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Reformulation of a semialgebraic set, without quantifiers

asked 2015-10-27 10:26:42 +0200

pang gravatar image

According to wikipedia:

The Tarski–Seidenberg theorem states that a set in (n + 1)-dimensional space defined by polynomial equations and inequalities can be projected down onto n-dimensional space, and the resulting set is still definable in terms of polynomial identities and inequalities.

That's great, and my question is if it can be done in practice, in a particular case. I have read that the original Tarski–Seidenberg algorithm is of little practical use, but maybe my problem is tractable with another algorithm:

We have a finite set of points in C^2 (z_i,w_i), and I want to project onto the first C coordinate the intersection of the cones {(z,w): |z-z_i|>|w-w_i|}. It would be awesome, for example for plotting the set, if I could actually get the polynomial inequalities that define the projection. Getting only the higher dimensional strata is fine (an open set in C).

Any idea? Thanks in advance.

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For the record: I couldn't get qepcad to make a projection of codimension 2 using even 4 points, but I did get some significant results by restricting to a three dimensional space, so that qepcad only projects along lines.

pang gravatar imagepang ( 2015-10-28 12:27:21 +0200 )edit

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answered 2015-10-27 13:20:23 +0200

tmonteil gravatar image

updated 2015-10-27 13:20:48 +0200

You should install and use qepcad. It has been updated recenty, so it should work, just install the latest version of Sage and run sage -i qepcad, see this page for more information.

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I downloaded the program and I'm calling it from the command line. I can't get it to work on any non-trivial example (it does work with all the examples from its documentation). The output is:

Failure occurred in: GCSI (final check) Reason for the failure: Too few cells reclaimed.

pang gravatar imagepang ( 2015-10-27 16:19:49 +0200 )edit

It just finished installing. Thanks for taking care of this packege, Thierry!

pang gravatar imagepang ( 2015-10-27 16:31:11 +0200 )edit

Great. Please do not hesitate to tell if there is some issue with this package.

tmonteil gravatar imagetmonteil ( 2015-10-27 16:38:56 +0200 )edit

I get a "RuntimeError: unable to start QEPCAD". I upgraded to sage 6.9 with

./sage -upgrade

and I installed qepcad with

./sage -i qepcad

The first upgrade reported errors when building the html docs, but nothing else, and everything seems to work fine. The installation of qepcad produced no errors (I think), and there are qepcad and saclib folders in sage/pkgs.

pang gravatar imagepang ( 2015-10-28 10:26:30 +0200 )edit

I have bug traces etc: should I open a ticket?

pang gravatar imagepang ( 2015-10-28 10:36:28 +0200 )edit

answered 2015-10-27 10:30:09 +0200

pang gravatar image

updated 2015-10-27 16:31:54 +0200

I found the optional package qepcad:

mentioned in this site:

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Asked: 2015-10-27 10:26:42 +0200

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Last updated: Oct 27 '15