Differential Resultants and Subresultants

Consider two differential operators L 1 = P a i d i and L 2 = P b j d 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 L 1 (y) = L 2 (y) = 0 is that the resultant in X of the polynomials (in C[X]) P a i X i and P b j X 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 theses 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..