  In many physical systems it is important to be aware of the crossings and
avoided crossings which occur when eigenvalues of a physical observable are
varied using an external parameter. We have discovered a powerful algebraic
method of finding such crossings via a mapping to the problem of locating the
roots of a polynomial in that parameter. We demonstrate our method on atoms and
molecules in a magnetic field, where it has implications in the search for
Feshbach resonances. In the atomic case our method allows us to point out a new
class of invariants of the Breit-Rabi Hamiltonian of magnetic resonance. In the
case of molecules, it enables us to find curve crossings with practically no
knowledge of the corresponding Born-Oppenheimer potentials.
