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Monotonicity of the first Dirichlet eigenvalue of regular polygons

This paper proves the 2006 Antunes-Freitas conjecture by demonstrating that the first Dirichlet eigenvalue of a regular polygon with fixed area strictly decreases as the number of sides $N$ increases for all $N \geq 3$ , along with the monotonicity of the ratios between consecutive eigenvalues.

Original authors: Joel Dahne, Javier Gómez-Serrano, Joana Pech-Alberich

Published 2026-01-26

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CC BY 4.0

Original authors: Joel Dahne, Javier Gómez-Serrano, Joana Pech-Alberich

Original paper licensed under CC BY 4.0 (http://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

Imagine you have a fixed amount of dough (let's say enough to make a circle with an area of $\pi$ ). You want to shape this dough into a regular polygon—a shape with straight sides and equal angles, like a stop sign (8 sides) or a triangle (3 sides).

The question this paper answers is: If you keep adding more sides to your polygon, does the "vibration frequency" of that shape go up or down?

In the world of math and physics, this "vibration frequency" is called the first Dirichlet eigenvalue (denoted as $\lambda_1$ ). You can think of it like the lowest musical note a drumhead shaped like your polygon would make if you plucked it. The paper proves that as you add more sides to your polygon (making it look more and more like a circle), this lowest note gets lower and lower.

Here is a breakdown of their findings using simple analogies:

1. The Main Discovery: The "Smoothing" Effect

The authors prove a long-standing guess (a conjecture from 2006) that was like a hunch among mathematicians but lacked a solid proof.

The Analogy: Imagine a drum. If the drum is a triangle, it has sharp corners. If it's a square, it's a bit rounder. If it's a 100-sided polygon, it's almost a perfect circle.
The Result: The paper shows that the "note" (the eigenvalue) played by the triangle is the highest. The square plays a lower note. The pentagon plays an even lower note. As you keep adding sides, the note keeps dropping until, finally, when you reach the perfect circle (the limit of infinite sides), the note hits its absolute lowest point.
The Math: They proved that $\lambda_1(\text{Triangle}) > \lambda_1(\text{Square}) > \lambda_1(\text{Pentagon}) > \dots > \lambda_1(\text{Circle})$ . It is a strictly decreasing staircase.

2. The Two-Part Strategy: "The Big Picture" and "The Close-Up"

Proving this for every number of sides (from 3 to infinity) is incredibly hard. So, the authors split the problem into two distinct zones, like a photographer using a wide-angle lens for the landscape and a macro lens for the flowers.

Part A: The "Big N" Zone (64 sides and up)

For shapes with many sides (64, 65, 66...), the polygon looks very much like a circle.

The Method: The authors used asymptotic expansions. Think of this as a mathematical "zoom lens" that approximates the shape of the polygon by looking at how it differs from a perfect circle. They wrote out a complex formula that predicts the note based on how many sides the shape has.
The Challenge: These formulas are messy and involve infinite series. To be sure their prediction was correct, they didn't just guess; they used computer-assisted proofs.
The Computer's Job: Instead of using standard numbers (which can have tiny rounding errors), they used "rigorous bounds." Imagine the computer saying, "The answer is definitely between 5.783 and 5.784," rather than "The answer is 5.78312." They proved that even with the worst possible error margins, the math still holds up: the note keeps dropping as sides are added.

Part B: The "Small N" Zone (3 to 63 sides)

For shapes with few sides (triangles, squares, hexagons), the "zoom lens" approximation doesn't work well because a triangle is very different from a circle.

The Method: They used a technique called the Method of Particular Solutions (MPS).
The Analogy: Imagine trying to guess the shape of a shadow. Instead of calculating the physics of light from scratch, you build a model out of Lego bricks (mathematical functions) that look like the shadow. You adjust the bricks until the shadow matches the real object as closely as possible.
The Result: They built these "Lego models" for every polygon from 3 to 63 sides, calculated their notes with extreme precision, and verified that the triangle is indeed the loudest, the square is quieter, and so on, all the way down to the 64-sided shape.

3. The "Ratio" Check

The paper also looked at the ratio between the notes of consecutive shapes (e.g., the note of a 10-sided polygon divided by the note of an 11-sided polygon).

They proved that this ratio also decreases as you add sides. This is like saying the "gap" between the notes gets smaller and smaller as the shapes get rounder, eventually settling into a steady rhythm as the shape becomes a circle.

4. Why Computers Were Essential

You might wonder, "Why not just do the math on paper?"

The Problem: The formulas involve complex numbers, integrals, and special functions (like Bessel functions and Polylogarithms) that are incredibly difficult to calculate by hand without making a tiny mistake.
The Solution: The authors treated the computer not as a calculator, but as a safety inspector. They programmed the computer to track every single possible error. If the computer said, "The result is safe," it meant that even if every single number in the calculation was slightly off in the worst possible direction, the final conclusion (that the note goes down) would still be true.

Summary

In short, this paper settles a 20-year-old debate. It confirms that if you have a fixed amount of material and you shape it into a regular polygon, the more sides you add, the "lower" the fundamental vibration of that shape becomes. The triangle is the most "tense" (highest pitch), and the circle is the most "relaxed" (lowest pitch). The authors proved this for every possible number of sides by combining high-level mathematical approximations for large numbers with rigorous computer-checked calculations for small numbers.

Technical Summary: Monotonicity of the First Dirichlet Eigenvalue of Regular Polygons

Problem Statement
This paper addresses the monotonicity of the first Dirichlet eigenvalue, denoted $\lambda_1$ , for regular $N$ -sided polygons ( $P_N$ ) of fixed area (specifically $\pi$ ). The central question is whether $\lambda_1(P_N)$ is a strictly decreasing function of $N$ for all $N \ge 3$ . This problem, originally conjectured by Antunes and Freitas in 2006, posits that as the number of sides increases, the regular polygon approaches the unit disk $D$ , and the eigenvalue decreases monotonically toward the disk's eigenvalue $\lambda_1(D)$ . Additionally, the authors investigate the monotonicity of the quotient $q_N = \lambda_1(P_N) / \lambda_1(P_{N+1})$ .

While the Pólya–Szegő conjecture (that the regular polygon minimizes $\lambda_1$ among all $N$ -gons of fixed area) implies the first part of this result, it has only been proven for $N=3$ and $N=4$ . Previous attempts to prove the Antunes–Freitas conjecture for large $N$ relied on asymptotic expansions but failed to quantify the threshold $N_0$ beyond which monotonicity holds, leaving the finite range of small $N$ unresolved.

Methodology
The authors employ a hybrid approach combining rigorous asymptotic analysis for large $N$ and computer-assisted proofs for small $N$ . The proof is split into two regimes: $N \ge N_0$ and $N < N_0$ , with $N_0 = 64$ .

Large $N$ Case ( $N \ge 64$ ): Asymptotic Analysis with Rigorous Bounds
- Approximate Eigenpair Construction: The authors utilize an asymptotic expansion of the eigenvalue and eigenfunction derived from the work of Berghaus et al. [5]. They construct an approximate eigenpair $(\lambda_{\text{app}}, u_{\text{app}})$ using the Schwarz-Christoffel map to transform the unit disk to the polygon. The expansion involves multiple polylogarithms and zeta values.
- A Posteriori Error Analysis: To prove the existence of a true eigenvalue near the approximation, they apply a lemma by Fox, Henrici, and Moler [28]. This requires bounding the "defect" (the residual of the approximate solution on the boundary) and the $L^2$ -norm of the approximate eigenfunction.
- Rigorous Numerics: The bounds for the defect and norm are computed using interval arithmetic and Taylor models (via the FLINT/Arb libraries). This ensures that all floating-point errors are rigorously controlled. The authors explicitly bound the error terms in the asymptotic expansions (up to order $O(N^{-6})$ ) and prove that the approximate eigenvalue is strictly decreasing and that the ratio $q_N$ is strictly decreasing for $N \ge 64$ .
Small $N$ Case ( $3 \le N < 64$ ): Method of Particular Solutions (MPS)
- Approximation: For smaller $N$ , the asymptotic expansion is insufficient. The authors use the Method of Particular Solutions (MPS), specifically the variant by Betcke and Trefethen [8]. They construct an approximate eigenfunction as a linear combination of Bessel functions centered at the polygon's vertices and the domain center.
- Validation: Similar to the large $N$ case, they compute rigorous upper bounds for the defect on the boundary and lower bounds for the $L^2$ -norm. They utilize the $N$ -fold symmetry of the domain to reduce the computational domain.
- Verification: Using these rigorous enclosures, they compute intervals for $\lambda_1(P_N)$ for $N=3$ to $N=64$ . They verify that the intervals are disjoint and strictly ordered, confirming the monotonicity chain $\lambda_1(P_3) > \lambda_1(P_4) > \dots > \lambda_1(P_{64})$ .

Key Contributions and Results

Proof of the Antunes–Freitas Conjecture: The paper provides a complete proof that $\lambda_1(P_N)$ is strictly monotonically decreasing for all $N \ge 3$ .
Monotonicity of Quotients: The authors prove that the ratio $q_N = \lambda_1(P_N) / \lambda_1(P_{N+1})$ is also strictly decreasing for all $N \ge 3$ , converging to 1 as $N \to \infty$ .
Quantitative Thresholds: Unlike previous asymptotic results, this work explicitly quantifies the threshold $N_0 = 64$ where the asymptotic behavior becomes dominant, bridging the gap between asymptotic theory and finite $N$ calculations.
Rigorous Computer-Assisted Proof: The paper demonstrates a robust framework for validating spectral properties of PDEs using computer-assisted proofs. It combines symbolic asymptotic expansions with interval arithmetic to handle error propagation, ensuring that the results are mathematically rigorous rather than merely numerical.

Significance
The paper settles a long-standing conjecture in spectral geometry regarding the behavior of Dirichlet eigenvalues on regular polygons. By rigorously connecting the asymptotic regime (large $N$ ) with the finite regime (small $N$ ), it eliminates the uncertainty regarding the transition point where asymptotic approximations become valid. The work also contributes to the methodology of computer-assisted proofs in partial differential equations, demonstrating how to handle complex special functions (polylogarithms, Bessel functions) and integral bounds with rigorous error control. The results confirm that the regular polygon is not only a local minimizer (as shown for specific $N$ by Bogosel–Bucur) but part of a globally monotonic sequence converging to the disk.