TropicalGeometry: new positive tropicalizations

Author: YueRen

docs/doc.main

@@ -246,6 +246,7 @@
      "TropicalGeometry/linear_space.md",
      "TropicalGeometry/groebner_theory.md",
      "TropicalGeometry/tropicalization.md",
+     "TropicalGeometry/positive_variety.md",
   ],
 
   "Noncommutative Algebra" => [

docs/oscar_references.bib

@@ -2290,6 +2290,18 @@ @Article{SV-D-V87
   zbmath        = {4069055}
 }
 
+@Article{SW05,
+  author        = {Speyer, David and Williams, Lauren},
+  title         = {The {{Tropical Totally Positive Grassmannian}}},
+  journal       = {Journal of Algebraic Combinatorics},
+  volume        = {22},
+  number        = {2},
+  pages         = {189--210},
+  year          = {2005},
+  month         = sep,
+  doi           = {10.1007/s10801-005-2513-3}
+}
+
 @Article{SY96,
   author        = {Shimoyama, Takeshi and Yokoyama, Kazuhiro},
   title         = {Localization and primary decomposition of polynomial ideals},

docs/src/TropicalGeometry/positive_variety.md

@@ -0,0 +1,9 @@
+# Positive tropicalizations of linear ideals
+
+## Introduction
+Positive tropial varieties (in OSCAR) are weighted polyhedral complexes and as per the definition in [SW05](@cite).  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.
+
+
+```@docs
+positive_tropical_variety(::MPolyIdeal, ::TropicalSemiringMap)
+```

src/TropicalGeometry/TropicalGeometry.jl

@@ -17,5 +17,6 @@ include("hypersurface.jl")
 include("curve.jl")
 include("linear_space.jl")
 include("variety.jl")
+include("positive_variety.jl")
 include("intersection.jl")
 include("groebner_fan.jl")

src/TropicalGeometry/positive_variety.jl

@@ -0,0 +1,73 @@
+@doc raw"""
+    positive_tropical_variety(I::MPolyIdeal,nu::TropicalSemiringMap)
+
+Return the positive tropical variety of `I` as a `PolyhedralComplex` as per the definition in [SW05](@cite).  Assumes that `I` is generated either by binomials or by linear polynomials and that `I` is defined either over
+(a) the rational numbers and that `nu` encodes the trivial valuation,
+(b) the rational function field over the rational numbers and that `nu` encodes the t-adic valuation.
+
+# Examples
+```jldoctest
+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
+
+```
+"""
+function positive_tropical_variety(I::MPolyIdeal,nu::TropicalSemiringMap)
+    if all(isequal(2),length.(gens(I)))
+        if all(isequal(-1),[prod([sign(c) for c in coefficients(g)]) for g in gens(I)])
+            # binomial ideal positive, return regular tropical variety
+            return tropical_variety(I,nu)
+        else
+            # binomial ideal not positive, return empty polyhedral complex in the correct ambient dimension
+            return polyhedral_complex(IncidenceMatrix(zeros(Int,0,0)),zero_matrix(QQ,0,ambient_dim(TropL)))
+        end
+    end
+
+    if all(isequal(1),total_degree.(gens(I)))
+        # Construct the tropicalization of I
+        TropL = tropical_linear_space(I,nu)
+
+        # find maximal polyhedra belonging to the positive part
+        # we check containment in the positive part by testing the initial ideal w.r.t. a relative interior point
+        positivePolyhedra = Polyhedron{QQFieldElem}[sigma for sigma in maximal_polyhedra(TropL) if is_initial_positive(I,nu,relative_interior_point(sigma))]
+
+        if isempty(positivePolyhedra)
+            # if there are no positive polyhedra,
+            # return empty polyhedral complex in the correct ambient dimension
+            return polyhedral_complex(IncidenceMatrix(zeros(Int,0,0)),zero_matrix(QQ,0,ambient_dim(TropL)))
+        end
+
+        Sigma = polyhedral_complex(positivePolyhedra)
+        mult = ones(ZZRingElem, n_maximal_polyhedra(Sigma))
+        minOrMax = convention(nu)
+        return tropical_variety(Sigma,mult,minOrMax)
+    end
+
+    error("input ideal not supported")
+end
+
+function is_initial_positive(I::MPolyIdeal, nu::TropicalSemiringMap, w::AbstractVector)
+    inI = initial(I,nu,w)
+    G = groebner_basis(inI; complete_reduction=true)
+
+    # the Groebner basis is binomial, check binomials have alternating signs
+    return all(isequal(-1),[prod([sign(c) for c in coefficients(g)]) for g in G])
+end

src/exports.jl

@@ -1317,6 +1317,7 @@ export positive_coroots
 export positive_hull
 export positive_root
 export positive_roots
+export positive_tropical_variety
 export possible_class_fusions
 export power_sum
 export powers_of_element

test/TropicalGeometry/positive_variety.jl

@@ -0,0 +1,14 @@
+@testset "src/TropicalGeometry/positive_variety.jl" begin
+
+    C = matrix(QQ,[[-3,1,-1,-2,2],[-1,1,-1,-1,1]])
+    R,x = polynomial_ring(QQ,ncols(C))
+    nu = tropical_semiring_map(QQ)
+    I = ideal(C*gens(R))
+    TropPlusI = positive_tropical_variety(I,nu)
+    @test n_maximal_polyhedra(TropPlusI) == 5
+
+    I = ideal([x[1]^2-x[2]^2,x[3]^3-x[4]^3])
+    TropPlusI = positive_tropical_variety(I,nu)
+    @test n_maximal_polyhedra(TropPlusI) == 1
+
+end