Algorithm for an optimal/minimal Carleman atlas

Develop an explicit algorithm that, given a nonlinear vector field on a manifold and the Carleman embedding using polynomial observables, constructs an optimal and/or minimal Carleman linearization atlas: a finite collection of Carleman charts (with specified centers and convergence radii) that cover the manifold while satisfying analyticity constraints, and that minimizes the number of charts required for the given vector field and manifold.

Background

The paper introduces the notion of a linearization atlas consisting of multiple local Carleman charts to overcome the limited radius of convergence of standard Carleman linearization. Topological bounds (e.g., Lusternik–Schnirelmann category and covering dimension) suggest lower and upper limits on the number of charts needed to cover a manifold, but analyticity constraints specific to Carleman (polynomial basis centered at chart basepoints) may require more charts.

The authors globalize Carleman linearization numerically but do not provide a method to optimize the atlas. They explicitly point out the absence of an explicit algorithm to construct an optimal or minimal Carleman atlas, motivating future work to systematize chart placement and sizing.