  Determining the global minimum of Higgs potentials with several Higgs fields
like the next-to-minimal supersymmetric extension of the Standard Model (NMSSM)
is a non-trivial task already at the tree level. The global minimum of a Higgs
potential can be found from the set of all its stationary points defined by a
multivariate polynomial system of equations. We introduce here the algebraic
Groebner basis approach to solve this system of equations. We apply the method
to the NMSSM with CP conserving as well as CP violating parameters. The results
reveal an interesting stationary-point structure of the potential. Requiring
the global minimum to give the electroweak symmetry breaking observed in Nature
excludes large parts of the parameter space.
