AI辅助证明了模空间庞加莱多项式的实根性，揭示隐藏的交错结构。
Real-rootedness of the Poincaré polynomials of $\overline{\mathcal M}_{0,n}$: an AI-assisted proof
- 构造双变量变形函数，揭示一元递推中隐藏的根交错规律。
- 证明庞加莱多项式所有根为实数，且贝蒂数呈超对数凹序列。
- 适合代数几何、组合数学及AI辅助数学证明研究者阅读。
本文证明了稳定n点有理曲线模空间$\overline{\mathcal M}_{0,n}$的庞加莱多项式$P_n(t)=\sum_{i=0}^{n-3} \dim H^{2i}(\overline{\mathcal M}_{0,n};\mathbb{Q})t^i$具有实根性，验证了Aluffi-Chen-Marcolli的猜想。证明基于Keel-Manin-Getzler递推，核心创新在于引入双变量变形$F_m(y,t)$，该变形显现出原一元递推中未见的根交错结构。对固定$t<0$，利用Sturm-Rolle理论控制$F_m$在$0<y<1-t$区间内的零点分布。原始多项式对应于$y=1$截面，移动根穿过此截面的有序交叉既保证了实根性，又导出严格交错。由此得出$\overline{\mathcal M}_{0,n}$的贝蒂数构成超对数凹序列。进一步地，我们证明了复射影直线退化中n个有序点的Fulton-MacPherson空间$\mathbb{P}^1[n]$的庞加莱多项式也具实根性与超对数凹性。$\overline{\mathcal M}_{0,n}$的证明通过与Google DeepMind开发的Co-Mathematician智能体协作的迭代式AI辅助流程完成，人类负责问题建模、验证尝试、识别漏洞、要求修正、比对文献并优化最终证明表述。额外的人类贡献在于发现类似残差变形策略可推广至$\mathbb P^1[n]$，从而得到相应实根性定理。
原文摘要 · Abstract (English)
We prove real-rootedness for the Poincaré polynomial \[ P_n(t)=\sum_{i=0}^{n-3} \dim H^{2i}(\overline{\mathcal M}_{0,n};\mathbb{Q})t^i \] of the Deligne--Mumford moduli space $\overline{\mathcal M}_{0,n}$ of stable $n$-pointed rational curves, proving a conjecture of Aluffi--Chen--Marcolli. The proof starts from the Keel--Manin--Getzler recurrence, but its main new idea is a bivariate deformation $F_m(y,t)$ of the Poincaré polynomial. This deformation reveals a hidden interlacing structure not visible in the one-variable recurrence. For fixed $t<0$, the zero set of $F_m$ in the $y$-direction is controlled by a Sturm--Rolle argument on the interval $0<y<1-t$. The original polynomial is recovered on the slice $y=1$, and the ordered crossings of the moving roots through this slice give both real-rootedness and strict interlacing. Consequently, the Betti numbers of $\overline{\mathcal M}_{0,n}$ form an ultra-log-concave sequence. We further prove real-rootedness and ultra-log-concavity for the Poincaré polynomial of the Fulton--MacPherson space $\mathbb{P}^1[n]$ of $n$ ordered points in degenerations of the complex projective line. The proof for $\overline{\mathcal M}_{0,n}$ was obtained through an iterative AI-assisted workflow with Co-Mathematician, an agentic frontier-model system developed by Google DeepMind. Our role was to formulate the problem, evaluate the proposed proof attempts, identify gaps and request corrections, compare the developing argument with the literature, and refine the presentation of the final proof. Our additional human contribution was to observe that a similar residual deformation strategy applies to the Fulton--MacPherson spaces $\mathbb P^1[n]$, yielding the corresponding real-rootedness theorem.
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