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Source. The definitions on pp. 1--2, Theorem 1.1 on p. 2, Definition 2.1 and Proposition 2.2 on pp. 2--3 and Corollary 6.3 on p. 10 of Venkata Siddharth Pendyala, Radial access for polynomial lemniscates: the cubic theorem and a degree-six obstruction, SSRN preprint (2026), doi:10.2139/ssrn.6850818, the edition named on the source card.

Statement

Setting (pp. 1--2). $\mathbb D$ is the open unit disk. A polynomial is admissible when it is monic, $f(z)=\prod_{k=1}^n(z-z_k)$, with every $z_k$ in $\overline{\mathbb D}$. Its filled lemniscate is $E_f=\{z\in\mathbb C:|f(z)|\le1\}$, which contains $0$ since $|f(0)|=\prod_k|z_k|\le1$, and $C_f$ is the component of $E_f$ containing $0$. $L(f)$ is the infimum of the lengths of rectifiable paths in $C_f$ that begin at $0$ and end on $\partial\mathbb D$, with $L(f)=+\infty$ when there is no such path, and $\Delta(f)=L(f)-1$. The radial-access threshold is

$$N^*=\min\{n\ge1:\text{some admissible monic polynomial of degree }n\text{ has }\Delta(f)>0\},$$

with $N^*=+\infty$ if no such $n$ exists.

Definition 2.1 (p. 2). An admissible $f$ has zero detour if $\Delta(f)=0$. A radius $[0,e^{i\theta}]=\{re^{i\theta}:0\le r\le1\}$ is safe for $f$ if $|f(re^{i\theta})|\le1$ for $0\le r\le1$.

Proposition 2.2 (Zero-detour criterion, p. 2). An admissible polynomial has zero detour if and only if it has a safe radius. With the radial-barrier functional (p. 3)

$$\mathcal B(f)=\min_{\theta\in\mathbb R}\ \max_{0\le r\le1}|f(re^{i\theta})|,$$

this reads: $\Delta(f)=0$ exactly when $\mathcal B(f)\le1$, and $\Delta(f)>0$ exactly when $\mathcal B(f)>1$.

Theorem 1.1 (Main theorem, p. 2). $4\le N^*\le6$: every admissible polynomial of degree at most three has a safe radius, and an explicit admissible polynomial of degree six has none.

Corollary 6.3 (p. 10). $N^*\in\{4,5,6\}$.

The paper leaves the exact value of $N^*$ open (abstract, p. 1).

For the problem page's $S(n)$. A path from $0$ to $\partial\mathbb D$ has length at least $1$, with equality only for a radius (p. 1), and a path in $E_f$ that starts at $0$ lies in $C_f$. So the worst-case length $S(n)$ of the problem page is $\sup_fL(f)=1+\sup_f\Delta(f)$ over admissible $f$ of degree $n$, and $N^*=\min\{n\ge1:S(n)>1\}$. Theorem 1.1 then gives $S(n)=1$ for $n\le3$ and, through Theorem 6.2, $S(6)>1$. This translation is the corpus's, not the paper's.

Read depth. Claims checked: the setting, Definition 2.1, Proposition 2.2, Theorem 1.1 and Corollary 6.3 were read clause by clause on the page images. The proof of Proposition 2.2 (pp. 2--3) was read through and not checked. The paper is an unrefereed preprint, and nothing here is independently reviewed.

Proof pointer

The paper combines (p. 10) Theorem 5.1, which gives $N^*\ge4$, with Theorem 6.2, which gives $N^*\le6$. Proposition 2.2 joins the radial and the metric notions (pp. 2--3): a safe radius is a path of length $1$; conversely, if $\Delta(f)=0$, paths in $C_f$ of lengths tending to $1$, parametrized proportionally to arclength, have a uniformly convergent subsequence whose limit $p$ satisfies $|p(t)|\le t$ and $|w-p(t)|\le1-t$ for a limit endpoint $w$ on the circle, which forces $p(t)=tw$; as $C_f$ is closed, the radius $[0,w]$ lies in it and is safe.

Dependencies

Theorem 5.1 (p. 7) and Theorem 6.2 (p. 9) of the same paper, with Proposition 2.2 (p. 2).

Bears on

• Problem 1120: in the problem page's worst-case reading $S(n)$, the theorem gives $S(n)=1$ for $n\le3$ and $S(6)>1$, so the least $n$ with $S(n)>1$ is $4$, $5$ or $6$. It says nothing about the growth of $S(n)$.