Difference indices of quasi-prime difference algebraic systems

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 for difference indices is given. As applications, an upper bound for the Hilbert-Levin regularity and an effective difference ideal membership theorem for quasi-prime difference algebraic systems are deduced.