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We consider the effective degree bound problem for Lasserre's hierarchy ofsum of squares (SOS) relaxations for the minimization of a polynomial function$f(x)$ of $n$ variables under $n$ polynomial equality constraints$g_1(x)=\cdots=g_n(x)=0$. Under the assumption that $g_1,\ldots,g_n$ have nonontrivial common complex zero locus at infinity, we establish an effectivedegree bound $\max\left(\sum_{i=1}^n {\mathrm{deg}}(g_i)-n,{\mathrm{deg}}(f)/2\right)$ for the exactness of Lasserre's hierarchy. Ourassumption holds on a Zariski open set in the space of $n$ polynomials of fixeddegrees, which is much weaker than the grid condition under which the sameeffective degree bound was previously known. As a direct application we obtainthe first explicit degree bound for gradient type SOS relaxation under ageneric condition.

An effective degree bound of Lasserre's relaxation for polynomial optimization with equality constraints