Differential resultants and subresultants

Abstract

Consider two differential operators £₁ = Σ a_idⁱ and £₂ = Σ b_jd^j with coefficients in a differential field, say C(t) with d = ∂/∂t for example.

If the a_i and b_j are constants, the condition for the existence of a solution y of £₁(y) = £₂(y) = 0 is that the resultant in X of the polynomials (in C[X]) Σ a_iXⁱ and Σ b_jX^j is zero.

A natural question is: how one could extend this for the case of non constant coefficients ?

One of the main motivation in the resultant techniques is the universality of the computation result. When you calculate the resultant, or the subresultants, you always stay in the ring where the coefficients lies and the result can be specialized to every particular case (if the equations depends on parameters for example) to get the information in these cases.

In this paper, we define all the differential equivalents of these classical objects. In theorem 4 we describe the universal properties of the differential subresultants. They are due to the fact that, as in the polynomial case, we can work out the problem by means of linear algebra.

As it was known by Ritt ([R]), the differential resultant is a polynomial in the a_i, b_j and their derivatives that tells us—in the general algebraic context of linear differential equations over a differential field—if they have a common solution.

In particular, if you know in a given context that there exists a base of solutions (e.g. you have C ^∞ coefficients and you work locally (Cauchy's theorem)), this universal calculation answers this existence problem in this particular context.

We here give a Sylvester style expression for the resultant and the subresultants. So, as theses objects are determinants, you can get size information on them if you have any reasonable notion of size in the differential ring and, as a corollary, complexity estimates for the calculation. At least three techniques may be used: Gauss elimination type techniques (e.g. the Bareiss algorithm) if the ring does not contain zero divisors, Berkowitz fast parallel algorithm ([Be]) or the natural extension of the subresultant algorithm.

As in the classical case, the differential resultant can be expressed in terms of the values of one of the operators on a base of the solutions of the other, we exhibit this formula and most of the differential equivalents of the classical ones.