An elementary approach to the polynomial $\tau$-functions of the KP hierarchy

We give an elementary construction of the solutions of the KP hierarchy associated with polynomial $\tau$-functions starting with a geometric approach to soliton equations based on the concept of a bi-Hamiltonian system. As a consequence, we establish a Wronskian formula for the polynomial $\tau$-functions of the KP hierarchy. This formula, known in the literature, is obtained very directly.