What was already known
Aluffi, Chen and Marcolli conjectured real-rootedness from computations and proved related positivity properties. The known recurrence alone had not established real-rootedness.
AI involvement
Co-Mathematician supplied the bivariate-deformation and interlacing strategy and repaired proof gaps. Bérczi and Kiem directed and reviewed the work and wrote the final proof; their Fulton–MacPherson result is separate.
The authors explicitly credit Co-Mathematician with the retained central deformation strategy and later proof repairs inside a human-led project.
Importance
Resolves the real-rootedness conjecture and strengthens it to simple roots and interlacing.
Limitations
The separately proved Fulton–MacPherson extension belongs to the human authors. No external proof audit or formalization of the all-n statement is documented.
Technical verification
The authors describe an explicit iterative human review of the full argument: they detected the missing recurrence/PDE justification and root-branch step, requested and assessed replacements, and present the resulting all-n Sturm and degree-count proof. Small symbolic checks were exploratory and are not a machine verification of the theorem. No independent external proof audit was found.
Formal checking
Iterative review of model proof attempts, identification of gaps, requested corrections, and final proof using PDE compatibility, weighted Sturm recurrence and crossing-degree count
- Checker
- Gergely Bérczi and Young-Hoon Kiem
- Scope
- Real-rootedness and strict interlacing of P_n for all n
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The authors explain the PDE/Sturm/crossing mechanism in the original paper. Their later geometric interpretation of F_m further documents human understanding; the initial lack of a geometric meaning did not leave the proof mechanism unexplained.
Correction and publication history
Resolves the real-rootedness conjecture and strengthens it to simple roots and interlacing.
Read this published revision