Borcea Variance Conjecture

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).

Codex generated an accompanying Lean formalization that Zhang (2026) reports was checked by the Lean kernel.

Bouthat and Fallat (2026) gave another proof of the bound. Their analytic argument treats polynomial degrees . For degrees 4 through 2047, directed integer arithmetic verifies finite certificates. This is an alternative proof of the quadratic case, not a resolution of the open case.