Differential Elimination for Analytic Functions

Abstract

The implicitization problem for certain parametrized sets of complex analytic functions is solved in this chapter by developing elimination methods based on Janet’s and Thomas’ algorithms. The first section discusses important elimination problems in detail, in particular, how to compute the intersection of a left ideal of an Ore algebra and a subalgebra which is generated by certain indeterminates, the intersection of a submodule of a finitely generated free module over an Ore algebra and the submodule which is generated by certain standard basis vectors, and the intersection of a radical differential ideal of a differential polynomial ring and a differential subring which is generated by certain differential indeterminates. These techniques allow, e.g., to determine all consequences of a given PDE system involving only certain of the unknown functions. Compatibility conditions for inhomogeneous linear systems are also addressed. The second section treats sets of complex analytic functions given by linear parametrizations, whereas the third section develops differential elimination techniques for multilinear parametrizations. Applications to symbolic solving of PDE systems are given. For instance, a family of exact solutions of the Navier-Stokes equations is computed.