  Using techniques of algorithmic algebraic geometry, we present a new and
efficient method for explicitly computing the vacuum space of N=1 gauge
theories. We emphasize the importance of finding special geometric properties
of these spaces in connecting phenomenology to guiding principles descending
from high-energy physics. We exemplify the method by addressing various
subsectors of the MSSM. In particular the geometry of the vacuum space of
electroweak theory is described in detail, with and without right-handed
neutrinos. We discuss the impact of our method on the search for evidence of
underlying physics at a higher energy. Finally we describe how our results can
be used to rule out certain top-down constructions of electroweak physics.
