The Borcea variance conjecture is a family of bounds on the distance from each root of a polynomial to its
nearest critical point, expressed in terms of centered
moments of the roots (Khavinson et al. 2011). If
has polynomial degree 
and roots 
, 
, ..., 
counted with multiplicity,
define
 |
(1)
|
for 
.
For 
,
use the radius of the smallest closed disk
containing the roots. The conjecture asserts that every
root 
has a critical point 
with 
.
For 
,
the minimizing center is the geometric centroid 
,
giving
 |
(2)
|
Thus 
is the standard deviation of the roots.
Zhang (2026) proved that every root 
has a critical point 
satisfying
 |
(3)
|
The constant 1 is sharp. Monotonicity of 
also establishes the conjecture for 
. The case 
is equivalent to the Sendov
conjecture, while 
remains open (Zhang 2026).
Zhang (2026) reports using ChatGPT for exploratory discussions and language refinement, while independently developing and checking the proof. Codex generated an accompanying Lean formalization that the author reports was checked by the Lean kernel.
