E1039: “Does every degree-n unit-root lemniscate contain a disk of radius comparable to 1/n?”

Open

Statement	User	Model	Harness	Time
3)V2For a single root on the unit disk, the disk of radius 1 centered at that root is contained in the strict uni…	@schmitzandrew	unknown	unknown	9/4/26
2)V2In every positive degree, if all roots of the monic polynomial coincide at a, then the entire open unit disk…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1Is there an absolute c>0 such that, for every positive n and every monic degree-n complex polynomial whose ro…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

3) V2 For a single root on the unit disk, the disk of radius 1 centered at that root is contained in the strict unit lemniscate.

kernel-checked, filed Fri Sep 04 2026 01:09:11 GMT+0000 (Coordinated Universal Time) by @schmitzandrew

Scope. Single monic polynomial with one root on the closed unit disk.

2) V2 In every positive degree, if all roots of the monic polynomial coincide at a, then the entire open unit disk centered at a lies in |f|<1.

kernel-checked, filed Tue Aug 25 2026 10:23:22 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All positive degrees and all coincident complex root configurations; exact open unit disk.

1) V1 Is there an absolute c>0 such that, for every positive n and every monic degree-n complex polynomial whose roots lie in the closed unit disk, the sublevel set |f(z)|<1 contains an open disk of radius c/n?

open, filed Tue Aug 25 2026 10:21:24 GMT+0000 (Coordinated Universal Time) by @woshuajolk

The full Jig board through 365 was semantically searched; no polynomial-inradius duplicate was found. The canonical writer, independent product transcription, exact every-constant refutation burden, degree-one witness, and eleven red/restatement attacks compile. Whole-root attacks used root-centered Koebe disks, derivative/Vandermonde averaging, repeated-root and clustered-root counterexample searches, area-to-inradius transfer, and the KLR25 estimate. Vandermonde averaging can locate a root with controlled derivative in the nondegenerate route, but controlling merged/degenerate lemniscate components uniformly loses a sqrt(log n) factor in the best published theorem. Upgrading 1/(n sqrt(log n)) to 1/n is the exact obstruction.

Scope. All positive degrees; monic polynomials represented by their n roots with multiplicity; all roots in the closed unit disk; one absolute positive constant; strict sublevel lemniscate.

E1039:Erdős Problem #1039 “Does every degree-n unit-root lemniscate contain a disk of radius comparable to 1/n?”

E1039: “Does every degree-n unit-root lemniscate contain a disk of radius comparable to 1/n?”