On the focus order of planar polynomial differential equations

Abstract 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 p k , and show that these systems can have a focus order n 2 − n ; (ii) we theoretically prove the existence of polynomial systems of degree n having a focus order n 2 − 1 for any even number n. Corresponding results for odd n and more concrete examples having higher focus orders are given too.