{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T00:34:02Z","timestamp":1767141242353,"version":"build-2238731810"},"reference-count":25,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,4,29]],"date-time":"2022-04-29T00:00:00Z","timestamp":1651190400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,4,29]],"date-time":"2022-04-29T00:00:00Z","timestamp":1651190400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Paris Lodron University of Salzburg"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Period Math Hung"],"published-print":{"date-parts":[[2023,3]]},"abstract":"Abstract<\/jats:title>\n \n Let\n \n \n $$ (G_n)_{n=0}^{\\infty } $$<\/jats:tex-math>\n \n \n \n (<\/mml:mo>\n \n G<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n \n n<\/mml:mi>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n \u221e<\/mml:mi>\n <\/mml:msubsup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n be a polynomial power sum, i.e. a simple linear recurrence sequence of complex polynomials with power sum representation\n \n \n $$ G_n = f_1\\alpha _1^n + \\cdots + f_k\\alpha _k^n $$<\/jats:tex-math>\n \n \n \n G<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n =<\/mml:mo>\n \n f<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n \n \u03b1<\/mml:mi>\n 1<\/mml:mn>\n n<\/mml:mi>\n <\/mml:msubsup>\n +<\/mml:mo>\n \u22ef<\/mml:mo>\n +<\/mml:mo>\n \n f<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n \n \u03b1<\/mml:mi>\n k<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msubsup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and polynomial characteristic roots\n \n \n $$ \\alpha _1,\\ldots ,\\alpha _k $$<\/jats:tex-math>\n \n \n \n \u03b1<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u2026<\/mml:mo>\n ,<\/mml:mo>\n \n \u03b1<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . For a fixed polynomial\n p<\/jats:italic>\n , we consider sets\n \n \n $$ \\left\\{ a,b,c \\right\\} $$<\/jats:tex-math>\n \n \n a<\/mml:mi>\n ,<\/mml:mo>\n b<\/mml:mi>\n ,<\/mml:mo>\n c<\/mml:mi>\n <\/mml:mfenced>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n consisting of three non-zero polynomials such that\n \n \n $$ ab+p, ac+p, bc+p $$<\/jats:tex-math>\n \n \n a<\/mml:mi>\n b<\/mml:mi>\n +<\/mml:mo>\n p<\/mml:mi>\n ,<\/mml:mo>\n a<\/mml:mi>\n c<\/mml:mi>\n +<\/mml:mo>\n p<\/mml:mi>\n ,<\/mml:mo>\n b<\/mml:mi>\n c<\/mml:mi>\n +<\/mml:mo>\n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n are elements of\n \n \n $$ (G_n)_{n=0}^{\\infty } $$<\/jats:tex-math>\n \n \n \n (<\/mml:mo>\n \n G<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n \n n<\/mml:mi>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n \u221e<\/mml:mi>\n <\/mml:msubsup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . We will prove that under a suitable dominant root condition there are only finitely many such sets if neither\n \n \n $$ f_1 $$<\/jats:tex-math>\n \n \n f<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n nor\n \n \n $$ f_1 \\alpha _1 $$<\/jats:tex-math>\n \n \n \n f<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n \n \u03b1<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is a perfect square.\n <\/jats:p>","DOI":"10.1007\/s10998-022-00460-y","type":"journal-article","created":{"date-parts":[[2022,4,29]],"date-time":"2022-04-29T05:04:00Z","timestamp":1651208640000},"page":"289-299","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A polynomial variant of diophantine triples in linear recurrences"],"prefix":"10.1007","volume":"86","author":[{"given":"Clemens","family":"Fuchs","sequence":"first","affiliation":[]},{"given":"Sebastian","family":"Heintze","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,4,29]]},"reference":[{"key":"460_CR1","doi-asserted-by":"publisher","first-page":"170","DOI":"10.1016\/j.jnt.2018.07.013","volume":"194","author":"M Bliznac Trebje\u0161anin","year":"2019","unstructured":"M. Bliznac Trebje\u0161anin, A. Filipin, Nonexistence of $$D(4)$$-quintuples. J. Number Theory 194, 170\u2013217 (2019)","journal-title":"J. Number Theory"},{"key":"460_CR2","unstructured":"N.C. Bonciocat, M. Cipu, M. Mignotte, There is no Diophantine $$D(-1)$$-quadruple. arXiv:2010.09200"},{"key":"460_CR3","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1016\/S0019-3577(98)80001-3","volume":"9","author":"P Corvaja","year":"1998","unstructured":"P. Corvaja, U. Zannier, Diophantine equations with power sums and Universal Hilbert Sets. Indag. Math. 9, 317\u2013332 (1998)","journal-title":"Indag. Math."},{"key":"460_CR4","unstructured":"A. Dujella, Diophantine $$ m $$-tuples. https:\/\/web.math.pmf.unizg.hr\/$$\\sim $$duje\/dtuples.html"},{"key":"460_CR5","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1017\/S0305004101005515","volume":"132","author":"A Dujella","year":"2002","unstructured":"A. Dujella, On the size of Diophantine $$ m $$-tuples. Math. Proc. Camb. Philos. Soc. 132, 23\u201333 (2002)","journal-title":"Math. Proc. Camb. Philos. Soc."},{"issue":"39","key":"460_CR6","doi-asserted-by":"publisher","first-page":"199","DOI":"10.3336\/gm.39.2.01","volume":"III","author":"A Dujella","year":"2004","unstructured":"A. Dujella, Bounds for the size of sets with the property $$D(n)$$. Glas. Mat. Ser. III(39), 199\u2013205 (2004)","journal-title":"Glas. Mat. Ser."},{"key":"460_CR7","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1142\/S1793042108001225","volume":"4","author":"A Dujella","year":"2008","unstructured":"A. Dujella, C. Fuchs, F. Luca, A polynomial variant of a problem of Diophantus for pure powers. Int. J. Number Theory 4, 57\u201371 (2008)","journal-title":"Int. J. Number Theory"},{"key":"460_CR8","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1023\/A:1022389711839","volume":"45","author":"A Dujella","year":"2002","unstructured":"A. Dujella, C. Fuchs, R.F. Tichy, Diophantine $$ m $$-tuples for linear polynomials. Period. Math. Hungar. 45, 21\u201333 (2002)","journal-title":"Period. Math. Hungar."},{"key":"460_CR9","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1016\/j.jnt.2005.12.005","volume":"120","author":"A Dujella","year":"2006","unstructured":"A. Dujella, C. Fuchs, P.G. Walsh, Diophantine $$ m $$-tuples for linear polynomials. II. Equal degrees. J. Number Theory 120, 213\u2013228 (2006)","journal-title":"J. Number Theory"},{"key":"460_CR10","doi-asserted-by":"publisher","first-page":"1449","DOI":"10.1142\/S1793042110003575","volume":"6","author":"A Dujella","year":"2010","unstructured":"A. Dujella, A. Jurasic, On the size of sets in a polynomial variant of a problem of Diophantus. Int. J. Number Theory 6, 1449\u20131471 (2010)","journal-title":"Int. J. Number Theory"},{"issue":"2","key":"460_CR11","doi-asserted-by":"publisher","first-page":"861","DOI":"10.18514\/MMN.2017.1533","volume":"17","author":"A Filipin","year":"2016","unstructured":"A. Filipin, A. Jurasi\u0107, On the size of Diophantine $$ m $$-tuples for linear polynomials. Miskolc Math. Notes 17(2), 861\u2013876 (2016)","journal-title":"Miskolc Math. Notes"},{"key":"460_CR12","doi-asserted-by":"crossref","unstructured":"C. Fuchs, S. Heintze, Perfect powers in polynomial power sums. in \u201cLie Groups, Number Theory, and Vertex Algebra\u201d (D. Adamovic, A. Dujella, A. Milas, P. Pandzic, eds.), Amer. Math. Soc., Contemp. Math. 768, 89\u2013104 (2021)","DOI":"10.1090\/conm\/768\/15456"},{"issue":"1","key":"460_CR13","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1017\/S0004972720001094","volume":"104","author":"C Fuchs","year":"2021","unstructured":"C. Fuchs, S. Heintze, On the growth of linear recurrences in function fields. Bull. Austr. Math. Soc. 104(1), 11\u201320 (2021)","journal-title":"Bull. Austr. Math. Soc."},{"key":"460_CR14","doi-asserted-by":"publisher","first-page":"853","DOI":"10.1515\/ms-2017-0015","volume":"67","author":"C Fuchs","year":"2017","unstructured":"C. Fuchs, C. Hutle, N. Irmak, F. Luca, L. Szalay, Only finitely many tribonacci Diophantine triples exist. Math. Slovaca 67, 853\u2013862 (2017)","journal-title":"Math. Slovaca"},{"key":"460_CR15","doi-asserted-by":"crossref","unstructured":"C. Fuchs, C. Hutle, F. Luca, Diophantine triples in linear recurrences of Pisot type. Res. Number Theory 4, Art. 29 (2018)","DOI":"10.1007\/s40993-018-0121-2"},{"key":"460_CR16","doi-asserted-by":"publisher","first-page":"1449","DOI":"10.1007\/s40840-016-0405-4","volume":"41","author":"C Fuchs","year":"2018","unstructured":"C. Fuchs, C. Hutle, F. Luca, L. Szalay, Diophantine triples and $$ k $$-generalized Fibonacci sequences. Bull. Malays. Math. Soc. 41, 1449\u20131465 (2018)","journal-title":"Bull. Malays. Math. Soc."},{"issue":"3","key":"460_CR17","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1007\/s00229-018-1070-8","volume":"159","author":"C Fuchs","year":"2019","unstructured":"C. Fuchs, C. Karolus, D. Kreso, Decomposable polynomials in second order linear recurrence sequences. Manuscr. Math. 159(3), 321\u2013346 (2019)","journal-title":"Manuscr. Math."},{"key":"460_CR18","first-page":"579","volume":"7","author":"C Fuchs","year":"2008","unstructured":"C. Fuchs, F. Luca, L. Szalay, Diophantine triples with values in binary recurrences. Ann. Sci. Norm. Super. Pisa Cl. Sci. (5) 7, 579\u2013608 (2008)","journal-title":"Ann. Sci. Norm. Super. Pisa Cl. Sci. (5)"},{"key":"460_CR19","doi-asserted-by":"publisher","first-page":"6665","DOI":"10.1090\/tran\/7573","volume":"371","author":"B He","year":"2019","unstructured":"B. He, A. Togb\u00e9, V. Ziegler, There is no Diophantine quintuple. Trans. Am. Math. Soc. 371, 6665\u20136709 (2019)","journal-title":"Trans. Am. Math. Soc."},{"issue":"46","key":"460_CR20","doi-asserted-by":"publisher","first-page":"283","DOI":"10.3336\/gm.46.2.02","volume":"III","author":"A Jurasic","year":"2011","unstructured":"A. Jurasic, Diophantine $$ m $$-tuples for quadratic polynomials. Glas. Mat. Ser. III(46), 283\u2013309 (2011)","journal-title":"Glas. Mat. Ser."},{"key":"460_CR21","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5740-0","volume-title":"Introduction to Algebraic and Abelian Functions. Grad Texts in Math","author":"S Lang","year":"1982","unstructured":"S. Lang, Introduction to Algebraic and Abelian Functions. Grad Texts in Math, vol. 89, 2nd edn. (Springer, New York, 1982)","edition":"2"},{"key":"460_CR22","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-1810-2","volume-title":"Fundamentals of Diophantine Geometry","author":"S Lang","year":"1983","unstructured":"S. Lang, Fundamentals of Diophantine Geometry (Springer, New York, 1983)"},{"key":"460_CR23","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-6046-0","volume-title":"Number Theory in Function Fields. Grad. Texts in Math.","author":"M Rosen","year":"2002","unstructured":"M. Rosen, Number Theory in Function Fields. Grad. Texts in Math., vol. 210 (Springer, New York, 2002)"},{"key":"460_CR24","doi-asserted-by":"crossref","unstructured":"W.M. Schmidt, Linear recurrence sequences, Diophantine approximation (Cetraro, 2000), 171\u2013247, Lecture Notes in Math, vol. 1819. (Springer, Berlin, 2003)","DOI":"10.1007\/3-540-44979-5_4"},{"issue":"1","key":"460_CR25","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1007\/s00222-008-0136-8","volume":"174","author":"U Zannier","year":"2008","unstructured":"U. Zannier, On composite lacunary polynomials and the proof of a conjecture of Schinzel. Invent. Math. 174(1), 127\u2013138 (2008)","journal-title":"Invent. Math."}],"updated-by":[{"DOI":"10.1007\/s10998-022-00468-4","type":"correction","label":"Correction","source":"publisher","updated":{"date-parts":[[2022,5,17]],"date-time":"2022-05-17T00:00:00Z","timestamp":1652745600000}}],"container-title":["Periodica Mathematica Hungarica"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10998-022-00460-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10998-022-00460-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10998-022-00460-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,2,27]],"date-time":"2023-02-27T10:29:38Z","timestamp":1677493778000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10998-022-00460-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,29]]},"references-count":25,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,3]]}},"alternative-id":["460"],"URL":"https:\/\/doi.org\/10.1007\/s10998-022-00460-y","relation":{},"ISSN":["0031-5303","1588-2829"],"issn-type":[{"value":"0031-5303","type":"print"},{"value":"1588-2829","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,4,29]]},"assertion":[{"value":"14 July 2021","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 April 2022","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 May 2022","order":3,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Correction","order":4,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"A Correction to this paper has been published:","order":5,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"https:\/\/doi.org\/10.1007\/s10998-022-00468-4","URL":"https:\/\/doi.org\/10.1007\/s10998-022-00468-4","order":6,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}}]}}