Sharp order in Erdős's minimum-area problem for polynomial lemniscates

Abstract

For a monic polynomial $p$, its filled unit lemniscate is the planar set $z:|p(z)|<1$. Let $\hat{I}{º}_n(K,1)$ 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 $c/\log n\le \hat{I}{º}_n(\overline{\mathbb{D}},1)\le \hat{I}{º}_n(\mathbb{T},1)\le C/\log n$. 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 $\mathbb{D}$.