  This paper is devoted to studying difference indices of quasi-regular
difference algebraic systems. We give the definition of difference indices
through a family of pseudo-Jacobian matrices. Some properties of difference
indices are proved. In particular, a Jacobi-type upper bound for the sum of the
order and the difference index is given. As applications, an upper bound of the
Hilbert-Levin regularity and an upper bound of the order for difference ideal
membership problem are deduced.
