---

Source. Theorem 5.1, p. 7, proof pp. 7--8, 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

Admissible polynomials, zero detour, safe radii and $N^*$ are as on the Theorem 1.1 page.

Theorem 5.1 (Cubic zero-detour theorem, p. 7). Every admissible monic polynomial of degree at most three has zero detour. Consequently $N^*\ge4$.

The proof exhibits a safe radius in every case, which gives zero detour by the easy half of Proposition 2.2. For the worst-case escape length $S(n)$ of the problem page, the theorem gives $S(n)=1$ for $n=1,2,3$: every such polynomial has an escape path of length exactly $1$, a radius.

Read depth. Claims checked: the statement was read clause by clause on the page image. The proof (pp. 7--8) was read through, following its case table (p. 7) to the result each case uses, and those results (Propositions 3.1 and 3.4, Lemma 3.2, Theorem 3.3 and Lemmas 4.1--4.4) were read in statement and proof; no step was checked. The paper is an unrefereed preprint, and nothing here is independently reviewed.

Proof pointer

Sections 3--5, pp. 3--8, in outline. Degrees one and two are Proposition 3.4 (p. 4). A cubic with a zero at $0$ is $zg(z)$ with $g$ an admissible quadratic, and a safe radius for $g$ is safe for it. For a cubic with three nonzero zeros:

• if their directions lie in a closed sector of opening $2\pi/3$, the bisecting radius is safe, since each zero is then within distance $1$ of both ends of that radius (Proposition 3.1, p. 3);
• if every zero has modulus at most $2/3$, Jensen's formula gives a point of the circle where $|f|\le1$ (Lemma 3.2), and the radius to it is safe (Theorem 3.3, p. 4);
• otherwise the zeros are spread, normalized as $s$, $ae^{i\alpha}$ and $be^{-i\beta}$ with $0<s,a,b\le1$ and $\cos(\alpha+\beta)\le-1/2$ (Lemma 4.1, p. 5), and a case table (p. 7) picks the ray. The ray through $s$ is safe when $s$ has the largest modulus, since then $|f(r)|\le|r^3-s^3|\le1$, and when $s\ge2/3$, by the bound $|r-\rho|(r^2+rq+p)\le1$ of Lemma 4.2 for an on-ray zero of modulus $\rho\ge2/3$, where $p$ is the modulus of the product and $q\le2-p$ the modulus of the sum of the other two zeros (Lemma 4.3(1)). Otherwise, with $a$ the largest modulus, the ray through $ae^{i\alpha}$ is safe when the same test passes or when $\cos\alpha\ge0$ (Lemma 4.4(2)). Failure of the test gives $s+b+sb>2$, hence $1/2<s<2/3$ and, by Lemma 4.4(1), $b>4/5$; when $\cos\alpha<0$ the ray through $be^{-i\beta}$ then passes the test by Lemma 4.3(2).

Dependencies

Proposition 2.2 (p. 2), stated on the Theorem 1.1 page, Propositions 3.1 (p. 3) and 3.4 (p. 4), Lemma 3.2 and Theorem 3.3 (p. 4) and Lemmas 4.1--4.4 (pp. 5--6) of the same paper; Lemma 3.2 rests on Jensen's formula.

Bears on

• Problem 1120: in the problem page's worst-case reading $S(n)$, the theorem gives $S(n)=1$ for $n\le3$, the trivial lower bound attained by a radius for every admissible polynomial of those degrees. It says nothing about degrees four and higher.