TL;DR
This paper surveys the properties, realization problems, and applications of volume polynomials, a special class of log-concave polynomials with significant combinatorial and analytic features, especially in algebraic matroids.
Contribution
It provides a comprehensive overview of volume polynomials, highlighting their inequalities, realization issues, and relevance to algebraic matroid combinatorics.
Findings
Review of fundamental inequalities for volume polynomials
Discussion of realization problems
Applications to algebraic matroid combinatorics
Abstract
Volume polynomials form a distinguished class of log-concave polynomials with remarkable analytic and combinatorial properties. I will survey realization problems related to them, review fundamental inequalities they satisfy, and discuss applications to the combinatorics of algebraic matroids. These notes are based on lectures given at the 2025 Summer Research Institute in Algebraic Geometry at Colorado State University.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Polynomial and algebraic computation · Computational Geometry and Mesh Generation