@misc{OSCAR,
key = {OSCAR},
organization = {The OSCAR Team},
title = {OSCAR -- Open Source Computer Algebra Research system,
Version 1.2.0-DEV},
year = {2024},
url = {https://www.oscar-system.org},
}
@book{OSCAR-book,
editor = {Decker, Wolfram and Eder, Christian and Fieker, Claus and Horn, Max and Joswig, Michael},
title = {The {C}omputer {A}lgebra {S}ystem {OSCAR}: {A}lgorithms and {E}xamples},
year = {2024},
publisher = {Springer},
series = {Algorithms and {C}omputation in {M}athematics},
volume = {32},
edition = {1},
url = {https://link.springer.com/book/9783031621260},
month = {8},
issn = {1431-1550},
}https://www.oscar-system.org[OSCAR]
OSCAR -- Open Source Computer Algebra Research system, Version 1.2.0-DEV,
The OSCAR Team, 2024. (https://www.oscar-system.org)
[OSCAR-book]
Wolfram Decker, Christian Eder, Claus Fieker, Max Horn, Michael Joswig, eds.
The Computer Algebra System OSCAR: Algorithms and Examples,
Algorithms and Computation in Mathematics, Springer, 2024.https://link.springer.com/book/97830316212608julia> C = Polymake.polytope.cube(3);
julia> Qx, x = polynomial_ring(QQ, [:x1,:x2])
(Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x1, x2])
julia> R = grade(Qx, [1,2])[1]
Multivariate polynomial ring in 2 variables over QQ graded by
x1 -> [1]
x2 -> [2]
julia> f = R(x[1]^2+x[2])
x1^2 + x2
julia> degree(f)
[2]
julia> F = free_module(R, 1)
Free module of rank 1 over R
julia> s = sub(F, [fF[1]])[1]
Submodule with 1 generator
1 -> (x1^2 + x2)e[1]
represented as with no relations.
julia> H, = hom(s, quo(F, s)[1])
(hom of (s, of
1 -> e[1]
by
1 -> (x1^2 + x2)e[1]), Map: H -> set of all homomorphisms from s to of Submodule with 1 generator
1 -> e[1]
by Submodule with 1 generator
1 -> (x1^2 + x2)e[1])
julia> (H[1])
Map with following data
Domain:
Submodule with 1 generator
1 -> (x1^2 + x2)*e[1]
represented as with no relations.
Codomain:
of Submodule with 1 generator
1 -> e[1]
by Submodule with 1 generator
1 -> (x1^2 + x2)*e[1]HOMOLOGY5https://docs.oscar-system.org21julia> using Pkg
julia> Pkg.add("Oscar")
julia> using Oscar
@misc{OSCAR, key = {OSCAR}, organization = {The OSCAR Team}, title = {OSCAR -- Open Source Computer Algebra Research system, Version 1.2.0-DEV}, year = {2024}, url = {https://www.oscar-system.org}, }
@book{OSCAR-book, editor = {Decker, Wolfram and Eder, Christian and Fieker, Claus and Horn, Max and Joswig, Michael}, title = {The {C}omputer {A}lgebra {S}ystem {OSCAR}: {A}lgorithms and {E}xamples}, year = {2024}, publisher = {Springer}, series = {Algorithms and {C}omputation in {M}athematics}, volume = {32}, edition = {1}, url = {https://link.springer.com/book/9783031621260}, month = {8}, issn = {1431-1550}, }https://www.oscar-system.org[OSCAR] OSCAR -- Open Source Computer Algebra Research system, Version 1.2.0-DEV, The OSCAR Team, 2024. (https://www.oscar-system.org) [OSCAR-book] Wolfram Decker, Christian Eder, Claus Fieker, Max Horn, Michael Joswig, eds. The Computer Algebra System OSCAR: Algorithms and Examples, Algorithms and Computation in Mathematics, Springer, 2024.https://link.springer.com/book/97830316212608julia> C = Polymake.polytope.cube(3);
julia> C.F_VECTOR pm::Vector
8 12 6
julia> RP2 = Polymake.topaz.real_projective_plane();
julia> RP2.HOMOLOGY pm::Array<topaz::HomologyGroup >
({} 0)
({(2 1)} 0)
({} 0)612julia> using Oscar
/ _ \ / | / | / \ | _ \ | Combining ANTIC, GAP, Polymake, Singular | | | |_ | | / \ | |) | | Type "?Oscar" for more information | || | ) | | / | _ < | Manual: https://docs.oscar-system.org _/ |__/ ___// __| _\ | Version 1.2.0-DEV julia> k, a = quadratic_field(-5) (Imaginary quadratic field defined by x^2 + 5, sqrt(-5))
julia> zk = maximal_order(k) Maximal order of Imaginary quadratic field defined by x^2 + 5 with basis AbsSimpleNumFieldElem[1, sqrt(-5)]
julia> factorizations(zk(6)) 2-element Vector{Fac{AbsSimpleNumFieldOrderElem}}: -1 -3 2 -1 (-sqrt(-5) - 1) (-sqrt(-5) + 1)
julia> Qx, x = polynomial_ring(QQ, [:x1,:x2]) (Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x1, x2])
julia> R = grade(Qx, [1,2])[1] Multivariate polynomial ring in 2 variables over QQ graded by x1 -> [1] x2 -> [2]
julia> f = R(x[1]^2+x[2]) x1^2 + x2
julia> degree(f) [2]
julia> F = free_module(R, 1) Free module of rank 1 over R
julia> s = sub(F, [fF[1]])[1] Submodule with 1 generator 1 -> (x1^2 + x2)e[1] represented as with no relations.
julia> H, = hom(s, quo(F, s)[1]) (hom of (s, of 1 -> e[1] by 1 -> (x1^2 + x2)e[1]), Map: H -> set of all homomorphisms from s to of Submodule with 1 generator 1 -> e[1] by Submodule with 1 generator 1 -> (x1^2 + x2)e[1])
julia> (H[1]) Map with following data Domain:
Submodule with 1 generator 1 -> (x1^2 + x2)*e[1] represented as with no relations. Codomain:
of Submodule with 1 generator 1 -> e[1] by Submodule with 1 generator 1 -> (x1^2 + x2)*e[1]HOMOLOGY5https://docs.oscar-system.org21julia> using Pkg julia> Pkg.add("Oscar") julia> using Oscar