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# gfanTropicalTraverse -- polyhedral data describing a tropical variety

## Synopsis

• Usage:
gfanTropicalTraverse(L)
• Inputs:
• Optional inputs:
• disableSymmetryTest
• nocones
• stable
• symmetry
• symsigns
• Outputs:
• F, an instance of the type Fan, describing the tropical variety of the given ideal

## Description

This method computes the tropical variety of a homogeneous ideal. Use gfanTropicalStartingCone to produce the pair of Groebner bases that gfanTropicalTraverse needs as input. It starts with the gfanTropicalStartingCone output which gives one of the Tropical Variety's cone, and traverses the rest of the Tropical Variety to find the remaining cones. It then outputs some key properties of the Tropical Variety. Cones gives the orbit of the cone if it exists. Multiplicities is the respective multiplicity of the maximal cones. And Maximal Cones refers to the cones of dimension DIM and DIM-1 and again gives the orbit of the cones. Changed in summer 2014: gfanTropicalTraverse now uses the min convention (so rays are -1 of the original gfan output). It currently returns a HashTable rather than a Fan.

 i1 : QQ[x,y,z] o1 = QQ[x..z] o1 : PolynomialRing i2 : P = gfanTropicalStartingCone {x+y+z} o2 = {{{z}, {y + z}}, {{z}, {x + y + z}}} o2 : List i3 : gfanTropicalTraverse P o3 = (Fan{...1...}, {1, 1, 1}) o3 : Sequence

gfan Documentation

This program computes a polyhedral fan representation of the tropical variety of a homogeneous prime ideal $I$. Let $d$ be the Krull dimension of $I$ and let $\omega$ be a relative interior point of $d$-dimensional Groebner cone contained in the tropical variety. The input for this program is a pair of marked reduced Groebner bases with respect to the term order represented by $\omega$, tie-broken in some way. The first one is for the initial ideal $in_\omega(I)$ the second one for $I$ itself. The pair is the starting point for a traversal of the $d$-dimensional Groebner cones contained in the tropical variety. If the ideal is not prime but with the tropical variety still being pure $d$-dimensional the program will only compute a codimension $1$ connected component of the tropical variety.
Options:
--symmetry:
Do computations up to symmetry and group the output accordingly. If this option is used the program will read in a list of generators for a symmetry group after the pair of Groebner bases have been read. Two advantages of using this option is that the output is nicely grouped and that the computation can be done faster.
--symsigns:
Specify for each generator of the symmetry group an element of ${-1,+1}^n$ which by its multiplication on the variables together with the permutation will keep the ideal fixed. The vectors are given as the rows of a matrix.
--nocones:
Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_ORBITS and MAXIMAL_CONES_ORBITS if --symmetry is used.
--disableSymmetryTest:
When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal.

--stable:
Traverse the stable intersection or, equivalently, pretend that the coefficients are genereric.
--interrupt value:
Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans).


## Ways to use gfanTropicalTraverse :

• gfanTropicalTraverse(List)

## For the programmer

The object gfanTropicalTraverse is .