  Solving large polynomial systems with coefficient parameters are ubiquitous
and constitute an important class of problems. We demonstrate the computational
power of two methods--a symbolic one called the Comprehensive Gr\"obner basis
and a numerical one called the cheater's homotopy-applied to studying both
potential energy landscapes and a variety of questions arising from geometry
and phenomenology. Particular attention is paid to an example in flux
compactification where important physical quantities such as the gravitino and
moduli masses and the string coupling can be efficiently extracted.
