Re: polytope over quadratic number fields
Posted: 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.
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.

That regular triangle can be treated like any polytope. The following computes the area, which results in \frac{3}{4}\sqrt{3}.
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.
Code: Select all
polytope> $t=new QuadraticExtension(0,1/2,3);
Code: Select all
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]]);
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polytope> print $triangle->VOLUME;
0+3/4r3