I'm looking for a couple good textbooks covering differential algebra. I'm a prospective Ph.D. student, and this is potentially applicable to my specialization. As such, I'm not afraid of depth; I've got a few years to work through it all.
Specifically, I'm interested in the connections between differential algebra and algebraic geometry; a focus on computation would be appreciated, as well. Any solid foundational books, along with any covering the above topics would be appreciated.
EDIT: I must clarify: I'm looking mostly for books covering the general theory of algebraic structures equipped with differential operators, with other books specifically supporting the topics given. When I says "connections to algebraic geometry" I speak about Grobner bases. I'm specifically looking to understand the f4 and f5 algorithms.
EDIT 2: Also, I am not a pure math student. I am studying the applications of algebraic techniques to problems in statistics. I'm fine with abstract topics, but only as long as they provide good insight towards concrete problems. I try to stay away from anything that can't be implemented on a computer.