  This paper discusses the notion of a deformation quantization for an
arbitrary polynomial Poisson algebra A. We examine the Hochschild cohomology
group H^3(A) and find that if a deformation of A exists it can be given by
bidifferential operators. We then compute an explicit third order deformation
quantization of A and show that it comes from a quantized enveloping algebra.
We show that the deformation extends to a fourth order deformation if and only
if the quantized enveloping algebra gives a fourth order deformation; moreover
we give an example where the deformation does not extend. A correction term to
the third order quantization given by the enveloping algebra is computed, which
precisely cancels the obstruction.
