From $\log 2$ to $\pi /2$: the sharp asymptotic inradius of polynomial lemniscates

YuJian Geng, Dong Qiu

Let $R_n$ be the infimum of the inradii of $\{z:|p(z)|<1\}$ over monic degree-$n$ polynomials whose zeros lie in the closed unit disk. We prove $n{R}_n\to \pi /2$, matching the asymptoti…