Tropical -- the main M2 package for tropical computations

Description

This is the main M2 package for all tropical computations. This uses Anders Jensen's package gfan, Michael Joswig's package Polymake, and also internal M2 computations.

The package defaults to using the min convention for tropical geometry. To switch to the max convention, reload the package using the command loadPackage("Tropical",Configuration=>{"tropicalMax"=>true});

The main command is tropicalVariety.

To use the Polymake commands see the Polymake interface instructions.

Contributors

The following people have also contributed to the package:

Authors

Version

This documentation describes version 1.0 of Tropical.

Source code

The source code from which this documentation is derived is in the file Tropical.m2. The auxiliary files accompanying it are in the directory Tropical/.

Exports

Types
- TropicalCycle -- a Type for working with tropical cycles
Functions and commands
- BergmanFan -- the Bergman fan of a matroid
- isBalanced -- checks whether a tropical cycle is balanced
- isTropicalBasis -- checks if a list of polynomials is a tropical basis for the ideal they generate
- multiplicities -- returns the list of multiplicities on maximal cones in a tropical cycle
- stableIntersection -- computes the stable intersection of two tropical varieties
- tropicalCycle -- constructs a TropicalCycle from a Fan and a list with multiplicities
- tropicalPrevariety -- the intersection of the tropical hypersurfaces
- tropicalVariety -- the tropical variety associated to an ideal
- visualizeHypersurface -- visualize the tropical hypersurface of the given polynomial
Methods
- ambDim(TropicalCycle) -- computes the dimension of the ambient space of a tropical cycle
- cones(ZZ,TropicalCycle) -- computes the cones of a tropical cycle
- dim(TropicalCycle) -- computes the dimension of a tropical cycle
- fan(TropicalCycle) -- outputs the fan associated to the tropical cycle
- fVector(TropicalCycle) -- computes the fVector of a tropical cycle
- "isBalanced(TropicalCycle)" -- see isBalanced -- checks whether a tropical cycle is balanced
- isPure(TropicalCycle) -- checks whether a tropical cycle is pure
- isSimplicial(TropicalCycle) -- checks whether a tropical cycle is simplicial
- "isTropicalBasis(List)" -- see isTropicalBasis -- checks if a list of polynomials is a tropical basis for the ideal they generate
- linealitySpace(TropicalCycle) -- computes the lineality space of a tropical cycle
- maxCones(TropicalCycle) -- computes the maximal cone of a tropical cycle
- "multiplicities(TropicalCycle)" -- see multiplicities -- returns the list of multiplicities on maximal cones in a tropical cycle
- rays(TropicalCycle) -- computes the rays of a tropical cycle
- "stableIntersection(TropicalCycle,TropicalCycle)" -- see stableIntersection -- computes the stable intersection of two tropical varieties
- "tropicalCycle(Fan,List)" -- see tropicalCycle -- constructs a TropicalCycle from a Fan and a list with multiplicities
- "tropicalPrevariety(List)" -- see tropicalPrevariety -- the intersection of the tropical hypersurfaces
- "tropicalPrevariety(List,List)" -- see tropicalPrevariety -- the intersection of the tropical hypersurfaces
- "tropicalVariety(Ideal)" -- see tropicalVariety -- the tropical variety associated to an ideal
- "visualizeHypersurface(RingElement)" -- see visualizeHypersurface -- visualize the tropical hypersurface of the given polynomial
Symbols
- ComputeMultiplicities -- option to compute the multiplicities in case they ideal is not prime
- IsHomogeneous -- option to declare if the input ideal is homogeneous
- Prime -- option to declare if the input ideal is prime
- Symmetry -- option to declare if the input ideal has symmetries
- "Valuation" -- see visualizeHypersurface -- visualize the tropical hypersurface of the given polynomial

For the programmer

The object Tropical is a package.