The Tropical Critical Points of an Affine Matroid

IF 1 3区数学 Q2 MATHEMATICS SIAM Journal on Discrete Mathematics Pub Date : 2024-06-24 DOI:10.1137/23m1556174

Federico Ardila-Mantilla, Christopher Eur, Raul Penaguiao

引用次数: 0

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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仿射矩阵的热带临界点

SIAM 离散数学杂志》，第 38 卷第 2 期，第 1930-1942 页，2024 年 6 月。摘要。我们证明仿射矩阵[math]的热带临界点数等于[math]的贝塔不变量。受最大似然度计算的启发，这个数被定义为[math]的伯格曼扇形与[math]的倒伯格曼扇形的交集度，其中[math]是[math]中既非循环也非coloop的元素。等价地，对于[math]上的一般权向量[math]，这是找到[math]上的权向量[math]和[math]上的权向量[math]与[math]，使得在[math]的每个回路（或，[math]）上，最小[math]权向量（或，[math]权向量）至少出现两次的方法的数目。这就回答了斯特姆费尔斯的一个问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Discrete Mathematics 数学-应用数学

CiteScore

1.90

自引率

0.00%

发文量

124

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Discrete Mathematics (SIDMA) publishes research papers of exceptional quality in pure and applied discrete mathematics, broadly interpreted. The journal''s focus is primarily theoretical rather than empirical, but the editors welcome papers that evolve from or have potential application to real-world problems. Submissions must be clearly written and make a significant contribution. Topics include but are not limited to: properties of and extremal problems for discrete structures combinatorial optimization, including approximation algorithms algebraic and enumerative combinatorics coding and information theory additive, analytic combinatorics and number theory combinatorial matrix theory and spectral graph theory design and analysis of algorithms for discrete structures discrete problems in computational complexity discrete and computational geometry discrete methods in computational biology, and bioinformatics probabilistic methods and randomized algorithms.

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