  We consider the effective degree bound problem for Lasserre's hierarchy of
sum of squares (SOS) relaxations for polynomial optimization with $n$
variables. Under the assumption that the first $n$ equality constraint defining
polynomials $g_1,\ldots,g_n$ have no nontrivial common complex zero locus at
infinity, and a nonsingularity condition, we establish an effective degree
bound for the exactness of Lasserre's hierarchy. Our assumption holds on a
Zariski open set in the space of polynomials of fixed degrees, which is much
weaker than the grid condition under which the same effective degree bound was
previously known. As a direct application we obtain the first explicit degree
bound for gradient type SOS relaxation under a generic condition.
