On the Index and the Order of Quasi-regular Implicit Systems of Differential Equations

TL;DR

This paper investigates the differentiation index and order of quasi-regular implicit differential algebraic equations, providing algebraic definitions, bounds, and solution existence results to advance understanding of such systems.

Contribution

It introduces an algebraic definition of the differentiation index, proves bounds for the sum of index and order, and establishes solution existence and uniqueness for implicit differential systems.

Findings

Established an algebraic definition of the differentiation index

Proved a Jacobi-type upper bound for the sum of order and index

Derived an upper bound for Hilbert-Kolchin regularity and solution uniqueness

Abstract

This paper is mainly devoted to the study of the differentiation index and the order for quasi-regular implicit ordinary differential algebraic equation (DAE) systems. We give an algebraic definition of the differentiation index and prove a Jacobi-type upper bound for the sum of the order and the differentiation index. Our techniques also enable us to obtain an alternative proof of a combinatorial bound proposed by Jacobi for the order. As a consequence of our approach we deduce an upper bound for the Hilbert-Kolchin regularity and an effective ideal membership test for quasi-regular implicit systems. Finally, we prove a theorem of existence and uniqueness of solutions for implicit differential systems.