Oscar.jl
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Welcome to the OSCAR project, a visionary new computer algebra system which combines the capabilities of four cornerstone systems: GAP, Polymake, Antic and Singular.
Installation
OSCAR requires Julia 1.6 or newer. In principle it can be installed and used like any other Julia package; doing so will take a couple of minutes:
julia> using Pkg
julia> Pkg.add("Oscar")
julia> using OscarHowever, some of OSCAR's components have additional requirements. For more detailed information, please consult the installation instructions on our website.
Contributing to OSCAR
Please read the introduction for new developers in the OSCAR manual to learn more on how to contribute to OSCAR.
Examples of usage
julia> using Oscar
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...combining (and extending) ANTIC, GAP, Polymake and Singular
Version 0.11.3-DEV ...
... which comes with absolutely no warranty whatsoever
Type: '?Oscar' for more information
(c) 2019-2022 by The OSCAR Development Team
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 nf_elem[1, sqrt(-5)]
julia> factorisations(zk(6))
2-element Vector{Fac{NfOrdElem}}:
-1 * (sqrt(-5) + 1) * (sqrt(-5) - 1)
-1 * 2 * -3
julia> Qx, x = PolynomialRing(QQ, [:x1,:x2])
(Multivariate Polynomial Ring in x1, x2 over Rational Field, fmpq_mpoly[x1, x2])
julia> R = grade(Qx, [1,2])[1]
Multivariate Polynomial Ring in x1, x2 over Rational Field graded by
x1 -> [1]
x2 -> [2]
julia> f = R(x[1]^2+x[2])
x1^2 + x2
julia> degree(f)
graded by [2]
julia> F = free_module(R, 1)
Free module of rank 1 over R
julia> s = sub(F, [f*F[1]])
Submodule with 1 generator
1 -> (x1^2 + x2)*e[1]
represented as subquotient with no relations.
julia> H, mH = hom(s, quo(F, s))
(hom of (s, Subquotient of Submodule with 1 generator
1 -> e[1]
by Submodule with 1 generator
1 -> (x1^2 + x2)*e[1]), Map from
H to Set of all homomorphisms from Submodule with 1 generator
1 -> (x1^2 + x2)*e[1]
represented as subquotient with no relations. to Subquotient of Submodule with 1 generator
1 -> e[1]
by Submodule with 1 generator
1 -> (x1^2 + x2)*e[1] defined by a julia-function with inverse)
julia> mH(H[1])
Map with following data
Domain:
=======
s
Codomain:
=========
Subquotient of Submodule with 1 generator
1 -> e[1]
by Submodule with 1 generator
1 -> (x1^2 + x2)*e[1]Of course, the cornerstones are also available directly:
julia> C = Polymake.polytope.cube(3);
julia> C.F_VECTOR
pm::Vector<pm::Integer>
8 12 6
julia> RP2 = Polymake.topaz.real_projective_plane();
julia> RP2.HOMOLOGY
PropertyValue wrapping pm::Array<polymake::topaz::HomologyGroup<pm::Integer>>
({} 0)
({(2 1)} 0)
({} 0)Citing OSCAR
If you have used OSCAR in the preparation of a paper please cite it as described below:
[OSCAR]
OSCAR -- Open Source Computer Algebra Research system, Version 0.11.3-DEV The OSCAR Team, 2022. (https://oscar.computeralgebra.de)
[OSCAR-book]
Christian Eder, Wolfram Decker, Claus Fieker, Max Horn, Michael Joswig, The OSCAR book, 2024.
If you are using BibTeX, you can use the following BibTeX entries:
@misc{OSCAR,
key = {OSCAR},
organization = {The OSCAR Team},
title = {OSCAR -- Open Source Computer Algebra Research system,
Version 0.11.3-DEV},
year = {2022},
url = {https://oscar.computeralgebra.de},
}
@Book{OSCAR-book,
editor = {Decker, Wolfram and Eder, Christian and Fieker, Claus and Horn, Max and Joswig, Michael},
title = {The OSCAR book},
year = {2024},
}
Funding
The development of this Julia package is supported by the Deutsche Forschungsgemeinschaft DFG within the Collaborative Research Center TRR 195.