TL;DR
This paper introduces a new facet ordering for simplicial trees based on a good leaf, enabling better combinatorial understanding of Betti numbers and providing refined recursive formulas and splitting techniques.
Contribution
It presents a novel facet ordering method for simplicial trees that improves the computation of Betti numbers and refines existing recursive formulas.
Findings
Provides a new ordering based on a good leaf
Enables Eliahou-Kervaire splitting of the facet ideal
Refines recursive formulas for Betti numbers
Abstract
Using the existence of a good leaf in every simplicial tree, we order the facets of a simplicial tree in order to find combinatorial information about the Betti numbers of its facet ideal. Applications include an Eliahou-Kervaire splitting of the ideal, as well as a refinement of a recursive formula of H\`a and Van Tuyl for computing the graded Betti numbers of simplicial trees.
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Taxonomy
TopicsCommutative Algebra and Its Applications · Polynomial and algebraic computation · Topological and Geometric Data Analysis