Symbolic Computation Group

It is a classical result that the coefficients of power series solutions of a linear differential equation obey a linear recurrence. This recurrence can be computed from simple ring morphism from linear differential operators to linear difference operators. However, the recurrence might not be minimal if the initial differential equations possesses apparent singularities, that is, if there are points where the differential equation is singular, but its solutions are not. The problem is to find a suitable multiple of the original equation, which may be an equation of higher order, which is free of apparent singularities.

In our approach to this problem, and in order to simultaneously consider the inverse problem, q-analogues and other related operators, we consider an equivalent problem in the framework of skew polynomial rings.

This talk will describe algorithms to remove apparent singularites and compute polynomial torsion modules in algebras of skew polynomials. In the differential case, we revisit algorithms by Tsai with a goal of greater efficiency. In the case of recurrences, $q$-recurrences, and Mahler equations, our algorithmic results seem new, and require a more involved machinery.

Joint work with F. Chyzak, Ph. Dumas, H. Le, J. Martins, M. Mishna, and B. Salvy.