The Differential Ideal [ P ] : M∞

Let F be a finite subset of the differential polynomial algebra k{ y 1, , yn } . In order to determine membership in the radical differential ideal { F }, one is led to express { F } as the intersection of differential ideals of the form [ P ] : M∞ for suitable subsets P and M of k { y 1, , yn } . One criterion for “suitability" is that the ideal [ P ] : M∞ should be radical; another is that the question of membership in this ideal should be reducible to the question of membership in its algebraic counterpart (P) : M∞. Lazard’s lemma provides sufficient conditions for the first criterion to hold; Rosenfeld’s lemma provides sufficient conditions for the second criterion to hold. In this paper, we prove substantially strengthened versions of both of these results, and apply them to Mansfield’s algorithms (Mansfield, 1993)for solving systems of PDEs.