  In this paper, the concept of sparse difference resultant for a Laurent
transformally essential system of difference polynomials is introduced and a
simple criterion for the existence of sparse difference resultant is given. The
concept of transformally homogenous polynomial is introduced and the sparse
difference resultant is shown to be transformally homogenous. It is shown that
the vanishing of the sparse difference resultant gives a necessary condition
for the corresponding difference polynomial system to have non-zero solutions.
The order and degree bounds for sparse difference resultant are given. Based on
these bounds, an algorithm to compute the sparse difference resultant is
proposed, which is single exponential in terms of the number of variables, the
Jacobi number, and the size of the Laurent transformally essential system.
Furthermore, the precise order and degree, a determinant representation, and a
Poisson-type product formula for the difference resultant are given.
