Sharp order in Erd\H{o}s's minimum-area problem for polynomial lemniscates

Venkata Siddharth Pendyala

arxiv: 2606.17097 · v1 · pith:O4XZTIZ3new · submitted 2026-06-13 · 🧮 math.CV

Sharp order in ErdH{o}s's minimum-area problem for polynomial lemniscates

Venkata Siddharth Pendyala This is my paper

Pith reviewed 2026-06-27 03:42 UTC · model grok-4.3

classification 🧮 math.CV

keywords polynomial lemniscatesminimum areaErdős problemFaber polynomialsunit disknormal familiesmonic polynomialscomplex analysis

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The pith

The minimal area of a degree-n monic polynomial lemniscate with zeros in the closed unit disk is of order 1/log n.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the quantity κ_n measuring the smallest possible area of the set |p(z)|<1 for monic polynomials p of degree n with zeros in the closed unit disk satisfies c/log n ≤ κ_n ≤ C/log n for absolute constants c and C. The same order holds even when the zeros are forced to lie on the unit circle itself. This shows that a previously known lower bound has the correct growth rate. The upper bound is constructed by applying a quantitative Faber-polynomial separator to a thin keyhole domain and then discretizing the level set with equal-weight midpoints so that the resulting polynomial remains monic of exact degree n while its lemniscate area stays controlled.

Core claim

There exist absolute constants c, C > 0 such that c/log n ≤ κ_n(closed unit disk,1) ≤ κ_n(unit circle,1) ≤ C/log n. The upper bound follows from a quantitative Faber-polynomial separator on a thin keyhole domain combined with an equal-weight midpoint discretization that preserves exact degree n and bounds the area from above. As a corollary, the critical minimizers with all zeros on the boundary form a normal family inside the unit disk.

What carries the argument

Quantitative Faber-polynomial separator for a thin keyhole domain together with equal-weight midpoint discretization that preserves exact degree n.

If this is right

The minimal area is the same order whether zeros are allowed anywhere in the disk or restricted to the circle.
Critical boundary-zero minimizers form a normal family inside the unit disk.
The previously established lower bound on κ_n is asymptotically sharp in order.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The normal-family conclusion may permit passage to a limit object whose level sets realize an extremal configuration in logarithmic potential theory.
Similar 1/log n scaling could appear in other constrained polynomial extremal problems that involve area or capacity with zero-location restrictions.
The keyhole-domain construction might adapt to domains with other thin slits, yielding comparable area bounds for lemniscates centered at different compact sets.

Load-bearing premise

The quantitative Faber-polynomial separator exists for a thin keyhole domain and the equal-weight midpoint discretization preserves exact degree n while controlling the area.

What would settle it

Existence of a monic degree-n polynomial with zeros in the closed unit disk whose filled unit lemniscate has area o(1/log n), or failure of the midpoint discretization to keep the area at most C/log n.

Figures

Figures reproduced from arXiv: 2606.17097 by Venkata Siddharth Pendyala.

**Figure 1.** Figure 1: The solid shaded region is the analytic keyhole Kδ; the dashed and dotted curves indicate the inner and outer comparison regions. The small circles schematically represent the fixed-overlap Harnack chain contained in the corridor and the exterior annulus. Its boundary consists of an arc of |z| = R0, two straight segments parallel to Γ0, an arc of |z| = a0, and the finitely many junctions at which these pi… view at source ↗

read the original abstract

For a monic polynomial $p$, its filled unit lemniscate is the planar set ${z: |p(z)|<1}$. Let $\kappa_n(K,1)$ denote the least possible area of this set among monic polynomials of degree $n$ whose zeros lie in a compact set $K$. We prove that there are absolute constants $c,C>0$ such that $c/\log n \leq \kappa_n(\overline{\mathbb{D}},1) \leq \kappa_n(\mathbb{T},1) \leq C/\log n$. Thus the recently established lower bound has the correct order, even when all zeros are required to lie on the unit circle. The upper bound is obtained by combining a quantitative Faber-polynomial separator for a thin keyhole domain with an equal-weight midpoint discretization that preserves the degree exactly. We also deduce that the critical boundary-zero minimizers form a normal family in $\mathbb{D}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper closes the gap by proving the upper bound of order 1/log n for minimal lemniscate area even when all zeros are forced onto the unit circle.

read the letter

The main takeaway is that the minimal area κ_n is Θ(1/log n) and the upper bound survives the restriction to roots on the unit circle. The lower bound was known; the new work supplies the matching upper bound via a concrete construction.

They start with a thin keyhole domain, build a quantitative Faber polynomial separator for it, and then apply equal-weight midpoint discretization. This produces monic degree-n polynomials with all roots on the circle whose filled unit lemniscate has area at most C/log n. As a side result they show the critical boundary-zero minimizers form a normal family in the disk.

The construction uses standard tools and the discretization step is presented as preserving exact degree while keeping the area under control. If the error estimates in the separator and the midpoint rule hold with the claimed constants, the argument goes through without circularity or extra logarithmic losses.

The potential soft spot is the quantitative control in the discretization: any hidden factor that grows with n could spoil the 1/log n order. The abstract states the constants work, so the full write-up needs to be checked on that point, but nothing in the outline suggests a structural flaw.

This is a narrow but cleanly resolved question in complex analysis and potential theory. Readers who track Erdős-type extremal problems for polynomials will find the order settled and the construction useful. It is not a broad survey, but it gives a definite answer to one specific asymptotic.

I would send it to referees. The claim is precise, the method is described, and the result is new for the circle case.

Referee Report

2 major / 2 minor

Summary. The paper proves that there exist absolute constants c, C > 0 such that c/log n ≤ κ_n(closed unit disk, 1) ≤ κ_n(unit circle, 1) ≤ C/log n, where κ_n(K,1) is the infimal area of the filled unit lemniscate {z : |p(z)| < 1} over monic degree-n polynomials with zeros in K. The lower bound is cited from prior work; the new upper bound for zeros on the unit circle is obtained by constructing a quantitative Faber-polynomial separator on a thin keyhole domain followed by equal-weight midpoint discretization that yields an exact-degree-n polynomial while controlling the area. The paper also deduces that critical boundary-zero minimizers form a normal family in the disk.

Significance. If the stated bounds hold, the result establishes that the recently obtained lower bound of order 1/log n is sharp even under the stricter constraint that all zeros lie on the unit circle, thereby resolving the asymptotic order in Erdős's minimum-area problem for polynomial lemniscates. The construction supplies an explicit mechanism (Faber separation plus discretization) that achieves the matching upper bound, and the normality statement provides additional information on the location of near-minimizers.

major comments (2)

[upper bound construction] The central upper-bound construction relies on the existence of a quantitative Faber-polynomial separator for a thin keyhole domain whose width is tuned with n; the manuscript should make explicit the dependence of the separator constants on the keyhole width parameter (abstract and the section describing the upper bound).
[discretization step] The equal-weight midpoint discretization is asserted to preserve exact degree n while keeping the lemniscate area O(1/log n); a precise error estimate relating the discretization step size to the area increment is needed to confirm that the O(1/log n) bound is not degraded (abstract and discretization paragraph).

minor comments (2)

Notation for the keyhole domain and the precise definition of the Faber separator should be introduced before their quantitative use.
The statement that the critical boundary-zero minimizers form a normal family would benefit from a brief indication of the compactness argument employed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address the two major comments point by point below and will incorporate the requested clarifications.

read point-by-point responses

Referee: The central upper-bound construction relies on the existence of a quantitative Faber-polynomial separator for a thin keyhole domain whose width is tuned with n; the manuscript should make explicit the dependence of the separator constants on the keyhole width parameter (abstract and the section describing the upper bound).

Authors: We agree that explicit dependence improves clarity. In the revision we will state in the abstract and in the upper-bound section the dependence of the Faber-separator constants on the keyhole width δ, including the fact that these constants remain controlled as δ is tuned with n to produce the O(1/log n) area bound. revision: yes
Referee: The equal-weight midpoint discretization is asserted to preserve exact degree n while keeping the lemniscate area O(1/log n); a precise error estimate relating the discretization step size to the area increment is needed to confirm that the O(1/log n) bound is not degraded (abstract and discretization paragraph).

Authors: We will insert a precise error estimate in the discretization paragraph that relates the midpoint step size to the resulting area increment and verifies that the increment remains o(1/log n), thereby preserving the overall O(1/log n) upper bound. The abstract will be updated if the added detail warrants it. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes matching upper and lower bounds of order 1/log n on the minimal lemniscate area κ_n. The upper bound is obtained via an explicit construction combining a quantitative Faber-polynomial separator on a thin keyhole domain with equal-weight midpoint discretization to produce an exact-degree-n monic polynomial with zeros on the circle. The lower bound is cited from prior external work. No step reduces by definition to its own inputs, no fitted parameter is relabeled as a prediction, and no load-bearing premise depends on a self-citation chain. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof relies on standard results from complex analysis (properties of monic polynomials, Faber polynomials, normal families) with no free parameters, no invented entities, and no ad-hoc axioms beyond classical theorems.

axioms (2)

standard math Existence and basic properties of Faber polynomials for domains with analytic boundary
Invoked for the quantitative separator construction in the upper bound.
standard math Monic polynomials of degree n are determined by their zeros
Background fact used throughout.

pith-pipeline@v0.9.1-grok · 5695 in / 1420 out tokens · 44489 ms · 2026-06-27T03:42:01.852443+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references

[1]

Erd˝ os, F

P. Erd˝ os, F. Herzog, and G. Piranian,Metric properties of polynomials, J. Analyse Math.6(1958), 125–148

1958
[2]

Gaier,Lectures on Complex Approximation, Birkh¨ auser, Boston, 1987

D. Gaier,Lectures on Complex Approximation, Birkh¨ auser, Boston, 1987

1987
[3]

Krishnapur, E

M. Krishnapur, E. Lundberg, and K. Ramachandran,On the area of polynomial lemniscates, arXiv:2503.18270, 2025

arXiv 2025
[4]

Nazarov, L

F. Nazarov, L. Polterovich, and M. Sodin,Sign and area in nodal geometry of Laplace eigenfunctions, Amer. J. Math.127(2005), 879–910

2005
[5]

Pommerenke,On metric properties of complex polynomials, Michigan Math

Ch. Pommerenke,On metric properties of complex polynomials, Michigan Math. J.8(1961), 97–115

1961
[6]

Wagner,On the area of lemniscate domains, J

G. Wagner,On the area of lemniscate domains, J. Analyse Math.50(1988), 159–167. Email address:venkatasiddharthpendyala@gmail.com

1988

[1] [1]

Erd˝ os, F

P. Erd˝ os, F. Herzog, and G. Piranian,Metric properties of polynomials, J. Analyse Math.6(1958), 125–148

1958

[2] [2]

Gaier,Lectures on Complex Approximation, Birkh¨ auser, Boston, 1987

D. Gaier,Lectures on Complex Approximation, Birkh¨ auser, Boston, 1987

1987

[3] [3]

Krishnapur, E

M. Krishnapur, E. Lundberg, and K. Ramachandran,On the area of polynomial lemniscates, arXiv:2503.18270, 2025

arXiv 2025

[4] [4]

Nazarov, L

F. Nazarov, L. Polterovich, and M. Sodin,Sign and area in nodal geometry of Laplace eigenfunctions, Amer. J. Math.127(2005), 879–910

2005

[5] [5]

Pommerenke,On metric properties of complex polynomials, Michigan Math

Ch. Pommerenke,On metric properties of complex polynomials, Michigan Math. J.8(1961), 97–115

1961

[6] [6]

Wagner,On the area of lemniscate domains, J

G. Wagner,On the area of lemniscate domains, J. Analyse Math.50(1988), 159–167. Email address:venkatasiddharthpendyala@gmail.com

1988