A new algorithm for computing regular representations for radicals of parametric differential ideals

The regular representation of the radical of a differential ideal has various applications such as solving the membership problem, computing Taylor expansion of solutions, finding the Lie symmetries, and solving dynamical systems. Presently, there is no algorithm giving all regular representations for all possible values of the parameters for a polynomial differential ideal with parametric coefficients. In this article, we propose a new algorithm that computes all different regular representations with respect to all possible states of the parameters. Also, we present an efficient criterion to reduce some ineffectual computations. Implementing the algorithm in Maple and several examples reported in this article demonstrate the high efficiency of the algorithm