GPT 6 Astra Ultra attempt on Math-5703

FORMAL RESTATEMENT

Erdős #511; reconstruction of the negative answer, 2026-09-24. For a monic polynomial $f$, let $E(f)=\{z\in\mathbb C:|f(z)|<1\}$. Is the number of components of $E(f)$ of diameter greater than a fixed $c>1$ bounded independently of the degree?

QUICK LITERATURE AND CONTEXT CHECK

Huang's Many Lemniscates with large diameter, version 2, gives counterexamples for every $1<c<4$ and attributes the earlier result to Pommerenke. We reconstruct the capacity argument, taking care that the question uses an open lemniscate.

OBJECTIVE AND ATTACK PLAN

Place many disjoint long rectangles in a thin ellipse of capacity one, approximate their union by a polynomial lemniscate, and make the polynomial monic by a dilation that cannot shrink distances.

WORK

We use three classical potential-theoretic inputs explicitly. The capacity of a full compact set with an exterior conformal map $az+O(1)$ is $|a|$. Capacity is monotone under inclusion. For a polynomial $q$ of degree $d$ and leading coefficient $a$, the compact lemniscate $\{|q|\leq1\}$ has capacity $|a|^{-1/d}$. We also use Hilbert's lemniscate approximation theorem: for a compact set $K$ with connected complement and every neighborhood $U$ of $K$, there is a polynomial $q$ with \[ K\subseteq\{|q|\leq1\}\subseteq U. \] These are the external inputs; the construction and all normalizations follow below.

Fix $1<c<4$ and a desired number $N$. Choose $0<s<1$ and $d_0$ with $c<d_0<2(1+s)$. The map $z\mapsto z+s/z$ conformally maps the exterior unit disk onto the exterior of the ellipse with semiaxes $1+s$ and $1-s$. Its leading coefficient is one, so the filled ellipse $Q$ has capacity one. Because its horizontal cross-sections near the center have length greater than $d_0$, its interior contains $N$ pairwise disjoint thin closed horizontal rectangles, each of length $d_0$. Let their union be $K$. The complement of this finite union of disjoint rectangles is connected. Choose a neighborhood $U$ consisting of disjoint neighborhoods of the rectangles, with closure in the interior of $Q$. Apply the approximation theorem with a still smaller neighborhood whose closure is in $U$, obtaining $q$ as above. There exists $\eta>0$ such that \[ L:=\{|q|\leq1+\eta\}\subset U. \] Indeed, on the complement of $U$ in a sufficiently large closed disk, $|q|$ has minimum strictly greater than one, and outside that disk it is greater than two. Choose $\eta$ smaller than these margins. Then $q_0=q/(1+\eta)$ satisfies \[ K\subset\{|q_0|<1\},\qquad \{|q_0|\leq1\}\subset Q. \] Every rectangle, being connected, lies in one component of the open lemniscate. Different rectangles lie in different components since these components stay in the disjoint pieces of $U$. Each of the selected $N$ components has diameter at least $d_0$.

If $a$ is the leading coefficient of $q_0$ and $d=\deg q_0$, monotonicity and the capacity formula give $|a|^{-1/d}\leq1$, hence $|a|\geq1$. Choose $w$ with $w^d=a$ and set $f(z)=q_0(z/w)$. This polynomial is monic, and $E(f)=wE(q_0)$. Thus the dilation multiplies diameters by $|w|\geq1$. It preserves the number of components and leaves at least $N$ components with diameter greater than $c$.

RESULT AND LIMITATIONS

Attempt status: solved. For each $1<c<4$ there is no degree-independent bound. This suffices to disprove the proposed assertion. The reconstruction depends on the explicitly stated classical capacity and lemniscate approximation theorems; their proofs and a classification at the endpoint $c=4$ are outside this attempt.

SOURCES

https://www.erdosproblems.com/511; W. Huang, Many Lemniscates with large diameter, version 2, https://arxiv.org/html/2509.11597v2.

Completion rate estimate: 100%.