Advances in Elimination Theory for Algebraic Differential and Difference Equations

We review our work on elimination theory for algebraic differential and difference equations over the past decade, highlighting some key developments. Our journey began with the initial establishment of the theory of differential Chow forms and differential Chow varieties. These basic concepts enable us to represent geometric objects such as solution sets of algebraic differential equations using their respective differential Chow coordinates. We then introduce the theory of sparse differential resultants for algebraic differential equations along with a single-exponential algorithm for efficiently computing sparse differential resultants. We further present the theory of sparse difference resultants by comparison with sparse differential resultants. Finally, we provide the effective differential-difference Nullstellensatz and effective elimination for algebraic differential-difference equations, demonstrating that the consistency checking of algebraic differential-difference equations in sequence rings is decidable.