A linear algebra approach to the differentiation index of generic DAE systems

TL;DR

This paper introduces a linear algebra approach to analyze the differentiation index of generic DAE systems, providing bounds and algorithmic insights that improve understanding of their structural properties.

Contribution

It presents a novel linear algebra framework for studying the differentiation index of DAE systems, including bounds on regularity and order, independent of characteristic sets.

Findings

Upper bounds for the regularity of the Hilbert-Kolchin function

Bounds on the order of the associated ideal

Algorithmic results on differential transcendence bases

Abstract

The notion of differentiation index for DAE systems of arbitrary order with generic second members is discussed by means of the study of the behavior of the ranks of certain Jacobian associated sub-matrices. As a by-product, we obtain upper bounds for the regularity of the Hilbert-Kolchin function and the order of the ideal associated to the DAE systems under consideration, not depending on characteristic sets. Some quantitative and algorithmic results concerning differential transcendence bases and induced equivalent explicit ODE systems are also established.