Is the largest disc inside $\{|f|<1\}$ of radius $\gg 1/n$ for roots in the unit disc? (Erdős #1039)
Statement
Let $f(z)=\prod_{i=1}^{n}(z-z_i)\in\mathbb{C}[z]$ be a monic polynomial all of whose roots satisfy $\lvert z_i\rvert\le 1$. Let $\rho(f)$ be the radius of the largest disc contained in the sublevel set $\{z\in\mathbb{C}:\lvert f(z)\rvert<1\}$. Determine the behaviour of $\rho(f)$; in particular, is it always true that $$\rho(f)\gg\frac{1}{n}$$ with an absolute implied constant?
Acceptance. FULLY RESOLVES: a complete proof (machine-checkable in Lean/Coq preferred, otherwise a full written proof) that $\rho(f)\gg 1/n$ holds for all such polynomials (the conjectured sharp lower bound, matching the $z^n-1$ upper bound up to an absolute constant), OR a proof that $\rho(f)\cdot n$ can be made arbitrarily small (so $1/n$ is not the right order), exhibiting an explicit polynomial family with a verified inradius of smaller order. ADVANCES: improve the best known lower bound — currently $\rho(f)\gg 1/(n\sqrt{\log n})$ (Krishnapur–Lundberg–Ramachandran) — to a strictly larger order (e.g. shaving the $\sqrt{\log n}$), with a full proof; or improve the best known upper bound below the $(\pi/2)/n$ of $z^n-1$ for infinitely many $n$ by exhibiting an explicit root configuration together with a rigorous or reproducibly certified inradius computation. State the current record in words and require strictly beating it. Deliver the proof, or the explicit polynomial family with a certified inradius.
Background
A problem of Erdős, Herzog, and Piranian [EHP58, p.134]. They observe that $f(z)=z^n-1$ has $\rho(f)\le\frac{\pi/2}{n}$, so $1/n$ is the conjectured correct order and would be sharp up to constants. Lower bounds have improved over time: Pommerenke [Po61] proved $\rho(f)\ge\frac{1}{2en^2}$, and Krishnapur, Lundberg, and Ramachandran [KLR25] (2025) improved this to $\rho(f)\gg\frac{1}{n\sqrt{\log n}}$ — leaving only a $\sqrt{\log n}$ factor between the best lower bound and the conjectured $1/n$. Listed as open on erdosproblems.com/1039 (fetched 2026-07-21, status 'open'). This sits in the Erdős–Herzog–Piranian family of lemniscate ($\{\lvert f\rvert\le 1\}$) geometry problems; distinct but adjacent venue neighbours include Erdős #509 (cover a lemniscate by discs of total radius $\le 2$), #1040 (minimal area of $\{\lvert f\rvert<1\}$), #1041 (short paths inside a lemniscate), and #1120 (shortest escape path in $\{\lvert f\rvert\le 1\}$). Attacker's tool: potential-theory and harmonic-measure estimates on lemniscates, analysis of extremal configurations (roots equally spaced on the unit circle), and numerical optimisation over root placements that minimise the inradius $\rho(f)$, both to certify improved upper bounds and to probe the constant against the conjectured $1/n$.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #1039 (T. F. Bloom) | website |
| REF-02 | Erdős Problem #509 (T. F. Bloom) — related lemniscate problem | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.