Let X be a closed subvariety of dimension d of the torus T of dimension n, defined by an ideal I in the Laurent polynomial ring. Its tropical variety Trop(X) is the set of weight vectors w whose initial ideal inw(I) contains no monomial. By the theorem of Bieri and Groves (1984) it is the support of a rational polyhedral fan of pure dimension d. Given a fan Σ in the cocharacter space, let X(Σ) denote the closure of X in the toric variety of Σ. Tevelev proved that X(Σ) is complete if and only if the support of Σ contains Trop(X), and called X(Σ) a tropical compactification when it is complete and the multiplication map from T × X(Σ) to the toric variety is faithfully flat. In that case the support of Σ equals Trop(X), and X(Σ) meets the orbit of each cone σ in a non-empty subvariety of pure dimension d − dim σ.

Two further facts make the definition computable. First, the intersection of X(Σ) with the orbit of σ is the quotient of the initial degeneration, the subvariety of T defined by inw(I) for w in the relative interior of σ, by the subtorus attached to σ (Maclagan and Sturmfels, 2015). Second, X is called schön when the multiplication map is smooth for one, equivalently for every, tropical compactification; this holds exactly when all initial degenerations are smooth, and Luxton and Qu (2011) proved that for a schön variety every fan supported on Trop(X) gives a tropical compactification. If in addition the fan is unimodular, the compactification is smooth with simple normal crossing boundary.

Gfan computes Gröbner fans and tropical varieties of polynomial ideals with rational coefficients, using the algorithms of Fukuda et al. (2007) and Bogart et al. (2007). For a prime homogeneous ideal it finds a starting cone and traverses the tropical variety through its facets, exploiting symmetry when a group action is supplied. polymake handles polyhedral fans, their refinements and their combinatorics (Gawrilow and Joswig, 2000), and its tropical component includes the intersection-theoretic tools of Hampe (2014). Initial ideals, saturations, primary decompositions and Jacobian ideals are available in Macaulay2 (Eisenbud et al., 2002) and Singular (Greuel and Pfister, 2008), both of which can call Gfan, and in OSCAR, which combines polyhedral and algebraic functionality in one system (Decker et al., 2025).

The input is a finite set of Laurent polynomials. Denominators are cleared and the resulting polynomial ideal is saturated with respect to the product of the variables, so that it defines the closure of X in affine space and has no component in the coordinate hyperplanes. Primality should be tested, because the traversal in the next step relies on the connectedness in codimension one of the tropical variety of an irreducible variety; a reducible input is decomposed and each component treated separately. The dimension d is recorded. If a torus acts on X by translation, the corresponding sublattice will appear later as the lineality space, and it is useful to know it in advance.

Gfan works with homogeneous ideals, so the ideal is homogenised with one new variable and saturated again; the tropical variety of X is recovered by intersecting with the hyperplane where the new coordinate vanishes, or equivalently by passing to the quotient by the all-ones vector. The output is a list of rays, a lineality space, the maximal cones and their multiplicities. Three checks cost almost nothing and catch most errors of input: every maximal cone has dimension d plus that of the lineality space, the weighted fan satisfies the balancing condition at each codimension-one cone, and the number of orbits of cones agrees with the symmetry group that was declared.

The fan returned by Gfan is the restriction of the Gröbner fan, on whose cones the initial ideal is constant. A tropical variety has no canonical coarsest fan structure in general, and the Gröbner structure may be finer or coarser than is convenient. After the quotient by the lineality space the fan is passed to polymake. If a smooth compactification is wanted, the fan is refined by stellar subdivisions until every cone is generated by part of a lattice basis (Cox et al., 2011); if only the boundary combinatorics matter, a simplicial refinement is enough. Any refinement of a fan that gives a tropical compactification again gives one, so refining is always safe, whereas coarsening has to be justified by the later steps.

For each cone of the Gröbner structure, or one representative per symmetry orbit, a weight vector in the relative interior is chosen and the initial ideal is computed. The initial degeneration is the subscheme of the torus that it defines, so smoothness is tested by the Jacobian criterion after saturating the ideal of minors by the product of the variables. Since the initial degeneration is invariant under the subtorus of the cone, the test can be run on a slice, which lowers the number of variables. If every test succeeds, X is schön and the theorem of Luxton and Qu applies to whatever fan was fixed in Step 3. If some fail, the failing cones are recorded; the variety may still have tropical compactifications, but they must be certified differently.

Let σ be a unimodular cone of dimension k with primitive generators extended to a basis of the cocharacter lattice. The basis defines a monomial change of coordinates under which the affine toric chart of σ becomes the product of affine k-space and a torus of dimension n − k. Substituting into the generators of I and saturating with respect to the first k coordinates gives the ideal of X(Σ) in that chart. Setting a subset of those coordinates to zero gives the strata lying over the faces of σ, and these can be compared with the initial ideals of Step 4 as a consistency check. Only maximal cones need to be processed, one per orbit.

In the schön case this step confirms what the theory predicts: each stratum is non-empty and smooth of dimension d − dim σ, and in a unimodular chart the closure is smooth and meets the coordinate hyperplanes transversally. In the general case the same computations serve as necessary conditions for flatness of the multiplication map. A stratum that is empty or of excessive dimension shows that the fan is not tropical for X along that cone, and the procedure returns to Step 3 to subdivide it. Passing these checks does not by itself prove flatness, and for varieties that are not schön the output should be reported as a candidate tropical compactification unless flatness is established by a separate argument.

The fan with its multiplicities, the weight vectors used, the initial ideals, the chart ideals and the exact software versions are written to files in the native formats of polymake or OSCAR. Because the computations are exact and deterministic once the term orders and weight vectors are fixed, a reader with these files can repeat every step. Intersection numbers on the compactification, when needed, can be computed from the fan through the Minkowski weights of Fulton and Sturmfels (1997).