Dual mean value problem for complex polynomials: the constant $L(d)$ and the conjecture $L(d)=1/d$

The dual mean value problem of Dubinin and Sugawa (2009), formulated independently by Ng, asks for the constant $L(d)$ governing how large the normalized difference quotient between a point and a critical point of a complex degree- $d$ polynomial must be: $L(d)=\inf_{P,z}\max_{\zeta\in\mathrm{Crit}(P)}|(P(z)-P(\zeta))/((z-\zeta)P'(z))|$ . The polynomial $(z+1)^d-1$ gives $L(d)\le 1/d$ , and the dual Smale mean value conjecture asserts $L(d)=1/d$ ; equivalently, every normalized polynomial $f$ (with $f(0)=0$ , $f'(0)=1$ ) has a critical point $\zeta$ with $|f(\zeta)/\zeta|\ge 1/d$ . The conjecture is proved for degrees $d\le 7$ , for odd polynomials, and for conservative polynomials, and the best uniform lower bound is of order $1/d^2$ ; the conjecture for general degree, the exact value of $L(d)$ , and the asymptotic behavior of $L(d)$ remain open.

Background

Smale's mean value conjecture grew out of the complexity analysis of Newton-type algorithms for finding roots of polynomials. For a complex polynomial $P$ of degree $d\ge 2$ , a point $\zeta$ with $P'(\zeta)=0$ is called a critical point and $P(\zeta)$ its critical value. Smale proved in 1981 that for every point $z$ that is not a critical point there exists a critical point $\zeta$ with $\Big|\frac{P(z)-P(\zeta)}{z-\zeta}\Big|\le 4\,|P'(z)|,$ a complex analogue of the real mean value theorem, and asked whether the factor 4 can be lowered to $1-1/d$ , the value attained by $P(z)=z^d-dz$ . It is convenient to abbreviate the normalized difference quotient by $Q(P,z,\zeta)=\frac{P(z)-P(\zeta)}{(z-\zeta)P'(z)}$ , an expression invariant under affine changes of the variable and of the target. Writing $S(P,z)=\min_{\zeta\in\mathrm{Crit}(P)}|Q(P,z,\zeta)|$ and $K(d)=\sup S(P,z)$ over all degree- $d$ polynomials and points $z$ , Smale's conjecture reads $K(d)=1-1/d$ ; it is confirmed for $d\le 4$ (Tischler; the case $d=4$ is attributed to Sikorav) and reported for $d=5$ by a computational proof, but is open in general.

In 2009 Dubinin and Sugawa posed the dual problem, formulated independently by Ng: instead of seeking a critical point at which the difference quotient from $z$ is small, seek one at which it is large. Setting $T(P,z)=\max_{\zeta\in\mathrm{Crit}(P)}|Q(P,z,\zeta)|$ and $L(d)=\inf T(P,z)$ over all degree- $d$ polynomials and all non-critical points $z$ , the polynomial $P^*(z)=(z+1)^d-1$ , whose single critical point $\zeta=-1$ has multiplicity $d-1$ , gives $T(P^*,0)=1/d$ , so $L(d)\le 1/d$ . The dual mean value conjecture asserts $L(d)=1/d$ for every $d\ge 2$ . By the affine invariance of $Q$ this is equivalent to a statement about normalized polynomials $f(z)=z+c_2z^2+\dots+c_dz^d$ (that is, $f(0)=0$ and $f'(0)=1$ ): every such $f$ of degree $d$ has a critical point $\zeta$ — necessarily $\zeta\ne 0$ since $f'(0)=1$ — with $|f(\zeta)/\zeta|\ge 1/d$ , the bound being attained by $f(z)=((z+1)^d-1)/d$ and its affine conjugates. Tischler's stronger conjecture $\min_{\zeta}|Q(P,z,\zeta)-1/2|\le 1/2-1/d$ would have implied both Smale's conjecture and $L(d)=1/d$ ; it holds for $d\le 4$ but was disproved by Tyson for every $d\ge 5$ , so it cannot serve as a route to the dual conjecture in general. A polynomial is called conservative if $P(\zeta)=\zeta$ at every critical point; conservative polynomials satisfy the dual conjecture by a theorem Tischler attributes to Yoccoz. Dubinin and Sugawa proved the first explicit lower bound $L(d)\ge 1/(d\,4^d)$ , which Dubinin later improved to $L(d)\ge (1/d)\tan(\pi/(4d))$ , noted improvable to $1/d^2$ — polynomial rather than exponential in $1/d$ , but still a factor of $d$ below the conjectured value. The dual conjecture is currently proved for $d\le 7$ (Hinkkanen–Kayumov–Khammatova), for odd polynomials (Tang), and for the conservative class; for degrees $d\ge 8$ and for the asymptotic behavior of $L(d)$ nothing sharp is known, and Hinkkanen–Kayumov–Khammatova showed that their method of selecting the critical point of minimal modulus, which handled $d\le 7$ , provably fails for $d>7$ .

Problem Statement

Determine the exact value of the dual mean value constant $L(d)=\inf_{P,z}\ \max_{\zeta\in\mathrm{Crit}(P)}\Big|\frac{P(z)-P(\zeta)}{(z-\zeta)P'(z)}\Big|,$ where, for each integer $d\ge 2$ , the infimum runs over all complex polynomials $P$ of degree exactly $d$ and all points $z$ with $P'(z)\ne 0$ . The central target is the dual mean value conjecture of Dubinin and Sugawa (formulated independently by Ng): prove or disprove that $L(d)=1/d$ for every $d\ge 2$ ; equivalently, that every normalized polynomial $f(z)=z+c_2z^2+\dots+c_dz^d$ with $c_d\ne 0$ (hence of degree exactly $d$ ) has a critical point $\zeta$ with $|f(\zeta)/\zeta|\ge 1/d$ . Resolving this central conjecture settles the task; since the cases $d\le 7$ are already settled, the task is equivalently settled by determining $L(d)$ exactly for the remaining degrees $d\ge 8$ . Related open questions are whether the uniform lower bound $L(d)\ge (1/d)\tan(\pi/(4d))$ , noted improvable to $1/d^2$ , can be improved toward the conjectured $1/d$ , and what the precise asymptotic behavior of $L(d)$ is as $d\to\infty$ . Results restricted to special polynomial classes (odd, conservative, $k$ -symmetric, real critical points, and the like) constitute progress but do not settle the problem as stated.

References

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