Integrability of polynomial vector fields and a dual problem

TL;DR

This paper explores the integrability of polynomial vector fields using duality concepts, analyzing formal solutions, invariants, and differential operators, with a detailed case study on quadratic vector fields.

Contribution

It introduces a duality framework for understanding polynomial vector field integrability, linking invariants, differential operators, and local integrals, especially for quadratic cases.

Findings

Duality provides new insights into integrability conditions.

Invariants relate closely to the integrability variety.

Quadratic vector fields exemplify the duality approach.

Abstract

We investigate the integrability of polynomial vector fields through the lens of duality in parameter spaces. We examine formal power series solutions annihilated by differential operators and explore the properties of the integrability variety in relation to the invariants of the associated Lie group. Our study extends to differential operators on affine algebraic varieties, highlighting the intrinsic connection between these operators and local analytic first integrals. To illustrate the duality the case of quadratic vector fields is considered in detail.