The maximal length of the Erdős–Herzog–Piranian lemniscate in high degree

What's new 2025-12-16

I’ve just uploaded to the arXiv my preprint The maximal length of the Erdős–Herzog–Piranian lemniscate in high degree. This paper asymptotically resolves an old question about the polynomial lemniscates

$\displaystyle \partial E_1(p) := \{ z: |p(z)| = 1\}$

attached to monic polynomials ${p}$ of a given degree ${n}$ , and specifically the question of bounding the arclength ${\ell(\partial E_1(p))}$ of such lemniscates. For instance, when ${p(z)=z^n}$ , the lemniscate is the unit circle and the arclength is ${2\pi}$ ; this in fact turns out to be the minimum possible length amongst all (connected) lemniscates, a result of Pommerenke. However, the question of the largest lemniscate length is open. The leading candidate for the extremizer is the polynomial

$\displaystyle p_0(z)= z^n-1$

whose lemniscate is quite convoluted, with an arclength that can be computed asymptotically as

$\displaystyle \ell(\partial E_1(p_0)) = B(1/2,n/2) = 2n + 4 \log 2 + O(1/n)$

where ${B}$ is the beta function.

(The images here were generated using AlphaEvolve and Gemini.) A reasonably well-known conjecture of Erdős, Herzog, and Piranian (Erdős problem 114) asserts that this is indeed the maximizer, thus ${\ell(\partial E_1(p)) \leq \ell(\partial E_1(p_0))}$ for all monic polynomials of degree ${n}$ .

There have been several partial results towards this conjecture. For instance, Eremenko and Hayman verified the conjecture when ${n=2}$ . Asympotically, bounds of the form ${\ell(\partial E_1(p)) \leq (C+o(1)) n}$ had been known for various ${C}$ such as ${\pi}$ , ${2\pi}$ , or ${4\pi}$ ; a significant advance was made by Fryntov and Nazarov, who obtained the asymptotically sharp upper bound

$\displaystyle \ell(\partial E_1(p)) \leq 2n + O(n^{7/8})$

and also obtained the sharp conjecture ${\ell(\partial E_1(p)) \leq \ell(\partial E_1(p_0))}$ for ${p}$ sufficiently close to ${p_0}$ . In that paper, the authors comment that the ${O(n^{7/8})}$ error could be improved, but that ${O(n^{1/2})}$ seemed to be the limit of their method.

I recently explored this problem with the optimization tool AlphaEvolve, where I found that when I assigned this tool the task of optimizing ${\ell(\partial E_1(p))}$ for a given degree ${n}$ , that the tool rapidly converged to choosing ${p}$ to be equal to ${p_0}$ (up to the rotation and translation symmetries of the problem). This suggested to me that the conjecture was true for all ${n}$ , though of course this was far from a rigorous proof. AlphaEvolve also provided some useful visualization code for these lemniscates which I have incorporated into the paper (and this blog post), and which helped build my intuition for this problem; I vew this sort of “vibe-coded visualization” as another practical use-case of present-day AI tools.

In this paper, we iteratively improve upon the Fryntov-Nazarov method to obtain the following bounds, in increasing order of strength:

(i) ${\ell(\partial E_1(p)) \leq 2n + O(n^{1/2})}$ .
(ii) ${\ell(\partial E_1(p)) \leq 2n + O(1)}$ .
(iii) ${\ell(\partial E_1(p)) \leq 2n + 4 \log 2 + o(1)}$ .
(iv) ${\ell(\partial E_1(p)) \leq \ell(\partial E_1(p_0))}$ for sufficiently large ${n}$ .

In particular, the Erdős–Herzog–Piranian conjecture is now verified for sufficiently large ${n}$ .

The proof of these bounds is somewhat circuitious and technical, with the analysis from each part of this result used as a starting point for the next one. For this blog post, I would like to focus on the main ideas of the arguments.

A key difficulty is that there are relatively few tools for upper bounding the arclength of a curve; indeed, the coastline paradox already shows that curves can have infinite length even when bounded. Thus, one needs to utilize some smooth or algebraic structure on the curve to hope for good upper bounds. One possible approach is via the Crofton formula, using Bezout’s theorem to control the intersection of the curve with various lines. This is already good enough to get bounds of the form ${\ell(\partial E_1(p)) \leq O(n)}$ (for instance by combining it with other known tools to control the diameter of the lemniscate), but it seems challenging to use this approach to get bounds close to the optimal ${\sim 2n}$ .

Instead, we follow Fryntov–Nazarov and utilize Stokes’ theorem to convert the arclength into an area integral. A typical identity used in that paper is

$\displaystyle \ell(\partial E_1(p)) = 2 \int_{E_1(p)} |p'|\ dA - \int_{E_1(p)} \frac{|p\varphi|}{\varphi} \psi\ dA$

where ${dA}$ is area measure, ${\varphi = p'/p}$ is the log-derivative of ${p}$ , ${\psi = p''/p'}$ is the log-derivative of ${p'}$ , and ${E_1(p)}$ is the region ${\{ |p| \leq 1 \}}$ . This and the triangle inequality already lets one prove bounds of the form ${\ell(\partial E_1(p)) \leq 2\pi n + O(n^{1/2})}$ by controlling ${\int_{E_1(p)} |\psi|\ dA}$ using the triangle inequality, the Hardy-Littlewood rearrangement inequality, and the Polya capacity inequality.

But this argument does not fully capture the oscillating nature of the phase ${\frac{|\varphi|}{\varphi}}$ on one hand, and the oscillating nature of ${E_1(p)}$ on the other. Fryntov–Nazarov exploited these oscillations with some additional decompositions and integration by parts arguments. By optimizing these arguments, I was able to establish an inequality of the form

$\displaystyle \ell(\partial E_1(p)) \leq \frac{1}{\pi} \int_{E_2(p)} |\psi|\ dA + O(X_1+X_2+X_3+X_4+X_5) \ \ \ \ \ (1)$

where ${E_2(p) = \{ |p| \leq 2\}}$ is an enlargement of ${E_1(p)}$ (that is significantly less oscillatory, as displayed in the figure below), and ${X_1,\dots,X_5}$ are certain error terms that can be controlled by a number of standard tools (e.g., the Grönwall area theorem).

One can heuristically justify (1) as follows. Suppose we work in a region where the functions ${\psi}$ , ${p}$ are roughly constant: ${\psi(z) \approx \psi_0}$ , ${p(z) \approx c_0}$ . For simplicity let us normalize ${\psi_0}$ to be real, and ${c_0}$ to be negative real. In order to have a non-trivial lemniscate in this region, ${c_0}$ should be close to ${-1}$ . Because the unit circle ${\partial D(0,1)}$ is tangent to the line ${\{ w: \Re w = -1\}}$ at ${-1}$ , the lemniscate condition ${|p(z)|=1}$ is then heuristically approximated by the condition that ${\Re (p(z) + 1) = 0}$ . On the other hand, the hypothesis ${\psi(z) \approx \psi_0}$ suggests that ${p'(z) \approx A e^{\psi_0 z}}$ for some amplitude ${z}$ , which heuristically integrates to ${p(z) \approx c_0 + \frac{A}{\psi_0} e^{\psi_0 z}}$ . Writing ${\frac{A}{\psi_0}}$ in polar coordinates as ${Re^{i\theta}}$ and ${z}$ in Cartesian coordinates as ${x+iy}$ , the condition ${\Re(p(z)+1)=0}$ can then be rearranged after some algebra as

$\displaystyle \cos (\psi_0 y + \theta) \approx -\frac{1+c_0}{Re^{\psi_0 x}}.$

If the right-hand side is much larger than ${1}$ in magnitude, we thus expect the lemniscate to be empty in this region; but if instead the right-hand side is much less than ${1}$ in magnitude, we expect the lemniscate to behave like a periodic sequence of horizontal lines of spacing ${\frac{\pi}{\psi_0}}$ . This makes the main terms on both sides of (1) approximately agree (per unit area).

A graphic illustration of (1) (provided by Gemini) is shown below, where the dark spots correspond to small values of ${\psi}$ that act to “repel” (and shorten) the lemniscate. (The bright spots correspond to the critical points of ${p}$ , which in this case consist of six critical points at the origin and one at both of ${+1/2}$ and ${-1/2}$ .)

By choosing parameters appropriately, one can show that ${X_1,\dots,X_5 = O(\sqrt{n})}$ and ${\frac{1}{\pi} \int_{E_2(p)} |\psi|\ dA \leq 2n + O(1)}$ , yielding the first bound ${\ell(\partial E_1(p)) \leq 2n + O(n^{1/2})}$ . However, by a more careful inspection of the arguments, and in particular measuring the defect in the triangle inequality

$\displaystyle |\psi(z)| = |\sum_\zeta \frac{1}{z-\zeta}| \leq \sum_\zeta \frac{1}{|z-\zeta|},$

where ${\zeta}$ ranges over critical points. From some elementary geometry, one can show that the more the critical points ${\zeta}$ are dispersed away from each other (in an ${\ell^1}$ sense), the more one can gain over the triangle inequality here; conversely, the ${L^1}$ dispersion ${\sum_\zeta |\zeta|}$ of the critical points (after normalizing so that these critical points have mean zero) can be used to improve the control on the error terms ${X_1,\dots,X_5}$ . Optimizing this strategy leads to the second bound

$\displaystyle \ell(\partial E_1(p)) \leq 2n + O(1).$

At this point, the only remaining cases that need to be handled are the ones with bounded dispersion: ${\sum_\zeta |\zeta| \leq 1}$ . In this case, one can do some elementary manipulations of the factorization

$\displaystyle p'(z) = n \prod_\zeta (z-\zeta)$

to obtain some quite precise control on the asymptotics of ${p(z)}$ and ${p'(z)}$ ; for instance, we will be able to obtain an approximation of the form

$\displaystyle p(z) \approx -1 + \frac{z p'(z)}{n}$

with high accuracy, as long as ${z}$ is not too close to the origin or to critical points. This, combined with direct arclength computations, can eventually lead to the third estimate

$\displaystyle \ell(\partial E_1(p)) \leq 2n + 4 \log 2 + o(1).$

The last remaining cases to handle are those of small dispersion, ${\sum_\zeta |\zeta| = o(1)}$ . An extremely careful version of the previous analysis can now give an estimate of the shape

$\displaystyle \ell(\partial E_1(p)) \leq \ell(\partial E_1(p_0)) - c \|p\| + o(\|p\|)$

for an absolute constant ${c>0}$ , where ${\|p\|}$ is a measure of how close ${p}$ is to ${p_0}$ (it is equal to the dispersion ${\sum_\zeta |\zeta|}$ plus an additional term ${n |1 + p(0)|^{1/n}}$ to deal with the constant term ${p(0)}$ ). This establishes the final bound (for ${n}$ large enough), and even shows that the only extremizer is ${p_0}$ (up to translation and rotation symmetry).