Bounds for elimination of unknowns in systems of differential-algebraic equations

TL;DR

This paper introduces a significantly improved upper bound for the number of differentiations needed to eliminate unknowns in differential-algebraic equations, with applications to algorithm design.

Contribution

It provides a new, tighter upper bound and a matching lower bound for elimination in DAEs, advancing theoretical understanding and practical algorithms.

Findings

New upper bound for unknowns elimination in DAEs

Lower bound demonstrating asymptotic tightness in low dimensions

Applications to designing efficient elimination algorithms

Abstract

Elimination of unknowns in systems of equations, starting with Gaussian elimination, is a problem of general interest. The problem of finding an a priori upper bound for the number of differentiations in elimination of unknowns in a system of differential-algebraic equations (DAEs) is an important challenge, going back to Ritt (1932). The first characterization of this via an asymptotic analysis is due to Grigoriev's result (1989) on quantifier elimination in differential fields, but the challenge still remained. In this paper, we present a new bound, which is a major improvement over the previously known results. We also present a new lower bound, which shows asymptotic tightness of our upper bound in low dimensions, which are frequently occurring in applications. Finally, we discuss applications of our results to designing new algorithms for elimination of unknowns in systems of DAEs.