Maximal Mediated Sets [polymake wiki]

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====== Maximal Mediated Sets ======

**max_mediated_sets** is a polymake extension allowing creation and inspection of the maximal mediated set associated to a polynomial. It accompanies [[paper_link|this paper]]. The parallel enumeration algorithm described there will be made available soon, too.

===== Prerequisites =====

You'll need a running ''polymake'' installation with the [[/external_software?#bundled_extensions_for_polymake|bundled extension]] ''libnormaliz'' for computing lattice points. ''libnormaliz'' comes bundled with ''polymake'', so if you did not do anything special like deliberately installing the minimal version, you should already have ''libnormaliz'' and do not have to do any extra work. See [[/download/start#get_latest_stable_release34|here]] for instructions on how to get and install ''polymake''.

===== Download =====

Use git to obtain your copy:
<code>
git clone https://git.polymake.org/extensions/max_mediated_sets
</code>

===== Installation =====

Assuming the directory you pulled the git to has the path ''~/max_mediated_sets''. To use the extension, fire up your ''polymake'' interactive shell and enter the following command:

<code>
>  import_extension "~/max_mediated_sets";
</code>

After this, you should be able to use the functionality provided by this extension.

===== Usage =====
This extension allows to compute maximal mediated sets from [[/documentation/polytope/pointconfiguration|PointConfigurations]], that is, all lattice points that are the midpoint of two all-even points.

=== Newton Configurations ===
This is interesting for Newton polytopes of polynomials, so there is a function producing the point configuration whose convex hull is the Newton polytope of a polynomial ''p''. For example, the following creates said point configuration for the polynomial $1+x_0^2+x_1^6$: 
<code>
> $p=new Polynomial('1+x_0^2+x_1^6');
> $n = newton_configuration($p);
> print $n->CONVEX_HULL->VERTICES;
1 0 0
1 0 1
1 2 0
</code>
Note that the vertices are handled in homogeneous coordinates. The polynomial gets stored in the attachment named ''POLYNOMIAL'':
<code>
> print $n->get_attachment("POLYNOMIAL");
x_0^2 + x_1^6 + 1
</code>
The coefficients of the polynomial get stored in the property ''TRIANGULATION.WEIGHTS''.
<code>
> print $n->TRIANGULATION->WEIGHTS;
1 1 1
</code>

=== Maximal Mediated Sets ===
You can compute the maximal mediated set of any point configuration whose convex hull has all-even vertices, like the one we created in the previous section.
<code>
> print $n->MAXIMAL_MEDIATED_SET;
1 0 0
1 0 2
1 0 4
1 0 6
1 2 0
1 1 3
1 1 2
1 0 3
1 1 0
1 0 1
1 1 1
1 0 5
</code>
You can look at the vanishing ideal, which in this case has two generators:
<code>
> print $n->VANISHING_IDEAL_GENERATORS->[1];
x_1^3 - x_3
> print $n->VANISHING_IDEAL_GENERATORS->[2];
x_0 - 1
</code>
You might also be interested in the dimension of the toric variety:
<code>
> print $n->TORIC_VARIETY_DIMENSION;
9
</code>