Does anybody know of any resources (books, classes, lecture notes, or anything) about the general theory of computer algebra systems (e.g. mathematica, sympy)?
"Introductory" materials are preferred, but I realize that with such a specialized subject anything is bound to be fairly advanced.
Here's one link from Wikipedia: Computer Algebra Systems
And another here: http://www.math.wpi.edu/IQP/BVCalcHist/calc5.html
I've found Algorithms for computer algebra by K.O. Geddes... to be pretty useful. I'm a junior undergrad with a light math background doing work on OpenAxiom (a CAS). Get ready for some heavy, heavy math though, my best advice is to have a couple books if only to have a different perspective if you get "stuck".
It might help if you suggest what you're looking into, what areas you're interested in, etc.
"General Theory" of CAS is a pretty huge scope for a question. That being said, I'll do my best to cover as much as I can in the hopes that something helps you find what you're looking for :)
The proceedings of the ISSAC and SIGSAM groups would no doubt have some good stuff about techniques for building CAS systems. A list of various topics in the general area of CAS building is available here: http://www.cs.berkeley.edu/~fateman/algebra.html
If you're more looking for information on how to code some of the math involved, I'm a fan of the "Numerical Recipes" series; it provides sample code and a reasonably decent explanation of math in a wide range of topics. Last I checked, an online version of an older revision of the book was available here: http://www.nrbook.com/a/bookcpdf.php (Note that this is the "Numerical Recipes in C" form of the book; there are versions in other languages as well).
For building a CAS in general, one place to start might be here: "Building a computer algebra environment by composition of collaborative tools" by Kajler and Safir; Another place you might check is here: http://www.math.wpi.edu/IQP/BVCalcHist/calc5.html where a high-level description of how a few folks implemented a CAS is listed.
The other thing you might try is diving into the code for a few of the open source CAS projects that exist: YACAS (Yet Another Computer Algebra System : Java), Axiom, etc. I like the list here: http://en.wikipedia.org/wiki/List_of_computer_algebra_systems
Hope something in there was useful!
The basics are nicely covered in PAIP; the source code is free online -- see particularly the source files with 'macsyma' in the name. Topics include rewrite-rule systems, simplification using canonical forms, integration and differentiation, and compiling and memoizing rewrite rules for speed.