Dual Smale's mean value conjecture (open degrees $n\ge8$)

The dual Smale mean value conjecture, formulated by Dubinin and Sugawa (2009) and independently by Ng, asserts that every complex polynomial $f$ of degree $n\ge2$ normalized by $f(0)=0$ and $f'(0)=1$ has a critical point $\zeta$ (a zero of $f'$) with $|f(\zeta)/\zeta|\ge1/n$; equivalently $\Lambda_n=\min_{f\in\mathcal P_n}\max_{f'(\zeta)=0}|f(\zeta)/\zeta|=1/n$, the sharp constant being attained by $f(z)=(1-(1-az)^n)/(na)$. The conjecture is proved for degrees $n\le7$ (the case $n=7$ by Hinkkanen, Kayumov, and Khammatova) and for special classes at all degrees (conservative polynomials, real critical points, critical points on one ray, odd polynomials), but it remains open for arbitrary polynomials of every degree $n\ge8$; the best general lower bound, due to Dubinin, is a factor $n$ below the conjectured constant.

Background

A critical point of a complex polynomial $P$ is a point $\zeta$ with $P'(\zeta)=0$, and $P(\zeta)$ is the corresponding critical value. Smale's mean value conjecture grew out of his 1981 study of the complexity of Newton's method for polynomial root-finding, where estimates on how close a critical value can lie to a regular value control the convergence of iteration. Smale proved that for every polynomial $P$ of degree $n\ge2$ and every noncritical point $z$ there is a critical point $\zeta$ with $|(P(z)-P(\zeta))/(z-\zeta)|\le4|P'(z)|$, and conjectured that the constant 4 can be replaced by 1 (the optimal constant is $1-1/n$). In the normalized class $\mathcal P_n=\{f(z)=z+\sum_{k=2}^nc_kz^k:\ c_n\ne0\}$ of degree-$n$ polynomials with $f(0)=0$ and $f'(0)=1$, writing $T(f)=\min_{f'(\zeta)=0}|f(\zeta)/\zeta|$, the classical conjecture reads $\max_{f\in\mathcal P_n}T(f)=1-1/n$; it remains open.

The dual Smale mean value conjecture, formulated by Dubinin and Sugawa (2009) and independently by Ng, reverses the order of the extremum: writing $S(f)=\max_{f'(\zeta)=0}|f(\zeta)/\zeta|$, it asserts that

$$\min_{f\in\mathcal P_n}S(f)=\frac1n\quad\text{for every }n\ge2,$$

that is, every normalized degree-$n$ polynomial has some critical point $\zeta$ with $|f(\zeta)/\zeta|\ge1/n$. The constant is sharp: $f(z)=(1-(1-az)^n)/(na)$ with $a\ne0$ has the single critical point $\zeta=1/a$ and $|f(\zeta)/\zeta|=1/n$.

The dual conjecture is known for degrees $2\le n\le4$ (Tischler 1989, via a strong form whose extension to $n\ge5$ was refuted by Tyson's 2005 counterexamples, which leave the settled cases $n\le4$ intact), for $n=5,6$ (Hinkkanen, Kayumov, Khammatova 2019), and for $n=7$ (Hinkkanen, Kayumov, Khammatova 2025, who also classify the near-extremal configurations). It is also known at all degrees for special classes: conservative polynomials — those for which $f(\zeta)/\zeta$ takes the same value at every critical point (Tischler 1989) — polynomials with real critical points (Hinkkanen and Kayumov 2010), polynomials whose critical points all lie on a single ray (Kayumov and Khammatova 2026, via a stronger conjectured inequality that implies the dual conjecture), and odd polynomials with nonzero linear term (Tang 2025). For general polynomials the best universal lower bound is Dubinin's: for every degree-$n$ polynomial $P$ and every point $z$ there is a critical point $\zeta$ with $|(P(z)-P(\zeta))/(z-\zeta)|\ge(1/n^2)|P'(z)|$, still a factor $n$ short of the conjectured sharp constant.

The proofs for $n\le7$ rest on one device: choose a critical point $\zeta$ of minimal modulus among the $n-1$ critical points, rescale so that $\zeta=1$ and all other critical points satisfy $|\zeta_j|\ge1$, and use the integral representation $f(w)/w=\int_0^1\prod_{j=1}^{n-1}(1-tw/\zeta_j)\,dt$ (valid since $f'(0)=1$) to reduce the extremal problem to a sharp inequality for a one-dimensional boundary integral — in the case $n=7$, the inequality $\int_0^1(1-t)\prod_{j=1}^{5}(1-z_jt)\,dt\ge1/7$ for points $|z_j|=1$. The source paper records that for $n>7$ this minimal-modulus approach 'does not work', so progress beyond $n=7$ requires a modified or new method. Adjacent lines of work — Smale-type mean value questions for finite Blaschke products (finite products of automorphisms of the unit disk, a rational analogue; Ng and Zhang 2016, Dubinin 2024) and C*-algebraic reformulations of both conjectures (Krishna 2022) — do not resolve the polynomial dual conjecture: the Ng–Zhang dual Blaschke result improves the Dubinin–Sugawa polynomial lower bound, but the improved bound remains of order $1/n^2$.

Problem Statement

Prove or refute the dual Smale mean value conjecture: for every integer $n\ge2$ and every complex polynomial $f(z)=z+\sum_{k=2}^{n}c_kz^k$ with $c_n\ne0$ (equivalently, every degree-$n$ polynomial with $f(0)=0$ and $f'(0)=1$), there exists a critical point $\zeta$ (a zero of $f'$; necessarily $\zeta\ne0$ since $f'(0)=1$) such that

$$\left|\frac{f(\zeta)}{\zeta}\right|\ge\frac1n.$$

Equivalently, determine $\Lambda_n=\min_{f\in\mathcal P_n}\max_{f'(\zeta)=0}|f(\zeta)/\zeta|$ and prove $\Lambda_n=1/n$ for every $n$. The constant is sharp, attained by the family $f(z)=(1-(1-az)^n)/(na)$, $a\ne0$; the source paper classifies the equality cases for $n=7$, where only this family attains equality. Since the cases $2\le n\le7$ are settled, the open content is arbitrary polynomials of degree $n\ge8$; the smallest open case is $n=8$. The problem is resolved by a proof covering all remaining degrees (directly or degree by degree), by a proof that the conjectured equality fails, or by an explicit counterexample: a polynomial of some degree $n\ge8$, normalized as above, with $\max_{f'(\zeta)=0}|f(\zeta)/\zeta|<1/n$.

References

1. Aimo Hinkkanen, Ilgiz R. Kayumov, Diana M. Khammatova, Dual Smale's mean value conjecture for $n=7$, Constructive Approximation 61(3):603–613 (2025), DOI 10.1007/s00365-025-09709-4; preprint arXiv:2303.17586, https://arxiv.org/abs/2303.17586
2. Ilgiz R. Kayumov, Diana M. Khammatova, On an inequality related to Dual Smale's mean value conjecture, Journal of Mathematical Analysis and Applications 556(1):130108 (2026), DOI 10.1016/j.jmaa.2025.130108, https://doi.org/10.1016/j.jmaa.2025.130108
3. Quanyu Tang, Dual Smale's mean value conjecture for odd polynomials, arXiv:2510.16875 (2025), https://arxiv.org/abs/2510.16875
4. Aimo Hinkkanen, Ilgiz R. Kayumov, Diana M. Khammatova, Dual Smale's mean value conjecture, Proceedings of the American Mathematical Society 147(12):5227–5237 (2019), DOI 10.1090/proc/14639, https://doi.org/10.1090/proc/14639
5. V. N. Dubinin, T. Sugawa, Dual mean value problem for complex polynomials, Proceedings of the Japan Academy, Series A, Mathematical Sciences 85(9):135–137 (2009), DOI 10.3792/pjaa.85.135, https://doi.org/10.3792/pjaa.85.135
6. David Tischler, Critical points and values of complex polynomials, Journal of Complexity 5(4):438–456 (1989), DOI 10.1016/0885-064X(89)90019-8, https://doi.org/10.1016/0885-064X(89)90019-8
7. Aimo Hinkkanen, Ilgiz R. Kayumov, On critical values of polynomials with real critical points, Constructive Approximation 32(2):385–392 (2010), DOI 10.1007/s00365-009-9079-6, https://doi.org/10.1007/s00365-009-9079-6
8. V. N. Dubinin, On the dual mean-value conjecture for complex polynomials, Mathematical Notes 106(1):133–135 (2019), DOI 10.1134/S0001434619070149, https://doi.org/10.1134/S0001434619070149
9. V. N. Dubinin, Blaschke products and Smale's conjecture on complex polynomials, Mathematical Notes 116(1):252–257 (2024), DOI 10.1134/S0001434624070198, https://doi.org/10.1134/S0001434624070198
10. K. Mahesh Krishna, C*-algebraic Smale Mean Value Conjecture and Dubinin-Sugawa Dual Mean Value Conjecture, arXiv:2206.08154 (2022), https://arxiv.org/abs/2206.08154
11. Stephen Smale, The fundamental theorem of algebra and complexity theory, Bulletin of the American Mathematical Society (New Series) 4(1):1–36 (1981), DOI 10.1090/S0273-0979-1981-14858-8, https://doi.org/10.1090/S0273-0979-1981-14858-8
12. Jeremy T. Tyson, Counterexamples to Tischler's strong form of Smale's Mean Value Conjecture, Bulletin of the London Mathematical Society 37(1):95–106 (2005), DOI 10.1112/S0024609304003613, https://doi.org/10.1112/S0024609304003613
13. Tuen Wai Ng, Smale's mean value conjecture for odd polynomials, Journal of the Australian Mathematical Society 75(3):409–412 (2003), DOI 10.1017/S1446788700008181, https://doi.org/10.1017/S1446788700008181
14. Tuen-Wai Ng, Yongquan Zhang, Smale's mean value conjecture for finite Blaschke products, The Journal of Analysis 24(2):331–345 (2016), DOI 10.1007/s41478-016-0007-4, https://doi.org/10.1007/s41478-016-0007-4