Year: 2026
Type: preprint
Abstract: 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 particular, gives a new proof of Sendov's conjecture. Two polynomial integration identities prove the result analytically for $n\ge2048$. 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 $150\,845$ certified three-dimensional regions; degrees $27$--$2047$ use a further $27\,429$ regions, with degree intervals allowed. The cases of degree 2 and 3 were already covered by Khavinson--Pereira--Putinar--Saff--Shimorin.
Source: arXiv (Cornell University)
Authors: Bouthat, Ludovick, Fallat, Shaun
Institutions: University of Delaware, University of Regina
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221565344