I have a finite abstract simplicial complex which I hope linearly embeds in $\mathbb{R}^3$. Is there a program/software that can determine if such a geometric realization exists, say just by feeding the program the complex's facets and it giving coordinates of an embedding?
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1$\begingroup$ I am not an expert in polymake but I would guess it can do this: see en.wikipedia.org/wiki/Polymake $\endgroup$Sam Hopkins– Sam Hopkins ♦2025-05-12 22:48:21 +00:00Commented May 12 at 22:48
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1$\begingroup$ I think you can do that with a program capable of solving first order theory of reals; after all, linear embedding of a simplicial complex is just a system of polynomial inequalities... But you'll probably need a script to generate those inequalities from incidence data, and the problem is known to be not much better than PSPACE — so if your complex is reasonably large, you're probably out of luck. $\endgroup$Denis T– Denis T2025-05-12 23:53:52 +00:00Commented May 12 at 23:53
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$\begingroup$ ...Also there's a paper by J. Renegar, "On the computational complexity and geometry of the first-order theory of the reals" in JSymbComp 1992, where you can (at least) learn how such a program should work. $\endgroup$Denis T– Denis T2025-05-12 23:58:08 +00:00Commented May 12 at 23:58
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