PreprintCC BY 4.0arXiv
Counterexamples to the Mu--Welker recursive decomposition in every degree
Abstract
The well-known open problem of Bell and Skandera asks whether a real-rooted polynomial f(t) with positive integer coefficients and constant term one is the f-polynomial of a simplicial complex. Mu and Welker proved that if the recursive decomposition f(t)=g(t)+th(t) satisfies the corresponding coefficient inequality hi<gi, then this open problem has an affirmative answer. Mu and Welker also conjectured that the real-rootedness of f(t) implies that of g(t) and h(t). We give counterexamples to the conjecture of Mu and Welker for every degree at least three, and prove that the assertion holds in degrees $1$ and $2$. Moreover, each polynomial we construct is the f-polynomial of a simplicial complex.
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BibTeX
@misc{liu2026counterexamples,
title = {{Counterexamples to the Mu--Welker recursive decomposition in every degree}},
author = {Feihu Liu and Ying Wang and Zihao Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.23694},
eprint = {2609.23694},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23694},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Counterexamples to the Mu--Welker recursive decomposition in every degree
AU - Feihu Liu
AU - Ying Wang
AU - Zihao Zhang
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2609.23694
UR - https://arxiv.org/abs/2609.23694
AB - The well-known open problem of Bell and Skandera asks whether a real-rooted polynomial f(t) with positive integer coefficients and constant term one is the f-polynomial of a simplicial complex. Mu and Welker proved that if the recursive decomposition f(t)=g(t)+th(t) satisfies the corresponding coefficient inequality h_i<g_i, then this open problem has an affirmative answer. Mu and Welker also conjectured that the real-rootedness of f(t) implies that of g(t) and h(t). We give counterexamples to the conjecture of Mu and Welker for every degree at least three, and prove that the assertion holds in degrees $1$ and $2$. Moreover, each polynomial we construct is the f-polynomial of a simplicial complex.
ER -