Modular absolute decomposition of equidimensional polynomial ideals

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).