{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T10:10:01Z","timestamp":1776766201240,"version":"3.51.2"},"reference-count":12,"publisher":"American Mathematical Society (AMS)","issue":"213","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Continuing the recent work of the second author, we prove that the diophantine equation\n \n \\[\n \n \n \n \n f<\/mml:mi>\n a<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n x<\/mml:mi>\n ,<\/mml:mo>\n y<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n \n x<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msup>\n \n \u2212\n \n <\/mml:mo>\n a<\/mml:mi>\n \n x<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n y<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n \n x<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \n y<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n +<\/mml:mo>\n a<\/mml:mi>\n x<\/mml:mi>\n \n y<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n +<\/mml:mo>\n \n y<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msup>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n f_a(x,y)=x^4-ax^3 y-x^2 y^2+axy^3+y^4=1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n \\]\n <\/disp-formula>\n for\n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n a<\/mml:mi>\n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \u2265\n \n <\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n |a|\\ge 3<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n has exactly 12 solutions except when\n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n a<\/mml:mi>\n \n |<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n |a|=4<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , when it has 16 solutions. If\n \n \n \n \n \n \u03b1\n \n <\/mml:mi>\n =<\/mml:mo>\n \n \u03b1\n \n <\/mml:mi>\n (<\/mml:mo>\n a<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\alpha =\\alpha (a)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n denotes one of the zeros of\n \n \n \n \n \n f<\/mml:mi>\n a<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n x<\/mml:mi>\n ,<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n f_a(x,1)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , then for\n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n a<\/mml:mi>\n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \u2265\n \n <\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n |a|\\ge 4<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n we also find all\n \n \n \n \n \n \u03b3\n \n <\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n \n Z<\/mml:mi>\n <\/mml:mrow>\n [<\/mml:mo>\n \n \u03b1\n \n <\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n \\gamma \\in \\Bbb Z[\\alpha ]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with\n \n \n \n \n \n Z<\/mml:mi>\n <\/mml:mrow>\n [<\/mml:mo>\n \n \u03b3\n \n <\/mml:mi>\n ]<\/mml:mo>\n =<\/mml:mo>\n \n Z<\/mml:mi>\n <\/mml:mrow>\n [<\/mml:mo>\n \n \u03b1\n \n <\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n \\Bbb Z[\\gamma ]=\\Bbb Z[\\alpha ]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/s0025-5718-96-00662-x","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"341-354","source":"Crossref","is-referenced-by-count":17,"title":["Complete solutions of a family of quartic Thue and index form equations"],"prefix":"10.1090","volume":"65","author":[{"given":"Maurice","family":"Mignotte","sequence":"first","affiliation":[]},{"given":"Attila","family":"Peth\u00f6","sequence":"additional","affiliation":[]},{"given":"Ralf","family":"Roth","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","unstructured":"R. Roth, LiPS\u2014Ein System f\u00fcr verteilte Anwendungen, Master Thesis, Dept. of Computer Science, Universit\u00e4t des Saarlandes, Germany, Jan. 1992."},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"18","DOI":"10.1016\/0022-314X(91)90090-X","article-title":"On the resolution of index form equations in biquadratic number fields. I, II","volume":"38","author":"Ga\u00e1l, I.","year":"1991","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","unstructured":"\\bysame, On the resolution of index form equations, Proc. ISSAC \u201991 (S. M. Watt, ed.), ACM Press, 1991 pp. 185\u2013186.","DOI":"10.1145\/120694.120720"},{"key":"4","unstructured":"E. Lee, Studies on Diophantine equations, Ph.D. Thesis, Cambridge University, 1992."},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"172","DOI":"10.1006\/jnth.1993.1043","article-title":"Verification of a conjecture of E. Thomas","volume":"44","author":"Mignotte, Maurice","year":"1993","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"1","key":"6","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1016\/0022-314X(91)90032-7","article-title":"On a family of cubics","volume":"39","author":"Mignotte, M.","year":"1991","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"key":"7","doi-asserted-by":"crossref","unstructured":"M. Mignotte and M. Waldschmidt, Linear forms in two logarithms and Schneider\u2019s method III, Ann. Fac. Sci. Toulouse Math. (5) 97 (1989), 43\u201375.","DOI":"10.5802\/afst.688"},{"issue":"196","key":"8","doi-asserted-by":"publisher","first-page":"777","DOI":"10.2307\/2938717","article-title":"Complete solutions to families of quartic Thue equations","volume":"57","author":"Peth\u0151, Attila","year":"1991","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"9","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1006\/jnth.1993.1024","article-title":"Solutions to certain families of Thue equations","volume":"43","author":"Thomas, Emery","year":"1993","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"1","key":"10","doi-asserted-by":"publisher","first-page":"176","DOI":"10.4153\/CJM-1993-010-1","article-title":"Minorations de combinaisons lin\u00e9aires de logarithmes de nombres alg\u00e9briques","volume":"45","author":"Waldschmidt, Michel","year":"1993","journal-title":"Canad. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0008-414X","issn-type":"print"},{"key":"11","unstructured":"\\bysame, Linear independence of logarithms of algebraic numbers, The Institute of Mathematical Sciences, IMSc. Report no. 116, Madras, 1992. [4]"},{"key":"12","unstructured":"M. Laurent, M. Mignotte, and Y. Nesterenko, Formes lin\u00e9aires en deux logarithmes et d\u00e9terminants d\u2019interpolation, J. Number Theory, to appear."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1996-65-213\/S0025-5718-96-00662-X\/S0025-5718-96-00662-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-213\/S0025-5718-96-00662-X\/S0025-5718-96-00662-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:02:28Z","timestamp":1776718948000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-213\/S0025-5718-96-00662-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996]]},"references-count":12,"journal-issue":{"issue":"213","published-print":{"date-parts":[[1996,1]]}},"alternative-id":["S0025-5718-96-00662-X"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-96-00662-x","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":[[1996]]}}}