We establish sharp degree bounds for Putinar sum-of-squares (SOS) certificates in terms of the positivity margin $δ$.
For any fixed Archimedean quadratic module with prescribed generators and any fixed bound on the degree of the objective, we prove a uniform certificate-degree upper bound of $O(δ^{-1/2}\log^2(2/δ))$ for normalized polynomial objectives.
We show that this bound is sharp by constructing a matching lower-bound example in three variables, thereby establishing that the dependence on $δ$ is optimal.
These matching bounds also yield sharp worst-case convergence rates for the Lasserre hierarchy.