  We show how combinatorial star products can be used to obtain strict
deformation quantizations of polynomial Poisson structures on $\mathbb R^d$,
generalizing known results for constant and linear Poisson structures to
polynomial Poisson structures of arbitrary degree. We give several examples of
nonlinear Poisson structures and construct explicit formal star products whose
deformation parameter can be evaluated to any real value of $\hbar$, giving
strict quantizations on the space of analytic functions on $\mathbb R^d$ with
infinite radius of convergence. We also address further questions such as
continuity of the classical limit $\hbar \to 0$, compatibility with
*-involutions, and the existence of positive linear functionals. The latter can
be used to realize the strict quantizations as *-algebras of operators on a
pre-Hilbert space which we demonstrate in a concrete example.
