Positive tropicalizations of linear ideals

Introduction

Positive tropial varieties (in OSCAR) are weighted polyhedral complexes and as per the definition in [SW05]. They may arise as tropicalizations of polynomial ideals over an ordered field. Currently, the only ideals supported are linear ideals over rational numbers or rational function fields over rational numbers.

positive_tropical_variety — Method

positive_tropical_variety(I::MPolyIdeal,nu::TropicalSemiringMap)

Return the positive tropical variety of I as a PolyhedralComplex as per the definition in [SW05].

Assumes that I is generated either by binomials or by linear polynomials and that I is defined either over

the rational numbers and that nu encodes the trivial valuation,
the rational function field over the rational numbers and that nu encodes the t-adic valuation.

Examples

julia> K,t = rational_function_field(QQ,"t")
(Rational function field over QQ, t)

julia> C = matrix(K,[[-3*t,1*t,-1*t,-2*t,2*t],[-1*t,1*t,-1*t,-1*t,1*t]])
[-3*t   t   -t   -2*t   2*t]
[  -t   t   -t     -t     t]

julia> R,x = polynomial_ring(K,ncols(C))
(Multivariate polynomial ring in 5 variables over K, AbstractAlgebra.Generic.MPoly{AbstractAlgebra.Generic.RationalFunctionFieldElem{QQFieldElem, QQPolyRingElem}}[x1, x2, x3, x4, x5])

julia> nu = tropical_semiring_map(K,t)
Map into Min tropical semiring encoding the t-adic valuation on Rational function field over QQ

julia> I = ideal(C*gens(R))
Ideal generated by
  -3*t*x1 + t*x2 - t*x3 - 2*t*x4 + 2*t*x5
  -t*x1 + t*x2 - t*x3 - t*x4 + t*x5

julia> TropPlusI = positive_tropical_variety(I,nu)
Min tropical variety

source