Limit cycles for planar semi-quasi-homogeneous polynomial vector fields

This paper is concerned with the limit cycles for planar semi-quasi-homogeneous polynomial systems. We give some explicit criteria for the nonexistence and existence of periodic orbits. A lower bound is given for the maximum number of limit cycles of such a system. The cyclicity and center problems are studied for some subfamilies of semi-quasi-homogeneous polynomial systems. Our results generalize those obtained for polynomial semi-homogeneous systems.