This method compute a pair of Groebner bases as needed for gfanTropicalTraverse. It heuristically finds a cone of the Tropical Variety. Its first output is the Groebner basis of the cone's monomial-free initial ideal. And the second output is the Groebner basis of the original ideal. Note that gfanTropicalStartingCone uses graded reverse lex order.
i1 : QQ[x,y,z] o1 = QQ[x..z] o1 : PolynomialRing |
i2 : gfanTropicalStartingCone{x+y+z} o2 = {{{z}, {y + z}}, {{z}, {x + y + z}}} o2 : List |
i3 : QQ[x,y] o3 = QQ[x..y] o3 : PolynomialRing |
i4 : I=ideal(x+y) o4 = ideal(x + y) o4 : Ideal of QQ[x..y] |
i5 : gfanTropicalStartingCone(I) o5 = {{{y}, {x + y}}, {{y}, {x + y}}} o5 : List |
gfan Documentation This program computes a starting pair of marked reduced Groebner bases to be used as input for gfan_tropicaltraverse. The input is a homogeneous ideal whose tropical variety is a pure d-dimensional polyhedral complex.Options:-g: Tell the program that the input is already a reduced Groebner basis.-d: Output dimension information to standard error.--stable: Find starting cone in the stable intersection or, equivalently, pretend that the coefficients are genereric.
The object gfanTropicalStartingCone is a method function with options.