Real algebraic geometry vs. algebraic geometry - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T00:49:54Z http://mathoverflow.net/feeds/question/29100 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/29100/real-algebraic-geometry-vs-algebraic-geometry Real algebraic geometry vs. algebraic geometry Noah Stein 2010-06-22T15:06:27Z 2010-08-19T05:17:40Z <p>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.</p> <p>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.</p> <p>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?</p> <p>I already have Basu, Pollack, and Roy's Algorithms in Real Algebraic Geometry but I'm looking for something less algorithmic.</p> http://mathoverflow.net/questions/29100/real-algebraic-geometry-vs-algebraic-geometry/29103#29103 Answer by lhf for Real algebraic geometry vs. algebraic geometry lhf 2010-06-22T15:30:30Z 2010-06-22T15:30:30Z <p>There's also <a href="http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9780521228459" rel="nofollow">Partially Ordered Rings and Semi-Algebraic Geometry</a> by Brumfiel and <a href="http://www.editions-hermann.fr/ficheproduit.php?lang=fr&amp;menu=&amp;ref=Math%25E9matiques+Real+algebraic+and+semi-algebraic+sets&amp;prodid=414" rel="nofollow">Real algebraic and semi-algebraic sets</a> by Benedetti and Risler.</p> http://mathoverflow.net/questions/29100/real-algebraic-geometry-vs-algebraic-geometry/29112#29112 Answer by Antongiulio for Real algebraic geometry vs. algebraic geometry Antongiulio 2010-06-22T16:24:17Z 2010-06-22T16:24:17Z <p>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.</p> http://mathoverflow.net/questions/29100/real-algebraic-geometry-vs-algebraic-geometry/36060#36060 Answer by Thierry Zell for Real algebraic geometry vs. algebraic geometry Thierry Zell 2010-08-19T05:17:40Z 2010-08-19T05:17:40Z <p>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 <em>X</em>, 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.</p> <p>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 <em>An Introduction to Semialgebraic Geometry</em> available on <a href="http://perso.univ-rennes1.fr/michel.coste/articles.html#books" rel="nofollow">his webpage</a> a very short treatment of some basic results, enough to give you a first impression.</p> <p>Other interesting books tend to be shorter and more focused than BCR, dealing with a specific aspect; e.g. Prestel's <em>Positive polynomials.</em> (dealing mostly with results such as Schmudgen's theorem), and Andradas-Brocker-Ruiz <em>Constructible sets in real geometry</em> (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.</p> <p>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.</p> <p>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.</p> <p>I'm being verbose as usual. Still, I hope it helps.</p>