New bounds and efficient algorithm for sparse difference resultant

TL;DR

This paper introduces new bounds and an efficient determinant-based algorithm for computing the sparse difference resultant, significantly improving the computational approach in difference elimination theory.

Contribution

It presents a novel method to compute the sparse difference resultant using algebraic resultants and provides new order bounds, enhancing efficiency and theoretical understanding.

Findings

The algorithm is efficient and based on determinants of coefficient matrices.

New order bounds for the sparse difference resultant are established.

Experimental results confirm the algorithm's effectiveness.

Abstract

The sparse difference resultant introduced in \citep{gao-2015} is a basic concept in difference elimination theory. In this paper, we show that the sparse difference resultant of a generic Laurent transformally essential system can be computed via the sparse resultant of a simple algebraic system arising from the difference system. Moreover, new order bounds of sparse difference resultant are found. 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 algebraic system. The complexity of the algorithm is analyzed and experimental results show the efficiency of the algorithm.