Difference Index of Quasi-regular Difference Algebraic Systems

TL;DR

This paper introduces a new way to measure the complexity of quasi-regular difference algebraic systems using difference indices, providing bounds and applications in algebraic computations.

Contribution

It defines difference indices via pseudo-Jacobian matrices and establishes properties and bounds, advancing the understanding of difference algebraic systems.

Findings

Established properties of difference indices.

Provided a Jacobi-type upper bound for order and difference index sum.

Derived upper bounds for Hilbert-Levin regularity and difference ideal membership.

Abstract

This paper is devoted to studying difference indices of quasi-regular difference algebraic systems. We give the definition of difference indices through a family of pseudo-Jacobian matrices. Some properties of difference indices are proved. In particular, a Jacobi-type upper bound for the sum of the order and the difference index is given. As applications, an upper bound of the Hilbert-Levin regularity and an upper bound of the order for difference ideal membership problem are deduced.