Center conditions: pull back of differential equations

TL;DR

This paper proves that pull back differential equations constitute an irreducible component in the space of polynomial differential equations with a center singularity, using Picard Lefschetz theory and related concepts.

Contribution

It establishes the irreducibility of pull back differential equations as a component in the space of polynomial equations with centers, extending previous methods.

Findings

Pull back differential equations form an irreducible component.

The method is inspired by Ilyashenko and Movasati's approach.

Uses Picard Lefschetz theory, Dynkin diagrams, and iterated integrals.

Abstract

The space of polynomial differential equations of a fixed degree with a center singularity has many irreducible components. We prove that pull back differential equations form an irreducible component of such a space. The method used in this article is inspired by Ilyashenko and Movasati s method. The main concepts are the Picard Lefschetz theory of a polynomial in two variables with complex coefficients, the Dynkin diagram of the polynomial and the iterated integral.