{"language":[{"iso":"eng"}],"license":"https://creativecommons.org/licenses/by/4.0/","month":"10","title":"Solubility of a resultant equation and applications","oa":1,"quality_controlled":"1","acknowledgement":"While working on this paper, the first author was supported by a FWF grant (DOI 10.55776/P36278).","status":"public","file_date_updated":"2026-02-24T07:56:34Z","date_published":"2025-10-21T00:00:00Z","OA_place":"publisher","OA_type":"gold","abstract":[{"lang":"eng","text":"The large sieve is used to estimate the density of quadratic polynomials Q ∈ Z[x],\r\nsuch that there exists an odd degree polynomial defined over Z which has resultant ±1 with Q.\r\nGiven a monic polynomial R ∈ Z[x] of odd degree, this is used to show that for almost all\r\nquadratic polynomials Q ∈ Z[x], there exists a prime p such that Q and R share a common\r\nroot in Fp. Using recent work of Landesman, an application to the average size of the odd part\r\nof the class group of quadratic number fields is also given"},{"text":" Le grand crible est utilisé pour estimer la densité des polynômes quadratiques Q ∈ Z[x] tels qu’il existe un polynôme de degré impair défini sur Z dont le résultant avec Q est égal à ±1. Étant donné un polynôme unitaire R ∈ Z[x] de degré impair, on s’en sert pour montrer que, pour presque tous les polynômes quadratiques Q ∈ Z[x], il existe un nombre premier p tel que Q et R aient une racine commune dans Fp. En utilisant des travaux récents de Landesman, on obtient également une application concernant la taille moyenne de la partie impaire du groupe de classe des corps quadratiques.","lang":"fre"}],"file":[{"date_updated":"2026-02-24T07:56:34Z","file_id":"21356","content_type":"application/pdf","date_created":"2026-02-24T07:56:34Z","success":1,"access_level":"open_access","creator":"dernst","file_name":"2025_JEP_Browning.pdf","relation":"main_file","file_size":1003689,"checksum":"828577ea48ac6109d3e9dd1aeddd45c4"}],"citation":{"ista":"Browning TD, Chan S. 2025. Solubility of a resultant equation and applications. Journal de l’ecole polytechnique mathematiques. 12, 1677–1691.","ieee":"T. D. Browning and S. Chan, “Solubility of a resultant equation and applications,” <i>Journal de l’ecole polytechnique mathematiques</i>, vol. 12. Ecole polytechnique, pp. 1677–1691, 2025.","apa":"Browning, T. D., &#38; Chan, S. (2025). Solubility of a resultant equation and applications. <i>Journal de l’ecole Polytechnique Mathematiques</i>. Ecole polytechnique. <a href=\"https://doi.org/10.5802/jep.320\">https://doi.org/10.5802/jep.320</a>","short":"T.D. Browning, S. Chan, Journal de l’ecole Polytechnique Mathematiques 12 (2025) 1677–1691.","chicago":"Browning, Timothy D, and Stephanie Chan. “Solubility of a Resultant Equation and Applications.” <i>Journal de l’ecole Polytechnique Mathematiques</i>. Ecole polytechnique, 2025. <a href=\"https://doi.org/10.5802/jep.320\">https://doi.org/10.5802/jep.320</a>.","ama":"Browning TD, Chan S. Solubility of a resultant equation and applications. <i>Journal de l’ecole polytechnique mathematiques</i>. 2025;12:1677-1691. doi:<a href=\"https://doi.org/10.5802/jep.320\">10.5802/jep.320</a>","mla":"Browning, Timothy D., and Stephanie Chan. “Solubility of a Resultant Equation and Applications.” <i>Journal de l’ecole Polytechnique Mathematiques</i>, vol. 12, Ecole polytechnique, 2025, pp. 1677–91, doi:<a href=\"https://doi.org/10.5802/jep.320\">10.5802/jep.320</a>."},"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"project":[{"name":"Rational curves via function field analytic number theory","grant_number":"P36278","_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3"}],"page":"1677-1691","arxiv":1,"ddc":["510"],"_id":"21343","year":"2025","has_accepted_license":"1","corr_author":"1","publisher":"Ecole polytechnique","oa_version":"Published Version","DOAJ_listed":"1","PlanS_conform":"1","external_id":{"arxiv":["2411.09264"]},"date_created":"2026-02-22T23:01:36Z","department":[{"_id":"TiBr"}],"volume":12,"date_updated":"2026-02-24T07:57:53Z","publication_status":"published","article_type":"original","scopus_import":"1","publication_identifier":{"eissn":["2270-518X"],"issn":["2429-7100"]},"intvolume":"        12","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","author":[{"orcid":"0000-0002-8314-0177","id":"35827D50-F248-11E8-B48F-1D18A9856A87","last_name":"Browning","first_name":"Timothy D","full_name":"Browning, Timothy D"},{"full_name":"Chan, Yik Tung","orcid":"0000-0001-8467-4106","id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1","last_name":"Chan","first_name":"Yik Tung"}],"day":"21","publication":"Journal de l'ecole polytechnique mathematiques","type":"journal_article","doi":"10.5802/jep.320","article_processing_charge":"Yes"}