  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. Let $N=N(p,q,m,n)$ be the
maximum number of limit cycles of such system. A lower bound is given for
$N=N(p,q,m,n)$. The cyclicity and center problem are studied for some
subfamilies of semi-quasi-homogeneous polynomial systems. Our results
generalize those obtained for polynomial semi-homogeneous systems.
