A perturbed differential resultant based implicitization algorithm for linear DPPEs

TL;DR

This paper introduces a new implicitization algorithm for linear differential polynomial parametric equations using a perturbed differential resultant, ensuring the existence of a nonzero resultant for effective implicitization.

Contribution

It presents a novel implicitization method based on a perturbed differential resultant, with necessary and sufficient conditions for the implicit ideal's dimension.

Findings

Existence of a linear perturbation making the differential resultant nonzero

Development of an implicitization algorithm using the lowest degree term of the differential resultant

Conditions for the implicit ideal to be of dimension n-1

Abstract

Let $\bbK$ be an ordinary differential field with derivation $\partial$. Let $\cP$ be a system of $n$ linear differential polynomial parametric equations in $n-1$ differential parameters with implicit ideal $\id$. Given a nonzero linear differential polynomial $A$ in $\id$ we give necessary and sufficient conditions on $A$ for $\cP$ to be $n-1$ dimensional. We prove the existence of a linear perturbation $\cP_{\phi}$ of $\cP$ so that the linear complete differential resultant $\dcres_{\phi}$ associated to $\cP_{\phi}$ is nonzero. A nonzero linear differential polynomial in $\id$ is obtained from the lowest degree term of $\dcres_{\phi}$ and used to provide an implicitization algorithm for $\cP$.