We consider a property of positive polynomials on a compact set with a small perturbation. When applied to a Polynomial Optimization Problem (POP), the property implies that the optimal value of the corresponding SemiDeﬁnite Programming (SDP) relaxation with suﬃciently large relaxation order is bounded from below by (f∗ − ) and from above by f∗ + (n + 1), where f∗ is the optimal value of the POP. We propose new SDP relaxations for POP based on modiﬁcations of existing sums-of-squares representation theorems. An advantage of our SDP relaxations is that in many cases they are of considerably smaller dimension than those originally proposed by Lasserre. We present some applications and the results of our computational experiments.