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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.

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  • $\begingroup$ Aren't F4 and F5 purely for polynomial rings? Or do they have differential analogues? $\endgroup$
    – arsmath
    Commented Dec 16, 2014 at 8:56
  • $\begingroup$ They are for polynomial rings. Specifically, they are for calculating Groebner bases for a given ideal. However, they use ideas drawn from differential algebra, specifically involutions (whatever that means). $\endgroup$
    – Benjamin Roycraft
    Commented Dec 16, 2014 at 18:34

5 Answers 5

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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.

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Another more recent book is Galois Theory of Linear Differential Equations by van der Put & Singer (Springer, 2003). You can find an online copy at one of the authors' home pages.

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Andy Magid's Lectures on Differential Galois Theory is pretty good. Ritt and Kaplansky have older introductions that are also not bad (I think I originally learned it from Kaplansky).

The more recent work, such as Hrushovski's proof of the Mordell-Lang conjecture for function fields, is heavily model-theoretic, so it would be worthwhile to do some general reading in model theory, if you already haven't. Boris Zilber has a lengthy manuscript on Zariski geometries on his webpage that talks about the big picture, though most of it is not particularly focused on differential algebra.

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Differential algebra and diophantine geometry by Buium, https://zbmath.org/?q=an:0870.12007.

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Besides the book by van der Put and Singer, which is mentioned above, I recommend that you have a look at Werner Seiler's Involution. This comprehensively covers many topics in the areas you list.

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