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Comp."],"published-print":{"date-parts":[[2021,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The notion of the capacity of a polynomial was introduced by Gurvits around 2005, originally to give drastically simplified proofs of the van der Waerden lower bound for permanents of doubly stochastic matrices and Schrijver\u2019s inequality for perfect matchings of regular bipartite graphs. Since this seminal work, the notion of capacity has been utilised to bound various combinatorial quantities and to give polynomial-time algorithms to approximate such quantities (e.g. the number of bases of a matroid). These types of results are often proven by giving bounds on how much a particular differential operator can change the capacity of a given polynomial. In this paper, we unify the theory surrounding such capacity-preserving operators by giving tight capacity preservation bounds for all nondegenerate real stability preservers. We then use this theory to give a new proof of a recent result of Csikv\u00e1ri, which settled Friedland\u2019s lower matching conjecture.<\/jats:p>","DOI":"10.1017\/s0963548321000122","type":"journal-article","created":{"date-parts":[[2021,4,30]],"date-time":"2021-04-30T10:21:42Z","timestamp":1619778102000},"page":"956-981","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":3,"title":["Counting matchings via capacity-preserving operators"],"prefix":"10.1017","volume":"30","author":[{"given":"Leonid","family":"Gurvits","sequence":"first","affiliation":[]},{"given":"Jonathan","family":"Leake","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2021,4,30]]},"reference":[{"key":"S0963548321000122_ref7","unstructured":"[7] B\u00fcrgisser, P. , Garg, A. , Oliveira, R. , Walter, M. and Wigderson, A. (2017) Alternating minimization, scaling algorithms, and the null-cone problem from invariant theory. arXiv preprint arXiv:1711.08039."},{"key":"S0963548321000122_ref10","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1989.136.241"},{"key":"S0963548321000122_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2007.05.011"},{"key":"S0963548321000122_ref30","doi-asserted-by":"publisher","DOI":"10.1145\/3055399.3055457"},{"key":"S0963548321000122_ref19","doi-asserted-by":"publisher","DOI":"10.1145\/1132516.1132578"},{"key":"S0963548321000122_ref2","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2018.00013"},{"key":"S0963548321000122_ref24","unstructured":"[24] Huh, J. , Schr\u00f6ter, B. and Wang, B. (2018) Correlation bounds for fields and matroids. arXiv preprint arXiv:1806.02675."},{"key":"S0963548321000122_ref27","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2015.182.1.8"},{"key":"S0963548321000122_ref14","first-page":"475","article-title":"Proof of the van der Waerden conjecture regarding the permanent of a doubly stochastic matrix","volume":"29","author":"Falikman","year":"1981","journal-title":"Math. Notes Acad. Sci. USSR"},{"key":"S0963548321000122_ref22","unstructured":"[22] Gurvits, L. (2011) Unleashing the power of Schrijver\u2019s permanental inequality with the help of the Bethe approximation. arXiv preprint arXiv:1106.2844."},{"key":"S0963548321000122_ref1","doi-asserted-by":"publisher","DOI":"10.1145\/3055399.3055469"},{"key":"S0963548321000122_ref31","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1922-1501220-0"},{"key":"S0963548321000122_ref17","unstructured":"[17] Garg, A. , Gurvits, L. , Oliveira, R. and Wigderson, A. (2015) Operator scaling: Theory and applications. arXiv preprint arXiv:1511.03730."},{"key":"S0963548321000122_ref32","unstructured":"[32] Zackrisson, S. (2017) Coefficients and zeros of mixed characteristic polynomials."},{"key":"S0963548321000122_ref28","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-004-0136-z"},{"key":"S0963548321000122_ref25","unstructured":"[25] Leake, J. (2017) A representation theoretic explanation of the Borcea-Br\u00e4nd\u00e9n characterization. arXiv preprint arXiv:1706.06168."},{"key":"S0963548321000122_ref12","unstructured":"[12] Csikv\u00e1ri, P. (2014) Lower matching conjecture, and a new proof of Schrijver\u2019s and Gurvits\u2019s theorems. arXiv preprint arXiv:1406.0766."},{"key":"S0963548321000122_ref20","doi-asserted-by":"publisher","DOI":"10.37236\/790"},{"key":"S0963548321000122_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/s00222-009-0189-3"},{"key":"S0963548321000122_ref18","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2016.95"},{"key":"S0963548321000122_ref3","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2018.188.2.1"},{"key":"S0963548321000122_ref26","unstructured":"[26] Marcus, A. , Spielman, D. and Srivastava, N. (2015) Finite free convolutions of polynomials. arXiv preprint arXiv:1504.00350."},{"key":"S0963548321000122_ref13","doi-asserted-by":"publisher","DOI":"10.1007\/BF00968054"},{"key":"S0963548321000122_ref29","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.1997.1798"},{"key":"S0963548321000122_ref6","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.20295"},{"key":"S0963548321000122_ref16","doi-asserted-by":"publisher","DOI":"10.37236\/834"},{"key":"S0963548321000122_ref11","doi-asserted-by":"publisher","DOI":"10.1016\/S0196-8858(03)00078-2"},{"key":"S0963548321000122_ref4","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(89)90009-8"},{"key":"S0963548321000122_ref23","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-10018-9_7"},{"key":"S0963548321000122_ref15","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548307008747"},{"key":"S0963548321000122_ref21","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-03562-3_4"},{"key":"S0963548321000122_ref8","doi-asserted-by":"crossref","unstructured":"[8] Br\u00e4nd\u00e9n, P. and Huh, J. (2019) Lorentzian polynomials. arXiv preprint arXiv:1902.03719.","DOI":"10.4007\/annals.2020.192.3.4"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548321000122","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,10,15]],"date-time":"2021-10-15T15:51:25Z","timestamp":1634313085000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548321000122\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,30]]},"references-count":32,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2021,11]]}},"alternative-id":["S0963548321000122"],"URL":"https:\/\/doi.org\/10.1017\/s0963548321000122","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,30]]},"assertion":[{"value":"\u00a9 The Author(s), 2021. 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 (http:\/\/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"}]}}