Self-study in real algebraic geometry

Question

I would like an introductory book, a pdf or an online course to self-study real algebraic geometry. My background is the most classical one: I've already studied this book and 80% of this book.

Thanks in advance

EDIT1

Of course, if my background is weak, which books should I read before in order to begin to study real algebraic geometry?

EDIT2

I know that real algebraic geometry is a huge area, but as I said above I'm looking for a book to introduce me the basic techniques. For example, Algebraic geometry is a very broad area, but Fulton is a good introduction to this subject, I would like a "fulton of real algebraic geometry" or something like that.

Related: (undergraduate) Algebraic Geometry Textbook Recommendations and Best Algebraic Geometry text book? (other than Hartshorne) — Mark Fantini, Feb 11 '14 at 00:44
Real algebraic geometry is very different from ordinary algebraic geometry; it's much more like model theory, and in particular I don't think references on ordinary algebraic geometry will be all that helpful. — Qiaochu Yuan, Feb 11 '14 at 00:46
@QiaochuYuan can you help me with the background? I have a notion of scheme theory also. — user42912, Feb 11 '14 at 00:50
@Fantini thank you for the recommendations, but I would like something more precise. — user42912, Feb 11 '14 at 00:52
Why specifically do you want to study real algebraic geometry? If you want to study it to do algebraic geometry with pictures or prove things about real geometric constructions (i.e. manifolds) using algebraic geometric techniques, I am unsure if it is what you desire. It's a very specific topic and as far as I know a relatively niche one. — PVAL-inactive, Feb 11 '14 at 01:05
@PVAL For example, Algebraic geometry is a very broad area, but Fulton is a good introduction to this subject, I would like a "fulton of real algebraic geometry" or something like that. — user42912, Feb 11 '14 at 01:34
Bochnak's "Real Algebraic Geometry" is a good book. However I don't think that some introductory book exists. You will have better luck if you ask some logicians, though. — user40276, Feb 11 '14 at 01:48
@user40276 logicians? I didn't understand what did you mean. — user42912, Feb 11 '14 at 01:53

score 14 · Answer 1 · edited Jul 02 '22 at 12:50

Here are a few options:

Real Algebraic Geometry by Bochnak, Coste and Roy.

This seems to be the standard reference for Real Algebraic Geometry. Most of the chapters(at least the first 5) should be accessible with a bit of work. Later chapters will require a bit more background.

Introduction to the Real Spectrum by P.L. Clark

This gives an overview of some of the ideas behind real algebraic geometry. It starts by defining what ordering of rings are and how they connect to geometry. Moves on something called the real spectrum of a ring together to results related to it.

Introduction to Real Algebra by T.Y. Lam

This is one my favorites intro papers. It is clearly written and presents the material well. It is a bit dated but I like how it treats the Real Spectrum. Might be a bit too advanced, specially if you have never seen scheme theoretic approach to algebraic geometry.

Real Algebraic Sets by M. Coste

Not sure if this is an intro to the subject but it gives a quick overview of semi-algebraic and real algebraic sets and discusses some topological ideas related to it. For example, how in low-dimension we can characterize real algebraic sets.

algorithmic approach:

Introduction to Semi-algebraic geometry by M. Coste.

Semi-algebraic geometry is often used as a synonym for real algebraic geometry. This gives you a quick intro together with some of its computational tools.

Algorithms in Real Algebraic Geometry by Basu, Pollack and Roy

Similar in spirit to the above, but a lot more comprehensive. Contains a lot of the background material. See also a more recent online version for updates and fixes.

Remark:

You might not be find a complete linear path to learning geometry. I am not sure if this is 100% sound advice but just get stuck in. If you then find some material which you haven't encountered you can take a small detour.