Breakthrough Tracker record
An Astra-assisted preprint claims the remaining case of Erdős Problem 1040
Ioannis Tzachristas proves that every compact plane set of logarithmic capacity one admits monic polynomials with zeros in the set whose unit lemniscates have arbitrarily small area. Combined with prior work and an unbounded-set argument, this answers the vanishing question for every closed infinite set of transfinite diameter at least one.
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- Stable ID
math-erdos-problem-1040-capacity-one-2026- Revision
math-erdos-problem-1040-capacity-one-2026.v1- Field
- Mathematics · Complex analysis, potential theory and polynomial lemniscates
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Provisional AI-assisted solution
- Last checked
AI role
The author says GPT-6 Astra, used through Codex, assisted proof exploration, mathematical reasoning, literature checks, drafting and parallel internal reviews.
Record details
- Problem or result
- Vanishing unit-lemniscate area in Erdős Problem 1040
- Authors
- Ioannis Tzachristas
- Institutions
- No affiliation stated in the manuscript
- Result date
- Submitted September 5 and announced September 9, 2026
Why it matters
The capacity-one case was the boundary left open by earlier results; the preprint would complete the affirmative vanishing classification at and above the sharp capacity threshold.
Limits
The author explicitly labels this an AI-assisted proposed result. It has no proof-assistant artifact, independent human verification or peer review, and it does not give degree bounds or explicit root sets.
Sources
Correction and revision history
- 2026-09-10 — Added after primary-source, scope, status, AI-role and limitation review.
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