bibkey: khavinson2011borcea
authors: D. Khavinson, R. Pereira, M. Putinar, E. B. Saff, S. Shimorin
year: 2011
title: "Borcea's variance conjectures on the critical points of polynomials"
doi: 10.1007/978-3-0348-0142-3_16
url: https://arxiv.org/abs/1010.5167
claim: Section 5, Conjecture 4 proposes a nonunivalence radius in terms of the minimum root moment; this assertion is refuted by the polynomial z minus z to the fourth power at order one.
strata_touched:
  - D5/S3/Zeros/CriticalPoints/BorceaNonunivalenceRadius
license: citation-only
triage: anchor

Borcea’s variance conjectures on the critical points of polynomials

Locator

• arXiv: https://arxiv.org/abs/1010.5167 ; specified version arXiv:1010.5167v2.
• DOI: 10.1007/978-3-0348-0142-3_16
• Location: arXiv:1010.5167v2, §5, Theorem 5.1 and Conjecture 4.

Source and scope

Theorem 5.1 (Alexander-Kakeya). If a polynomial F of degree n has no critical points in a closed disk of radius R, then it is univalent in the concentric disk of radius R sin(π/n).

If F is an nth degree polynomial with F(0) = 0, then F is not univalent in any closed disk of radius larger than σ_p(F) sin(π/n), p ≥ 1, centered at zero.

Of course, if the conjecture is true for p = 1, then it follows for all p ≥ 1. By the Alexander-Kakeya Theorem, the truth of this conjecture implies the truth of Conjecture 1 for the same value of p. It can be easily seen that the strongest case p = 1 of this conjecture is true for n = 2.

Here the zeros $z_j$ include algebraic multiplicity and

$${\sigma }_p(F)=\underset{c\in \mathbb{C}}{\inf }{\left(\frac{1}{n}\sum _{j=1}^n|z_j-c{|}^p\right)}^{1/p}.$$

The definitions and the conjecture are attributed to the source. The counterexample and the exact estimates below are repository derivations.

Counterexample

For $F(z)=z-z^4$, the roots are $0,1,\omega ,\overline{\omega }$, where $\omega =-1/2+i\sqrt{3}/2$. Therefore ${\sigma }_1(F)\le 3/4$ and

$${\sigma }_1(F)\sin (\pi /4)\le 3\sqrt{2}/8<5/8.$$

For $|z|,|w|\le 5/8$, the divided difference is

$$F(z)-F(w)=(z-w)(1-(z^3+z^{2}w+z{w}^2+w^3)),$$

and the sum in parentheses has modulus at most $125/128<1$. Thus $F$ is injective on that closed disk, contradicting Conjecture 4.

Literature boundary

Searches for “1010.5167” with “Conjecture 4”, “Borcea univalence counterexample”, and the source title with “erratum” found no refutation. The two 2026 preprints proving Borcea’s 2-variance conjecture, arXiv:2610.02035 and arXiv:2610.12499, do not mention univalence. Citation indexes have not been exhaustively checked; worldwide priority and absence of later settlements are not established. The counterexample concerns Conjecture 4’s univalence route, and does not refute Borcea’s variance Conjecture 1.

License and formalization boundary

This citation-only note incorporates no source software. The Lean result is the literal negation of the fully quantified Conjecture 4, witnessed at degree four, order one and radius five eighths. The all-finite-order degree-six extension in the problem note is an analytic calculation outside the Lean result.