Real-rootedness of genus-zero moduli-space Poincaré polynomials

For the moduli space of stable n-pointed genus-zero curves, is the even-degree Poincaré polynomial real-rooted for every n≥3? A stronger target asks that its n−3 roots be simple and negative for n≥4 and strictly interlace those at consecutive n.

Why it matters and what counts as progress

Why it matters: Real-rootedness and interlacing impose strong structure on the cohomological counting polynomials of genus-zero moduli spaces.

What counts as progress: Prove real-rootedness uniformly in n, establish simple roots and interlacing or extend the phenomenon to related moduli spaces.

How the problem developed

2024 · Conjecture [1][2]
Paolo Aluffi, Stephanie Chen, and Matilde Marcolli Conjectured real zeros for the Poincaré polynomials of the genus-zero stable-curve moduli spaces, supported by asymptotic ultra-log-concavity, γ-positivity, and numerical evidence.
11 Jun 2026 · Reported claim [2]
Gergely Bérczi and Young-Hoon Kiem; Google DeepMind Co-Mathematician The revised preprint proves real-rootedness for P_n(t) of M̄₀,ₙ and ultra-log-concavity of its Betti numbers; the authors separately extend the method to P¹[n].

AI contribution

Real-rootedness of the Poincaré polynomials of M̄₀,ₙ: an AI-assisted proof

Gergely Bérczi, Young-Hoon Kiem

For every n≥3, the even-degree Poincaré polynomial of the moduli space of stable n-pointed genus-zero curves is real-rooted. For n≥4, its n−3 roots are simple and negative and strictly interlace those for consecutive n.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (human)

Human understandingFull