Pendyala: Radial access for polynomial lemniscates: the cubic theorem and a degree-six obstruction

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theorem_1_1: Pendyala's main theorem, as claimed in an unrefereed preprint: every monic polynomial of degree at most three with all zeros in the closed unit disk has a radius from 0 to the unit circle on which its modulus stays at most 1, and an explicit degree-six one has none, so the least degree at which length-one escape can fail is 4, 5 or 6.

theorem_5_1: Pendyala's cubic zero-detour theorem, as claimed in an unrefereed preprint: every monic polynomial of degree at most three with all zeros in the closed unit disk has a radius from 0 to the unit circle on which its modulus is at most 1, so the worst-case shortest escape length S(n) is 1 for n <= 3.

theorem_6_2: Pendyala's degree-six obstruction, as claimed in an unrefereed preprint with a rational certificate: the monic polynomial with double zeros at -rho e^{2 pi i 927/960} and -rho e^{2 pi i 317/960} and simple zeros at -rho e^{2 pi i 697/960} and -rho e^{2 pi i 547/960}, rho = 1999999/2000000, has modulus greater than 1 somewhere on every radius, so the worst-case shortest escape length S(6) exceeds 1.

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Venkata Siddharth Pendyala, Radial access for polynomial lemniscates: the cubic theorem and a degree-six obstruction, SSRN preprint (2026), 14 pp., doi:10.2139/ssrn.6850818. The print has no author line, no date, no affiliation and no acknowledgment, and says nothing about AI assistance; its information dictionary leaves the author field empty and gives a creation timestamp of 28 May 2026, 05:17 UTC. The attribution is the author's own: on 18 June 2026, in the erdosproblems.com thread of Problem 1120, Venkata Siddharth Pendyala announced it as "a secondary paper I have written", with the DOI link, as the claim page records. No notice is printed in the fourteen pages, and SSRN's terms were not read for this card; the term is unstated.

The copy read for this card is that PDF, downloaded from SSRN, which hosts it under the DOI; it carries no version marker and is identified by its fourteen pages and the timestamp above. SSRN lets an author replace a paper's file under the same DOI, so the card describes the copy downloaded on 2026-10-14. An admissible polynomial is a monic $f(z)=\prod_{k=1}^n(z-z_k)$ with every $z_k$ in the closed unit disk; $E_f=\{z:|f(z)|\le1\}$ contains $0$, and $C_f$ is its component containing $0$ (p. 1). The paper asks when $C_f$ contains a whole radius $[0,e^{i\theta}]$. It sets $L(f)$, the infimum of the lengths of rectifiable paths in $C_f$ from $0$ to the unit circle ($+\infty$ if there is none), $\Delta(f)=L(f)-1$, and the threshold $N^*$, the least $n\ge1$ for which some admissible $f$ of degree $n$ has $\Delta(f)>0$ ($+\infty$ if there is none) (p. 2). Proposition 2.2 (pp. 2--3) shows that $\Delta(f)=0$ exactly when some radius is safe, $|f(re^{i\theta})|\le1$ for $0\le r\le1$. Theorem 1.1 (p. 2), the main theorem, asserts $4\le N^*\le6$. Its lower half, Theorem 5.1 (p. 7), gives every admissible polynomial of degree at most three a safe radius, through a sector theorem (Proposition 3.1, p. 3), a small-radius theorem for zeros of modulus at most $2/3$ (Theorem 3.3, p. 4), the quadratic case (Proposition 3.4, p. 4) and a case analysis of the spread cubic (Sections 4 and 5, pp. 5--8). Its upper half, Theorem 6.2 (p. 9), is an explicit monic polynomial of degree six, all of whose zeros have modulus $1999999/2000000$, with no safe radius: by Lemma 6.1 (p. 9), in every direction its modulus exceeds $1$ at distance $1$, $1/300$ or $439/1000$ from $0$. Corollary 6.3 (p. 10) restates the main theorem as $N^*\in\{4,5,6\}$, and the exact value is left open.

Appendix A (pp. 10--14) reduces Lemma 6.1 to 8192 inequalities between rational numbers, one for each angle interval of length $1/8192$ of a turn, and prints the nine-block table of radii that is the certificate (Table 1, p. 11), an audit of the smallest margin in each block with SHA-256 hashes of the exact margins (Table 2, p. 12), and a Wolfram Language listing of a checker that recomputes every inequality in exact rational arithmetic (pp. 12--14). The listing is part of the paper; it was not run for this card.

Relation to the author's arXiv paper. Pendyala announced this note together with the arXiv preprint on shortest paths in lemniscate sublevel sets, which claims $c\sqrt{\log n}\le S(n)\le\pi n$ for all sufficiently large $n$ for the worst-case escape length $S(n)$. This note treats the length-one end of the same quantity: $S(n)=\sup_fL(f)=1+\sup_f\Delta(f)$ over admissible $f$ of degree $n$, and $\Delta(f)\ge0$ because every path from $0$ to the unit circle has length at least $1$, so $S(n)=1$ exactly when every such $f$ has zero detour, and $N^*$ is the least $n$ with $S(n)>1$. This translation is the corpus's, not the paper's. The note works in the whole set $E_f$ and the arXiv paper in its part in the closed disk; the lengths agree, since a path from $0$ stays in the closed disk until it first meets the circle.

Read status: claims checked. The definitions of pp. 1--2, Theorem 1.1, Definition 2.1 and Proposition 2.2 (pp. 2--3), Theorem 5.1 (p. 7), the construction of the degree-six polynomial and (6.1) (pp. 8--9), Lemma 6.1 and Theorem 6.2 (p. 9), Corollary 6.3 and Remark 6.4 (p. 10), and the definitions and Tables 1 and 2 of Appendix A (pp. 10--12) were read clause by clause on the page images. The proofs of Proposition 2.2 and Theorem 5.1, with the results Theorem 5.1 uses, were read through and not checked. Of the certificate only its bookkeeping was recomputed here, in exact arithmetic and without the printed checker: the leading coefficient of the product, the directions of the zeros, the coverage of the angle intervals by the blocks of Table 1, and the radii those blocks give to the intervals containing the zero directions. None of the 8192 inequalities was checked, the printed checker was not run, and nothing here is independently reviewed.

Contents

• Theorem 1.1 (p. 2), with the definitions of pp. 1--2, Proposition 2.2 and Corollary 6.3 (p. 10): $4\le N^*\le6$.
• Theorem 5.1 (p. 7): every admissible polynomial of degree at most three has zero detour, so $N^*\ge4$.
• Theorem 6.2 (p. 9), with Lemma 6.1 (p. 9) and the certificate of Appendix A (pp. 10--14): an explicit admissible polynomial of degree six, with all zeros in the open disk, has no safe radius, so $N^*\le6$.

Source: https://doi.org/10.2139/ssrn.6850818.

Bears on. #1120: in the problem page's worst-case reading $S(n)$, Theorem 5.1 gives $S(n)=1$ for $n\le3$ and Theorem 6.2 gives $S(6)>1$, so the least $n$ with $S(n)>1$ is $4$, $5$ or $6$. The note gives no lower bound for $S(6)-1$ and says nothing about $S(n)$ for $n=4$, $n=5$ or $n\ge7$, or about its growth.

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