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A counterexample to the complex polynomial Koebe radius conjecture

This paper disproves the 2018 complex-coefficient polynomial Koebe radius conjecture by constructing a specific normalized cubic polynomial that is univalent on a neighborhood of the unit disk yet omits a point with a modulus strictly smaller than the conjectured radius, thereby establishing a strict separation between real and complex cubic Koebe radii and proving that degree three is the minimal degree for such a counterexample.

Original authors: Aran S. Ziegler

Published 2026-10-26

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Aran S. Ziegler

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of complex mathematics, there is a special class of shapes created by stretching and twisting a simple circle. Imagine taking a flexible rubber disk and pulling it in various directions without ever letting the material tear or fold back on itself. In this field, known as complex analysis, such a transformation is called "univalent," meaning it maps every point on the original circle to a unique, distinct point on the new shape. Mathematicians have long been fascinated by a specific question regarding these shapes: no matter how you stretch the rubber disk, as long as you follow the rules, how large a perfect, centered circle can you guarantee will always fit inside the resulting shape? This guaranteed safe zone is known as the Koebe radius. It represents a fundamental limit of safety; if you stay within this radius, you are certain to be inside the shape, regardless of the specific way the circle was distorted.

For decades, researchers believed they had found the exact size of this safe zone for a particular type of distortion involving polynomials, which are mathematical expressions built from adding together powers of a variable. In 2018, a group of mathematicians proposed a precise formula for this radius, suggesting that the limit was the same whether the stretching rules were simple and real, or more complicated and complex. They believed the answer was a specific number derived from trigonometry, roughly equal to 0.382. This idea seemed robust, supported by proofs for simpler cases and a logical extension to more complex ones. The mathematical community accepted this as a likely truth, assuming the rules governing these shapes were uniform across different types of coefficients.

However, a new study by Aran S. Ziegler from the Auckland University of Technology has shattered this assumption. The researcher has constructed a specific, carefully designed example that proves the proposed formula is incorrect when applied to complex numbers. The paper does not merely suggest a flaw; it provides a concrete, undeniable counterexample. Ziegler created a specific polynomial, a three-term equation with precise numerical coefficients, that behaves differently than the theory predicted. This polynomial is injective, meaning it successfully stretches the circle without folding, but it fails to cover the entire area that the old theory claimed it must.

The heart of the discovery lies in the behavior of this specific polynomial. The researcher demonstrated that this equation is perfectly valid and univalent on a disk slightly larger than the standard unit circle. Yet, when the researcher calculated the image of a specific point on the edge of this disk, the result was a value that fell just outside the safe zone predicted by the 2018 conjecture. The distance of this point from the center is approximately 0.38125, which is strictly smaller than the conjectured limit of roughly 0.38196. Because this point is omitted from the shape created by the polynomial, the guaranteed safe zone for this specific case must be smaller than the theory allowed. This single, explicit example is enough to disprove the general rule that the complex and real limits are identical.

To ensure this result was not a calculation error or a fluke, the author subjected the proof to rigorous, machine-verified scrutiny. The entire argument was translated into a formal language that a computer could check step-by-step, verifying every inequality and arithmetic operation without any room for human error. The computer confirmed that the polynomial is indeed univalent on the required disk and that the omitted point truly exists at the calculated distance. The study also utilized a method involving rational numbers to certify that the polynomial meets all the necessary conditions for being a valid counterexample. This level of verification gives the result a high degree of certainty, moving it from a hypothesis to a proven fact within the field.

The implications of this finding are clear and specific. It establishes a strict separation between the behavior of polynomials with real coefficients and those with complex coefficients. While the older theory held true for the simpler, real-number cases, the complex world operates under different constraints. The study confirms that the degree of the polynomial matters; for the simplest cases of one and two terms, the old theory still holds, but the moment the complexity increases to three terms, the rule breaks. The researcher did not claim to find the new, correct limit for the complex case, only that the old limit was too high. The exact new boundary remains unknown, but the path forward is now clear: the mathematical landscape for these shapes is more intricate than previously thought, and the safety zone for complex distortions is smaller than the elegant formula once suggested.

Technical Summary: A Counterexample to the Complex Polynomial Koebe Radius Conjecture

Problem Statement
The paper addresses the determination of the Koebe radius $\rho(\mathcal{F})$ for the class of univalent (holomorphic and injective) polynomials $\mathcal{F}$ normalized as $f(z) = z + \sum_{j=2}^N a_j z^j$ . The Koebe radius is defined as the supremum of radii $r$ such that the disk $D_r$ is contained in the image $f(D)$ for every $f \in \mathcal{F}$ .

In 2018, Dmitrishin, Smorodin, and Stokolos conjectured that for both the class of real-coefficient polynomials ( $\mathcal{U}_{N,\mathbb{R}}$ ) and complex-coefficient polynomials ( $\mathcal{U}_N$ ), the Koebe radius is given by the formula:
$\rho(\mathcal{U}_N) = \rho(\mathcal{U}_{N,\mathbb{R}}) = \frac{1}{4 \cos^2(\pi/(N + 2))}$
While the real cubic case ( $N=3$ ) was proven to satisfy this formula with a radius of $(3-\sqrt{5})/2 \approx 0.381966$ , the complex case remained an open conjecture. This paper aims to disprove the equality $\rho(\mathcal{U}_3) = \rho(\mathcal{U}_{3,\mathbb{R}})$ by constructing a specific complex cubic polynomial that violates the conjectured radius.

Methodology
The author employs a constructive approach combined with rigorous analytic verification and formal proof:

Construction of a Counterexample: The author defines a specific normalized cubic polynomial with rational coefficients:
$p(z) = z + \left( \frac{9}{10} + i\frac{1}{60} \right)z^2 + \frac{2}{7}z^3$
Univalence Verification: To prove $p(z)$ $p (z)$ is univalent on a disk $D_R$ $D_{R}$ with $R = 1001/1000$ $R = 1001/1000$ , the paper utilizes the classical Brannan mechanism. This involves:
- Reducing the injectivity condition to a "quadratic zero-exclusion" problem.
- Defining a family of quadratic polynomials $q_x(w) = 1 + 2axw + b(4x^2-1)w^2$ and proving they have no zeros in $|w| \le R$ for $x \in [0,1]$ .
- Applying a lemma (Lemma 2.1) which states that if $|A - cA| < 1 - c^2$ , the quadratic $1 + Aw + cw^2$ has no zeros in the unit disk.
Exact Rational Certification: The verification of the zero-exclusion condition relies on exact arithmetic rather than floating-point approximation. The author expresses the relevant inequality condition as a polynomial $H(y)$ $H (y)$ and prove $H(y) > 0$ $H (y) > 0$ on $[0,1]$ $[0, 1]$ using Bernstein coefficients.
- The interval $[0,1]$ is subdivided, and exact rational lower bounds for the Bernstein coefficients of $H(y)$ are calculated.
- Since Bernstein basis functions are non-negative and sum to 1, positive lower bounds for the coefficients guarantee the positivity of the function.
Formal Verification: The entire proof, including the polynomial definition, normalization, univalence on the enlarged disk, and the strict bounds, is verified using the Lean 4 theorem prover (version 4.34.1) with the mathlib library. The proof relies only on standard axioms (propext, Classical.choice, Quot.sound) without unproven hypotheses.
Omitted Value Calculation: The author identifies a specific point $\zeta = (-24 + 7i)/25$ on the boundary of the unit disk (where $|\zeta|=1$ ). They calculate its image $w_0 = p(\zeta)$ and prove that $|w_0|$ is strictly less than the conjectured radius.

Key Results

Disproof of Conjecture: The polynomial $p(z)$ is injective on $D_{1001/1000}$ , implying it is univalent on the closed unit disk. However, the image of the unit disk under $p$ omits the point $w_0$ , where:
$|w_0| = \sqrt{\frac{641029}{4410000}} \approx 0.3812585$
Strict Separation of Radii: The calculated omitted modulus satisfies:
$|w_0| < \frac{763}{2000} < \frac{3-\sqrt{5}}{2}$
This establishes the strict inequality:
$\rho(\mathcal{U}_3) \le \sqrt{\frac{641029}{4410000}} < \rho(\mathcal{U}_{3,\mathbb{R}})$
Thus, the complex Koebe radius for degree 3 is strictly smaller than the real Koebe radius, refuting the conjecture that they are equal.
Least Degree of Failure: The paper verifies that for degrees $N=1$ and $N=2$ , the real and complex Koebe radii are identical ( $\rho=1$ and $\rho=1/2$ , respectively). Therefore, degree 3 is the minimal degree at which the conjecture fails.

Significance and Claims
The paper claims to provide the first explicit counterexample to the complex polynomial Koebe radius conjecture for $N=3$ . Its significance lies in:

Refuting the Complex Equality: It demonstrates that the extremal behavior of real-coefficient polynomials does not extend to the complex domain for cubic polynomials.
Establishing an Upper Bound: It provides a concrete, rigorous upper bound for the complex cubic Koebe radius ( $\approx 0.3812585$ ), distinguishing it from the real value ( $\approx 0.3819660$ ).
Methodological Rigor: The work highlights the utility of combining classical complex analysis (Dieudonné criterion, argument principle) with modern formal verification (Lean) and exact rational arithmetic (Bernstein coefficients) to settle problems involving strict inequalities.

The author explicitly states that while they provide an upper bound, they do not determine the optimal (exact) value of the complex cubic Koebe radius, nor do they claim that the conjecture fails for all degrees $N > 3$ . The result is specific to the cubic case.

Technical Summary: A Counterexample to the Complex Polynomial Koebe Radius Conjecture

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