Support bound for differential elimination in polynomial dynamical systems

TL;DR

This paper extends bounds on differential equations in polynomial dynamical systems to more general functions and parameters, improving elimination techniques and computational reach.

Contribution

It generalizes previous bounds to arbitrary polynomial observation functions and parametric systems, enhancing differential elimination methods.

Findings

New bounds improve elimination accuracy

Approach handles more complex systems

Outperforms existing software in certain cases

Abstract

We study an important special case of the differential elimination problem: given a polynomial parametric dynamical system $\mathbf{x}' = \mathbf{g}(\boldsymbol{\mu}, \mathbf{x})$ and a polynomial observation function $y = f(\boldsymbol{\mu}, \mathbf{x})$, find the minimal differential equation satisfied by $y$. In our previous work, for the case $y = x_1$, we established a bound on the support of such a differential equation for the non-parametric case and shown that it can be turned into an algorithm via the evaluation-interpolation approach. The main contribution of the present paper is a generalization of the aforementioned result in two directions: to allow any polynomial function $y = f(\mathbf{x})$, not just a single coordinate, and to allow $\mathbf{g}$ and $f$ depend on unknown symbolic parameters. We conduct computation experiments to evaluate the accuracy of our new bound and show that the approach allows to perform elimination for some cases out of reach for the state of the art software.