# Oscar.jl

Documentation |
Build Status |
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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 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
Version 0.12.1-DEV ...
... which comes with absolutely no warranty whatsoever
Type: '?Oscar' for more information
(c) 2019-2023 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 = polynomial_ring(QQ, [:x1,:x2])
(Multivariate Polynomial Ring in x1, x2 over Rational Field, QQMPolyRingElem[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 0.12.1-DEV, The OSCAR Team, 2023. (https://www.oscar-system.org)
[OSCAR-book]
Wolfram Decker, Christian Eder, 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 0.12.1-DEV},
year = {2023},
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 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.