  We present new results and an algorithm for standard basis computations of a
0-dimensional ideal I in a power series ring or in the localization of a
polynomial ring in finitely many variables over a field K. The algorithm
provides a significant speed up if K is the quotient field of a Noetherian
integral domain A, when coefficient swell occurs. The most important special
cases are perhaps when A is the ring of integers resp. when A is a polynomial
ring over some field in finitely many parameters. Given I as an ideal in the
polynomial ring over A, we compute first a standard basis modulo a prime number
p, resp. by specializing the parameter to a constant. We then use the "highest
corner" of the specialized ideal to cut off high order terms from the
polynomials during the standard basis computation over K to get the speed up.
An important fact is that we can choose p as an arbitrary prime resp. as an
arbitrary constant, not just a "lucky" resp. "random" one. Correctness of the
algorithm will be deduced from a general semicontinuity theorem due to the
first two authors. The computer algebra system Singular provides already the
functionality to realize the algorithm and we present several examples
illustrating its power.
