Normalization of rationally integrable systems
Nguyen Tien Zung
TL;DR
The paper proves that rationally integrable analytic vector fields $X$ on $(\mathbb{C}^n,0)$ with $X(0)=0$ possess a local analytic Poincaré-Birkhoff normalization, extending Zung’s torus-action method from analytic integrability to rational integrability. Central to the approach is the intrinsic torus action generated by the semisimple part $X^S=\sum_i \gamma_i Z_i$ and the toric degree $\tau$, together with a toric conservation principle that propagates rational first integrals and commuting vector fields to torus-invariant structures. The proof combines a geometric approximation scheme, near-conserving torus actions on annular domains outside the singular set $\mathcal{S}$, projection to level sets, and a sharp-horn holomorphic extension to obtain a global analytic torus action that yields the normalization. This establishes a robust geometric pathway to analytic normalization for a broad class of non-analytic first integrals and opens avenues toward extending Galoisian obstructions to rationally integrable systems. The work also delineates limitations with Darboux-type integrals and outlines future directions involving Diophantine-type conditions and further generalizations.
Abstract
In two previous papers we showed that any analytically integrable vector field admits a local analytic Poincaré-Birkhoff normalization in the neighborhood of a singular point. The aim of this paper is to extend this analytic normalization result to the case of rationally integrable systems, where the first integrals and commuting vector fields are not required to be analytic, but just rational (i.e., quotients of analytic functions or vector fields by analytic functions).