The dual Smale mean value conjecture, formulated by Dubinin and Sugawa (2009) and independently by Ng, asserts that every complex polynomial normalized by f(0)=0f(0)=0 and f′(0)=1f'(0)=1 has a critical point ζ\zeta (a zero of f′f') with ∣f(ζ)/ζ∣≥1/n|f(\zeta)/\zeta|\ge1/n, where nn is the degree. The constant 1/n1/n is sharp and is attained when all n−1n-1 critical points coalesce at one point. The conjecture is proved for degrees n≤7n\le7, for conservative polynomials, for odd polynomials of any degree with nonzero linear term (and, more generally, for polynomials invariant under a root-of-unity symmetry), and for polynomials whose critical points lie on a single ray, while the best bound valid for all polynomials, due to Dubinin, is a factor nn weaker than the conjectured constant. The method that settled n≤7n\le7, selecting a critical point of minimal modulus, provably fails from degree 8 on. The problem is to prove the conjecture for arbitrary degree — the open core being unrestricted polynomials of degree n≥8n\ge8 — or to refute it by an explicit counterexample.

Background

In 1981, Steve Smale, studying the complexity of Newton-type methods for finding roots of complex polynomials, proved a quantitative mean value bound: if PP is a complex polynomial of degree n≥2n\ge2 and zz is a point with P′(z)≠0P'(z)\ne0, then among the critical points of PP — the zeros ζ\zeta of the derivative P′P', which number n−1n-1 when counted with multiplicity — there is one with ∣(P(z)−P(ζ))/(z−ζ)∣≤4∣P′(z)∣|(P(z)-P(\zeta))/(z-\zeta)|\le4|P'(z)|. Smale conjectured that the constant 4 can be replaced by 1−1/n1-1/n; the polynomial z+cznz+cz^n shows no smaller constant can work, and this classical conjecture remains open (Beardon, Minda and Ng (2002) survey the background and partial estimates).

By affine changes of variable the general statement reduces to the class Pn\mathcal P_n of normalized polynomials f(z)=z+c2z2+⋯+cnznf(z)=z+c_2z^2+\cdots+c_nz^n with cn≠0c_n\ne0, that is, f(0)=0f(0)=0 and f′(0)=1f'(0)=1. Writing T(f)=min⁡f′(ζ)=0∣f(ζ)ζ∣,S(f)=max⁡f′(ζ)=0∣f(ζ)ζ∣,T(f)=\min_{f'(\zeta)=0}\left|\frac{f(\zeta)}{\zeta}\right|,\qquad S(f)=\max_{f'(\zeta)=0}\left|\frac{f(\zeta)}{\zeta}\right|, Smale's conjecture reads max⁡f∈PnT(f)=1−1/n\max_{f\in\mathcal P_n}T(f)=1-1/n: some critical point makes the difference quotient from the origin small. The dual problem, formulated by Dubinin and Sugawa (2009) and independently by Ng, asks the reverse question: is there always a critical point where the quotient is large? The conjecture is min⁡f∈PnS(f)=1/n\min_{f\in\mathcal P_n}S(f)=1/n for every n≥2n\ge2; equivalently, every f∈Pnf\in\mathcal P_n has a critical point ζ\zeta with ∣f(ζ)/ζ∣≥1/n|f(\zeta)/\zeta|\ge1/n. The same translation-and-scaling reduction shows this is equivalent to the invariant form: for every polynomial PP of degree n≥2n\ge2 and every noncritical point zz there is a critical point ζ\zeta with ∣(P(z)−P(ζ))/(z−ζ)∣≥∣P′(z)∣/n|(P(z)-P(\zeta))/(z-\zeta)|\ge|P'(z)|/n, a lower mean value bound. The constant cannot be improved: for the mono-critical polynomial f(z)=(1−(1−az)n)/(na)f(z)=(1-(1-az)^n)/(na), a≠0a\ne0, all n−1n-1 critical points coalesce at ζ=1/a\zeta=1/a and f(ζ)/ζ=1/nf(\zeta)/\zeta=1/n exactly.

Partial results. Tischler (1989) proved the conjecture for conservative polynomials — those for which f(ζ)/ζf(\zeta)/\zeta takes the same value at all critical points — and established the stronger two-sided bound ∣f(ζ)/ζ−1/2∣≤1/2−1/n|f(\zeta)/\zeta-1/2|\le1/2-1/n for 2≤n≤42\le n\le4; Tyson (2005) showed this stronger bound fails for n≥5n\ge5, refuting only the strengthening and not the dual conjecture itself. Dubinin and Sugawa (2009) proved the first general bound, of exponential type in nn, later refined by Ng and Zhang (2016), and Dubinin (2019) obtained the best known universal bound: for every degree-nn polynomial PP and every point zz there is a critical ζ\zeta with ∣(P(z)−P(ζ))/(z−ζ)∣≥(1/n2)∣P′(z)∣|(P(z)-P(\zeta))/(z-\zeta)|\ge(1/n^2)|P'(z)| — still a factor nn below the conjectured constant. Degree by degree, Hinkkanen, Kayumov and Khammatova proved the conjecture for n=5n=5 and n=6n=6 (2019) and for n=7n=7 (2025). Their method selects a critical point of minimal modulus, scales it to 11, and uses the integral representation f(w)/w=∫01∏j=1n−1(1−tw/ζj) dtf(w)/w=\int_0^1\prod_{j=1}^{n-1}(1-tw/\zeta_j)\,dt over the critical points ζj\zeta_j, reducing the problem to sharp multivariate trigonometric inequalities on a torus, verified in part by computer algebra. However, the 2019 paper also exhibits an explicit degree-8 polynomial, with critical points 11 and six copies of e−2πi/5e^{-2\pi i/5} (all seven of modulus 11), for which the critical point ζ=1\zeta=1 gives ∣f(ζ)/ζ∣≈0.0873<1/8|f(\zeta)/\zeta|\approx0.0873<1/8, while a critical point of the sextuple at ζ=e−2πi/5\zeta=e^{-2\pi i/5} gives ≈0.1384>1/8\approx0.1384>1/8: the seven critical points tie for minimal modulus, and taking an arbitrary such point can return a value below 1/n1/n, so the selection heuristic that settled n≤7n\le7 provably fails as a universal method from degree 8 on. Most recently, Tang (2025) verified the conjecture for all odd polynomials (those with P(−z)≡−P(z)P(-z)\equiv-P(z), hence of odd degree) with nonzero linear term, and more generally for polynomials satisfying P(λz)≡λP(z)P(\lambda z)\equiv\lambda P(z) for a root of unity λ\lambda; Kayumov and Khammatova (2026) proposed a stronger conjectured inequality and verified it when all critical points lie on a single ray; and Khammatova (2026) proved an analog of the inequality for kk-symmetric polynomials with 3k3k critical points. What remains is the general case: unrestricted polynomials of degree n≥8n\ge8, and with it a proof or refutation valid for all degrees.

Problem Statement

Prove or refute the dual Smale mean value conjecture: for every integer n≥2n\ge2 and every polynomial f(z)=z+c2z2+⋯+cnznf(z)=z+c_2z^2+\cdots+c_nz^n with cn≠0c_n\ne0, there exists a critical point ζ\zeta (a zero of f′f') such that ∣f(ζ)ζ∣≥1n.\left|\frac{f(\zeta)}{\zeta}\right|\ge\frac{1}{n}. Equivalently, min⁡f∈PnS(f)=1/n\min_{f\in\mathcal P_n}S(f)=1/n for every nn, where S(f)=max⁡f′(ζ)=0∣f(ζ)/ζ∣S(f)=\max_{f'(\zeta)=0}|f(\zeta)/\zeta|; or in translation-invariant form, for every complex polynomial PP of degree n≥2n\ge2 and every noncritical point zz there is a critical point ζ\zeta with ∣(P(z)−P(ζ))/(z−ζ)∣≥∣P′(z)∣/n|(P(z)-P(\zeta))/(z-\zeta)|\ge|P'(z)|/n.

Degrees n≤7n\le7 are already settled, as are the special classes described in the background (conservative polynomials, odd polynomials with nonzero linear term and, more generally, polynomials with root-of-unity symmetry P(λz)≡λP(z)P(\lambda z)\equiv\lambda P(z), and polynomials with all critical points on one ray). The open core is unrestricted polynomials of degree n≥8n\ge8. A refutation consists of a single explicit normalized polynomial of some degree n≥8n\ge8 for which every critical point ζ\zeta satisfies ∣f(ζ)/ζ∣<1/n|f(\zeta)/\zeta|<1/n. The equality cases are known for the settled degrees: for n≤7n\le7, S(f)=1/nS(f)=1/n holds only for the mono-critical family f(z)=(1−(1−az)n)/(na)f(z)=(1-(1-az)^n)/(na), a∈C∖{0}a\in\mathbb C\setminus\{0\}, and the same is expected in general.

References

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