Problem 1120

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claims/: The 2 claim pages of Problem 1120, one per claimant's result; the problem's standing derives from them.

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Statement. Let $f\in \mathbb{C}[z]$ be a monic polynomial of degree $n$, all of whose roots satisfy $\lvert z\rvert\leq 1$. Let

$$E= \{ z : \lvert f(z)\rvert \leq 1\}.$$

What is the shortest length of a path in $E$ joining $z=0$ to $\lvert z\rvert =1$?

Formulation. The question is read as asking for the worst case: the largest, over admissible $f$ of degree $n$, of the shortest length of such a path, as a function $S(n)$ of $n$. This is how the site's commentary reads it. The trivial lower bound is $S(n)\geq 1$, with equality for $f(z)=z^n$.

Status. The site labels the problem OPEN (page last edited 30 December 2025). Two pending partial claims by Pendyala bear on it: an arXiv preprint asserting $c\sqrt{\log n}\leq S(n)\leq \pi n$ for large $n$ (claim page), and an SSRN preprint asserting $S(n)=1$ for $n\leq 3$ and $S(6)>1$ (claim page).

Source. erdosproblems.com/1120, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1120, https://www.erdosproblems.com/1120.

References.

• [Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155-180.

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• pendyala_2026_shortest_paths_polynomial_lemniscate_sublevel_sets
• pendyala_2026_shortest_paths_polynomial_lemniscate_sublevel_sets / proposition_9_7
• pendyala_2026_shortest_paths_polynomial_lemniscate_sublevel_sets / theorem_1_2