Federico Ardila-Mantilla, Christopher Eur, Raul Penaguiao
{"title":"The Tropical Critical Points of an Affine Matroid","authors":"Federico Ardila-Mantilla, Christopher Eur, Raul Penaguiao","doi":"10.1137/23m1556174","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1930-1942, June 2024. <br/> Abstract. We prove that the number of tropical critical points of an affine matroid [math] is equal to the beta invariant of [math]. Motivated by the computation of maximum likelihood degrees, this number is defined to be the degree of the intersection of the Bergman fan of [math] and the inverted Bergman fan of [math], where [math] is an element of [math] that is neither a loop nor a coloop. Equivalently, for a generic weight vector [math] on [math], this is the number of ways to find weights [math] on [math] and [math] on [math] with [math] such that, on each circuit of [math] (resp., [math]), the minimum [math]-weight (resp., [math]-weight) occurs at least twice. This answers a question of Sturmfels.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/23m1556174","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1930-1942, June 2024. Abstract. We prove that the number of tropical critical points of an affine matroid [math] is equal to the beta invariant of [math]. Motivated by the computation of maximum likelihood degrees, this number is defined to be the degree of the intersection of the Bergman fan of [math] and the inverted Bergman fan of [math], where [math] is an element of [math] that is neither a loop nor a coloop. Equivalently, for a generic weight vector [math] on [math], this is the number of ways to find weights [math] on [math] and [math] on [math] with [math] such that, on each circuit of [math] (resp., [math]), the minimum [math]-weight (resp., [math]-weight) occurs at least twice. This answers a question of Sturmfels.
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