In this paper, the concept of sparse difference resultant for a Laurent transformally essential system of Laurent difference polynomials is introduced and its properties are proved. In particular, order and degree bounds for the 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 system. Also, the precise order, degree, a determinant representation, and a Poisson-type product formula for the difference resultant are given.

In this paper, the concept of sparse differential resultant for a differentially essential system of differential polynomials is introduced and its properties are proved. In particular, a degree bound for the sparse differential resultant is given. Based on the degree bound, an algorithm to compute the sparse differential resultant is proposed, which is single exponential in terms of the order, the number of variables, and the size of the differentially essential system.