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..

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    $\begingroup$ 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, 2014 at 10:15
  • $\begingroup$ @bournol Thanks! Will take a look... $\endgroup$
    – Igor Rivin
    Mar 10, 2014 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, 2014 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, 2014 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, 2014 at 14:38


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