We study exactness condition for Lasserre’s relaxation method for polynomial optimization problem with n variables under equality constraints defined by n polynomials. Under the assumption that the quotient ring has dimension equal to the product of the degrees of the n equality defining polynomials, we obtain an explicit bound on the order of Lasserre’s relaxation which guarantees exactness. When the common zero locus are real and all of multiplicity one, we describe the exact region as the convex hull of the moment map image of a vector subspace. For the relaxation of order equal to the explicit bound minus one, the convex hull coincides with the moment map image, and is diffeomorphic to its amoeba. Based on the theory of amoeba, we obtain an explicit description of the exact region, from which we further derive error estimations for relaxation of this specific order.