  We study rational solutions of the Boussinesq equation, which is a soliton
equation solvable by the inverse scattering method. These rational solutions,
which are algebraically decaying and depend on two arbitrary parameters, are
expressed in terms of special polynomials that are derived through a bilinear
equation, have a similar appearance to rogue-wave solutions of the focusing
nonlinear Schr\"odinger (NLS) equation and have an interesting structure.
Further rational solutions of the Kadomtsev-Petviashvili I (KPI) equation are
derived in two ways, from rational solutions of the NLS equation and from
rational solutions of the Boussinesq equation. It is shown that the two
families of rational solutions of the KPI equation are fundamentally different.
