Real algebraic geometry vs. algebraic geometry

Question

This question is predicated on my understanding that real algebraic geometry (henceforth RAG) is the version of algebraic geometry (AG) one gets when replacing (esp. algebraically closed) fields with formally real (esp. real closed) fields. This makes for substantial differences in the theory because such fields can be ordered, and with order comes the notion of a semialgebraic set and a stronger topology.

I am aware that there is a notion of "real spectrum" analogous to the traditional spectrum of a commutative ring, though I'm not terribly familiar with either. I assume this allows one to glue things together and define "real schemes" or some such thing. Or if not, I assume the reason this doesn't work is something one would learn in the study of RAG.

My question: Given the differences in the theories, how well does one need to understand "traditional" AG to study RAG? Are there references (preferably books) which introduce RAG at an abstract level without assuming much knowledge of AG? Or is asking for this like when people ask how they can learn about motives without knowing about AG first?

I already have Basu, Pollack, and Roy's Algorithms in Real Algebraic Geometry but I'm looking for something less algorithmic.

Answer 1

Real algebraic geometry comes with its own set of methods. While keeping in mind the complex picture is sometimes useful (e.g. for any real algebraic variety X, the Smith-Thom inequality asserts that $b(X(\mathbb{R})) \leq b(X(\mathbb{C}))$, where $b(\cdot)$ denotes the sum of the topological Betti numbers with mod 2 coefficients), most of the technique used are either built from scratch or borrow from other areas, such as singularity theory or model theory.

The literature is a lot smaller for RAG than for traditional AG; the basic reference is the book by Bochnak, Coste and Roy (preferably the English-language edition which is more recent by more than 10 years, and has been greatly expanded). The book covers in particular the real spectrum, the transfer principle (which makes non-standard methods really easy), stratifications and Nash manifolds, among other topics. Michel Coste also has An Introduction to Semialgebraic Geometry available on his webpage a very short treatment of some basic results, enough to give you a first impression.

Other interesting books tend to be shorter and more focused than BCR, dealing with a specific aspect; e.g. Prestel's Positive polynomials. (dealing mostly with results such as Schmudgen's theorem), and Andradas-Brocker-Ruiz Constructible sets in real geometry (dealing mostly with the minimum number of inequalities required to define basic sets). The book by Benedetti and Risler is very interesting and concrete; I found some passages very useful and some results are hard to find in other books (the sections on additive complexity of polynomials are very thorough), but it is a bit scatterbrained for my taste.

As the name indicates, the book by Basu Pollack and Roy is entirely focused on the algorithmic aspects. It's a very good book, and you may still pick up some of the theory in there, but it does not sound like what you are after right now.

As for o-minimality, there again, Michel Coste's webpage contains an introduction that nicely complements van den Dries's book. I would hesitate to bundle o-minimality with real algebraic geometry. In some respects, the two domains are undoubtedly close cousins, and o-minimality can be seen as a wide-ranging generalization of real algebraic structures; on the other hand, each disciplines has also its own aspects and problems that do not translate all that well into the other.

I'm being verbose as usual. Still, I hope it helps.

Answer 2

There's also Partially Ordered Rings and Semi-Algebraic Geometry by Brumfiel and Real algebraic and semi-algebraic sets by Benedetti and Risler.

Answer 3

For an easy introduction to RAG, you could read van den Dries book "Tame topology and o-minimal structures": he treats the more general notion of o-minimal structures instead of real closed fields, and he does not uses any tool from AG.

Answer 4

When I first studied real algebraic geometry. I was more concerned with the algebra (probably even now). Similar to the classical algebraic geometry where students get to first learn commutative algebra before they do some algbraic geometry, I learned first real algebra (without the geometry). Real algebra alone is a big field and by the time I started real algebraic geometry it was a little late (so I practically did only real algebra during my PhD years). Still, if you do want to get the fundamentals of real algebra (before doing real algebraic and analytic geometry) and if you know some German, I would highly recommend the book of Knebusch and Scheiderer also available for free here. You will learn more about convex valuations, preorderings, partial orderings, real closed fields, cones, Artin Schreier theorem, the real spectra, the Harrison topology and constructible topology, than you would in a typical commutative algebra book. If you don't know any German then of course you could still read BCR (Bochnak, Coste and Roy). My PhD thesis has a preliminaries section that covers some basics you would need to go through. I could also recommend Brumfiel Partially ordered rings and semialgebraic geometry (which I fond sometimes more helpful than BCR).. and of course one could not forget the introductory material written by T.Y.Lam An introduction to Real Algebra (Rocky Mountain Journal of Mathematics, 1984, Vol.14, No.4, p.767-814).

Answer 5

The book Real Algebraic Varieties by Frederic Mangolte is worth checking out. It develops "traditional" algebraic geometry minding always the existence of $\mathbb{R}$.

In the introduction he comments the standard books of the field, such as Bochnak, Coste and Roy, and talks about the phenomenology of real algebraicity. It was very clarifying to me.

He does not talk about the semialgebraic side of things and mentions the book An introduction to semialgebraic geometry by Michel Coste which is freely available.