A proof of Borcea's quadratic variance conjecture

Abstract

Let $p$ be a complex polynomial of degree $n\ge 2$, and let ${\sigma }_2(p)$ be the root mean square distance of its zeros from their centroid, including multiplicities. We prove that every zero of $p$ lies within distance ${\sigma }_2(p)$ of a critical point. This proves Borcea's quadratic conjecture and, in particular, gives a new proof of Sendov's conjecture. Two polynomial integration identities prove the result analytically for $n\ge 2048$. For the remaining degrees, we derive explicit necessary inequalities, including a variable integration endpoint and an exact centered coefficient identity. Directed integer arithmetic certifies a finite covering of the resulting parameter domains. Degrees $4$--$26$ use $150845$ certified three-dimensional regions; degrees $27$--$2047$ use a further $27429$ regions, with degree intervals allowed. The cases of degree 2 and 3 were already covered by Khavinson--Pereira--Putinar--Saff--Shimorin.

arXiv comment22 pages, 2 tables. Verifier and certificates: https://github.com/LudovickBouthat/borcea-verifier, archived at https://doi.org/10.5281/zenodo.23025895