Does anyone have suggestions/experience for any software packages to study real algebraic varieties (for example, counting connected components of hypersurfaces, figuring out the topological type of these components, etc)? In principle, all this should be possible using Cylindrical Algerbaic Decomposition and variants thereof, but..

3$\begingroup$ Take a loot at the package "qepcad" (it implements quantifier elimination) for sage. It is available as an experimental package in the default installation (run "sage experimental" to see all available packages in your installation). You can get an idea of what you can do reading the beginning of flask.sagenb.org/src/interfaces/qepcad.py $\endgroup$ – boumol Mar 10 '14 at 10:15

$\begingroup$ @bournol Thanks! Will take a look... $\endgroup$ – Igor Rivin Mar 10 '14 at 14:09

$\begingroup$ In case you use it let us know (in the future) how useful is this software for doing the computations you are interested. I know the software but I have just used it for some toy examples while lecturing about Tarski's theorem on real closed fields. I have always wondered what is its potential from the perspective of "algebraic geometers" $\endgroup$ – boumol Mar 11 '14 at 8:51

$\begingroup$ @bournol I would not describe myself as an algebraic geometer, but there are some interesting statistical questions on the topology of random real algebraic varieties. These sorts of studies always stress the software, so I will certainly report... $\endgroup$ – Igor Rivin Mar 11 '14 at 12:19

$\begingroup$ For what it's worth, I see that recent versions of Maple, especially Maple 16 onwards, have expanded abilities in finding real solutions to polynomial systems. See, for instance, the RegularChains package. $\endgroup$ – J W Apr 25 '14 at 14:38