Uniform access to roots for relaxed Newton maps

Definitions and mathematical statement

Let $f\in\mathbb C[z]$ have degree $d\geq1$, with all roots in the unit disk, and let $\alpha$ be a root of multiplicity $m\in\{1,\ldots,d\}$. Use the usual relaxed Newton map \[ N_{h,f}(z)=z-h\frac{f(z)}{f'(z)},\qquad 0<h\leq m, \] viewed as a rational map after cancellation of removable factors; in particular $N_{h,f}(\alpha)=\alpha$. Let $A_h^*(\alpha)$ be the connected component containing $\alpha$ of the basin $\{z:N_{h,f}^{\circ n}(z)\to\alpha\}$. Define the common immediate-basin set \[ B(f,\alpha)=\bigcap_{0<h\leq m}A_h^*(\alpha),\qquad C_R=\{z\in\mathbb C:|z|=R\}. \] Prove that there is an absolute constant $c>0$, independent of $f,d,\alpha,m,R$, such that for every such polynomial and root and every real $R\geq3$, the set $C_R\cap B(f,\alpha)$ contains arcs of total Euclidean arclength at least \[ \frac{2\pi R}{cd}. \] A literal arc formulation is the existence of pairwise disjoint arcs $J_j\subseteq C_R\cap B(f,\alpha)$, finite or countably many, with $\sum_j\operatorname{length}(J_j)\geq2\pi R/(cd)$. The same points on these arcs must lie in the immediate basin for every real relaxation parameter in the whole interval $(0,m]$; separate large arcs depending on $h$ would not suffice. The supplied item does not define $N_{h,f}$ explicitly, so its normalization should be checked against the usual formula above.

Short English statement

For every root of a polynomial with roots in the unit disk, find uniformly large arcs on every circle of radius at least three that lie in its immediate basin for all relaxation parameters $0<h\leq m$ simultaneously. The arc fraction should be bounded below by an absolute constant times $1/d$.

Sources

• Scott Sutherland, 1992 problem list, item C2, as collected in Stony Brook, Open Problems in Dynamical Systems, https://www.math.stonybrook.edu/open-problems-dynamical-systems; accessed 2026-07-29, with source-TeX and PDF-text extraction recorded in the provenance.
• The supplied research summary mentions Smale's alpha theory, hybrid Newton methods, and large seeds as related context, but supplies no theorem citation establishing this bound.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $5300071$, source code catalogue entry, Uniform access to roots for relaxed Newton maps. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.