Sparse Difference Resultant

TL;DR

This paper introduces the concept of sparse difference resultant for difference polynomial systems, providing criteria for existence, bounds, algorithms, and explicit formulas, advancing the algebraic understanding of such systems.

Contribution

It defines sparse difference resultants, establishes existence criteria, and develops algorithms with explicit formulas for their computation.

Findings

Provides a simple criterion for the existence of sparse difference resultants.

Establishes order and degree bounds for the resultants.

Proposes an efficient algorithm with single exponential complexity.

Abstract

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.