Purdy's Inequality for Hyperplane Arrangements

For an arrangement of $n$ hyperplanes in $\mathbb{P}^3_{\mathbb{C}}$ with $\ell$ intersection lines and $p$ intersection points where at least three hyperplanes meet, the refined form of Purdy's inequality expects $p - \ell + n + 2 \ge 0$ . An explicit arrangement built from roots of unity violates it.

Result: Disproved(see note)
Status: Resolved
AI contribution: AI-assisted
Method: Computation
Field: Discrete geometry
Posed by: George Purdy
Year posed: —
Years open: —
Solved: 2026-07-09
Model: ChatGPT
Vendor: OpenAI
Collaborators: Mateusz Michalek, Piotr Pokora
Verification: Site-confirmed
Publication: Preprint
Significance: 15 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

the refined form for essential arrangements in projective three-space

What the AI did

The acknowledgement credits the model with generating the computer programs used to carry out the symbolic computations over configurations of points and planes, which is how the violating arrangement was checked.

Verification

The counterexample is an explicit root-of-unity arrangement whose point, line and plane counts the paper works out in closed form, so the violation is a finite check. arXiv preprint, not yet peer-reviewed.

Source

arXiv:2607.08463 - A counterexample to Purdy's inequality for hyperplane arrangements

Purdy's Inequality for Hyperplane Arrangements

What was actually shown

What the AI did

Verification

Source

Discussion