Tropicalization of polynomial ideals
Introduction
Tropical varieties can arise as tropicalizations of polynomial ideals. For a general introduction, see
- Chapter 3 in [MS15]
For algorithmic details, see
Main function
tropical_variety — Functiontropical_variety(I::MPolyIdeal, nu::Union{TropicalSemiringMap,Nothing}=nothing; weighted_polyhedral_complex_only::Bool=false, skip_saturation::Bool=false, skip_primary_decomposition::Bool=false)Return the tropicalization of I with respect to nu as a Vector{TropicalVariety}. If nu==nothing, will compute with respect to the trivial valuation and min convention. If weighted_polyhedral_complex_only==true, will not cache any additional information. If skip_saturation==true, will not saturate I with respect to the product of all variables. If skip_primary_decomposition==true, will not decompose I.
tropical_variety is currently under development and only works for ideals that primary decompose into principal, linear, and binomial ideals.
Examples
julia> R,(x,y) = QQ[:x, :y];
julia> I = ideal([(x^2+y)*(x+y^2)*(x+y)]);
julia> tropical_variety(I)
3-element Vector{TropicalVariety}:
Min tropical variety
Min tropical variety
Min tropical variety