A Differential Algebra Sampler
University of Pennsylvania
We discuss several problems involving differential algebraic varieties and ideals in differential polynomial rings. The first one is the completeness of projective differential varieties. We consider examples showing the failure to generalize (even in the finite-rank case) of the classical “fundamental theorem of elimination theory”. We also treat identification of complete differential varieties and a connection to the differential catenary problem. We finish by examining what proof-theoretic techniques have to say about the constructive content of results such as the Ritt-Raudenbush basis theorem and differential Nullstellensatz. Our remarks include work with James Freitag and Omar León-Sánchez as well as ongoing work with Henry Towsner.