Maximal multiplicity of the line at infinity in a sextic polynomial differential system

Abstract

We investigate planar polynomial differential systems of degree six having an invariant straight line at infinity with high algebraic multiplicity in the sense of the Poincaré compactification. By analyzing the determinant associated with the invariant line at infinity, we show that the maximal multiplicity attainable in this class is 16. The computations involved are extremely large, and only representative cases are presented. The obtained results support the conjecture that the bound M_∞(n)=3n-2 is sharp for polynomial differential systems of degree n ≥ 2.