Symmetry in the core of a zero-dimensional monomial ideal

The core of an ideal is the intersection of all of its reductions. We have shown that under certain conditions, the exponent set of the core of a zero-dimensional monomial ideal exhibits translational symmetry. In addition, in two dimensions, the core of a monomial ideal is often the core of a reduction number one ideal. We provide an algorithm for obtaining that reduction number one ideal and, subsequently, its core.