A proof of Borcea's quadratic variance conjecture

Let $p$ be a complex polynomial of degree $n\ge2$, and let $σ_2(p)$ be the root mean square distance of its zeros from their centroid, including multiplicities. We prove that every zero of $p$ lies within distance $σ_2(p)$ of a critical point. This proves Borcea's quadratic conjecture and, in partic