The OSCAR Book Companion
The OSCAR Book Companion
This page contains the code samples from the corresponding chapter in the OSCAR book. You can access the full chapter here.
Interested readers can find more information in this article.
julia> B = [[1,2],[1,3],[1,4],[2,3],[2,4]];
julia> M = matroid_from_bases(B,4)
Matroid of rank 2 on 4 elements
julia> independent_sets(M)
10-element Vector{Vector{Int64}}:
[]
[1]
[2]
[3]
[4]
[1, 4]
[2, 4]
[1, 3]
[2, 3]
[1, 2]
julia> flats(M)
5-element Vector{Vector{Int64}}:
[]
[1]
[2]
[3, 4]
[1, 2, 3, 4]
julia> circuits(M)
3-element Vector{Vector{Int64}}:
[3, 4]
[1, 2, 4]
[1, 2, 3]
julia> X = matrix(GF(2), [1 0 1 0 1 0 1; 0 1 1 0 0 1 1; 0 0 0 1 1 1 1])
[1 0 1 0 1 0 1]
[0 1 1 0 0 1 1]
[0 0 0 1 1 1 1]
julia> F = matroid_from_matrix_columns(X)
Matroid of rank 3 on 7 elements
julia> is_isomorphic(F, fano_matroid())
true
julia> G = automorphism_group(F);
julia> describe(G)
"PSL(3,2)"
julia> nonbases(vamos_matroid())
5-element Vector{Vector{Int64}}:
[1, 2, 3, 4]
[1, 2, 5, 6]
[1, 2, 7, 8]
[3, 4, 5, 6]
[5, 6, 7, 8]
julia> M = moebius_kantor_matroid();
julia> nonbases(M)
8-element Vector{Vector{Int64}}:
[1, 2, 3]
[1, 4, 5]
[1, 7, 8]
[2, 4, 6]
[2, 5, 8]
[3, 4, 7]
[3, 6, 8]
[5, 6, 7]
julia> realization_space(M)
The realization space is
[1 0 1 0 1 0 1 1]
[0 1 1 0 0 1 1 x1]
[0 0 0 1 1 -x1 -x1 + 1 1]
in the multivariate polynomial ring in 1 variable over ZZ
within the vanishing set of the ideal
Ideal (x1^2 - x1 + 1)
avoiding the zero loci of the polynomials
RingElem[x1, x1 - 1, x1^2 + 1]
julia> X = matrix(GF(2), [1 0 1 0 1 0 1; 0 1 1 0 0 1 1; 0 0 0 1 1 1 1])
[1 0 1 0 1 0 1]
[0 1 1 0 0 1 1]
[0 0 0 1 1 1 1]
julia> F = matroid_from_matrix_columns(X)
Matroid of rank 3 on 7 elements
julia> realization_space(F, simplify=false)
The realization space is
[0 1 1 1 1 0 0]
[1 0 1 x2 0 1 0]
[1 0 x1 0 x3 0 1]
in the multivariate polynomial ring in 3 variables over ZZ
within the vanishing set of the ideal
Ideal (2, x3 + 1, x2 - 1, x1 + 1)
avoiding the zero loci of the polynomials
RingElem[x2, x3, x1, x1*x2 - x2*x3 + x3, x1 - x3 - 1, x1 + x2 - 1]
julia> realization_space(F, simplify=true)
The realization space is
[0 1 1 1 1 0 0]
[1 0 1 1 0 1 0]
[1 0 1 0 1 0 1]
in the integer ring
within the vanishing set of the ideal
2ZZ
julia> is_realizable(F, char=5)
false
julia> realization_space(non_fano_matroid())
The realization space is
[1 1 0 0 1 1 0]
[0 1 1 1 1 0 0]
[0 1 1 0 0 1 1]
in the integer ring
avoiding the zero loci of the polynomials
RingElem[2]
julia> is_realizable(vamos_matroid())
false
julia> is_realizable(M, q=9)
true
julia> qs = [2,3,4,5,7,8,9,11,13];
julia> println([is_realizable(M, q=a) for a in qs])
Bool[0, 1, 1, 0, 1, 0, 1, 0, 1]
julia> M = cycle_matroid(complete_graph(4));
julia> realization_space(M)
The realization space is
[1 0 1 0 1 0]
[0 1 1 0 0 1]
[0 0 0 1 1 -1]
in the integer ring
julia> db = Polymake.Polydb.get_db();
julia> collection = db["Matroids.Small"];
julia> query = Dict("RANK"=>3,"N_ELEMENTS"=>9,"SIMPLE"=>true);
julia> results = Polymake.Polydb.find(collection, query);
julia> oscar_matroids = [Matroid(pm) for pm in results];
julia> length(oscar_matroids)
383
julia> char_0_matroids = [M for M in oscar_matroids
if is_realizable(M, char=0)];
julia> length(char_0_matroids)
370
julia> RS = realization_space(pappus_matroid(), char=0)
The realization space is
[1 0 1 0 x2 x2 x2^2 1 0]
[0 1 1 0 1 1 -x1*x2 + x1 + x2^2 1 1]
[0 0 0 1 x2 x1 x1*x2 x1 x2]
in the multivariate polynomial ring in 2 variables over QQ
avoiding the zero loci of the polynomials
RingElem[x1 - x2, x2, x1, x2 - 1, x1 + x2^2 - x2, x1 - 1, x1*x2 - x1 - x2^2]
julia> realization(RS)
One realization is given by
[1 0 1 0 2 2 4 1 0]
[0 1 1 0 1 1 1 1 1]
[0 0 0 1 2 3 6 3 2]
in the rational field
julia> results = collection = db = 0; # allow connection cleanup
julia> M = cycle_matroid(complete_graph(4))
Matroid of rank 3 on 6 elements
julia> tutte_polynomial(M)
x^3 + 3*x^2 + 4*x*y + 2*x + y^3 + 3*y^2 + 2*y
julia> char_poly = characteristic_polynomial(M)
q^3 - 6*q^2 + 11*q - 6
julia> factor(char_poly)
1 * (q - 2) * (q - 1) * (q - 3)
julia> A = chow_ring(M);
julia> GR, _ = graded_polynomial_ring(QQ,symbols(base_ring(A)));
julia> AA = chow_ring(M,ring=GR);
julia> vol_map = volume_map(M,AA)
#3605 (generic function with 1 method)
julia> e = matroid_groundset(M)[1];
julia> proper_flats = flats(M)[2:length(flats(M))-1];
julia> a = sum([AA[i] for i in 1:length(proper_flats) if e in proper_flats[i]])
x_{Edge(2, 1)} + x_{Edge(2, 1),Edge(3, 1),Edge(3, 2)} + x_{Edge(2, 1),Edge(4, 1),Edge(4, 2)} + x_{Edge(2, 1),Edge(4, 3)}
julia> b = sum([AA[i] for i in 1:length(proper_flats) if !(e in proper_flats[i])])
x_{Edge(3, 1)} + x_{Edge(3, 2)} + x_{Edge(4, 1)} + x_{Edge(4, 2)} + x_{Edge(4, 3)} + x_{Edge(3, 1),Edge(4, 2)} + x_{Edge(3, 1),Edge(4, 1),Edge(4, 3)} + x_{Edge(3, 2),Edge(4, 1)} + x_{Edge(3, 2),Edge(4, 2),Edge(4, 3)}
julia> k = 1
1
julia> R = base_ring(AA);
julia> g = grading_group(R)[1];
julia> PD1, mapPD1 = homogeneous_component(AA,k*g);
julia> basis_PD1 = [mapPD1(x) for x in gens(PD1)];
julia> PD2, mapPD2 = homogeneous_component(AA,(rank(M)-k-1)*g);
julia> basis_PD2 = [mapPD2(x) for x in gens(PD2)];
julia> Mat1 = matrix(QQ,[[vol_map(b1*b2) for b1 in basis_PD1] for b2 in basis_PD2])
[-1 0 0 0 0 0 0 1]
[ 0 -1 0 0 0 0 0 0]
[ 0 0 -1 0 0 0 0 1]
[ 0 0 0 -1 0 0 0 0]
[ 0 0 0 0 -1 0 0 1]
[ 0 0 0 0 0 -1 0 0]
[ 0 0 0 0 0 0 -1 0]
[ 1 0 1 0 1 0 0 -2]
julia> Mat2 = matrix(QQ,[[vol_map(b1*b^(rank(M)-2k-1)*b2) for b1 in basis_PD1] for b2 in basis_PD1]);
julia> Mat1 == Mat2
true
julia> RR, _ = graded_polynomial_ring(QQ,"y_#" => 1:length(basis_PD1));
julia> map = hom(RR,AA,basis_PD1);
julia> K = kernel(hom(RR,AA,[b^(rank(M)-2k)*b3 for b3 in basis_PD1]));
julia> basis_HR = [map(h) for h in gens(K) if degree(h).coeff==k*g.coeff]
7-element Vector{MPolyQuoRingElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}:
x_{Edge(4, 3)}
-x_{Edge(2, 1),Edge(3, 1),Edge(3, 2)} + x_{Edge(2, 1),Edge(4, 1),Edge(4, 2)}
-x_{Edge(2, 1),Edge(3, 1),Edge(3, 2)} + 2*x_{Edge(2, 1),Edge(4, 3)}
-x_{Edge(2, 1),Edge(3, 1),Edge(3, 2)} + 2*x_{Edge(3, 1),Edge(4, 2)}
-x_{Edge(2, 1),Edge(3, 1),Edge(3, 2)} + x_{Edge(3, 1),Edge(4, 1),Edge(4, 3)}
-x_{Edge(2, 1),Edge(3, 1),Edge(3, 2)} + 2*x_{Edge(3, 2),Edge(4, 1)}
-x_{Edge(2, 1),Edge(3, 1),Edge(3, 2)} + x_{Edge(3, 2),Edge(4, 2),Edge(4, 3)}
julia> Mat3 = matrix(QQ,[[(-1)^k*vol_map(b1*b^(rank(M)-2k-1)*b2) for b1 in basis_HR] for b2 in basis_HR])
[ 2 0 -2 0 -1 0 -1]
[ 0 2 1 1 1 1 1]
[-2 1 5 1 1 1 1]
[ 0 1 1 5 1 1 1]
[-1 1 1 1 2 1 1]
[ 0 1 1 1 1 5 1]
[-1 1 1 1 1 1 2]
julia> is_positive_definite(matrix(ZZ,[ZZ(i) for i in Mat3]))
true
julia> reduced_characteristic_polynomial(M)
q^2 - 5*q + 6
julia> [vol_map(a^(rank(M)-j-1)*b^j) for j in range(0,rank(M)-1)]
3-element Vector{QQFieldElem}:
1
5
6