Fix Grassmann Pluecker (#3382)

* fixed

* fix for symmetric intersections test

* back to the roots

* revert

* Apply suggestions from code review

Co-authored-by: Benjamin Lorenz <benlorenz@users.noreply.github.com>

---------

Co-authored-by: Tommy Hofmann <thofma@gmail.com>
Co-authored-by: Benjamin Lorenz <benlorenz@users.noreply.github.com>

src/Rings/mpoly-ideals.jl

@@ -1922,7 +1922,15 @@ small_generating_set(I::MPolyIdeal{<:MPolyDecRingElem}) = minimal_generating_set
 #
 ################################################################################
 function grassmann_pluecker_ideal(subspace_dimension::Int, ambient_dimension::Int)
-    return convert(MPolyIdeal{QQMPolyRingElem}, Polymake.ideal.pluecker_ideal(subspace_dimension, ambient_dimension))
+  I = convert(MPolyIdeal{QQMPolyRingElem},
+              Polymake.ideal.pluecker_ideal(subspace_dimension, ambient_dimension))
+  base = base_ring(I)
+  o = degrevlex(base)
+
+  return ideal(IdealGens(base, gens(I), o;
+                   keep_ordering=true,
+                   isReduced=true,
+                   isGB=true))
 end
 
 @doc raw"""
@@ -1944,33 +1952,31 @@ julia> grassmann_pluecker_ideal(2, 4)
 Ideal generated by
   x[1]*x[6] - x[2]*x[5] + x[3]*x[4]
 
-julia> R, x = polynomial_ring(residue_ring(ZZ, 7)[1], "x" => (1:2, 1:3), ordering=:degrevlex)
+julia> R, x = polynomial_ring(residue_ring(ZZ, 7)[1], "x" => (1:2, 1:3))
 (Multivariate polynomial ring in 6 variables over ZZ/(7), zzModMPolyRingElem[x[1, 1] x[1, 2] x[1, 3]; x[2, 1] x[2, 2] x[2, 3]])
 
 julia> grassmann_pluecker_ideal(R, 2, 4)
 Ideal generated by
-  x[1, 2]*x[2, 2] + 6*x[2, 1]*x[1, 3] + x[1, 1]*x[2, 3]
+  x[1, 1]*x[2, 3] + 6*x[2, 1]*x[1, 3] + x[1, 2]*x[2, 2]
 ```
 
 """
 function grassmann_pluecker_ideal(ring::MPolyRing,
                                  subspace_dimension::Int,
                                  ambient_dimension::Int) 
-    pluecker_ideal = grassmann_pluecker_ideal(subspace_dimension, ambient_dimension)
-    converted_generators = elem_type(ring)[]
-    coeff_ring = base_ring(ring)
-    for g in gens(pluecker_ideal)
-        converted_coeffs = [coeff_ring(numerator(c)) for c in AbstractAlgebra.coefficients(g)]
-        polynomial = MPolyBuildCtx(ring)
-
-        for (i, exponent) in enumerate(AbstractAlgebra.exponent_vectors(g))
-            c = converted_coeffs[i]
-            push_term!(polynomial, c, exponent)
-        end
-        converted_g = finish(polynomial)
-        push!(converted_generators, converted_g)
-    end
-    return ideal(ring, converted_generators)
+  I = grassmann_pluecker_ideal(subspace_dimension, ambient_dimension)
+  coeff_ring = base_ring(ring)
+  o = degrevlex(ring)
+  coeffmap = c -> begin
+    @assert is_integral(c)
+    return coeff_ring(numerator(c))
+  end
+  h = hom(base_ring(I), ring, coeffmap, gens(ring))
+  converted_generators = elem_type(ring)[h(g) for g in groebner_basis(I; ordering = degrevlex(base_ring(I)))]
+  ideal(IdealGens(ring, converted_generators, o;
+                   keep_ordering=true,
+                   isReduced=true,
+                   isGB=true))
 end
 
 ################################################################################