Zero products of Toeplitz operators with n-harmonic symbols

On the Bergman space of the unit polydisk in the complex n-space, we solve the zero-product problem for two Toeplitz operators with n-harmonic symbols that have local continuous extension property up to the distinguished boundary. In the case where symbols have additional Lipschitz continuity up to the whole distinguished boundary, we solve the zero-product problem for products with four factors. We also prove a local version of this result for products with three factors.