Mu--Welker递归分解在每一度上的反例

中文摘要 AI 辅助

Bell和Skandera的著名开放问题询问：一个具有正整数系数的首一实根多项式$f(t)$是否是一个单纯复形的$f$-多项式。Mu和Welker证明了如果递归分解$f(t)=g(t)+th(t)$满足相应的系数不等式$h_i<g_i$，那么这个开放问题有肯定答案。Mu和Welker还猜想$f(t)$的实根性蕴含$g(t)$和$h(t)$的实根性。我们给出了Mu和Welker猜想在每一度至少为三时的反例，并证明了该断言在度1和度2成立。此外，我们构造的每个多项式都是某个单纯复形的$f$-多项式。

英文摘要

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.

发表机构

• Center for Combinatorics, LPMC,Nankai University（南开大学组合数学中心）
• School of Mathematics and Statistics,North China University of Water Resources and Electric Power（华北水利水电大学数学与统计学院）
• School of Mathematics and Statistics,Beijing Institute of Technology（北京理工大学数学与统计学院）

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