  In this paper, we present a modular strategy which describes key properties
of the absolute primary decomposition of an equidimensional polynomial ideal
defined by polynomials with rational coefficients. The algorithm we design is
based on the classical technique of elimination of variables and colon ideals
and uses a tricky choice of prime integers to work with. Thanks to this
technique, we can obtain the number of absolute irreducible components, their
degree, multiplicity and also the affine Hilbert function of the reduced
components (namely, their initial ideal w.r.t. a degree-compatible term
ordering) .
