{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T01:18:13Z","timestamp":1778635093487,"version":"3.51.4"},"reference-count":0,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2024,9,18]],"date-time":"2024-09-18T00:00:00Z","timestamp":1726617600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2024,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We present a new explicit formula for the determinant that contains superexponentially fewer terms than the usual Leibniz formula. As an immediate corollary of our formula, we show that the tensor rank of the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000233_inline1.png\"\/><jats:tex-math>\n$n \\times n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> determinant tensor is no larger than the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000233_inline2.png\"\/><jats:tex-math>\n$n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-th Bell number, which is much smaller than the previously best-known upper bounds when <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000233_inline3.png\"\/><jats:tex-math>\n$n \\geq 4$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Over fields of non-zero characteristic we obtain even tighter upper bounds, and we also slightly improve the known lower bounds. In particular, we show that the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000233_inline4.png\"\/><jats:tex-math>\n$4 \\times 4$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> determinant over <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000233_inline5.png\"\/><jats:tex-math>\n${\\mathbb{F}}_2$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> has tensor rank exactly equal to <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000233_inline6.png\"\/><jats:tex-math>\n$12$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Our results also improve upon the best-known upper bound for the Waring rank of the determinant when <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000233_inline7.png\"\/><jats:tex-math>\n$n \\geq 17$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and lead to a new family of axis-aligned polytopes that tile <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000233_inline8.png\"\/><jats:tex-math>\n${\\mathbb{R}}^n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1017\/s0963548324000233","type":"journal-article","created":{"date-parts":[[2024,9,18]],"date-time":"2024-09-18T05:56:12Z","timestamp":1726638972000},"page":"769-794","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":4,"title":["A new formula for the determinant and bounds on its tensor and Waring ranks"],"prefix":"10.1017","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9200-488X","authenticated-orcid":false,"given":"Robin","family":"Houston","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2532-6950","authenticated-orcid":false,"given":"Adam P.","family":"Goucher","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7456-1447","authenticated-orcid":false,"given":"Nathaniel","family":"Johnston","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2024,9,18]]},"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548324000233","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,11,11]],"date-time":"2024-11-11T13:25:02Z","timestamp":1731331502000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548324000233\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,18]]},"references-count":0,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2024,11]]}},"alternative-id":["S0963548324000233"],"URL":"https:\/\/doi.org\/10.1017\/s0963548324000233","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,9,18]]},"assertion":[{"value":"\u00a9 The Author(s), 2024. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}