Paths in Polynomial Lemniscates: A Degree-Seven Counterexample and Radial Connections

Will Cook

Long paper

Paths in Polynomial Lemniscates: A Degree-Seven Counterexample and Radial Connections

Erdős #1041 By Will Cook 29 September 2026 Authorship, AI use and citation

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Summary. The record gives the rational estimates and geometric proof for ani's degree-seven polynomial: every connected set in its strict unit lemniscate joining two roots has Hausdorff measure greater than two. It develops contained paths for trinomials, collinear roots and two sparse families. Ordinary analytic arguments for critical-value criteria retain their unformalised inputs beside the statements. Critical-value moments, counterexamples to prescribed paths and questions about flow-strip geometry complete the account. The one-polynomial formal statement and correspondence with the historical wording remain distinct.

This paper covers the exact geometry of ani's one-polynomial counterexample, proved positive families and critical-value arguments with their unformalised inputs identified; formal-statement status and historical correspondence remain owned by the claims registry.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or the current status of Erdős #1041, including the later one-polynomial Formal Conjectures path-image Hausdorff-length check and its unreviewed historical curve-length correspondence; those distinctions belong to docs/claims.json in the mathematics repository.

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In this paper

Collinear roots and Chebyshev comparison

We scale the extreme zeros of $T_{n}$ to $\pm 1$ and make the polynomial monic. Its endpoint zeros then agree with those of the polynomial under study, while subtraction lowers the degree. This is the comparison needed for the alternation argument. Put

r_{n} = \cos \frac{π}{2 n}, C_{n} = \frac{1}{2^{n - 1} r_{n}^{n}} .

Theorem 9.1 (Chebyshev comparison). Let $m \geq 0$ , let $p \in R [X]$ be monic of degree $m + 2$ , and let

- 1 < c_{0} < \dots < c_{m} < 1, | c_{i} | \leq 1.

Suppose $p (- 1) = p (1) = 0$ and $p (c_{i}) p (c_{i + 1}) < 0$ for $0 \leq i < m$ . Then

min_{0 \leq i \leq m} | p (c_{i}) | \leq C_{m + 2} .

This theorem is the finite alternation inequality; it does not construct the segment between two roots.

All real-rooted polynomials, and their images under rotations and translations, satisfy the collinearity hypothesis. Three noncollinear roots already fall outside it. The gain is an explicit sharp bound for the modulus along one adjacent-root segment, not a bound obtained by assuming that segment is contained.

Theorem 9.2 (a sharp bound for collinear roots). Let $f$ be a monic polynomial of degree $n \geq 2$ whose zero occurrences are collinear, and let $D$ be their diameter. Some two adjacent zero occurrences are joined by a segment of length at most $D$ on which

\begin{matrix} (9) & | f (z) | \leq \frac{(D / 2)^{n}}{2^{n - 1} \cos^{n} (π / (2 n))} . \end{matrix}

The constant in (9) is best possible in every degree. Equality is attained by affine images of the zeros of $T_{n}$ whose extreme zeros have distance $D$ .

Proof. A repeated zero gives the constant path, so assume the zero values are distinct. Let $m$ be the midpoint of the two extreme zeros, and choose $θ \in R$ so that their line is $m + e^{i θ} R$ . Put $R = D / 2$ . Translation by $m$ , rotation by $e^{i θ}$ and scaling by $R$ give the polynomial

q (w) = R^{- n} e^{- i n θ} f (m + R e^{i θ} w),

which is monic with real zeros $- 1 = y_{1} < \dots < y_{n} = 1$ , and $| f (m + R e^{i θ} w) | = R^{n} | q (w) |$ .

Compare $q$ with the monic endpoint-normalized Chebyshev polynomial

q_{*} (x) = \frac{T_{n} (r_{n} x)}{2^{n - 1} r_{n}^{n}} .

Both $q$ and $q_{*}$ vanish at $\pm 1$ , and $| q_{*} | \leq C_{n}$ on $[- 1, 1]$ . For each gap $[y_{i}, y_{i + 1}]$ , choose $c_{i}$ at which $| q |$ is maximal. The maximum lies in the interior: $q$ vanishes at the endpoints and nowhere inside the gap. Since every root is simple, the signs at the $n - 1$ chosen points alternate. If every gap maximum were larger than $C_{n}$ , then $q - q_{*}$ would have the same nonzero signs there. The intermediate value theorem would give $n - 2$ zeros between consecutive $c_{i}$ , in addition to the two zeros at $\pm 1$ . These $n$ zeros are distinct, but $q - q_{*}$ is nonzero and has degree at most $n - 1$ because its leading terms cancel. This contradiction selects a gap on which $| q | \leq C_{n}$ . Scaling back proves (9).

For sharpness, take

y_{k} = \frac{\cos ((2 k - 1) π / (2 n))}{r_{n}}, 1 \leq k \leq n .

These are the zeros of $q_{*}$ , have extremes $\pm 1$ , and every adjacent gap contains a scaled Chebyshev extremum where $| q_{*} | = C_{n}$ . No smaller universal constant can work. ◻

Corollary 9.3 (collinear Erdős case). If the zero occurrences of a monic polynomial of degree $n \geq 2$ lie on one line in the open unit disc, two of them are joined by a curve of length strictly below $2$ inside ${| f | < 1}$ .

Proof. A repeated zero gives the constant path, so assume the zeros are distinct. Their diameter satisfies $0 < D < 2$ . Since $\cos (π / (2 n)) \geq 1 / \sqrt{2}$ , the defining formula gives $C_{n} \leq 2^{1 - n / 2} \leq 1$ , with equality possible only at $n = 2$ . Thus (9) is strictly below one, and the selected segment has length at most $D < 2$ . ◻

Corollary 9.3 already follows from a theorem of Erdős, Herzog and Piranian [4]: if the zeros of a monic polynomial of degree $n$ are real, lie in $[- 1, 1]$ and have centroid in $[0, 1]$ , then ${| f | < 1} \cap R$ contains an interval holding at least $n / 2$ of the zeros. Apply it to the normalized polynomial $q$ , replacing it by the monic polynomial $(- 1)^{n} q (- w)$ if its centroid is negative. This reverses the root line without changing modulus bounds. For $n \geq 3$ two consecutive zeros in that interval are joined by a segment of length at most $D$ inside ${| f | < (D / 2)^{n}}$ , and for $n = 2$ the segment between the zeros lies in ${| f | \leq (D / 2)^{2}}$ . Theorem 9.2 adds the sharp level and its equality configurations.

Critical sequences and the extremal configuration.

For distinct real zeros, each gap between consecutive zeros contains exactly one zero of $q^{'}$ , and $| q |$ attains its maximum over the gap only there. The points $c_{i}$ chosen in the proof of Theorem 9.2 are therefore the critical points of $q$ , and the gap maxima are the moduli of its ordered critical sequence $(q (c_{1}), \dots, q (c_{n - 1}))$ , with $c_{1} < \dots < c_{n - 1}$ . In this distinct-root setting, the ordered sequence determines the real polynomial up to an increasing real affine change of variable [17]. The proof above uses only the alternation count, not this classification theorem. In this language, (9) says that a real polynomial of degree $n$ with $n$ distinct real zeros, leading coefficient $a$ and zero diameter $D$ has a critical value of modulus at most $C_{n} | a | (D / 2)^{n}$ .

The comparison polynomial $q_{*}$ is the monic polynomial of least deviation from zero on $[- 1 / r_{n}, 1 / r_{n}]$ , characterised by the equioscillation conditions recalled in [17]. All $n - 1$ of its critical values have modulus $C_{n}$ , so every critical point of $q_{*}$ lies on the level curve ${| q_{*} | = C_{n}}$ . Extremals with this property also occur in the level-curve length problem, where some extremal polynomial has all its critical points on ${| p | = 1}$ [14], and in the sharp bound for $| f^{'} |$ on a connected sublevel set ${| f | \leq 1}$ of a monic polynomial, proved by Eremenko and Lempert [18], whose equality cases are $e^{- i n θ} T_{n} (2^{1 / n - 1} e^{i θ} z + b)$ with $θ$ real [19].

Scope.

The sharpness assertion concerns the maximum of $| f |$ on the selected adjacent-root segment, and the displayed configurations attain it; no claim is made that they exhaust the equality cases, and no claim is made that the selected segment is a shortest path in ${| f | < 1}$ .

About this paper

Authorship and AI use. Will Cook built and directed the research infrastructure and maintains the public release. He reviewed claims when he could. AI agents did most of the research and drafting. Cook did not independently verify every claim.

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