polytope over quadratic number fields

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joswig
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Joined: 24 Dec 2010, 11:10

Re: polytope over quadratic number fields

Postby joswig » 23 Aug 2019, 12:08

Indeed, this is a gap in the documentation. Thanks for pointing this out. I just made an addition to https://polymake.org/doku.php/user_guid ... oordinates, which will become visible not before tomorrow (after the automated update). Here it comes.

polymake has limited support for other ordered fields for coefficients of polytopes (and linear programs). An example are quadratic extensions to the rationals. As an example we show how to construct a regular triangle.

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polytope> $t=new QuadraticExtension(0,1/2,3);
This gives the number \sqrt{3}/2. More generally, you can construct the value a+b(\sqrt r) via QuadraticExtension(a, b, r). Note that currently polymake is restricted to compute with single extensions, i.e., you cannot mix distinct square roots.

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polytope> print $t*$t; 3/4

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polytope> $triangle = new Polytope<QuadraticExtension>(POINTS=>[[1,0,1],[1,$t,-1/2],[1,-$t,-1/2]]);
That regular triangle can be treated like any polytope. The following computes the area, which results in \frac{3}{4}\sqrt{3}.

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polytope> print $triangle->VOLUME; 0+3/4r3


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