Problem 7 — Give a similar universal estimate of the radius of convergence for multidimensional Newton’s method of…

Definitions and mathematical statement

Problem (Victor Vassiliev, Problem 7). Give a similar universal estimate for the radius of convergence of the multidimensional Newton method of [Shub and Smale1993]. The preceding estimate to which "similar" refers is not included in the statement.

In the conventional analytic square-system setting, let $f:\Omega\subseteq\mathbb K^m\to\mathbb K^m$, $\mathbb K={\mathbb{R}}$ or ${\mathbb{C}}$, have a zero $x^*$ with invertible derivative. Newton iteration is \[ x_{j+1}=x_j-Df(x_j)^{-1}f(x_j) \] whenever the inverse exists. Relative to a specified norm and domain, a guaranteed convergence radius $r$ means that every $x_0$ with $\|x_0-x^*\|<r$ has all iterates defined in the domain and converging to $x^*$. The request is for a universal lower estimate for such a radius in terms of the intended data. The original data, allowed function class, field, norm, domain, convergence rate, and dependence on dimension or degree are not supplied; this conventional model does not resolve those missing choices.

The research summary attributes a solution to Shub--Smale $\alpha$-theory (1993), describes later refinements and implementations including alphaCertified and interval-arithmetic certification, and proposes the numerical bound $1/(2\gamma(f,x^*))$ or an $\alpha$-criterion threshold, analogous to an earlier $d/(2n-1)$ estimate. In a usual analytic convention, \[ \gamma(f,x^*)=\sup_{k\geq2} \left\|Df(x^*)^{-1}\frac{D^k f(x^*)}{k!}\right\|^{1/(k-1)}. \] The quoted numerical constant and its required hypotheses are not established by the supplied record and are not asserted as a universal theorem here. The summary does not define the $d,n$ of its comparison, state a precise $\alpha$-criterion, or reconcile these data with the source's intended meaning of "similar.

Short English statement

Seek a universal guaranteed convergence neighborhood for multidimensional Newton iteration, analogous to an omitted preceding estimate. The registry labels the problem solved by $\alpha$-theory, but its proposed numerical $\gamma$-bound and the intended comparison require verification.

Sources

• Local UnsolvedMath registry, permanent ID 10800018, source code catalogue entry; proposer Victor Vassiliev, Problem 7.
• Vassiliev, A Few Problems on Monodromy and Discriminants, Problem 7, with the supplied citation [Shub and Smale1993]. Full bibliographic details of that citation are not provided. Supplied article URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/; recorded access 2026-07-29; extraction html-structure. The article contains eighteen explicitly labeled theorem-style problem environments.
• The supplied record has status solved and SOLVED-IN-LITERATURE, crediting Shub--Smale $\alpha$-theory and subsequent certification work. Its generic background instead describes the source as presenting an open problem and retains NEEDS_REVIEW for a current primary-literature audit. The specific numerical bound and claimed resolution are supplied assessments, not independently verified facts. Redistribution terms for the publicly accessible author/publisher copy also require review. The registry assigns L2 whereas background records default L3.