diff --git a/src/TropicalGeometry/TropicalGeometry.jl b/src/TropicalGeometry/TropicalGeometry.jl
index 2fae5858027c..3ce5a308bb27 100644
--- a/src/TropicalGeometry/TropicalGeometry.jl
+++ b/src/TropicalGeometry/TropicalGeometry.jl
@@ -20,6 +20,6 @@ include("variety_binomial.jl")
 include("variety_principal.jl")
 include("variety_affine_linear.jl")
 include("variety_zerodimensional.jl")
-include("variety_equidimensional.jl")
+include("variety_prime.jl")
 include("intersection.jl")
 include("groebner_fan.jl")
diff --git a/src/TropicalGeometry/variety.jl b/src/TropicalGeometry/variety.jl
index ca7b0a9471bf..f4f78b0837bc 100644
--- a/src/TropicalGeometry/variety.jl
+++ b/src/TropicalGeometry/variety.jl
@@ -24,10 +24,10 @@ end
 #
 ################################################################################
 
-function Base.show(io::IO, tv::TropicalVariety{typeof(min), true})
+function Base.show(io::IO, ::TropicalVariety{typeof(min), true})
     print(io, "Min tropical variety")
 end
-function Base.show(io::IO, tv::TropicalVariety{typeof(max), true})
+function Base.show(io::IO, ::TropicalVariety{typeof(max), true})
     print(io, "Max tropical variety")
 end
 
@@ -40,7 +40,7 @@ end
 ################################################################################
 
 @doc raw"""
-    tropical_variety(Sigma::PolyhedralComplex, mult, minOrMax::Union{typeof(min),typeof(max)}=min)
+    tropical_variety(Sigma::PolyhedralComplex, mult::Vector{ZZRingElem}, minOrMax::Union{typeof(min),typeof(max)}=min)
 
 Return the `TropicalVariety` whose polyhedral complex is `Sigma` with multiplicities `mult` and convention `minOrMax`. Here, `mult` is optional can be specified as a `Vector{ZZRingElem}` which represents a list of multiplicities on the maximal polyhedra in the order of `maximal_polyhedra(Sigma)`.  If `mult` is unspecified, then all multiplicities are set to one.
 
@@ -133,21 +133,37 @@ If `nu==nothing`, will compute with respect to the trivial valuation and min con
 If `weighted_polyhedral_complex_only==true`, will not cache any additional information.
 
 !!! warning
-    Assumes that `I` is equi-dimensional.  Only special cases supported:
-    - any valuation: `I` principal, binomial, affine linear
-    - trivial and p-adic valuation only: `I` general
+    Experimental feature, only special cases supported:
+    - any coefficient field and any valuation: `I` principal, binomial, or affine linear
+    - QQ and trivial / p-adic valuation only: `I` prime
 
 # Examples
 ```jldoctest
-julia> R,(x,y) = QQ["x","y"];
+julia> R,(x,y,z) = QQ["x","y","z"];
 
-julia> I = ideal([(x^2+y)*(x+y^2)*(x+y)]);
+julia> nu_2 = tropical_semiring_map(QQ,2)
+Map into Min tropical semiring encoding the 2-adic valuation on Rational field
 
-julia> tropical_variety(I)
-3-element Vector{TropicalVariety}:
- Min tropical variety
- Min tropical variety
- Min tropical variety
+julia> f1 = 8*x^2 + x*y + x*z + x + 8*y^2 + y*z + y + 8*z^2 + z + 8;
+
+julia> f2 = x + 1;
+
+julia> I = ideal([f1,f2]);
+
+julia> TropI_0 = tropical_variety(I)
+Min tropical variety
+
+julia> vertices(TropI_0)
+1-element SubObjectIterator{PointVector{QQFieldElem}}:
+ [0, 0, 0]
+
+julia> TropI_2 = tropical_variety(I,nu_2)
+Min tropical variety
+
+julia> vertices(TropI_2)
+2-element SubObjectIterator{PointVector{QQFieldElem}}:
+ [0, -3, 3]
+ [0, 3, -3]
 
 ```
 """
@@ -171,7 +187,7 @@ function tropical_variety(I::MPolyIdeal, nu::Union{TropicalSemiringMap,Nothing}=
     end
 
     # I general
-    return tropical_variety_equidimensional(I,nu,weighted_polyhedral_complex_only=weighted_polyhedral_complex_only)
+    return tropical_variety_prime(I,nu,weighted_polyhedral_complex_only=weighted_polyhedral_complex_only)
 end
 
 
@@ -206,13 +222,14 @@ function homogenize_pre_tropicalization(I::MPolyIdeal)
 end
 
 
-function dehomogenize_post_tropicalization(Sigma::PolyhedralComplex)
-    @req lineality_dim(Sigma)>0 "dehomogenizing polyhedral complex without lineality"
+function dehomogenize_post_tropicalization(TropV::TropicalVarietySupertype)
+
+    @req lineality_dim(TropV)>0 "dehomogenizing polyhedral complex without lineality"
 
     ###
     # Construct hyperplane {first coord = 0}
     ###
-    n = ambient_dim(Sigma)
+    n = ambient_dim(TropV)
     zerothUnitRowVector = zeros(Int,1,n)
     zerothUnitRowVector[1,1] = 1
     dehomogenisingHyperplane = polyhedron((zeros(Int,0,n),zeros(Int,0)), (zerothUnitRowVector,[0]))
@@ -224,7 +241,7 @@ function dehomogenize_post_tropicalization(Sigma::PolyhedralComplex)
     dehomogenizedVertices = Vector{QQFieldElem}[]
     incidenceMatrixRays = Vector{Int}[]
     dehomogenizedRays = Vector{QQFieldElem}[]
-    for sigma in maximal_polyhedra(Sigma)
+    for sigma in maximal_polyhedra(TropV)
         sigmaDehomogenized = intersect(sigma,dehomogenisingHyperplane)
         incidenceVectorVertices = Int[]
         V,_ = minimal_faces(sigmaDehomogenized)
@@ -267,12 +284,17 @@ function dehomogenize_post_tropicalization(Sigma::PolyhedralComplex)
     ###
     # Dehomogenize lineality space
     ###
-    sigma = first(maximal_polyhedra(Sigma))
+    sigma = first(maximal_polyhedra(TropV))
     sigmaDehomogenized = intersect(sigma,dehomogenisingHyperplane)
     dehomogenizedLineality = [linealityVector[2:end] for linealityVector in lineality_space(sigmaDehomogenized)]
 
-    return polyhedral_complex(incidenceMatrixVerticesAndRays,
-                              dehomogenizedVerticesAndRays,
-                              collect(length(dehomogenizedVertices)+1:length(dehomogenizedVertices)+length(dehomogenizedRays)),
-                              dehomogenizedLineality)
+    SigmaDehom =  polyhedral_complex(incidenceMatrixVerticesAndRays,
+                                     dehomogenizedVerticesAndRays,
+                                     collect(length(dehomogenizedVertices)+1:length(dehomogenizedVertices)+length(dehomogenizedRays)),
+                                     dehomogenizedLineality)
+
+    TropVDehom = tropical_variety(SigmaDehom,multiplicities(TropV),convention(TropV))
+    # TropVDehom.__attrs = deepcopy(TropV.__attrs) # TODO: not working, how to copy all attributes?
+    return TropVDehom
+
 end
diff --git a/src/TropicalGeometry/variety_affine_linear.jl b/src/TropicalGeometry/variety_affine_linear.jl
index 7a84aacc1a5b..f8b6516cd842 100644
--- a/src/TropicalGeometry/variety_affine_linear.jl
+++ b/src/TropicalGeometry/variety_affine_linear.jl
@@ -7,32 +7,16 @@
 ################################################################################
 
 function tropical_variety_affine_linear(I::MPolyIdeal,nu::TropicalSemiringMap; weighted_polyhedral_complex_only::Bool=false)
-    ###
-    # Compute reduced Groebner basis (usually already cached),
-    # and check whether the linear polynomials have a constant term
-    ###
-    R = base_ring(I)
+    # compute a reduced GB to check whether I is homogeneous
     G = groebner_basis(I,complete_reduction=true)
-    if min(total_degree.(Iterators.flatten(collect.(terms.(G))))...)==1
-        # input homogneeous, construct TropicalVariety via TropicalLinearSpace
-        TropV = tropical_variety(tropical_linear_space(I,nu,weighted_polyhedral_complex_only=weighted_polyhedral_complex_only))
-        if !weighted_polyhedral_complex_only
-            set_attribute!(TropV,:algebraic_ideal,I)
-            set_attribute!(TropV,:tropical_semiring_map,nu)
-        end
-        return TropV
-    else
-        # input inhomogeneous, homogenise first
-        Ih = homogenize_pre_tropicalization(I)
-        TropLh = tropical_linear_space(Ih,nu,weighted_polyhedral_complex_only=true)
-        Sigma = dehomogenize_post_tropicalization(polyhedral_complex(TropLh))
 
-        multiplicities = ones(ZZRingElem, n_maximal_polyhedra(Sigma))
-        TropV = tropical_variety(Sigma,multiplicities)
-        if !weighted_polyhedral_complex_only
-            set_attribute!(TropV,:algebraic_ideal,I)
-            set_attribute!(TropV,:tropical_semiring_map,nu)
-        end
-        return TropV
+    if all(Oscar._is_homogeneous.(G))
+        # input homogneeous, construct TropicalVariety via TropicalLinearSpace
+        return tropical_variety(tropical_linear_space(I,nu,weighted_polyhedral_complex_only=weighted_polyhedral_complex_only))
     end
+
+    # input inhomogeneous, homogenise first
+    Ih = homogenize_pre_tropicalization(I)
+    TropLh = tropical_linear_space(Ih,nu,weighted_polyhedral_complex_only=true)
+    return dehomogenize_post_tropicalization(TropLh)
 end
diff --git a/src/TropicalGeometry/variety_equidimensional.jl b/src/TropicalGeometry/variety_prime.jl
similarity index 82%
rename from src/TropicalGeometry/variety_equidimensional.jl
rename to src/TropicalGeometry/variety_prime.jl
index 12a1a2842f51..6c9280882b2f 100644
--- a/src/TropicalGeometry/variety_equidimensional.jl
+++ b/src/TropicalGeometry/variety_prime.jl
@@ -1,17 +1,26 @@
 ################################################################################
 #
-#  Tropicalization of equi-dimensional ideals
+#  Tropicalization of prime ideals
 #
-#  WARNING: assumes without test that `I` is equi-dimensional
+#  WARNING: assumes without test that `I` is prime
 #
 ################################################################################
 
-function tropical_variety_equidimensional(I::MPolyIdeal, nu::TropicalSemiringMap{QQField,Nothing,<:Union{typeof(min),typeof(max)}}; weighted_polyhedral_complex_only::Bool=false)
-    return tropical_variety_equidimensional_singular(I,nu; weighted_polyhedral_complex_only=weighted_polyhedral_complex_only)
+function tropical_variety_prime(I::MPolyIdeal, nu::TropicalSemiringMap; weighted_polyhedral_complex_only::Bool=false)
+    # compute a reduced GB to check whether I is homogeneous
+    G = groebner_basis(I,complete_reduction=true)
+
+    if all(Oscar._is_homogeneous.(G))
+        return tropical_variety_prime_singular(I,nu; weighted_polyhedral_complex_only=weighted_polyhedral_complex_only)
+    end
+
+    Ih = homogenize_pre_tropicalization(I)
+    TropIh = tropical_variety_prime_singular(Ih,nu,weighted_polyhedral_complex_only=weighted_polyhedral_complex_only)
+    return dehomogenize_post_tropicalization(TropIh)
 end
 
-# trivial valuation case
-function tropical_variety_equidimensional_singular(I::MPolyIdeal, nu::TropicalSemiringMap{QQField,Nothing,<:Union{typeof(min),typeof(max)}}; weighted_polyhedral_complex_only::Bool=false)
+# trivial valuation
+function tropical_variety_prime_singular(I::MPolyIdeal, nu::TropicalSemiringMap{QQField,Nothing,<:Union{typeof(min),typeof(max)}}; weighted_polyhedral_complex_only::Bool=false)
     R = base_ring(I)
     singularCommand = join(["ring r=0,("*join(string.(symbols(R)),",")*"),dp;",
                             "ideal I = "*join(string.(gens(I)), ",")*";",
@@ -29,7 +38,8 @@ function tropical_variety_equidimensional_singular(I::MPolyIdeal, nu::TropicalSe
     return TropI
 end
 
-function tropical_variety_equidimensional_singular(I::MPolyIdeal, nu::TropicalSemiringMap{QQField,ZZRingElem,<:Union{typeof(min),typeof(max)}}; weighted_polyhedral_complex_only::Bool=false)
+# p-adic valuation
+function tropical_variety_prime_singular(I::MPolyIdeal, nu::TropicalSemiringMap{QQField,ZZRingElem,<:Union{typeof(min),typeof(max)}}; weighted_polyhedral_complex_only::Bool=false)
     R = base_ring(I)
     singularCommand = join(["ring r=0,("*join(string.(symbols(R)),",")*"),dp;",
                             "ideal I = "*join(string.(gens(I)), ",")*";",
diff --git a/test/TropicalGeometry/variety.jl b/test/TropicalGeometry/variety.jl
index a7d3a5776d4a..bfe139340aff 100644
--- a/test/TropicalGeometry/variety.jl
+++ b/test/TropicalGeometry/variety.jl
@@ -19,11 +19,11 @@
         R,(x,y,z,w) = QQ["x","y","z","w"]
         I = ideal(R,[x+2*y,z+4*w])
         nu = tropical_semiring_map(QQ,2)
-        TropV = first(tropical_variety(I,nu))
+        TropV = tropical_variety(I,nu)
         TropL = tropical_linear_space(I,nu)
         @test issetequal(maximal_polyhedra(TropV),maximal_polyhedra(TropL))
         nu = tropical_semiring_map(QQ,2,max)
-        TropV = first(tropical_variety(I,nu))
+        TropV = tropical_variety(I,nu)
         TropL = tropical_linear_space(I,nu)
         @test issetequal(maximal_polyhedra(TropV),maximal_polyhedra(TropL))
     end
@@ -33,12 +33,36 @@
         f = x*y*z+2
         nu = tropical_semiring_map(QQ,2)
         TropH = tropical_hypersurface(f,nu)
-        TropV = first(tropical_variety(ideal(R,f),nu))
+        TropV = tropical_variety(ideal(R,f),nu)
         @test issetequal(maximal_polyhedra(TropH),maximal_polyhedra(TropV))
         nu = tropical_semiring_map(QQ,2,max)
-        TropV = first(tropical_variety(ideal(R,f),nu))
+        TropV = tropical_variety(ideal(R,f),nu)
         TropH = tropical_hypersurface(f,nu)
         @test issetequal(maximal_polyhedra(TropH),maximal_polyhedra(TropV))
     end
 
+    # running tropical_variety and all its subroutines
+    @testset "testing tropical_variety" begin
+
+        # principal ideals
+        R,(x,y,z) = QQ["x","y","z"]
+        f = x^2+y^2+z^2+1
+        TropV = tropical_variety(ideal(R,f))
+        @test f_vector(TropV) == [1,4,6]
+
+        # binomial ideals
+        f = x^2+1
+        g = y^2+1
+        TropV = tropical_variety(ideal(R,[f,g]))
+        @test f_vector(TropV) == [0,1]
+
+        # affine linear ideals
+        f = x+z+1
+        g = y+z+1
+        TropV = tropical_variety(ideal(R,[f,g]))
+        @test f_vector(TropV) == [1,3]
+
+        # general ideals, see doctests
+
+    end
 end