Completion of overdetermined parabolic PDEs

In this paper we apply methods of commutative algebra to analysis of systems of PDEs. More precisely, we show that systems which are parabolic in a generalized sense are equivalent to certain completed systems which are parabolic in the standard sense. We also propose a constructive method for getting this completion, and Gröbner basis methods, via symbol modules of the systems, play a central role in practical computations. Moreover, we can easily construct systems which are not parabolic in the generalized sense but nevertheless become parabolic when completed.