Resultants: Algebraic and Differential

This report summarises ongoing discussions of the authors on the topic of differential resultants which have three goals in mind. First, we aim to try to understand existing literature on the topic. Second, we wish to formulate some interesting questions and research goals based on our understanding of the literature. Third, we would like to advance the subject in one or more directions, by pursuing some of these questions and research goals. Both authors have somewhat more background in nondifferential, as distinct from differential, computational algebra. For this reason, our approach to learning about differential resultants has started with a careful review of the corresponding theory of resultants in the purely algebraic (polynomial) case. We try, as far as possible, to adapt and extend our knowledge of purely algebraic resultants to the differential case. Overall, we have the hope of helping to clarify, unify and further develop the computational theory of differential resultants.