New bounds and efficient algorithm for sparse difference resultant

Let $\mathbb{P}=\{\mathbb{P}_0,\mathbb{P}_1,\dots,\mathbb{P}_n\}$ be a generic Laurent transformally essential system and $\mathbb{P}_{\mathbb{T}}=\{\mathbb{P}_0,\mathbb{P}_1,\dots,\mathbb{P}_m\} (m\leq n)$ be its super essential system. We show that the sparse difference resultant of a simplified system of $\mathbb{P}_{\mathbb{T}}$ by setting the selected $n-m$ variables to one is the same to the one of $\mathbb{P}$. Moreover, new order bounds of sparse difference resultant are obtained. Then we propose an efficient algorithm to compute sparse difference resultant which is the quotient of two determinants whose elements are the coefficients of the polynomials in the strong essential system. We analyze complexity of the algorithm. Experimental results show the efficiency of the algorithm