AI 中文总结
本文将特征零代数闭域上三维余维Artinian Gorenstein代数的弱Lefschetz性质结论推广到任意特征的无限域，同时证明了相关的公因子结论。
AI 中文摘要
设𝖐为域，S=𝖐[x,y,z]，R=S/I是标准分次的三维余维Artinian Gorenstein𝖐-代数。这类代数的h-向量已知是对称且单峰的。Miró-Roig证明，若𝖐是特征零的代数闭域且R的h-向量至少有三个峰，则R具有弱Lefschetz性质。本文将该结果推广到任意特征的无限域，采用不同的、初等且更直接的论证。特别地，我们无需𝖐为代数闭域的假设即可得到Miró-Roig定理。在此过程中，我们还证明了一个适用于任意域的独立有意义的结论：若R的h-向量至少有两个峰，且s是峰的最大次数，则I中次数不超过s的元素没有公因子。
英文摘要
Let $\mathsf k$ be a field, $S=\mathsf k[x,y,z]$, and $R=S/I$ be a standard graded Artinian Gorenstein $\mathsf k$-algebra of codimension three. The $h$-vector of such an algebra is known to be symmetric and unimodal. Miró-Roig proved that if $\mathsf k$ is algebraically closed of characteristic zero and the $h$-vector of $R$ has at least three peaks, then $R$ has the weak Lefschetz property. In this article, we extend this result to any infinite field of arbitrary characteristic, using a different, elementary, and more direct argument. In particular, we recover Miró-Roig's theorem without the hypothesis that $\mathsf k$ is algebraically closed. Along the way, we also prove a statement of independent interest that holds over any field: if the $h$-vector of $R$ has at least two peaks, and if $s$ is the largest degree of a peak, then the elements of $I$ of degree at most $s$ have no common factor.
Comments11 pages. Comments welcome!
