{"external_id":{"arxiv":["2307.05712"]},"publisher":"EMS Press","corr_author":"1","oa_version":"Published Version","year":"2025","doi":"10.4171/jems/1697","type":"journal_article","publication":"Journal of the European Mathematical Society","article_processing_charge":"No","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","day":"06","author":[{"last_name":"Yao Xiao","first_name":"Stanley","full_name":"Yao Xiao, Stanley"},{"full_name":"Yamagishi, Shuntaro","id":"0c3fbc5c-f7a6-11ec-8d70-9485e75b416b","first_name":"Shuntaro","last_name":"Yamagishi"}],"article_type":"original","publication_status":"epub_ahead","publication_identifier":{"issn":["1435-9855"],"eissn":["1435-9863"]},"date_created":"2026-02-16T16:00:02Z","date_updated":"2026-02-19T09:25:12Z","department":[{"_id":"TiBr"}],"OA_place":"publisher","OA_type":"diamond","abstract":[{"lang":"eng","text":"We prove that there does not exist F∈Q[x,y] of degree 4 such that F(Z^2 )=Z ≥0. In particular, this answers a question by John S. Lew and Bjorn Poonen for quartic polynomials."}],"quality_controlled":"1","date_published":"2025-08-06T00:00:00Z","status":"public","acknowledgement":"The first author would like to thank Samir Siksek for introducing the problem\r\nto him. The material in Section 6 is a result of discussing with many people, and the second author is very grateful to Tim Browning, Stephanie Chan, Jakob Glas, Jakub Löwit, Mirko Mauri, Marta Pieropan, Mike Roth, Matteo Verzobio and Victor Wang for taking the time to answer his questions and for their valuable suggestions. We thank Tim Browning, Yijie Diao, Ana Marija Vego and the anonymous referee for their helpful comments.\r\nThe first author was supported by NSERC Discovery Grant RGPIN-2024-06810. The\r\nsecond author was supported by the NWO Veni Grant 016.Veni.192.047 during his time at\r\nUtrecht University and by a FWF grant (DOI 10.55776/P32428) at the Institute of Science and\r\nTechnology Austria while working on this paper.","oa":1,"language":[{"iso":"eng"}],"title":"Quartic polynomials in two variables do not represent all non-negative integers","month":"08","_id":"21260","arxiv":1,"main_file_link":[{"url":"https://doi.org/10.4171/JEMS/1697","open_access":"1"}],"project":[{"_id":"26AEDAB2-B435-11E9-9278-68D0E5697425","name":"New frontiers of the Manin conjecture","grant_number":"P32428","call_identifier":"FWF"}],"citation":{"mla":"Yao Xiao, Stanley, and Shuntaro Yamagishi. “Quartic Polynomials in Two Variables Do Not Represent All Non-Negative Integers.” <i>Journal of the European Mathematical Society</i>, EMS Press, 2025, doi:<a href=\"https://doi.org/10.4171/jems/1697\">10.4171/jems/1697</a>.","short":"S. Yao Xiao, S. Yamagishi, Journal of the European Mathematical Society (2025).","chicago":"Yao Xiao, Stanley, and Shuntaro Yamagishi. “Quartic Polynomials in Two Variables Do Not Represent All Non-Negative Integers.” <i>Journal of the European Mathematical Society</i>. EMS Press, 2025. <a href=\"https://doi.org/10.4171/jems/1697\">https://doi.org/10.4171/jems/1697</a>.","ama":"Yao Xiao S, Yamagishi S. Quartic polynomials in two variables do not represent all non-negative integers. <i>Journal of the European Mathematical Society</i>. 2025. doi:<a href=\"https://doi.org/10.4171/jems/1697\">10.4171/jems/1697</a>","ieee":"S. Yao Xiao and S. Yamagishi, “Quartic polynomials in two variables do not represent all non-negative integers,” <i>Journal of the European Mathematical Society</i>. EMS Press, 2025.","ista":"Yao Xiao S, Yamagishi S. 2025. Quartic polynomials in two variables do not represent all non-negative integers. Journal of the European Mathematical Society.","apa":"Yao Xiao, S., &#38; Yamagishi, S. (2025). Quartic polynomials in two variables do not represent all non-negative integers. <i>Journal of the European Mathematical Society</i>. EMS Press. <a href=\"https://doi.org/10.4171/jems/1697\">https://doi.org/10.4171/jems/1697</a>"}}