Difference Index of Quasi-Prime Difference Algebraic Systems

TL;DR

This paper introduces the concept of difference indices for quasi-prime difference algebraic systems, establishing properties, bounds, and applications to regularity and ideal membership problems.

Contribution

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

Findings

Defined the quasi dimension polynomial.

Established properties and bounds of difference indices.

Applied results to regularity and ideal membership bounds.

Abstract

This paper is devoted to studying difference indices of quasi-prime difference algebraic systems. We define the quasi dimension polynomial of a quasi-prime difference algebraic system. Based on this, we give the definition of the difference index of a quasi-prime difference algebraic system through a family of pseudo-Jacobian matrices. Some properties of difference indices are proved. In particular, an upper bound of difference indices is given. As applications, an upper bound of the Hilbert-Levin regularity and an upper bound of orders for difference ideal membership problem are deduced.