AI 中文总结
本文推导了$\bar{\frak{M}}_{0,n}$的庞加莱多项式等的两个新闭式，得到Betti数新公式与欧拉示性数新递推，精炼了其渐近估计。
AI 中文摘要
本文重新研究了稳定n点有理曲线的Deligne-Mumford模空间$\boldsymbol{\bar{\frak{M}}_{0,n}}$的庞加莱多项式、Betti数与欧拉示性数。我们对两个最新的庞加莱多项式闭式公式给出了初等推导，这两个公式分别来自Aluffi-Marcolli-Nascimento（arXiv:2406.13095）与Eur-Ferroni-Matherne-Pagaria-Vecchi（arXiv:2504.16776）。我们的方法表明，这两个公式已隐含在Getzler和Manin的生成级数结果中，可通过生成函数、二项式级数及斯特林数的标准恒等式的初等操作从中提取。除这些新推导外，相同方法还为与这些庞加莱多项式相关的精炼不变量（即特殊求和项与双变量精炼）产生了新的线性递推关系。作为进一步结果，我们得到了两个此前未以该形式记录的Betti数新公式。我们还研究了欧拉示性数$\boldsymbol{\bar{\frak{M}}_{0,n}}$，利用Lambert W函数的合适分支的泰勒展开，证明其序列可通过将完全贝尔多项式在显式辅助整数序列上求值得到。该贝尔多项式表示给出了Hessenberg行列式公式与不同于著名二次Keel-Manin递推的新线性递推，还提供了从Aluffi-Marcolli-Nascimento考虑的Lambert W函数表达式中显式提取欧拉示性数的方法。最后，我们通过计算完整渐近展开，对这些欧拉示性数的Manin-Zagier渐近估计进行了精炼。
英文摘要
In this paper, we revisit the Poincaré polynomials, Betti numbers, and Euler characteristics of the Deligne-Mumford moduli spaces $\overline{\mathcal M}_{0,n}$ of stable $n$-pointed rational curves. We give elementary derivations of two recent closed formulas for their Poincaré polynomials, due respectively to Aluffi-Marcolli-Nascimento (arXiv:2406.13095) and to Eur-Ferroni-Matherne-Pagaria-Vecchi (arXiv:2504.16776). Our approach shows that both formulas are already implicit in the generating-series results of Getzler and Manin, and can be extracted from them by elementary manipulations of generating functions, the binomial series, and standard identities for Stirling numbers. Beyond these new derivations, the same method also yields new linear recurrence relations for refined invariants associated with these Poincaré polynomials, namely distinguished summands and a bivariate refinement. As a further consequence, we obtain two additional formulas for the Betti numbers, not previously recorded in this form. We also study the Euler characteristics $χ(\overline{\mathcal M}_{0,n})$. Using the Taylor expansion of a suitable branch of the Lambert $W$-function, we show that their sequence is obtained by evaluating complete Bell polynomials at an explicit auxiliary integer sequence. This Bell-polynomial representation yields Hessenberg determinantal formulas and a new linear recursion, distinct from the well-known quadratic Keel-Manin recursion. It also provides an explicit extraction of the Euler characteristics from the Lambert $W$-function expression considered by Aluffi-Marcolli-Nascimento. Finally, we refine the Manin-Zagier asymptotic estimate for these Euler characteristics by computing the full asymptotic expansion.
Comments28 pages
