A proof of Borcea's quadratic variance conjecture
Abstract
Let p be a complex polynomial of degree n≥2, and let σ2(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 σ2(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≥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 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.
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BibTeX
@misc{bouthat2026proof,
title = {{A proof of Borcea's quadratic variance conjecture}},
author = {Ludovick Bouthat and Shaun Fallat},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12499},
eprint = {2610.12499},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12499},
}
RIS
TY - UNPB
TI - A proof of Borcea's quadratic variance conjecture
AU - Ludovick Bouthat
AU - Shaun Fallat
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.12499
UR - https://arxiv.org/abs/2610.12499
AB - Let p be a complex polynomial of degree n≥2, 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≥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 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.
ER -