Sparse Differential Resultant for Laurent Differential Polynomials

TL;DR

This paper introduces the concept of Laurent differentially essential systems, defines their sparse differential resultants, and provides bounds and an algorithm for their computation, advancing symbolic differential algebra.

Contribution

It presents the first criterion for Laurent differentially essential systems, defines their sparse differential resultants, and develops an efficient algorithm for computing them.

Findings

Order and degree bounds for the sparse differential resultant are established.

An algorithm with single exponential complexity for computing the sparse differential resultant is proposed.

The basic properties of the sparse differential resultant are proved.

Abstract

In this paper, we first introduce the concept of Laurent differentially essential systems and give a criterion for Laurent differentially essential systems in terms of their supports. Then the sparse differential resultant for a Laurent differentially essential system is defined and its basic properties are proved. In particular, order and degree bounds for the sparse differential resultant are given. Based on these bounds, an algorithm to compute the sparse differential resultant is proposed, which is single exponential in terms of the number of indeterminates, the Jacobi number of the system, and the size of the system.