  We study the algebraic and geometric structures that underly the space of
vacua of N=1 super Yang-Mills theories at the non-perturbative level. Chiral
operators are shown to satisfy polynomial equations over appropriate rings, and
the phase structure of the theory can be elegantly described by the
factorization of these polynomials into irreducible pieces. In particular, this
idea yields a powerful method to analyse the possible smooth interpolations
between different classical limits in the gauge theory. As an application in
U(Nc) theories, we provide a simple and completely general proof of the fact
that confining and Higgs vacua are in the same phase when fundamental flavors
are present, by finding an irreducible polynomial equation satisfied by the
glueball operator. We also derive the full phase diagram for the theory with
one adjoint when Nc is less than or equal to 7 using computational algebraic
geometry programs.
