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 1.10 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 Oscar

However, 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

/ _ \ / \ / _\ / \ | _ \ | Combining and extending ANTIC, GAP, | || |__ | | / ^ \ | ´ / | Polymake and Singular _/ _/ __/// _|_|_\ | Type "?Oscar" for more information o--------o-----o-----o--------o | Documentation: https://docs.oscar-system.org S Y M B O L I C T O O L S | Version 1.5.0-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 subquotient with no relations

julia> H, mH = hom(s, quo(F, s)[1]) (hom of (s, Subquotient of submodule with 1 generator 1: e[1] by submodule with 1 generator 1: (x1^2 + x2)e[1]), Map: H -> set of all homomorphisms from s to subquotient of submodule with 1 generator 1: e[1] by submodule with 1 generator 1: (x1^2 + x2)e[1])

julia> mH(H[1]) Module homomorphism from s to 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. For example:

``` julia> C = Polymake.polytope.cube(3);

julia> C.F_VECTOR pm::Vectorpm::Integer 8 12 6

julia> RP2 = Polymake.topaz.realprojectiveplane();

julia> RP2.HOMOLOGY pm::Arraytopaz::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 on our website.

Funding

The development of this Julia package is supported by the German Research Foundation (DFG) within the Collaborative Research Center TRR 195.