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TL;DR

This paper explores the structure of fully non-linear PDEs in holomorphic functions, providing new existence theorems, a refined implicit function theorem, and results on center manifolds and normal forms for 3D vector fields.

Contribution

It introduces an implicit function theorem that avoids Nash-Moser perturbation conditions and establishes optimal existence results for certain singular PDEs.

Findings

New implicit function theorem without Nash-Moser conditions

Optimal existence results for bi-dimensional singularities

Corollaries on center manifolds and normal forms for 3D vector fields

Abstract

We investigate the structure of fully non-linear P.D.E.'s in holomorphic functions, with emphasis on the functorial generalisation of so called "irregular" O.D.E.'s. Highlights are an implicit function theorem removing the perturbation conditions of Nash-Moser type, best possible existence results when the singularity of the linearised P.D.E. is at worst bi-dimensional, and various, again optimal, corollaries on existence of centre manifolds and conjugation to normal form of 3-dimensional vector fields.