Testing Equality in Differential Ring Extensions Defined by PDE's and Limit Conditions

Abstract.

  We present here methods to test equality in differential extensions of effective rings. The extensions considered are obtained by adjunction of formal power series solutions of given (non linear) systems of PDE's with initial/limit conditions. The equality test is indeed the only operation that is not trivial in such extensions, and is hence a central problem for formally manipulating solutions of differential systems without computing or studying them completely. The problem has been solved in [9] for ordinary differential systems and in [17] for solutions of some partial differential systems with a finite set of initial conditions at the origin. We first recall and refine the results of [17] and then give a method to handle some partial differential systems with limit conditions on the axis x ₁=0. In particular, we reduce the study of the regular cases of the latter differential algebraic problem to a purely (computable) algebraic one, and the singular cases to a topological one, the Ritt problem concerning the distribution of the singular zeros of a differential system among its irreducible components.