---
id: "2026-10-12-chow-polynomials-matroids-not-log-concave"
url: "https://postcutoff.com/e/2026-10-12-chow-polynomials-matroids-not-log-concave/"
as_of: "2026-10-14T07:26:00+02:00"
date: "2026-10-12"
date_precision: day
category: science
importance: 3
confidence: medium
status: [Result confirmed]
verification: Preprint only
sources: 1
editor: human editor
human_review: null
version: null
---

As of: 2026-10-14 07:26 CEST.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-10-12-chow-polynomials-matroids-not-log-concave/

# Chow polynomials of matroids are not log-concave

Full title: 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

arXiv 2610.17135 (12 Oct 2026) by Gergely Bérczi, Basile Coron, Christopher Eur and Shiyue Li constructs explicit infinite families of matroids from finite geometries whose multichain counts fail to be log-concave. This breaks log-concavity of Chow, augmented Chow, chain and h-polynomials and disproves the real-rootedness conjectures of Ferroni–Schröter, Huh–Stevens and Athanasiadis–Kalampogia-Evangelinou from the lowest possible ranks. The authors say "LLMs were substantially used in the searches": Eur found a non-constructive disproof with ChatGPT Astra on 23 Sep, and on 4 Oct a launched LLM search found the explicit families with Claude Opus 5.5 within 5–12 hours.

## 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 2610.17135](https://arxiv.org/abs/2610.17135). 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.

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 165 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 104 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 195 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 134 days after its cutoff

## Sources

1. [arXiv 2610.17135: Chow polynomials and chain polynomials of matroids are not log-concave](https://arxiv.org/abs/2610.17135) (arxiv.org, paper)

## Changes

- 2026-10-14 (filed): Created from the arXiv paper

## Related

- 2026-10-13: [October 2026: the arXiv stream of conjectures settled with disclosed AI help continues](https://postcutoff.com/e/2026-10-13-ai-assisted-results-october-2026/index.md)