AI 中文总结
研究五个变量的单项式理想贝蒂数与基域特征的关系，通过将其简化为无平方因子孪生理想并应用霍赫斯特公式，证明其贝蒂数与特征无关，给出相关计算结果，且表明佩娃六变量例子的界限是精确的。
AI 中文摘要
亚历山德罗尼证明了至多四个变量的单项式理想的贝蒂数与基域的特征无关，而佩娃展示了六个变量中依赖于特征的贝蒂数。我们证明了五个变量的情况是与特征无关的。具体而言，若\(S = k[x_1,\ldots,x_5]\)且\(M\subseteq S\)是单项式理想，则\(S/M\)的多重分次、分次和总贝蒂数与\(\text{char}(k)\)无关。证明将任意单项式理想简化为无平方因子的孪生理想，然后应用霍赫斯特公式。拓扑学依据是至多五个顶点的单纯复形具有无挠整同调。佩娃由\(\mathbb{RP}^2\)的六顶点三角剖分得出的例子表明该界限是精确的。我们还记录了五个变量的无平方因子单项式理想的分次贝蒂表的计算结果（直至重新标记）。
英文摘要
Alesandroni proved that Betti numbers of monomial ideals in at most four variables are independent of the characteristic of the base field, while characteristic-dependent Betti numbers occur in six variables. We prove that the five-variable case is characteristic-independent. More precisely, if $S=k[x_1,\ldots,x_5]$ and $M\subseteq S$ is a monomial ideal, then the multigraded, graded, and total Betti numbers of $S/M$ are independent of $\operatorname{char}(k)$. The proof reduces arbitrary monomial ideals to squarefree twin ideals and then applies Hochster's formula. The topological input is that simplicial complexes on at most five vertices have torsion-free integral homology. The six-variable example arising from the six-vertex triangulation of $\mathbb{RP}^{2}$ shows that the bound is sharp. We also record a computation of the graded Betti tables of squarefree monomial ideals in five variables up to relabeling.
Comments16 pages, 1 figure, 2 tables. Revised to match the published version; updated attribution, references, and minor copyediting. Code and data available at https://github.com/voltroom0606/fivevariablebettinumbers
Journal refJournal of Pure and Applied Algebra 230 (2026), 108354
DOI：10.1016/j.jpaa.2026.108354
