Pendyala 2026 shortest paths polynomial lemniscate sublevel sets

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proposition_9_7: Pendyala's linear upper bound, as claimed in an unrefereed preprint: for every degree n at least 1, every monic polynomial with all zeros in the closed unit disk has a path from 0 to the unit circle of length at most pi n inside the part of the closed disk where its modulus is at most 1.

theorem_1_2: Pendyala's main theorem, as claimed in an unrefereed preprint: the largest shortest length S(n) of a path from 0 to the unit circle inside the part of the closed disk where a monic degree-n polynomial with zeros in the closed disk has modulus at most 1 satisfies c sqrt(log n) <= S(n) <= pi n for all sufficiently large n, so S(n) tends to infinity.

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Venkata Siddharth Pendyala, Shortest paths in polynomial lemniscate sublevel sets and a problem of Erdős. arXiv preprint (2026). arXiv:2606.19178, doi:10.48550/arXiv.2606.19178. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2606.19178), every other right reserved.

The copy read for this card is the arXiv preprint arXiv:2606.19178v1 (17 June 2026). For a monic degree-n polynomial f with all zeros in the closed unit disk, let E_f be the part of the disk where the modulus of f is at most 1, and let S(n) be the largest possible shortest path length inside E_f from the origin to the unit circle (Definition 1.1). Theorem 1.2, the main theorem, asserts that for all sufficiently large n one has c sqrt(log n) at most S(n) at most pi n for an absolute constant c > 0, so that S(n) tends to infinity. This would confirm the qualitative unboundedness that Erdos expected (the introduction cites Problem 4.22 of Hayman's 1974 list, which reports that Erdos expected slow growth to infinity), while the paper makes no claim that the order sqrt(log n) is sharp. In the abstract's words (p. 1), "The proof combines an explicit geometric maze, Green-function and Faber-polynomial estimates, analytic quantization of circle measures, and a reciprocal-sweeping upper bound." For problem 1120 this is the relevant recent claim, but it remains an unverified single-author preprint whose provenance, including possible AI assistance, was questioned on the problem forum.

Read status: claims checked. Definition 1.1 and Theorem 1.2 (p. 3), Propositions 8.1 (p. 25) and 9.7 (p. 31), Remark 10.1 (p. 32) and the introduction (p. 2) were read clause by clause on the print; the proofs were read in outline only and were not checked.

Contents

• Theorem 1.2 (p. 3, with Definition 1.1): $c\sqrt{\log n}\le S(n)\le\pi n$ for all sufficiently large $n$, with an absolute constant $c>0$, so $S(n)\to\infty$; the lower half is Proposition 8.1 (p. 25).
• Proposition 9.7 (p. 31): $S(n)\le\pi n$ for every $n\ge1$, by reflecting the zeros into $g(z)=\prod_j(1-\overline{a_j}z)$ and taking the shorter of two edge-disjoint arcs of the curve $\lvert g\rvert=1$ from $0$ to the circle.

Source: https://arxiv.org/abs/2606.19178.

Bears on. #1120: in the problem page's worst-case reading $S(n)$, Theorem 1.2 claims $c\sqrt{\log n}\le S(n)\le\pi n$ for all sufficiently large $n$, hence $S(n)\to\infty$, and Proposition 9.7 claims $S(n)\le\pi n$ for every $n\ge1$; neither determines the order of $S(n)$.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.