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In this study a simplified analytic test of stability of linear discrete systems is obtained. This test also yields the necessary and sufficient conditions for a real polynomial in the variable z to have all its roots inside the unit circle in the z plane. The new stability constraints require the evaluation of only half the number of Schur-Cohn determinants , . It is shown that for the test of a fourth-order system only a third-order determinant is required and for the fifth-order, one second-order and one fourth-order determinant are required. The test is applied directly in the z plane and yields the minimum number of constraint terms. Stability constraints up to the sixth-order case are obtained and for the nth-order case are formulated. The simplicity of this criterion is similar to that of the Lienard-Chipard criterion  for the continuous case which has a decisive advantage over the Routh-Hurwitz criterion , . Finally, general conditions on the number of roots inside the unit circle for n even and odd are also presented in this paper.