AI-Assisted Proof Establishes Real-Rootedness of Poincaré Polynomials in Algebraic Geometry
Mathematicians using Google DeepMind's Co-Mathematician AI system have proven the Aluffi–Chen–Marcolli conjecture on the real-rootedness of Poincaré polynomials of the Deligne–Mumford moduli space of stable rational curves. The proof introduces a novel bivariate deformation of the Poincaré polynomial that exposes a hidden interlacing structure, resolved through a Sturm–Rolle argument. The result establishes that the Betti numbers of this moduli space form an ultra-log-concave sequence, and the AI-assisted workflow represents a notable example of frontier AI contributing substantively to original mathematical research.
A team of mathematicians, working iteratively with Co-Mathematician — an agentic AI system developed by Google DeepMind — has proven that the Poincaré polynomials of the Deligne–Mumford compactification of the moduli space of stable n-pointed rational curves are real-rooted, confirming a conjecture by Aluffi, Chen, and Marcolli. The central innovation is a bivariate deformation of the Poincaré polynomial that makes visible an interlacing structure hidden in the standard one-variable Keel–Manin–Getzler recurrence. For fixed negative values of one variable, a Sturm–Rolle argument controls the zero set in the other variable over a specific interval, and recovering the original polynomial on a particular slice yields both real-rootedness and strict interlacing of roots. A direct consequence is that the Betti numbers of the moduli space form an ultra-log-concave sequence. The human researchers' role included problem formulation, evaluating and correcting AI-proposed proof attempts, and situating the argument within the existing literature; they also independently extended the deformation strategy to prove analogous results for Fulton–MacPherson spaces of n ordered points. The 16-page preprint was posted to arXiv in May 2026 and has not yet undergone formal peer review.
What's missing
The paper is a preprint and has not yet been peer-reviewed, so independent verification of the proof's correctness is pending. The extent to which the AI system generated genuinely novel mathematical ideas versus recombining known techniques is not fully characterized. Additionally, the paper does not detail the number of AI-proposed proof attempts that failed before a correct argument was found, which would be relevant to assessing the AI's actual contribution.
What different sources said
- arXiv cs.AICenter
Real-rootedness of the Poincar\'e polynomials of $\overline{\mathcal M}_{0,n}$: an AI-assisted proof
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