Commutator and Semicommutator of Two Monomial-Type Toeplitz Operators on the Unit Polydisk

In this paper, we completely characterize the finite rank commutator and semicommutator of two monomial-type Toeplitz operators on the weighted Bergman space or on the weighted pluriharmonic Bergman space of the unit polydisk. As a consequence, some interesting new phenomenons appear in operator theory on the unit polydisk which are different from that on the unit disk and the unit ball.