Smale's mean value conjecture for complex polynomials

Smale's mean value conjecture, posed in 1981 in connection with the complexity analysis of Newton's method for polynomial root-finding, asserts that for every complex polynomial $p$ of degree $n\ge 2$ and every point $z$ with $p'(z)\ne 0$, there is a critical point $\zeta$ (a zero of $p'$) with $|p(\zeta)-p(z)|/|\zeta-z|\le (1-1/n)|p'(z)|$; equivalently, every polynomial $P$ of degree $n\ge 2$ with $P(0)=0$ and $P'(0)=1$ has a critical point $\zeta$ with $|P(\zeta)/\zeta|\le 1-1/n$. The constant $1-1/n$ is sharp, attained by the binomials $a_1 z+a_n z^n$. Smale proved the bound with constant $4$; despite successive improvements — hyperbolic-metric and Bieberbach-coefficient bounds of the form $4-O(1/n)$, and Crane's 2007 bound $M_n<4-2.263/\sqrt n$ for $n\ge 8$ — all known bounds stay near $4$ as $n$ grows, while the conjectured optimal constants tend to $1$. The conjecture is proved only for degrees $n\le 5$ and for special classes of polynomials. The problem remains open: the surviving open core is the conjecture for general polynomials (in particular every degree $n\ge 6$) and the associated asymptotic question of whether the optimal constants behave as $1-1/n$ rather than $4-O(1/\sqrt n)$. The task is to prove or refute the conjecture, i.e. the assertion that $M_n=(n-1)/n$ for every degree $n\ge 2$.

Background

For a polynomial $p$ with complex coefficients, a critical point is a point $\zeta$ with $p'(\zeta)=0$, and $p(\zeta)$ is the corresponding critical value. The real mean value theorem has no direct complex analogue: given a point $z$, there is in general no $\zeta$ at which $p'$ equals the chord gradient $|p(\zeta)-p(z)|/|\zeta-z|$. In his 1981 study of the cost of approximating zeros of a polynomial by Newton's method, Smale proved the following complex substitute: for every polynomial $p$ of degree $n\ge 2$ and every point $z$ with $p'(z)\ne 0$, there exists a critical point $\zeta$ with $\frac{|p(\zeta)-p(z)|}{|\zeta-z|}\le 4\,|p'(z)|.$ Smale observed (his Problem 1A) that the constant $4$ might be reducible to $1$, and the degree-sharp value $1-1/n$ is the natural target: if $M_n$ denotes the optimal constant for polynomials of degree $n$ (the supremum over all degree-$n$ polynomials $p$ and non-critical points $z$ of the minimum of the above ratio over the critical points of $p$), then the binomials $p(z)=a_1 z+a_n z^n$ with $a_1a_n\ne 0$ at $z=0$ show that $M_n\ge(n-1)/n$, since every critical point $\zeta$ of such a binomial satisfies $|p(\zeta)/\zeta|=((n-1)/n)|p'(0)|$ exactly. Smale's mean value conjecture is the assertion that these examples are extremal: $M_n=(n-1)/n$ for every $n\ge 2$. By affine changes of the variable and of the polynomial, the question has an equivalent normalized form: for every polynomial $P$ of degree $n\ge 2$ with $P(0)=0$ and $P'(0)=1$, there is a critical point $b$ with $|P(b)/b|\le 1-1/n$. The gap between what is known and what is conjectured is striking. Smale's original argument uses a classical coefficient estimate of Loewner for univalent (one-to-one holomorphic) functions. Later bounds came from geometric function theory: Beardon, Minda and Ng (2002) used the hyperbolic metric of suitable punctured domains (the natural conformally invariant metric on plane domains other than the whole plane) to get $M_n\le 4^{(n-2)/(n-1)}$; Conte, Fujikawa and Lakic (2007) used Bieberbach's coefficient inequality $|a_2|\le 2$ for univalent functions to get $M_n\le 4(n-1)/(n+1)$; Fujikawa and Sugawa (2006) combined the two methods; and Dubinin (2006) improved the hyperbolic-metric bound by a further factor $(n-1)/n$, that is, $M_n\le \frac{n-1}{n}\,4^{(n-2)/(n-1)}$. All of these are $4-O(1/n)$. Crane (2006) proved that extremal polynomials for the degree-$n$ problem exist and satisfy a geometric property, and Crane (2007) obtained the first bound that departs from $4$ at a square-root rate: $M_n<4-2.263/\sqrt n$ for $n\ge 8$, a proof that lower-bounds the logarithmic capacity (the conformally invariant measure of the size of a compact set arising in planar potential theory) of certain configurations using numerically validated estimates. Every known bound thus approaches $4$ as $n$ grows, whereas the conjectured optimal constants approach $1$. The conjecture has been confirmed in restricted settings. It is proved for degrees $n\le 5$ (the case $n=5$ is due to Sendov and Marinov, 2006), and Sendov and Marinov (2007) reported a numerical verification for $n\le 10$. It holds for polynomials whose zeros all lie on a circle centered at the origin and for conservative polynomials (those whose critical values are proportional to the corresponding critical points), both by Tischler (1989); however, a stronger pointwise form of the inequality that Tischler proposed was disproved by Tyson (2005), so only the minimum-over-critical-points form survives. Odd polynomials with zero constant term admit the bound with constant $2$ (Ng, 2003), and polynomials with real critical points satisfy the conjecture (Hinkkanen and Kayumov, 2010). Adjacent but distinct questions — the dual conjecture of Dubinin–Sugawa (2009) asking for a critical point with $|P(\zeta)/\zeta|\ge 1/n$, analogs for finite Blaschke products (rational self-maps of the unit disk built from polynomials whose zeros all lie on the unit circle; Ng–Zhang 2016, Dubinin 2024), a Möbius-invariant version for rational maps (Crane, 2004), and a dynamical strengthening verified for degrees $2$ and $3$ (Miles-Leighton and Pilgrim, 2012) — form an active surrounding literature but leave the primal inequality untouched.

Problem Statement

Prove or refute Smale's mean value conjecture in its sharp, degree-dependent form. For each integer $n\ge 2$, let $M_n=\sup_{p,z}\;\min_{\zeta:\,p'(\zeta)=0}\frac{|p(\zeta)-p(z)|}{|\zeta-z|\,|p'(z)|},$ where the supremum runs over all complex polynomials $p$ of degree exactly $n$ and all points $z$ with $p'(z)\ne 0$, and the minimum runs over the critical points of $p$. Prove, or disprove, that $M_n=\frac{n-1}{n}\quad\text{for every }n\ge 2,$ that is, that for every complex polynomial $p$ of degree $n\ge 2$ and every point $z$ with $p'(z)\ne 0$ there exists a critical point $\zeta$ of $p$ such that $\frac{|p(\zeta)-p(z)|}{|\zeta-z|}\le\left(1-\frac{1}{n}\right)|p'(z)|.$ Equivalently, in normalized form: every polynomial $P$ of degree $n\ge 2$ with $P(0)=0$ and $P'(0)=1$ has a critical point $b$ with $|P(b)/b|\le 1-1/n$. The constant $(n-1)/n$ cannot be lowered: for every binomial $p(z)=a_1 z+a_n z^n$ with $a_1a_n\ne 0$ and $z=0$, every critical point gives exactly the value $(n-1)/n$. A complete answer determines whether $M_n=(n-1)/n$ holds for every $n\ge 2$, and in particular settles the asymptotic question left open by Crane (2007): whether the true behavior is the conjectured $1-1/n$, rather than the best known $4-O(1/\sqrt n)$. The weaker assertion that the degree-independent constant can be reduced from Smale's $4$ to $1$ (Smale's original Problem 1A) is subsumed by, and implied by, the sharp conjecture; a proof of a weaker bound such as $M\le 1$ or an improved $4-c/\sqrt n$ estimate is progress but not a resolution of the stated question.

References

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