  A general criterion is given for when the center of a Taft algebra smash
product is the fixed ring. This is applied to the study of the noncommutative
discriminant. Our method relies on the Poisson methods of Nguyen, Trampel, and
Yakimov, but also makes use of Poisson Ore extensions. Specifically, we fully
determine the inner faithful actions of Taft algebras on quantum planes and
quantum Weyl algebras. We compute the discriminant of the corresponding smash
product and apply it to compute the Azumaya locus and restricted automorphism
group.
