Quadratic vector fields with invariant algebraic curve of large degree

TL;DR

This paper constructs explicit quadratic planar vector fields with large degree invariant algebraic curves, demonstrating the existence of such systems with invariant curves of degrees nine, twelve, fifteen, and eighteen.

Contribution

The paper provides explicit examples of quadratic vector fields with high-degree invariant algebraic curves, advancing understanding of polynomial vector fields with complex invariant structures.

Findings

Explicit quadratic vector fields with invariant curves of degrees 9, 12, 15, 18

Construction methods for polynomial vector fields with large degree invariants

Demonstration of existence of high-degree invariant algebraic curves in quadratic systems

Abstract

We construct a polynomial planar vector field of degree two with one invariant algebraic curves of large degree. We exhibit an explicit quadratic vector fields which invariant curves of degree nine, twelve, fifteen and eighteen degree.