A solution to the Erd\H{o}s Problem #1040
Quick summary
arXiv:2609.06050v1 Announce Type: cross Abstract: For a compact set $K\subset\mathbb{C}$, let $\vartheta(K)$ be the infimum of the planar areas of the unit lemniscates of all monic polynomials with zeros in $K$, allowing arbitrary degree and repeated zeros. We prove that $\vartheta(K)=0$ whenever $\operatorname{cap}(K)=1$, with no regularity assumption on $K$. The proof uses a centered harmonic polynomial that is positive on all but a set of arbitrarily small area in the polynomial hull of $K$. A Fourier average of exterior harmonic measures realizes this polynomial as the logarithmic potentia
Key takeaways
- arXiv:2609.06050v1 Announce Type: cross Abstract: For a compact set $K\subset\mathbb{C}$, let $\vartheta(K)$ be the infimum of the planar areas of the unit lemniscates of all monic polynomials with zeros in $K$, allowing arbitrary degree and repeated zeros.
- We prove that $\vartheta(K)=0$ whenever $\operatorname{cap}(K)=1$, with no regularity assumption on $K$.
- The proof uses a centered harmonic polynomial that is positive on all but a set of arbitrarily small area in the polynomial hull of $K$.
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