ag.algebraica geometry – Non-compact example where Putinar's Positivstellensatz fails

One way to establish Putinar's positivstellensatz is as follows: a compact set of polynomial inequalities $ mathcal {P} = {P_1 (x) geq 0, ldots, P_m (x) geq 0 } $ it is not satisfactory if and only if there are sum of squares polynomials $ Q_0 (x), ldots, Q_m (x) in Sigma ^ 2 $ such that $ -1 = Q_0 + sum_ {i in [m]} Q_i (x) P_i (x) $.
I was wondering if we know an explicit, non-compact set of polynomial inequalities on which the generalization of Putit's Positivstellensatz, where the requirement of compactness is omitted, is known to fail.
Thanks for the help.