Doubly-resonant saddle-nodes in $(\mathbb{C}^{3},0)$ and the fixed singularity at infinity in the Painlev\'e equations (part II): sectorial normalization

TL;DR

This paper establishes sectorial normalization for doubly-resonant saddle-node singularities in complex three-dimensional vector fields, with applications to irregular singularities at infinity in Painlevé equations, extending classical results.

Contribution

It generalizes the classical sectorial normalization theorem to three-dimensional doubly-resonant saddle-nodes, with uniqueness and Gevrey-1 summability results.

Findings

Proves analytic normalization over sectorial domains for the singularities.

Shows the normalizing map is essentially unique.

Demonstrates weak Gevrey-1 summability of the normalization.

Abstract

In this work, following [Bit15], we consider analytic singular vector fields in $(\mathbb{C}^{3},0)$ with an isolated and doubly-resonant singularity of saddle-node type at the origin. Such vector fields come from irregular two-dimensional differential systems with two opposite non-zero eigenvalues, and appear for instance when studying the irregular singularity at infinity in Painlev{\'e} equations (P\_j), j=I...V , for generic values of the parameters. Under suitable assumptions, we prove a theorem of analytic nor-malization over sectorial domains, analogous to the classical one due to Hukuhara-Kimura-Matuda [HKM61] for saddle-nodes in $(\mathbb{C}^{2},0)$. We also prove that the normalizing map is essentially unique and weakly Gevrey-1 summable.