On the focus order of planar polynomial differential equations

This paper is devoted to finding the highest possible focus order of planar polynomial differential equations. The results consist of two parts: (i) we explicitly construct a class of concrete systems of degree n, where n+1 is a prime p or a power of a prime pk, and show that these systems can have a focus order n2−n; (ii) we theoretically prove the existence of polynomial systems of degree n having a focus order n2−1 for any even number n. Corresponding results for odd n and more concrete examples having higher focus orders are given too.