  In a first contribution, we revisit two certificates of positivity on
(possibly non-compact) basic semialgebraic sets due to Putinar and Vasilescu
[Comptes Rendus de l'Acad\'emie des Sciences-Series I-Mathematics, 328(6)
(1999) pp. 495-499]. We use Jacobi's technique from [Mathematische Zeitschrift,
237(2) (2001) pp. 259-273] to provide an alternative proof with an effective
degree bound on the sums of squares multipliers in such certificates. As a
consequence, it allows one to define a hierarchy of semidefinite relaxations
for a general polynomial optimization problem. Convergence of this hierarchy to
a neighborhood of the optimal value as well as strong duality and analysis are
guaranteed. In a second contribution, we introduce a new numerical method for
solving systems of polynomial inequalities and equalities with possibly
uncountably many solutions. As a bonus, one may apply this method to obtain
approximate global optimizers in polynomial optimization.
