  We present a general method for computing discriminants of noncommutative
algebras. It builds a connection with Poisson geometry and expresses the
discriminants as products of Poisson primes. The method is applicable to
algebras obtained by specialization from families, such as quantum algebras at
roots of unity. It is illustrated with the specializations of the algebras of
quantum matrices at roots of unity and more generally all quantum Schubert cell
algebras.
