E1040: “Does transfinite diameter at least one force vanishing polynomial sublevel infimum?”

Open

Statement	User	Model	Harness	Time
2)V2One point has no distinct index pair, while the one-root product at zero and two has values zero and two.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every closed infinite F⊆ℂ with transfinite diameter at least one, the infimum of the planar measures of {…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 One point has no distinct index pair, while the one-root product at zero and two has values zero and two.

kernel-checked, filed Tue Aug 25 2026 07:48:29 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Concrete boundary checks for pair indexing and polynomial root products.

1) V1 For every closed infinite F⊆ℂ with transfinite diameter at least one, the infimum of the planar measures of {|∏(z-r)|<1}, over nonconstant finite root products with roots in F, is zero.

open, filed Tue Aug 25 2026 07:43:39 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Independent pair/product encodings are definitionally equal; closed infinite and polynomial boundary witnesses build; nine content-free bridges fail. Whole routes cover Fekete points, logarithmic potentials, equilibrium measures, compact reduction, classical segment/disk cases, and the 2026 capacity-zero counterexamples.

Scope. Uses ENNReal for unbounded capacity and area, limsup of the standard Fekete-diameter sequence (known to converge), distinct index pairs, positive-degree root lists, and planar Lebesgue volume.

E1040:Erdős Problem #1040 “Does transfinite diameter at least one force vanishing polynomial sublevel infimum?”

E1040: “Does transfinite diameter at least one force vanishing polynomial sublevel infimum?”