Borcea’s variance conjecture

Prove that for all 1 ≤ p < ∞ and n ≥ 2, the optimal constant C_Borcea(p,n) equals 1; that is, under the moment constraint (1/n)∑_{i=1}^n |z_i|^p ≤ 1 for the zeros z_1,…,z_n of a degree-n polynomial, each zero ζ has an associated critical point ξ with |ξ − ζ| ≤ 1.

Background

Borcea’s conjecture generalizes Sendov’s setting by replacing unit-disk containment with an Lp moment constraint on zero locations. It implies the classical Sendov bound and has known confirmations in specific low-degree cases.

A complete proof would unify several distance-to-critical-point phenomena under a broader geometric moment framework.

References

It was conjectured by Borcea\footnote{In the notation of , the condition zip implies that $\sigma_p(F) \leq 1$, where $F(z) \coloneqq (z-z_1) \dots (z-z_n)$, and the claim that a critical point lies within distance $1$ of any zero is the assertion that $h(F,F') \leq 1$. Thus, the statement of Borcea's conjecture given here is equivalent to that in Conjecture 1 after normalizing the set of zeroes by a dilation and translation.} Conjecture 1 that C_{\ref{Borcea}(p,n)=1 for all $1 \leq p < \infty$ and $n \geq 2$.

zip:

1n∑i=1n∣zi∣p≤1,\frac{1}{n} \sum_{i=1}^n |z_i|^p \leq 1,

— Mathematical exploration and discovery at scale  (2511.02864 - Georgiev et al., 3 Nov 2025) in Subsection “Sendov’s conjecture and its variants” (Section 4.10)

Motivated by his work on Sendov's conjecture , Borcea proposed a family of p-variance conjectures in which the spread of the zeros is measured by centered moments rather than by the radius of a containing disk; see Conjecture~1. \begin{conjecture}[Borcea p-variance conjectures]\label{conj:borcea} Let $P$ be a complex polynomial of degree $n\ge 2$ and $1\le p\le \infty$. For every zero a of $P$, there exists a critical point \zeta of $P$ such that

|a-\zeta|\le \sigma_p(P).

\end{conjecture}

— Borcea's 2-variance conjecture  (2610.02035 - Zhang, 1 Oct 2026) in Section 1, Introduction; Conjecture 1 (labeled Conjecture~\ref{conj:borcea})