","author":[{"given":"Andries","family":"Brouwer","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jan","family":"Draisma","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2010,9,9]]},"reference":[{"issue":"11","key":"1","doi-asserted-by":"publisher","first-page":"5171","DOI":"10.1090\/S0002-9947-07-04116-5","article-title":"Finite generation of symmetric ideals","volume":"359","author":"Aschenbrenner, Matthias","year":"2007","journal-title":"Trans. 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Buchberger\u2019s algorithm, and polynomials in infinitely many variables","author":"Cohen, Daniel E.","year":"1987","ISBN":"https:\/\/id.crossref.org\/isbn\/3540181709"},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"406","DOI":"10.1006\/jabr.1994.1160","article-title":"Gr\u00f6bner bases of ideals of minors of a symmetric matrix","volume":"166","author":"Conca, Aldo","year":"1994","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"},{"issue":"3","key":"6","doi-asserted-by":"publisher","first-page":"409","DOI":"10.1007\/BF01299745","article-title":"Gr\u00f6bner bases and triangulations of the second hypersimplex","volume":"15","author":"de Loera, Jes\u00fas A.","year":"1995","journal-title":"Combinatorica","ISSN":"https:\/\/id.crossref.org\/issn\/0209-9683","issn-type":"print"},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1016\/j.aim.2009.08.008","article-title":"Finiteness for the \ud835\udc58-factor model and chirality varieties","volume":"223","author":"Draisma, Jan","year":"2010","journal-title":"Adv. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0001-8708","issn-type":"print"},{"issue":"7","key":"8","doi-asserted-by":"publisher","first-page":"835","DOI":"10.1016\/j.jsc.2006.04.004","article-title":"Gr\u00f6bner bases of ideals invariant under endomorphisms","volume":"41","author":"Drensky, Vesselin","year":"2006","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"issue":"3-4","key":"9","doi-asserted-by":"publisher","first-page":"463","DOI":"10.1007\/s00440-006-0033-2","article-title":"Algebraic factor analysis: tetrads, pentads and beyond","volume":"138","author":"Drton, Mathias","year":"2007","journal-title":"Probab. Theory Related Fields","ISSN":"https:\/\/id.crossref.org\/issn\/0178-8051","issn-type":"print"},{"key":"10","unstructured":"Mathias Drton and Han Xiao. Finiteness of small factor analysis models. personal communication, 2008."},{"key":"11","unstructured":"Chris J. Hillar and Seth Sullivant. Finite Gr\u00f6bner bases in infinite dimensional polynomial rings and applications. Preprint, available from \\verb+http:\/\/arxiv.org\/abs\/0908.1777+, 2009."},{"issue":"1","key":"12","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1007\/BF02571229","article-title":"Gr\u00f6bner bases and Stanley decompositions of determinantal rings","volume":"205","author":"Sturmfels, Bernd","year":"1990","journal-title":"Math. Z.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5874","issn-type":"print"},{"issue":"3","key":"13","doi-asserted-by":"publisher","first-page":"867","DOI":"10.4310\/PAMQ.2006.v2.n3.a12","article-title":"Combinatorial secant varieties","volume":"2","author":"Sturmfels, Bernd","year":"2006","journal-title":"Pure Appl. Math. Q.","ISSN":"https:\/\/id.crossref.org\/issn\/1558-8599","issn-type":"print"},{"issue":"2","key":"14","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1216\/JCA-2009-1-2-327","article-title":"A Gr\u00f6bner basis for the secant ideal of the second hypersimplex","volume":"1","author":"Sullivant, Seth","year":"2009","journal-title":"J. Commut. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/1939-0807","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2011-80-274\/S0025-5718-2010-02415-9\/S0025-5718-2010-02415-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2011-80-274\/S0025-5718-2010-02415-9\/S0025-5718-2010-02415-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:48:02Z","timestamp":1776790082000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2011-80-274\/S0025-5718-2010-02415-9\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,9,9]]},"references-count":14,"journal-issue":{"issue":"274","published-print":{"date-parts":[[2011,4]]}},"alternative-id":["S0025-5718-2010-02415-9"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2010-02415-9","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":[[2010,9,9]]}}}