Tropical Basis Computation

[satya123999]

Dear all,

I was wondering if there exists any tutorial for computing the tropical basis in polymake ?

For example:
I have two tropical hypersurfaces defined by:
min(a1,a2,4*a1+1);
min(a1,a2,2*a2+1);

and I am interested to compute the tropical basis (whose intersections define the tropical variety)

Best Regards,
Satya

[hampe]

Hello Satya!

Polymake definitely cannot do this at the moment (In any case, since this is an algebraic computation rather than a polyhedral one, this would be a task for Singular or GFan).

Polymake has interfaces to these two software systems, but as far as I know, neither currently has a function for computing general tropical bases (gfan can do this in the case where the resulting variety is a curve). I suppose the computation of a tropical basis is implicit in some of the algorithms used to compute tropicalizations - but there is no direct access.

Do you really need the tropical basis or do you just need the tropicalization? It is also possible, that the computation may be easier in the particular cases you want to compute. In many cases, a universal Gröbner basis can serve as a tropical basis.

[satya123999]

Hello,

Thank you for your quick reply.

Do you really need the tropical basis or do you just need the tropicalization? It is also possible, that the computation may be easier in the particular cases you want to compute. In many cases, a universal Gröbner basis can serve as a tropical basis.

By tropicalization, you meant computing the tropical variety ? If yes, then is it possible to compute it for two tropical hypersurfaces (which I mentioned earlier) ? I may have to install gfan to enable necessary algebraic computations ?

Best Regards,
Satya

[hampe]

Yes indeed, tropicalization = tropical variety of an ideal. For computing that, I do recommend installing gfan (even if you wanted to use it through our interface, you'd still need to install it separately). Singular should also have its own tropicalization functions soon, but I don't know if that is publicly available yet.

You can find the source and a manual here: http://home.math.au.dk/jensen/software/gfan/gfan.html. Some Linux distributions have a gfan package, but that's often an old version, so I'd recommend trying to install the newest version 0.5 by hand.

Gfan can compute tropicalizations of arbitrary (homogeneous) ideals, though it may be slow if your example is large. If you only want the set-theoretic intersection of your two tropical hypersurfaces, you can actually do that in polymake:

• Compute the two tropical hypersurfaces as described in the tutorial https://polymake.org/doku.php/tutorial/apps_tropical, e.g.:

  Code: Select all

  $h1 = new Hypersurface<Min>(POLYNOMIAL=>toTropicalPolynomial("min(a1+3a0,a2+3a0,4*a1+1)")); $h2 = new Hypersurface<Min>(POLYNOMIAL=>toTropicalPolynomial("min(a1+a0,a2+a0,2*a2+1)"));
  Note that you need to homogenize the tropical polynomials!
• Compute

  Code: Select all

  $s = set_theoretic_intersection($h1,$h2);
• It will tell you that it consists of a point and a half-line:

  Code: Select all

  print $s->VERTICES; 0 1 1 1 -1/3 -1/3 1 -1/2 -1 print $s->MAXIMAL_POLYTOPES; {0 1} {2}

[satya123999]

Thank you very much.

So, with polymake, I can compute the tropical prevariety using the command "set_theoretic_intersection" and for variety I need gfan in addition. I will try it

Best Regards,
Satya

[hampe]

Indeed. Note that Gfan also has a command to compute the prevariety, called

Code: Select all

gfan_tropicalintersection