The location of the zeros of discrete systems characteristic polynomials with respect to the unit circle is investigated. A new sequence of symmetric polynomials of descending degrees is defined for the characteristic polynomial and the number of zeros inside and outside the unit circle is shown to be related to a certain sign variation pattern of the polynomials in the sequence. A stability table based on this sequence is presented to obtain the sought distribution of zeros. The new table is close in appearance, size, and number of arithmetic operations to the Routh table used for continuous-time system polynomials. By comparison with the table of Jury, based on the theory of Marden, Cohn, and Schur, the new table involves about half the number of entries and a corresponding significant saving in computations. The study includes a detailed consideration of all possible cases of singular conditions so that the complete information on the number of zeros inside, outside, and on the unit circle is always obtained (including some additional information on possible reciprocal pairs of zeros). Necessary and sufficient conditions for a polynomial to have all its zeros inside the unit circle are obtained as a special outcome. Other additional necessary conditions, that are useful to shorten procedures when the table is used only for testing the stability of the system, are also given.