{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:47:32Z","timestamp":1776797252701,"version":"3.51.2"},"reference-count":13,"publisher":"American Mathematical Society (AMS)","issue":"290","license":[{"start":{"date-parts":[[2015,2,25]],"date-time":"2015-02-25T00:00:00Z","timestamp":1424822400000},"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>In the first (theoretical) part of this paper, we prove a number of constraints on hypothetical counterexamples to the Casas-Alvero conjecture, building on ideas of Graf von Bothmer, Labs, Schicho and van de Woestijne<\/p>\n                  <p>\n                    that were recently reinterpreted by Draisma and de Jong in terms of\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 valuations. In the second (computational) part, we present ideas improving upon Diaz-Toca and Gonzalez-Vega\u2019s Gr\u00f6bner basis approach to the Casas-Alvero conjecture. One application is an extension of the proof of Graf von Bothmer et al. to the cases\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"5 p Superscript k\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>5<\/mml:mn>\n                            <mml:msup>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">5p^k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"6 p Superscript k\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>6<\/mml:mn>\n                            <mml:msup>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">6p^k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"7 p Superscript k\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>7<\/mml:mn>\n                            <mml:msup>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">7p^k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    (that is, for each of these cases, we determine the finite list of primes\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                    to which their proof is not applicable). Finally, by combining both parts, we settle the Casas-Alvero conjecture in degree\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"12\">\n                        <mml:semantics>\n                          <mml:mn>12<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">12<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    (the smallest open case).\n                  <\/p>","DOI":"10.1090\/s0025-5718-2014-02809-3","type":"journal-article","created":{"date-parts":[[2014,2,25]],"date-time":"2014-02-25T10:55:28Z","timestamp":1393325728000},"page":"3017-3037","source":"Crossref","is-referenced-by-count":1,"title":["Constraints on counterexamples to the Casas-Alvero conjecture and a verification in degree 12"],"prefix":"10.1090","volume":"83","author":[{"given":"Wouter","family":"Castryck","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Robert","family":"Laterveer","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Myriam","family":"Ouna\u00efes","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2014,2,25]]},"reference":[{"key":"1","unstructured":"Shi Bai, Pierrick Gaudry, Alexander Kruppa, Fran\u00e7ois Morain, Emmanuel Thom\u00e9 and Paul Zimmerman, CADO-NFS 1.1, available at \\verb\"http:\/\/cado-nfs.gforge.inria.fr\/\"."},{"issue":"3-4","key":"2","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1006\/jsco.1996.0125","article-title":"The Magma algebra system. I. The user language","volume":"24","author":"Bosma, Wieb","year":"1997","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"issue":"1","key":"3","doi-asserted-by":"publisher","first-page":"326","DOI":"10.1006\/jabr.2000.8727","article-title":"Higher order polar germs","volume":"240","author":"Casas-Alvero, Eduardo","year":"2001","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"},{"issue":"1-2","key":"4","first-page":"54","article-title":"La conjecture de Casas Alvero pour les degr\u00e9s 5\ud835\udc5d^{\ud835\udc52}","volume":"4(35)","author":"Chellali, Mustapha","year":"2012","journal-title":"An. Univ. Dun\\u{a}rea de Jos Gala\\c{t}i Fasc. II Mat. Fiz. Mec. Teor.","ISSN":"https:\/\/id.crossref.org\/issn\/2067-2071","issn-type":"print"},{"key":"5","unstructured":"Rosa de Frutos, Perspectivas aritm\u00e9ticas para la conjetura de Casas-Alvero, Ph.D. thesis, Universidad de Valladolid, (2013)."},{"key":"6","unstructured":"Gema M. Diaz-Toca and Laureano Gonzalez-Vega, On analyzing a conjecture about univariate polynomials and their roots by using maple, Proceedings of the Maple Conference 2006, Waterloo (Canada), July 23-26, 2006, pp. 81\u201398 (2006)."},{"issue":"80","key":"7","first-page":"29","article-title":"On the Casas-Alvero conjecture","author":"Draisma, Jan","year":"2011","journal-title":"Eur. Math. Soc. Newsl.","ISSN":"https:\/\/id.crossref.org\/issn\/1027-488X","issn-type":"print"},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"224","DOI":"10.1016\/j.jalgebra.2007.06.017","article-title":"The Casas-Alvero conjecture for infinitely many degrees","volume":"316","author":"Graf von Bothmer, Hans-Christian","year":"2007","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"},{"issue":"4","key":"9","doi-asserted-by":"publisher","first-page":"963","DOI":"10.2307\/1990996","article-title":"Sharp effective Nullstellensatz","volume":"1","author":"Koll\u00e1r, J\u00e1nos","year":"1988","journal-title":"J. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0894-0347","issn-type":"print"},{"key":"10","unstructured":"Adrien-Marie Legendre, Th\u00e9orie des nombres, Firmin Didot Fr\u00e8res, Paris (1830)."},{"key":"11","series-title":"Algorithms and Computation in Mathematics","isbn-type":"print","volume-title":"Polynomials","volume":"11","author":"Prasolov, Victor V.","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9783642039799"},{"key":"12","series-title":"Springer Monographs in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0551-7","volume-title":"The theory of classical valuations","author":"Ribenboim, Paulo","year":"1999","ISBN":"https:\/\/id.crossref.org\/isbn\/0387985255"},{"key":"13","unstructured":"Hendrik Verhoek, Some remarks about a polynomial conjecture of Casas-Alvero, S\u00e9minaire Bourbakettes, Paris (2009)."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2014-83-290\/S0025-5718-2014-02809-3\/S0025-5718-2014-02809-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2014-83-290\/S0025-5718-2014-02809-3\/S0025-5718-2014-02809-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:08:07Z","timestamp":1776794887000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2014-83-290\/S0025-5718-2014-02809-3\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,2,25]]},"references-count":13,"journal-issue":{"issue":"290","published-print":{"date-parts":[[2014,11]]}},"alternative-id":["S0025-5718-2014-02809-3"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2014-02809-3","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":[[2014,2,25]]}}}