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

Exactness and Effective Degree Bound of Lasserre's Relaxation for Polynomial Optimization over Finite Variety