  The Flory-Huggins theory describes the phase separation of solutions
containing polymers. Although it finds widespread application from polymer
physics to materials science to biology, the concentrations that coexist in
separate phases at equilibrium have not been determined analytically, and
numerical techniques are required that restrict the theory's ease of
application. In this work, we derive an implicit analytical solution to the
Flory-Huggins theory of one polymer in a solvent by applying a procedure that
we call the implicit substitution method. While the solutions are implicit and
in the form of composite variables, they can be mapped explicitly to a phase
diagram in composition space. We apply the same formalism to multicomponent
polymeric systems, where we find analytical solutions for polydisperse mixtures
of polymers of one type. Finally, while complete analytical solutions are not
possible for arbitrary mixtures, we propose computationally efficient
strategies to map out coexistence curves for systems with many components of
different polymer types.
