  This paper is devoted to studying difference indices of quasi-prime
difference algebraic systems. We define the quasi dimension polynomial of a
quasi-prime difference algebraic system. Based on this, we give the definition
of the difference index of a quasi-prime difference algebraic system through a
family of pseudo-Jacobian matrices. Some properties of difference indices are
proved. In particular, an upper bound of difference indices is given. As
applications, an upper bound of the Hilbert-Levin regularity and an upper bound
of orders for difference ideal membership problem are deduced.
