H-vectors of simplicial complexes with Serre's conditions

TL;DR

This paper investigates the properties of h-vectors of simplicial complexes satisfying Serre's condition (Sr), establishing non-negativity results for initial and tail segments of the h-vector.

Contribution

It proves that for complexes satisfying Sr, the initial h-vector entries and the sum of tail entries are non-negative, extending understanding of combinatorial invariants under Serre's conditions.

Findings

h_k(Δ) ≥ 0 for k=0,...,r

Sum of h_k(Δ) for k≥r ≥ 0

Provides conditions linking Serre's properties to h-vector positivity

Abstract

We study $h$-vectors of simplicial complexes which satisfy Serre's condition ($S_r$). We say that a simplicial complex $\Delta$ satisfies Serre's condition ($S_r$) if $\tilde H_i(\lk_\Delta(F);K)=0$ for all faces $F \in \Delta$ and for all $i < \min \{r-1,\dim \lk_\Delta(F)\}$, where $\lk_\Delta(F)$ is the link of $\Delta$ with respect to $F$ and where $\tilde H_i(\Delta;K)$ is the reduced homology groups of $\Delta$ over a field $K$. The main result of this paper is that if $\Delta$ satisfies Serre's condition ($S_r$) then (i) $h_k(\Delta)$ is non-negative for $k =0,1,...,r$ and (ii) $\sum_{k\geq r}h_k(\Delta)$ is non-negative.