  In this paper we establish new renormalized oscillation theorems for discrete
symplectic eigenvalue problems with Dirichlet boundary conditions. These
theorems present the number of finite eigenvalues of the problem in arbitrary
interval $(a,b]$ using number of focal points of a transformed conjoined basis
associated with Wronskian of two principal solutions of the symplectic system
evaluated at the endpoints $a$ and $b.$ We suppose that the symplectic
coefficient matrix of the system depends nonlinearly on the spectral parameter
and that it satisfies certain natural monotonicity assumptions. In our
treatment we admit possible oscillations in the coefficients of the symplectic
system by incorporating their nonconstant rank with respect to the spectral
parameter.
