# gfanInterface -- a Macaulay2 interface to gfan

## Description

gfanInterface is an interface to Anders Jensen's Gfan software (available at http://home.imf.au.dk/jensen/software/gfan/gfan.html), which is a C++ program to compute the Groebner fan (i.e. all the initial ideals) of an ideal.

The main function in this package is gfan which computes all of the Groebner bases and initial ideals of a given ideal. A useful feature of this function is that it can handle symmetries in the ideal. If you want the geometric information of this list of Groebner basis, see gfanGroebnerCone.

Most of the functions in gfanInterface require MarkedPolynomialList marked polynomial lists as input. In a marked polynomial list, the leading term of each polynomial is distinguished. New users should read the the guide Conventions for calling methods with options. Since gfan is distributed with Macaulay2, one rarely needs to consult the guide for Installation and Configuration of gfanInterface.

Most of functions in the gfan package are accessible through this interface. If you wish to use one whose interface is not included here send a message to the package author. Also, please feel free to suggest changes to the parameter types and return types of each method.

## Version

This documentation describes version 0.4 of gfanInterface.

## Source code

The source code from which this documentation is derived is in the file gfanInterface.m2.

## Exports

• Types
• Functions and commands
• Methods
• "gfan(Ideal)" -- see gfan -- all reduced Groebner bases of a polynomial ideal
• "gfan(List)" -- see gfan -- all reduced Groebner bases of a polynomial ideal
• "gfan(MarkedPolynomialList)" -- see gfan -- all reduced Groebner bases of a polynomial ideal
• "gfanBuchberger(Ideal)" -- see gfanBuchberger -- reduced Groebner basis with respect to some monomial order
• "gfanBuchberger(List)" -- see gfanBuchberger -- reduced Groebner basis with respect to some monomial order
• "gfanBuchberger(MarkedPolynomialList)" -- see gfanBuchberger -- reduced Groebner basis with respect to some monomial order
• "gfanDoesIdealContain(Ideal,List)" -- see gfanDoesIdealContain -- check ideal membership by the division algorithm
• "gfanDoesIdealContain(List,List)" -- see gfanDoesIdealContain -- check ideal membership by the division algorithm
• "gfanDoesIdealContain(MarkedPolynomialList,List)" -- see gfanDoesIdealContain -- check ideal membership by the division algorithm
• "gfanFanCommonRefinement(Fan,Fan)" -- see gfanFanCommonRefinement -- find the common refinement of two polyheadral fans
• "gfanFanProduct(Fan,Fan)" -- see gfanFanProduct -- computes the product of polyhedral fans
• "gfanGroebnerCone(Ideal)" -- see gfanGroebnerCone -- polyhedral information about a Groebner cone
• "gfanGroebnerCone(Ideal,Ideal)" -- see gfanGroebnerCone -- polyhedral information about a Groebner cone
• "gfanGroebnerCone(Ideal,List)" -- see gfanGroebnerCone -- polyhedral information about a Groebner cone
• "gfanGroebnerCone(Ideal,MarkedPolynomialList)" -- see gfanGroebnerCone -- polyhedral information about a Groebner cone
• "gfanGroebnerCone(List)" -- see gfanGroebnerCone -- polyhedral information about a Groebner cone
• "gfanGroebnerCone(List,Ideal)" -- see gfanGroebnerCone -- polyhedral information about a Groebner cone
• "gfanGroebnerCone(List,List)" -- see gfanGroebnerCone -- polyhedral information about a Groebner cone
• "gfanGroebnerCone(List,MarkedPolynomialList)" -- see gfanGroebnerCone -- polyhedral information about a Groebner cone
• "gfanGroebnerCone(MarkedPolynomialList)" -- see gfanGroebnerCone -- polyhedral information about a Groebner cone
• "gfanGroebnerCone(MarkedPolynomialList,Ideal)" -- see gfanGroebnerCone -- polyhedral information about a Groebner cone
• "gfanGroebnerCone(MarkedPolynomialList,List)" -- see gfanGroebnerCone -- polyhedral information about a Groebner cone
• "gfanGroebnerCone(MarkedPolynomialList,MarkedPolynomialList)" -- see gfanGroebnerCone -- polyhedral information about a Groebner cone
• "gfanHomogeneitySpace(Ideal)" -- see gfanHomogeneitySpace -- homogeneity space of a list of polynomials
• "gfanHomogeneitySpace(List)" -- see gfanHomogeneitySpace -- homogeneity space of a list of polynomials
• "gfanHomogeneitySpace(MarkedPolynomialList)" -- see gfanHomogeneitySpace -- homogeneity space of a list of polynomials
• "gfanHomogenize(Ideal,Symbol)" -- see gfanHomogenize -- homogenize a list of polynomials with respect to a weight vector
• "gfanHomogenize(List,Symbol)" -- see gfanHomogenize -- homogenize a list of polynomials with respect to a weight vector
• "gfanHomogenize(MarkedPolynomialList,Symbol)" -- see gfanHomogenize -- homogenize a list of polynomials with respect to a weight vector
• "gfanInitialForms(Ideal,List)" -- see gfanInitialForms -- initial forms of polynomials with respect to a weight vector
• "gfanInitialForms(List,List)" -- see gfanInitialForms -- initial forms of polynomials with respect to a weight vector
• "gfanInitialForms(MarkedPolynomialList,List)" -- see gfanInitialForms -- initial forms of polynomials with respect to a weight vector
• "gfanIsMarkedGroebnerBasis(Ideal)" -- see gfanIsMarkedGroebnerBasis -- checks if a list of polynomials are a Groebner basis
• "gfanIsMarkedGroebnerBasis(List)" -- see gfanIsMarkedGroebnerBasis -- checks if a list of polynomials are a Groebner basis
• "gfanIsMarkedGroebnerBasis(MarkedPolynomialList)" -- see gfanIsMarkedGroebnerBasis -- checks if a list of polynomials are a Groebner basis
• "gfanKrullDimension(Ideal)" -- see gfanKrullDimension -- krull dimension
• "gfanKrullDimension(List)" -- see gfanKrullDimension -- krull dimension
• "gfanKrullDimension(MarkedPolynomialList)" -- see gfanKrullDimension -- krull dimension
• "gfanLatticeIdeal(List)" -- see gfanLatticeIdeal -- computes the lattice ideal of a lattice
• "gfanLeadingTerms(List)" -- see gfanLeadingTerms -- leading terms of a list (or list of lists) of marked polynomials
• "gfanLeadingTerms(MarkedPolynomialList)" -- see gfanLeadingTerms -- leading terms of a list (or list of lists) of marked polynomials
• "gfanMarkPolynomialSet(List,List)" -- see gfanMarkPolynomialSet -- mark the initial terms of a list of polynomials with respect to a weight vector
• "gfanMinkowskiSum(Ideal)" -- see gfanMinkowskiSum -- the Minkowski sum of Newton polytopes
• "gfanMinkowskiSum(List)" -- see gfanMinkowskiSum -- the Minkowski sum of Newton polytopes
• "gfanMinors(ZZ,ZZ,ZZ)" -- see gfanMinors -- minors of a matrix of indeterminates
• "gfanMixedVolume(List)" -- see gfanMixedVolume -- mixed volume of a list of polynomials
• "gfanOverIntegers(Ideal)" -- see gfanOverIntegers -- all reduced Groebner bases of a poynomial ideal with coefficients in ZZ
• "gfanOverIntegers(Ideal,List)" -- see gfanOverIntegers -- all reduced Groebner bases of a poynomial ideal with coefficients in ZZ
• "gfanPolynomialSetUnion(List,List)" -- see gfanPolynomialSetUnion -- union of two lists of polynomials
• "gfanPolynomialSetUnion(List,MarkedPolynomialList)" -- see gfanPolynomialSetUnion -- union of two lists of polynomials
• "gfanPolynomialSetUnion(MarkedPolynomialList,List)" -- see gfanPolynomialSetUnion -- union of two lists of polynomials
• "gfanPolynomialSetUnion(MarkedPolynomialList,MarkedPolynomialList)" -- see gfanPolynomialSetUnion -- union of two lists of polynomials
• "gfanRender(List)" -- see gfanRender -- render an image of a Groebner fan
• "gfanRender(String,List)" -- see gfanRender -- render an image of a Groebner fan
• "gfanRenderStaircase(List)" -- see gfanRenderStaircase -- render the staircase of a monomial initial ideal
• "gfanRenderStaircase(String,List)" -- see gfanRenderStaircase -- render the staircase of a monomial initial ideal
• "gfanResultantFan(List)" -- see gfanResultantFan -- Tropical variety of the sparse (toric) resultant variety.
• "gfanSaturation(Ideal)" -- see gfanSaturation -- saturation of an ideal
• "gfanSecondaryFan(List)" -- see gfanSecondaryFan -- computes the secondary fan of a vector configuration
• "gfanStats(List)" -- see gfanStats -- display certain properties of a list of polynomials
• "gfanSubstitute(Ideal,PolynomialRing)" -- see gfanSubstitute -- rename the variables of a list of polynomials
• "gfanSubstitute(List,PolynomialRing)" -- see gfanSubstitute -- rename the variables of a list of polynomials
• "gfanSubstitute(MarkedPolynomialList,PolynomialRing)" -- see gfanSubstitute -- rename the variables of a list of polynomials
• "gfanToLatex(List)" -- see gfanToLatex -- convert a list of polynomials to LaTeX
• "gfanToLatex(MarkedPolynomialList)" -- see gfanToLatex -- convert a list of polynomials to LaTeX
• "gfanToPolyhedralFan(List)" -- see gfanToPolyhedralFan -- polyhedral data about the fan of a list of Groebner bases
• "gfanTropicalBasis(Ideal)" -- see gfanTropicalBasis -- tropical basis of an ideal
• "gfanTropicalBruteForce(List)" -- see gfanTropicalBruteForce -- computes the tropical variety of an ideal
• "gfanTropicalEvaluation(RingElement,List)" -- see gfanTropicalEvaluation -- evaluates a tropical polynomial function
• "gfanTropicalFunction(RingElement)" -- see gfanTropicalFunction -- tropicalizes a polynomial
• "gfanTropicalHyperSurface(RingElement)" -- see gfanTropicalHyperSurface -- the tropical hypersurface of a principal ideal
• "gfanTropicalIntersection(List)" -- see gfanTropicalIntersection -- polyhedral data describing intersection of tropical hypersurfaces
• "gfanTropicalLinearSpace(List,ZZ,ZZ)" -- see gfanTropicalLinearSpace -- equations of a tropical linear space from Pluecker coordinates
• "gfanTropicalMultiplicity(List)" -- see gfanTropicalMultiplicity -- multiplicity of a tropical cone
• "gfanTropicalRank(Matrix)" -- see gfanTropicalRank -- the tropical rank of a matrix
• "gfanTropicalStartingCone(Ideal)" -- see gfanTropicalStartingCone -- a pair of Groebner bases for use with gfanTropicalTraverse
• "gfanTropicalStartingCone(List)" -- see gfanTropicalStartingCone -- a pair of Groebner bases for use with gfanTropicalTraverse
• "gfanTropicalTraverse(List)" -- see gfanTropicalTraverse -- polyhedral data describing a tropical variety
• "gfanTropicalWeilDivisor(Fan,Fan)" -- see gfanTropicalWeilDivisor -- the tropical Weil divisor of a piecewise linear function
• "MPLConverter(Ideal)" -- see MPLConverter -- Converts lists or ideals into markedPolynomialLists
• "MPLConverter(List)" -- see MPLConverter -- Converts lists or ideals into markedPolynomialLists
• Symbols
• polymakeConeToCone (missing documentation)
• polymakeFanToFan (missing documentation)
• Other things
• gfanFunctions (missing documentation)

## For the programmer

The object gfanInterface is .