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Liouvillian integrability of vector fields in higher dimensions

Waleed Aziz, Colin Christopher, Chara Pantazi, Sebastian Walcher

TL;DR

The paper advances the theory of Liouvillian integrability for complex rational vector fields in dimensions higher than two by proving that, whenever a Liouvillian first integral exists, it can be realized after passing to a finite algebraic extension and performing two successive integrations from suitable 1-forms. The authors extend Singer's 2D result to n≥3, utilizing Puiseux-series methods to simplify computations and to obtain constructive criteria expressed via ω and α with dω = α ∧ ω and dα = 0. In the three-dimensional case, they provide a dichotomy: either a Darboux-type integrating factor exists over the base field, or an inverse Jacobi multiplier of Darboux type exists after extending the field. The paper also uses the Puiseux framework to give elementary proofs of Singer's theorem for rational 1-forms and of the Prelle-Singer theorem on elementary integrability, highlighting a unified method for higher-dimensional integrability problems.

Abstract

We consider complex rational vector fields in dimension $n>2$ (equivalently, differential forms of degree $n-1$ in $n$ variables) which admit a Liouvillian first integral. Extending a classical result by Singer for $n=2$, our main result states that there exists a first integral which is obtained by two successive integrations from one-forms with coefficients in a finite algebraic extension of the rational function field. The proof uses Puiseux series in a novel way to simplify computations. We also apply this method to give elementary proofs of Singer's theorem for rational one-forms, and of the Prelle-Singer theorem on elementary integrability of rational vector fields.

Liouvillian integrability of vector fields in higher dimensions

TL;DR

The paper advances the theory of Liouvillian integrability for complex rational vector fields in dimensions higher than two by proving that, whenever a Liouvillian first integral exists, it can be realized after passing to a finite algebraic extension and performing two successive integrations from suitable 1-forms. The authors extend Singer's 2D result to n≥3, utilizing Puiseux-series methods to simplify computations and to obtain constructive criteria expressed via ω and α with dω = α ∧ ω and dα = 0. In the three-dimensional case, they provide a dichotomy: either a Darboux-type integrating factor exists over the base field, or an inverse Jacobi multiplier of Darboux type exists after extending the field. The paper also uses the Puiseux framework to give elementary proofs of Singer's theorem for rational 1-forms and of the Prelle-Singer theorem on elementary integrability, highlighting a unified method for higher-dimensional integrability problems.

Abstract

We consider complex rational vector fields in dimension (equivalently, differential forms of degree in variables) which admit a Liouvillian first integral. Extending a classical result by Singer for , our main result states that there exists a first integral which is obtained by two successive integrations from one-forms with coefficients in a finite algebraic extension of the rational function field. The proof uses Puiseux series in a novel way to simplify computations. We also apply this method to give elementary proofs of Singer's theorem for rational one-forms, and of the Prelle-Singer theorem on elementary integrability of rational vector fields.

Paper Structure

This paper contains 10 sections, 11 theorems, 36 equations.

Table of Contents

  1. Introduction and overview of results
  2. Background and some known results
  3. Liouvillian extensions
  4. Singer's theorem for one-forms
  5. Extension of Singer's theorem to vector fields in higher dimensions
  6. The main result
  7. Three dimensional vector fields with Liouvillian first integrals
  8. A proof of the Prelle-Singer theorem
  9. Appendix
  10. Laurent and Puiseux expansions

Key Result

Theorem 1

Let $\omega$ be a rational 1--form over $K=\mathbb C(x_1,\ldots,x_n)$. Then $\omega$ is Liouvillian integrable if and only if there exists a closed 1--form $\alpha \in \mathbb C(x_1,\ldots,x_n)'$ such that $d\omega=\alpha \wedge \omega$.

Theorems & Definitions (28)

  • Definition 1
  • Remark 1
  • Remark 2
  • Definition 2
  • Remark 3
  • Theorem 1: Singer's Theorem for 1-forms
  • proof
  • Theorem 2
  • proof
  • Remark 4
  • ...and 18 more