Smale's Mean Value Conjecture: the Sharp Degree-Dependent Constant $K(n)$ and Its Extremal Polynomials

For a complex polynomial $p$ of degree $n\ge 2$ and a point $x$ with $p'(x)\ne 0$, let $S(p,x)$ denote the minimum of $|(p(\zeta)-p(x))/((\zeta-x)p'(x))|$ over the critical points $\zeta$ of $p$, and let $K(n)$ be the supremum of $S(p,x)$ over all degree-$n$ polynomials. Smale's mean value conjecture (1981) asserts $K(n)=(n-1)/n$ for every $n$, the bound being exactly attained by $z^n-\lambda z$ at the origin; it is rigorously settled only for $n\le 4$, while the best published upper bound for $n\ge 8$ is $K(n)<4-2.263/\sqrt n$. The problem is to prove or refute $K(n)=(n-1)/n$ for all $n$; whether the extremal is unique up to affine equivalence (the orbit of $z^n+z$) is a related open question from Crane's 2006 study, and strict monotonicity $K(n+1)>K(n)$ is an immediate consequence of the conjectured value.

Background

For a real differentiable function, the mean value theorem produces an intermediate point where the derivative equals the slope of the chord. No intermediate-point statement of this kind survives when a complex polynomial $p$ is viewed as a map of the plane to itself, but Stephen Smale, analyzing Newton's method for polynomial root-finding in his 1981 paper on the fundamental theorem of algebra and complexity theory, proved a finite substitute: if $x$ is not a critical point (a critical point is a zero of the derivative $p'$), then among the finitely many critical points $\zeta$ of $p$ there is one with $|p(\zeta)-p(x)|/|\zeta-x|\le 4|p'(x)|$. The constant 4 comes from the Koebe one-quarter theorem, a classical covering theorem for univalent (that is, injective holomorphic) functions. Writing

$S(p,x)=\min_{p'(\zeta)=0}\left|\frac{p(\zeta)-p(x)}{(\zeta-x)\,p'(x)}\right|,\qquad K(n)=\sup\{S(p,x):\deg p=n,\ p'(x)\ne 0\},$

Smale conjectured that the optimal constant valid for all degrees is 1 and, more sharply, that $K(n)=(n-1)/n$ for every $n\ge 2$. This is the strongest possible claim, because for $p(z)=z^n-\lambda z$ ($\lambda\ne 0$) at $x=0$ every critical point gives the ratio exactly $(n-1)/n$, so $K(n)\ge(n-1)/n$ with equality conjectured.

Equivalently, the conjecture asserts that for every complex polynomial $p$ of degree $n\ge 2$ and every $x$ with $p'(x)\ne 0$, at least one critical point $\zeta$ satisfies $|p(\zeta)-p(x)|\le\frac{n-1}{n}\,|p'(x)|\,|\zeta-x|$. Under the standard normalization $P(0)=0$, $P'(0)=1$ (the quantity $S$ is invariant under pre- and post-composition with complex-affine maps $z\mapsto\alpha z+\beta$, $w\mapsto\gamma w+\delta$), it takes the form $\min_j |P(b_j)/b_j|\le(n-1)/n$, where $b_1,\dots,b_{n-1}$ are the critical points. The problem has been open since 1981.

Known results fall far short of $(n-1)/n$. The conjecture is proved only for degrees $n=2,3,4$ (recorded in the Rahman–Schmeisser monograph on the analytic theory of polynomials). For general degree the best upper bounds are: $K(n)\le 4^{(n-2)/(n-1)}$ (Beardon, Minda and Ng, using the hyperbolic metric, the conformally invariant metric of plane domains), $K(n)\le 4(n-1)/(n+1)$ (Conte, Fujikawa and Lakic, via the Bieberbach coefficient inequality for univalent functions), a sharpened combination of the two (Fujikawa and Sugawa), and $K(n)<4-2.263/\sqrt n$ for $n\ge 8$ (Crane), still the best bound for large $n$. All of these tend to 4 as $n\to\infty$. For restricted classes much more is known: the uniform constant 2 holds for odd polynomials with nonzero linear term (Ng); the constant $2/3$ holds when all critical points are real, $1/2$ when they lie in a small sector, with further results when they lie on a union of two rays (Hinkkanen–Kayumov). A stronger closed-set form proposed by Tischler was refuted by Tyson — a warning that plausible strengthenings of Smale's statement can fail even when the original conjecture stands.

Crane's 2006 study of the extremal polynomials of the conjecture proved that extremizers exist (after normalization, the supremum defining $K(N)$ is attained) and established a geometric property of them, and he reformulated the question algebraically: fixing $N$ and normalizing, the condition that a polynomial together with a base point achieve $S(p,x)>c$ translates into symmetric polynomial equations in the critical points, cut out by an irreducible symmetric polynomial $Q_N$ on an algebraic hypersurface $X_N$; real quantifier elimination (an exact symbolic algorithm for deciding systems of polynomial equations and inequalities over the reals) then makes $K(N)$ decidable for each fixed $N$, but only infeasibly slowly as $N$ grows. His paper leaves open: the value $K(N)$; uniqueness of the candidate extremal $\chi_N(z)=z^N+z$ up to affine equivalence; the strict monotonicity hypotheses $K(N+1)>K(N)$ assumed for some of his stronger conclusions; and explicit formulas for $Q_N$. Strict monotonicity follows from the conjectured value since $\frac{n}{n+1}>\frac{n-1}{n}$.

Problem Statement

Let $p$ be a complex polynomial of degree $n\ge 2$, let $x\in\mathbb C$ satisfy $p'(x)\ne 0$, and let $\zeta_1,\dots,\zeta_{n-1}$ be the critical points of $p$ (the zeros of $p'$, counted with multiplicity). Define

$S(p,x)=\min_{1\le j\le n-1}\left|\frac{p(\zeta_j)-p(x)}{(\zeta_j-x)\,p'(x)}\right|,\qquad K(n)=\sup\{S(p,x):\deg p=n,\ p'(x)\ne 0\}.$

Prove or refute Smale's mean value conjecture (Smale, 1981; posed as the sharp open question in Crane's 2006 study of the extremal polynomials):

$K(n)=\frac{n-1}{n}\quad\text{for every }n\ge 2.$

Since $K(n)\ge(n-1)/n$ is attained by $p(z)=z^n-\lambda z$ at $x=0$ (any $\lambda\ne 0$), the conjecture is equivalent to the statement that for every complex polynomial $p$ of degree $n\ge 2$ and every $x$ with $p'(x)\ne 0$, at least one critical point $\zeta$ of $p$ satisfies $|p(\zeta)-p(x)|\le\frac{n-1}{n}|p'(x)|\,|\zeta-x|$.

The conjecture is proved for degrees $n=2,3,4$ and remains open for $n\ge 5$ (Rahman–Schmeisser 2002; Ng–Zhang 2016). The conjectured value $K(n)=(n-1)/n$ immediately implies strict monotonicity $K(n+1)>K(n)$ for all $n$ (a hypothesis assumed for some stronger conclusions in Crane's study). A related open companion question from Crane's study of the extremal polynomials is whether every extremal is affine-equivalent to $\chi_n(z)=z^n+z$, that is, whether equality $K(n)=(n-1)/n$ is attained, up to pre- and post-composition with complex-affine maps $z\mapsto\alpha z+\beta$, $w\mapsto\gamma w+\delta$ (under which $S$ is invariant), only by the orbit of $z^n-\lambda z$, $\lambda\ne 0$, with base point $0$; in Crane's program this uniqueness is conjectured.

References

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