We present an algorithm to determine the set of essential singular solutions of a differential equation. Essential singular solutions can be informally described as follows: the general solution of a differential equation is usually described as a solution depending on a number of arbitrary constants equal to the order of the differential equation. The essential singular solutions are those tha...

Let k be a field of characteristic 0. Given a polynomial mapping f = (f1, . . . , fp) from kn to kp, the local Bernstein–Sato ideal of f at a point a ∈ kn is defined as an ideal of the ring of polynomials in s = (s1, . . . , sp). We propose an algorithm for computing local Bernstein–Sato ideals by combining Gröbner bases in rings of differential operators with primary decomposition in a polynom...

We consider reasoning and minimization in systems of polynomial ordinary differential equations (odes). The ring of multivariate polynomials is employed as a syntax for denoting system behaviours. We endow polynomials with a transition system structure based on the concept of Lie derivative, thus inducing a notion ofL-bisimulation. Two states (variables) are provenL-bisimilar if and only if the...

The aim of this work is to study some noncommutative differential operators. We take an algorithmic approach as well as further developing the mathematics, and design and analyse algorithms to solve fundamental problems. We also give applications to differential equations. First we consider how to factor skew polynomials. These are polynomials in a differential or difference operator. Using the...

As was observed by Castro and Galligo [3] , [5], the Buchberger algorithm for computing Gröbner bases of ideals of the polynomial ring applies also to the Weyl algebra, i.e., the ring of differential operators with polynomial coefficients. This generalization of the Buchberger algorithm has turned out to be very fruitful in the computational approach to the theory of Dmodules, which aims at an ...