slug: khavinson-pereira-putinar-saff-shimorin-2011-nonunivalence-radius
bibkey: khavinson2011borcea
doi: 10.1007/978-3-0348-0142-3_16
url: https://arxiv.org/abs/1010.5167
triage: theorem
motivation_gids:
  - D5/S3/Zeros/CriticalPoints/BorceaNonunivalenceRadius.result

Khavinson–Pereira–Putinar–Saff–Shimorin: nonunivalence radius

Problem

D. Khavinson, R. Pereira, M. Putinar, E. B. Saff and S. Shimorin, Borcea’s variance conjectures on the critical points of polynomials, arXiv:1010.5167v2, §5, Theorem 5.1 and Conjecture 4. See khavinson2011borcea.

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.

For every $n\ge 1$, every complex polynomial $F$ of degree $n$ vanishing at zero, every real $p\ge 1$ and every real $r>{\sigma }_p(F)\sin (\pi /n)$, the assertion forbids injectivity on the closed disk centered at zero with radius $r$. Zeros are counted with algebraic multiplicity, and ${\sigma }_p$ is the infimum over all complex centers of the normalized root moment. The formal definition includes every stated quantifier and imposes no simple-root or geometric restrictions.

Motivation

The conjecture proposes an analytic route from a root-spread statistic to a critical-point bound. The external assertion and proposed counterexample are identified in issue #18950.

Gap

The proposed radius can be strictly smaller than a disk on which the polynomial remains injective. A root moment controls a global root configuration; the divided difference can still stay away from zero throughout that disk.

Route

Set $F(z)=z-z^4$, $p=1$, and $r=5/8$. Its four roots with multiplicity are $0,1,\omega ,\overline{\omega }$, with $\omega =-1/2+i\sqrt{3}/2$ and all three nonzero roots of modulus one. Evaluation of the moment at center zero gives ${\sigma }_1(F)\le 3/4$. Nonnegativity of every root moment supplies the lower bound needed for this infimum comparison.

Since $\sin (\pi /4)=\sqrt{2}/2$,

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

For $|z|,|w|\le 5/8$,

$$F(z)-F(w)=(z-w)(1-z^3-z^{2}w-z{w}^2-w^3),|z^3+z^{2}w+z{w}^2+w^3|\le 4(5/8{)}^3=125/128<1.$$

The second factor never vanishes, proving injectivity on the closed disk.

Falsifier

The exact degree-four polynomial, order one and radius five eighths refute the universal assertion. The roots, strict radius inequality and closed-disk injectivity must all hold for the same polynomial and radius; none is substituted by an open-disk estimate or a numerical approximation.

Evidence

D5/S3/Zeros/CriticalPoints/BorceaNonunivalenceRadius.result : ¬ conjecture proves the refutation with the full conjecture as its target. The only public declarations are the root-moment definition, its infimum, the conjecture and its negation. All auxiliary calculations are private.

Triage

REFUTED — Conjecture 4, by $z-z^4$ at $p=1$. The conjectured radius is at most $3\sqrt{2}/8$, while the polynomial is injective on the closed disk of radius $5/8$.

COMPUTED (analytic calculation, not Lean-verified here) — failure at every finite real $p\ge 1$. For $G(z)=z-z^6$, the roots are zero and the five fifth roots of unity. Thus ${\sigma }_p(G)\le (5/6{)}^{1/p}\le 1$, and ${\sigma }_p(G)\sin (\pi /6)\le 1/2<3/5$. The divided-difference sum is $z^5+z^{4}w+z^3{w}^2+z^2{w}^3+z{w}^4+w^5$. Its six terms each have modulus at most $(3/5{)}^5$, so its modulus is at most $6(3/5{)}^5=1458/3125<1$. Hence $G$ is injective on the closed disk $|z|\le 3/5$ for every such order.

UNAFFECTED — Borcea’s variance Conjecture 1. Only the proposed nonunivalence route fails; no counterexample to the critical-point variance statement follows.

OPEN — a correct universal radius replacing ${\sigma }_p(F)\sin (\pi /n)$. For $n\ge 2$, one can ask for the sharp constants $C_{n,p}$ such that every degree-$n$ polynomial vanishing at zero fails injectivity beyond $C_{n,p}{\sigma }_p(F)$. The counterexample rules out the proposed constants but does not determine valid sharp replacements. For $z-z^4$, the same estimate proves injectivity whenever $4r^3<1$.

ASSUMED-UNVERIFIED

Absence of subsequent settlements and worldwide priority are not established. The degree-six calculation and any proposed sharp universal replacement are not Lean theorems in this module. Theorem 5.1 is contextual literature, not an additional formalized claim.