Chow polynomials of matroids are not log-concave
Counterexamples found in an LLM search (Claude Opus 5.5, ChatGPT Astra) refute three real-rootedness conjectures
Result confirmed
The takeaway
Matroid theorists ran a targeted LLM search after a workshop; within hours Claude Opus 5.5 found explicit families of matroids that break log-concavity of Chow and chain polynomials, disproving conjectures of Ferroni–Schröter, Huh–Stevens and Athanasiadis–Kalampogia-Evangelinou.
Status
- Claim
Result confirmed
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- 3 of 5
- Last verified
- 14 October 2026
Your AI and this story
- GPT-6 Astra165 days after its cutoff
- Claude Opus 5.5104 days after its cutoff
- Gemini 4 Argon195 days after its cutoff
- Grok 4.7134 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 104 days before it.
Key facts
- Paper: ‘Chow polynomials and chain polynomials of matroids are not log-concave’, Bérczi, Coron, Eur, Li, arXiv 2610.17135 [math.CO], v1, 12 Oct 2026
- Disproves: real-rootedness conjectures of Ferroni–Schröter (Chow polynomials), Huh–Stevens and Athanasiadis–Kalampogia-Evangelinou; the failures start ‘from the lowest possible ranks’
- Construction: ‘hyperplane coalescence’ plus classical operations of projective geometry and combinatorial designs
- AI timeline (paper): 23 Sep, Eur ‘found a non-constructive disproof of the real-rootedness conjecture for Chow polynomials using ChatGPT Astra’; 30 Sep, Coron found with ChatGPT Astra a counterexample to log-concavity of h-polynomials of graph associahedra ‘of very high vertex count’; 4 Oct, Bérczi, Coron and Li ‘orchestrated and launched an LLM-search’; ‘About 5 hours and 12 hours later, SL found using Claude Opus 5.5 three explicit families of matroids as counterexamples’; Bérczi then found another family with ChatGPT Astra
- AI use (general): ‘LLMs were substantially used in the searches of these counterexamples … The mathematical content was digested and reshaped from the LLM search reports, checked, and written by the authors’; a detailed AI-methodology analysis is promised for an update
- Origin: the BIRS-Oaxaca workshop ‘Algebra and Geometry of Matroids’ (27 Sep – 2 Oct 2026), where Eur told Li about the LLM-generated disproof
What happened
On 12 Oct 2026 Gergely Bérczi, Basile Coron, Christopher Eur and Shiyue Li posted arXiv 2610.171351. Chow polynomials (Hilbert–Poincaré series of the Chow rings of matroids) and chain polynomials had been conjectured to have only real zeros, which would imply log-concavity. The paper gives explicit infinite families of matroids, built from finite geometries by “hyperplane coalescence”, for which even log-concavity fails.
The AI disclosure gives a dated account. Eur found a non-constructive disproof of real-rootedness with ChatGPT Astra on 23 Sep and mentioned it at a BIRS-Oaxaca matroid workshop. On 30 Sep Coron found a (very large) counterexample for graph associahedra with ChatGPT Astra. On 4 Oct three of the authors launched a targeted LLM search; about 5 and 12 hours later Li “found using Claude Opus 5.5 three explicit families of matroids as counterexamples … of lowest ranks possible”. The authors “digested and reshaped” the search reports into the paper.
Because the counterexamples are explicit, they can be checked by computation; the paper is still unrefereed. No outside reactions were found on 14 Oct (web search; X not searched directly).
Why it matters
Log-concavity of matroid invariants is the area of June Huh’s Fields-Medal work, and real-rootedness conjectures there were widely studied. This is a clear example of a now-common pattern: humans state the target, LLM searches produce explicit counterexamples within hours, and the paper is the human digest.
Sources
1 source from 1 site. Numbers match the chips in the text.
1 source: 1 primary
Primary
Changes
- Filed from the arXiv paper