  We give three algebraic equations which allow a geometric classification of
all spectral types of equilibria of a given $m$-dimensional dynamical system,
and we analyse them thoroughly in dimension 3 and 4. The loci defined by these
equations correspond to definite types of bifurcations. The complement of such
loci give a geometric decomposition of the space of invariants in open domains
in which the equilibrium has a given spectral types. The usefulness of this
approach lays in the fact that, when dealing with a parameter-dependent
dynamical system, the pull-back of the loci from the space of invariants to the
parameter space gives the bifurcation-decomposition of parameter space for the
dynamical system at hand. We also give effective methods to explicitly compute
the spectral indices.
