Discrete-time linear periodic SISO systems having uniform relative degree are considered. A closed-form expression of the blocking input is derived and exploited to obtain a computationally advantageous characterisation of the structural zeros. Indeed, it suffices to compute the eigenvalues of a suitably defined matrix, where is the system order. It is shown that, differently from the general case studied in previous papers, the number of zeros of linear periodic SISO systems with uniform relative degree is always time-invariant and equal to the difference between the system order and the relative degree. The new characterisation is also used to provide a simple expression for the zeros of linear periodic systems described by input-output difference equations.