  The Gr\"obner basis detection (GBD) is defined as follows: Given a set of
polynomials, decide whether there exists -and if "yes" find- a term order such
that the set of polynomials is a Gr\"obner basis. This problem was shown to be
NP-hard by Sturmfels and Wiegelmann. We show that GBD when studied in the
context of zero dimensional ideals is also NP-hard. An algorithm to solve GBD
for zero dimensional ideals is also proposed which runs in polynomial time if
the number of indeterminates is a constant.
