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.
Best Answer
Except for Buium's book these suggestions mostly cover only algebraic versions of linear differential equations and this is only a limited view of the theory developed by Kolchin and others. Kaplansky remains, I think, the best introduction to the basic algebra in rings with differential operators. There is also Kolchin's book "Differential Algebra and Algebraic Groups" although the latter part of this book is an exposition of algebraic groups Kolchin developed that is hard to follow. The first three chapters are useful though. Finally I would suggest looking at the sequence of papers by Jerry Kovacic who took on, until his untimely death, the task of reformulating differential algebra geoemetry from a scheme theoretic perspective.