Update: Functionality for matroid realization spaces (oscar-system#2925)

- Add functionality for realization space of matroids. The realization space contains all vector sets that realize a given matroid.
- Add several standard constructions.

docs/src/Combinatorics/matroids.md

@@ -38,9 +38,12 @@ matroid_from_revlex_basis_encoding(rvlx::String, r::IntegerUnion, n::IntegerUnio
 ```@docs
 uniform_matroid(r::IntegerUnion,n::IntegerUnion)
 fano_matroid()
+moebius_kantor_matroid()
 non_fano_matroid()
 non_pappus_matroid()
 pappus_matroid()
+perles_matroid()
+R10_matroid()
 vamos_matroid()
 all_subsets_matroid(r::Int)
 projective_plane(q::Int)
@@ -109,46 +112,15 @@ reduced_characteristic_polynomial(M::Matroid)
 revlex_basis_encoding(M::Matroid)
 is_isomorphic(M1::Matroid, M2::Matroid)
 is_minor(M::Matroid, N::Matroid)
+automorphism_group(M::Matroid)
+matroid_base_polytope(M::Matroid)
 ```
 
+
 ### Chow Rings
 ```@docs
 chow_ring(M::Matroid; ring::Union{MPolyRing,Nothing}=nothing, extended::Bool=false)
 augmented_chow_ring(M::Matroid)
 ```
 
-## Matroid strata and realization spaces
-
-For a matroid ``M``, of rank ``d`` on a ground set ``E`` of size ``n`` realizable over a ring ``K``, 
-its *matroid realization space* ``\mathsf{Gr}(M)`` is the locally closed subscheme of the Grassmannian
-``\mathsf{Gr}(d,K^n)`` of ``d``-dimensional subspaces of ``K^n`` realizing ``M``. Precisely, 
-
-```math
-    \mathsf{Gr}(M) = \{F \in \mathsf{Gr}(d,K^n) \, : \, p_{I}(F) \neq 0 \text{ iff } I \in \mathcal{B}(M)\}
-```
-
-where ``\mathcal{B}(M)`` denotes the set of bases of ``M`` and  ``p_{I}(F)`` denotes the ``I``th Pl\"ucker 
-coordinate of the linear subspace ``F \subset K^n``. The coordinate ring of ``\mathsf{Gr}(M)`` can be computed
-using matrix coordinates, see Construction 2.2 of [Cor21](@cite). The following function computes this ring. 
-
-```@docs
-matroid_stratum_matrix_coordinates(M::Matroid, B::GroundsetType, F::AbstractAlgebra.Ring = ZZ)
-```
-
-When the matroid ``M`` is connected, the diagonal torus of ``\mathsf{PGL}(n)`` acts freely on 
-``\mathsf{Gr}(M)`` and its quotient is the *realization space* ``R(M)`` of ``M``. There are two main differences between the coordinate rings ``K[\mathsf{Gr}(M)]`` and ``K[R(M)]``.
-
- - To comupte ``K[R(M)]``, we  assume that ``d+1`` columns ``A = \{a_1,\ldots,a_{d+1}\}``  of the reference matrix ``X`` are the unit vectors ``e_1,\ldots,e_n`` and ``e_1+ \cdots + e_n``. Note that every ``d`` element subset of ``A`` must be a basis of ``M``. Disconnected matroids do not satisfy this property.
- - The columns of the reference matrix ``X`` may be treated as projective coordinates.
 
-The coordinate ring of ``R(M)`` is computed in the following function. 
-
-
-```@docs
-matroid_realization_space(M::Matroid, A::GroundsetType, F::AbstractAlgebra.Ring=ZZ)
-```
-
-
-```@docs
-    automorphism_group(m::Matroid)
-```

docs/src/DeveloperDocumentation/styleguide.md

@@ -33,6 +33,8 @@ deviate from them in some cases; in that case just do so.
   create a point-type (or use the one already there) and then use this.
   For user-facing functions, please do not use re-purposed lists, arrays,
   matrices...
+- Input sanity checks should be enabled by default, they can then be disabled
+  internally if they are known to be true, and manually by users.
 
 
 ## Naming conventions

experimental/MatroidRealizationSpaces/docs/doc.main

@@ -0,0 +1,5 @@
+[
+   "Matroid Realization Spaces" => [
+      "introduction.md",
+   ]
+]

experimental/MatroidRealizationSpaces/docs/src/introduction.md

@@ -0,0 +1,34 @@
+```@meta
+CurrentModule = Oscar
+```
+
+# Matroid Realization Spaces
+
+
+Let ``M`` be a matroid of rank ``d`` on a ground set ``E`` of size ``n``. Its
+*realization space* ``\mathcal{R}(M)`` is an affine scheme that parameterizes
+all hyperplane arrangements that realize the matroid ``M`` (up to the action of
+``PGL(r)``).  We provide functions that determine the affine coordinate ring of
+``\mathcal{R}(M)``. 
+
+
+```@docs
+is_realizable(M::Matroid; char::Union{Int,Nothing}=nothing, q::Union{Int,Nothing}=nothing)
+defining_ideal(RS::MatroidRealizationSpace)
+inequations(RS::MatroidRealizationSpace)
+ambient_ring(RS::MatroidRealizationSpace)
+realization_space(M::Matroid; B::Union{GroundsetType,Nothing} = nothing, 
+    F::AbstractAlgebra.Ring = ZZ, saturate::Bool=false, reduce::Bool=true,
+    char::Union{Int,Nothing}=nothing, q::Union{Int,Nothing}=nothing)
+realization(M::Matroid; B::Union{GroundsetType,Nothing} = nothing, 
+    F::AbstractAlgebra.Ring = ZZ, saturate::Bool=false, reduce::Bool=true,
+    char::Union{Int,Nothing}=nothing, q::Union{Int,Nothing}=nothing)
+realization(RS::MatroidRealizationSpace)
+```
+
+If ``B`` is the polynomial ring `ambient_ring(RS)`, ``I`` the ideal
+`defining_ideal(RS)`, and ``U`` the multiplicative semigroup generated by
+`inequations(RS)`, then the coordinate ring of the realization space
+``\mathcal{R}(M)`` is isomorphic to ``U^{-1}B/I``.  
+
+

experimental/MatroidRealizationSpaces/src/MatroidRealizationSpaces.jl

@@ -0,0 +1,13 @@
+include("realization_space.jl")
+
+### Methods for realization spaces of matroids
+function underlying_scheme(RS::MatroidRealizationSpace{BRT, RT}) where {BRT<:Ring, RT<:MPolyQuoLocRing}
+  isdefined(RS, :underlying_scheme) && return RS.underlying_scheme::Spec{BRT, RT}
+
+  P = ambient_ring(RS)::MPolyRing
+  I = defining_ideal(RS)::MPolyIdeal
+  U = MPolyPowersOfElement(P, P.(inequations(RS)))::MPolyPowersOfElement
+  RS.underlying_scheme = Spec(P, I, U)
+  return RS.underlying_scheme::Spec{BRT, RT}
+end
+