# Invariant algebraic surfaces of polynomial vector fields in dimension three

## TL;DR

This paper explores criteria for the existence and nonexistence of invariant algebraic surfaces in three-dimensional polynomial vector fields, providing degree bounds and explicit constructions, thus extending classical planar results to higher dimensions.

## Contribution

It introduces new criteria and degree bounds for invariant algebraic surfaces in 3D polynomial vector fields, including explicit constructions and generalizations to higher dimensions.

## Key findings

- Degree bounds for irreducible semi-invariants are established.
- Existence criteria for algebraic Jacobi multipliers are provided.
- Explicit examples show generic conditions do not exclude invariant surfaces.

## Abstract

We discuss criteria for the nonexistence, existence and computation of invariant algebraic surfaces for three-dimensional complex polynomial vector fields, thus transferring a classical problem of Poincar\'e from dimension two to dimension three. Such surfaces are zero sets of certain polynomials which we call semi-invariants of the vector fields. The main part of the work deals with finding degree bounds for irreducible semi-invariants of a given polynomial vector field that satisfies certain properties for its stationary points at infinity. As a related topic, we investigate existence criteria and properties for algebraic Jacobi multipliers. Some results are stated and proved for polynomial vector fields in arbitrary dimension and their invariant hypersurfaces. In dimension three we obtain detailed results on possible degree bounds. Moreover by an explicit construction we show for quadratic vector fields that the conditions involving the stationary points at infinity are generic but they do not a priori preclude the existence of invariant algebraic surfaces. In an appendix we prove a result on invariant lines of homogeneous polynomial vector fields.