Linearizability of homogeneous quartic polynomial systems with 1:−2 resonance

In this paper, we consider the linearizability problem of complex planar polynomial systems of the form x ̇ = x + P 4 , y ̇ = − 2 y − Q 4 , where P 4 and Q 4 are homogeneous quartic polynomials. We obtain the necessary and sufficient conditions for linearizing the systems.