From boundary geometry to counterexamples in Smale's mean value problem

Tom Adamczewski; Matthew J. Colbrook

Home/Archive/arXiv

arXiv · 2610.36158

From boundary geometry to counterexamples in Smale's mean value problem

Tom Adamczewski·Matthew J. Colbrook

View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper provides a human digestion of the recent counterexample to Smale's mean value problem found by GPT-6 Astra. We give a geometric mechanism for constructing counterexamples. An outward normal that re-enters an analytic Jordan domain, subject to a curvature condition, yields a holomorphic model whose normalized divided differences exceed one in modulus on the boundary. Polynomial perturbations then concentrate every critical point near the boundary whilst preserving the strict inequality. This turns boundary geometry into counterexamples in every sufficiently large degree. We also locate these critical points to first order and show that their limiting distribution is the logarithmic equilibrium measure on the boundary. We end with an explicit example of degree $29\,984$, verified by interval arithmetic.

Explore related subjects

math.CV

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Tom Adamczewski, Matthew J. Colbrook. 2026-10-26. From boundary geometry to counterexamples in Smale's mean value problem. https://arxiv.org/abs/2610.36158

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On quaternionic analysis and a certain generalized fractal-fractional $ψ$-Fueter operator

This paper introduces a fractal $ψ$-Fueter operator in the quaternionic context inspired by the concepts of proportional derivative and Hausdorff derivative ({fractal derivative}) of a function with respect to a fractal measure. Moreover, we establish the corresponding Stokes and Borel-Pompeiu formulas associated to this generalized fractal $ψ$-Fueter operator with respect to a vector valued function induced by truncated exponential functions.

math.CV↗

arXiv2026

Modulus estimates and cavitation in higher dimensions

We explore the phenomenon of cavitation in higher-dimensional elasticity, defining it as the mapping of a punctured ball onto a non-degenerate ring domain. Crucially, for the class of locally quasiconformal mappings (or more general mappings) defined on the punctured ball $0<|x|<1$ in $\mathbb R^n$ that we examine, cavitation is equivalent to a failure of continuous extension to the origin. While existing modulus estimates prove insufficient for reliably detecting cavitation in this setting, our study establishes refined modulus bounds. This is achieved by introducing a novel directional dilatation which, in conjunction with the known angular dilatation, overcomes the limitations of previous methods. We illustrate our theoretical findings with several examples that demonstrate both cavitation occurrence and its absence.

math.CV↗

arXiv2026

Fractional Besov-Sobolev Spaces on Chord-arc Curves

Let $Γ$ be a bounded rectifiable Jordan curve, with complementary domains $Ω_i$ and $Ω_e$. For $1<p<\infty$ and $0<s<1$, we study the spaces $\mathcal B_{p,p}^s(Ω)$ of harmonic functions satisfying \[ \iint_Ω|\nabla u(z)|^p \operatorname{dist}(z,Γ)^{(1-s)p-1}\,dA(z)<\infty, \qquad Ω\in\{Ω_i,Ω_e\}. \] When $Γ$ is the unit circle, the traces of these two spaces coincide with the fractional Besov-Sobolev space $B^s_{p,p}(Γ)$. The equivalence of boundary seminorms and harmonic energies on the circle raises a geometric question: which rough curves preserve these estimates? For every radial-Lipschitz curve $Γ$, we prove \[ \operatorname{Tr}_Γ\mathcal B_{p,p}^s(Ω_i) =B_{p,p}^s(Γ) =\operatorname{Tr}_Γ\mathcal B_{p,p}^s(Ω_e) \] whenever either $1 < p < 2$ and $2/p - 1 \leq s < 1$, $2 < p < \infty$ and $0 < s \leq 1/2 + 1/p$ or $p=2$ and $0<s<1$. For arbitrary chord-arc curves, the same trace identities hold when $1/p\le s<1/2+1/(2p)$. Taking $s=1-1/p$ yields an almost-Dirichlet principle for the unweighted $p$-energy on radial-Lipschitz domains when $3/2\le p\le4$. Here, the upper bound $4$ is sharp. In particular, every Riemann map $g:Ω_i\to\mathbb D_i$ from a bounded radial-Lipschitz domain satisfies $g'\in L^4(Ω_i)$, and the exponent $4$ is optimal uniformly over this class. This gives a sharp endpoint counterpart of Brennan's integrability conclusion for radial-Lipschitz domains.

math.CV↗