Tropical geometry in computations

From CRCG-Wiki

Tropical geometry in computations

A problem which has recently gained lots of attention is the computation of the implicit ideal of the image of a polynomial map with combinatorial methods. Given a map from an affine variety we are interested in the ideal defining its image. Such a computation can be done by elimination, which involves Gröbner bases. But Gröbner bases algorithms are in general very hard and a computation might fail for large examples. Therefore a new approach is to understand the combinatorics of the implicit ideal first. If the image is a hypersurface, the implicit ideal is generated by one polynomial and we can aim at understanding its Newton polytope. For more general ideals the tropical variety plays the role of the (dual fan of the) Newton polytope. There is an algorithm by Bernd Sturmfels, Jenia Tevelev and Josephine Yu which allows to compute implicit ideals via tropical geometry (see here).

Tropical computational methods have been applied to find equilibria of the gravitational n-body problem for n=4. Such an equilibrium is a configuration of n point masses which can rotate rigidly such that the outward centrifugal forces and the gravitational attractions cancel exactly. An equilibrium is a solution of a certain system of polynomial equations which encode that the centrifugal force and the gravitation cancel. But already for small n these equations become very complicated. Even whether the number of solutions is finite has been unknown for a long time, and is still unknown for arbitrary n. For n=4 a solution has been found by Marshall Hampton and Richard Moeckel using computational tropical arguments (see here).

This picture shows an equilibrium for 4 collinear bodys with weights 1, 2, 3 and 4.