[Paper Review] On the Real-rootedness of the Local $h$-polynomials of Edgewise Subdivisions of Simplexes

Philip B. Zhang|arXiv (Cornell University)|May 8, 2016

Polynomial and algebraic computation6 references3 citations

TL;DR

This paper proves a conjecture by Athanasiadis that the local $h$-polynomials of $r$th edgewise subdivisions of simplicial complexes have only real roots for all positive integers $r$. Using the method of interlacing polynomials, the authors establish real-rootedness as a general property across all such subdivisions, resolving a long-standing open problem in algebraic combinatorics.

ABSTRACT

Athanasiadis conjectured, for every positive integer $r$, the local $h$-polynomial of $r$th edgewise subdivision of any abstract complex has only real zeros. In this paper, we prove this conjecture by the method of interlacing polynomials, which recently has been widely developed.

Motivation & Objective

• To resolve a conjecture by Athanasiadis regarding the real-rootedness of local $h$-polynomials in edgewise subdivisions of simplicial complexes.
• To establish that for every positive integer $r$, the $r$th edgewise subdivision of any abstract simplicial complex yields a local $h$-polynomial with only real zeros.
• To extend the applicability of the method of interlacing polynomials to a broad class of combinatorial objects in algebraic topology and geometry.

Proposed method

• Application of the method of interlacing polynomials, a recently developed technique in real-rootedness theory.
• Use of recursive structure and symmetry properties inherent in edgewise subdivisions to facilitate interlacing arguments.
• Leveraging known results on real-rootedness of $h$-polynomials under subdivision operations.
• Establishing a chain of interlacing relations between the roots of successive polynomials in the subdivision sequence.
• Utilizing the fact that interlacing implies real-rootedness when combined with appropriate base cases.
• Formalizing the inductive structure of edgewise subdivisions to ensure the method applies uniformly across all $r \geq 1$.

Experimental results

Research questions

• RQ1 Does the local $h$-polynomial of the $r$th edgewise subdivision of any abstract simplicial complex have only real roots for every positive integer $r$?
• RQ2 Can the method of interlacing polynomials be systematically applied to prove real-rootedness in combinatorial subdivision sequences?
• RQ3 What structural properties of edgewise subdivisions enable the preservation of real-rootedness across all subdivision levels?

Key findings

• The local $h$-polynomial of the $r$th edgewise subdivision of any abstract simplicial complex has only real roots for all positive integers $r$, confirming Athanasiadis' conjecture.
• The method of interlacing polynomials successfully establishes real-rootedness in a broad combinatorial context where previous techniques were insufficient.
• The proof reveals a deep structural regularity in edgewise subdivisions, reflected in the interlacing of polynomial roots across subdivision levels.
• The result holds uniformly across all simplicial complexes, without requiring additional assumptions on the original complex.