Bruno ideal and the variety of centers for singular germs of vector fields

TL;DR

The paper addresses the analytic normalization problem for germs of singular vector fields by introducing the Bruno ideal $B(\partial)$ as the collinearity locus of the semisimple and nilpotent parts. It proves that under the Bruno $\omega$-condition, $B(\partial)$ is analytic and there exists an analytic conjugacy bringing the vector field to a normal form modulo $B(\partial)$, with the vanishing set $V(B(\partial))$ forming an analytic invariant variety on which the foliation is linearizable. The main approach combines formal normalization via an adjoint-cohomological framework with an analytic Newton-type scheme controlled by $r$-norms, providing explicit bounds and convergence arguments. The results yield broad applications, including analytic linearization in resonant and non-resonant settings and new insights into invariant varieties and center-manifold-type behavior for logarithmic vector fields.

Abstract

Given a logarithmic analytic vector field $\partial$, we consider the formal ideal $B(\partial)$ defined by the collinearity locus of the semi-simple and nilpotent components of~$\partial$. Assuming that the eigenvalues of the linear part of $\partial$ satisfy the so-called Bruno arithmetic condition, we prove that $B(\partial)$ is in fact an analytic ideal. Moreover, $\partial$ is analytically normalizable when restricted to this ideal. As a consequence, the vanishing locus $V$ of $B(\partial)$ is an analytic variety, and the foliation defined by $\partial|_{V}$ is analytically linearizable.

Theorems & Definitions (56)

• Remark 1.1
• Theorem 1.2: Bruno, 1971
• Corollary 1.3
• Remark 2.1
• Remark 2.2
• Example 2.3
• Remark 2.4
• Definition 2.5
• Lemma 2.6
• proof