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Planar vector fields without invariant algebraic curves

Gabriel Fazoli, Paulo Santana

TL;DR

This work strengthens the link between planar polynomial vector fields and projective foliations to show that, under suitable fixed invariant nodal curves, the space of degree $n$ vector fields with no invariant algebraic curves beyond the prescribed one is generically large: open/dense in the complex setting and residual with full measure in the real setting. It extends the Lins Neto–Sad–Scárdua framework using Camacho–Sad obstructions and analytic geometry of configuration spaces to produce general and robust nonexistence results for invariant algebraic curves, including for Kolmogorov foliations where the coordinate axes are the only invariant curves. The paper also refines perspectives on Hilbert’s 16th problem by proving that, if a uniform bound on the number of limit cycles exists for degree $n$, there are vector fields attaining this bound with no algebraic limit cycles, emphasizing the nonrigidity of algebraic cycles in generic dynamical systems. Overall, the results yield a broad, generic-structure picture: most planar polynomial vector fields avoid extraneous invariant algebraic curves, while explicit families (e.g., Kolmogorov) illustrate the sharpness and applicability of the method to classical problems in foliation theory and dynamical systems.

Abstract

In this work we revisit and extend the method introduced by Lins Neto, Sad and Scárdua for detecting the non-existence of invariant algebraic curves other than some prescribed invariant nodal curve. We prove that, under the existence of a suitable example, the space of polynomial vector fields whose elements have the prescribed curve as their unique invariant algebraic curve is residual and of full measure. We apply this framework to Kolmogorov vector fields, showing that generically the coordinate axes are the unique invariant algebraic curves. Finally, we also refine existing characterizations related to Hilbert's 16th problem, showing that if there exists a bound for the number of limit cycles of a vector field of degree n, then it can be attained by a vector field without algebraic limit cycles.

Planar vector fields without invariant algebraic curves

TL;DR

This work strengthens the link between planar polynomial vector fields and projective foliations to show that, under suitable fixed invariant nodal curves, the space of degree vector fields with no invariant algebraic curves beyond the prescribed one is generically large: open/dense in the complex setting and residual with full measure in the real setting. It extends the Lins Neto–Sad–Scárdua framework using Camacho–Sad obstructions and analytic geometry of configuration spaces to produce general and robust nonexistence results for invariant algebraic curves, including for Kolmogorov foliations where the coordinate axes are the only invariant curves. The paper also refines perspectives on Hilbert’s 16th problem by proving that, if a uniform bound on the number of limit cycles exists for degree , there are vector fields attaining this bound with no algebraic limit cycles, emphasizing the nonrigidity of algebraic cycles in generic dynamical systems. Overall, the results yield a broad, generic-structure picture: most planar polynomial vector fields avoid extraneous invariant algebraic curves, while explicit families (e.g., Kolmogorov) illustrate the sharpness and applicability of the method to classical problems in foliation theory and dynamical systems.

Abstract

In this work we revisit and extend the method introduced by Lins Neto, Sad and Scárdua for detecting the non-existence of invariant algebraic curves other than some prescribed invariant nodal curve. We prove that, under the existence of a suitable example, the space of polynomial vector fields whose elements have the prescribed curve as their unique invariant algebraic curve is residual and of full measure. We apply this framework to Kolmogorov vector fields, showing that generically the coordinate axes are the unique invariant algebraic curves. Finally, we also refine existing characterizations related to Hilbert's 16th problem, showing that if there exists a bound for the number of limit cycles of a vector field of degree n, then it can be attained by a vector field without algebraic limit cycles.

Paper Structure

This paper contains 16 sections, 19 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Preliminary section
  3. From differential equations to foliations
  4. Homogeneous coordinates of foliations
  5. Foliations with the line at infinity invariant
  6. Invariant algebraic sets
  7. Singular set and separatrices
  8. Foliations without algebraic solutions
  9. More on singularities and existence of separatrices
  10. Foliations without algebraic solutions
  11. Generalization
  12. Foliation with a fixed invariant nodal curve
  13. Foliations without algebraic solution other than a fixed nodal curve
  14. Kolmogorov foliations
  15. Proof of the main theorems
  16. ...and 1 more sections

Key Result

Theorem A

Let $F\in\mathbb{C}[x,y]$ be nodal curve taking the line at infinity into account, and let $n\in \mathbb N$. Suppose there exists $\mathcal{X}_0\in\mathfrak{X}_n^F(\mathbb{\mathbb C}^2)$ with only Poincaré singularities, such that $\mathcal{X}_0$ does not satisfy any Camacho-Sad obstructions. Then: Moreover, suppose that there exists $\mathcal{X}_1\in\mathfrak{X}_n^F(\mathbb{\mathbb R}^2)$ with o

Theorems & Definitions (34)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem 1: Theorem $2$ of GasSan2025
  • Theorem D
  • Remark 1
  • Example 1
  • Proposition 1: Lemma $2$ of ALN1988
  • Proposition 2
  • proof
  • ...and 24 more