{"has_accepted_license":"1","ec_funded":1,"publication":"Mathematische Annalen","publisher":"Springer Nature","tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"abstract":[{"lang":"eng","text":"We prove upper and lower bounds on the number of pairs of commuting n x n matrices with integer entries in [-T, T], as T -> . Our work uses Fourier analysis and leads to an analysis of exponential sums involving matrices over finite fields. These are bounded by combining a stratification result of Fouvry and Katz with a new result about the flatness of the commutator Lie bracket."}],"main_file_link":[{"url":"https://doi.org/10.1007/s00208-025-03285-5","open_access":"1"}],"PlanS_conform":"1","acknowledgement":"The authors are very grateful to Alina Ostafe, Matthew Satriano and Igor Shparlinski for drawing their attention to this problem and for useful comments, and to Michael Larsen and Peter Sarnak for their helpful correspondence. We also thank the referee for their valuable input. While working on this paper the first author was supported by a FWF grant (DOI 10.55776/P36278), the second author by a Sloan Research Fellowship, and the third author by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 101034413. Open access funding provided by Institute of Science and Technology (IST Austria).","corr_author":"1","month":"09","arxiv":1,"oa_version":"Published Version","OA_place":"publisher","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","external_id":{"arxiv":["2409.01920"]},"title":"Pairs of commuting integer matrices","scopus_import":"1","OA_type":"hybrid","publication_status":"epub_ahead","oa":1,"article_type":"original","_id":"20367","year":"2025","date_created":"2025-09-21T22:01:31Z","date_updated":"2025-09-23T08:13:39Z","ddc":["510"],"doi":"10.1007/s00208-025-03285-5","department":[{"_id":"TiBr"}],"publication_identifier":{"eissn":["1432-1807"],"issn":["0025-5831"]},"citation":{"short":"T.D. Browning, W. Sawin, V. Wang, Mathematische Annalen (2025).","ista":"Browning TD, Sawin W, Wang V. 2025. Pairs of commuting integer matrices. Mathematische Annalen.","ama":"Browning TD, Sawin W, Wang V. Pairs of commuting integer matrices. <i>Mathematische Annalen</i>. 2025. doi:<a href=\"https://doi.org/10.1007/s00208-025-03285-5\">10.1007/s00208-025-03285-5</a>","apa":"Browning, T. D., Sawin, W., &#38; Wang, V. (2025). Pairs of commuting integer matrices. <i>Mathematische Annalen</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00208-025-03285-5\">https://doi.org/10.1007/s00208-025-03285-5</a>","chicago":"Browning, Timothy D, Will Sawin, and Victor Wang. “Pairs of Commuting Integer Matrices.” <i>Mathematische Annalen</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s00208-025-03285-5\">https://doi.org/10.1007/s00208-025-03285-5</a>.","ieee":"T. D. Browning, W. Sawin, and V. Wang, “Pairs of commuting integer matrices,” <i>Mathematische Annalen</i>. Springer Nature, 2025.","mla":"Browning, Timothy D., et al. “Pairs of Commuting Integer Matrices.” <i>Mathematische Annalen</i>, Springer Nature, 2025, doi:<a href=\"https://doi.org/10.1007/s00208-025-03285-5\">10.1007/s00208-025-03285-5</a>."},"date_published":"2025-09-10T00:00:00Z","article_processing_charge":"Yes (via OA deal)","author":[{"orcid":"0000-0002-8314-0177","id":"35827D50-F248-11E8-B48F-1D18A9856A87","first_name":"Timothy D","full_name":"Browning, Timothy D","last_name":"Browning"},{"first_name":"Will","last_name":"Sawin","full_name":"Sawin, Will"},{"last_name":"Wang","full_name":"Wang, Victor","id":"76096395-aea4-11ed-a680-ab8ebbd3f1b9","first_name":"Victor","orcid":"0000-0002-0704-7026"}],"type":"journal_article","project":[{"grant_number":"P36278","name":"Rational curves via function field analytic number theory","_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3"},{"_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","name":"IST-BRIDGE: International postdoctoral program","grant_number":"101034413","call_identifier":"H2020"}],"day":"10","language":[{"iso":"eng"}],"status":"public","quality_controlled":"1"}