# 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 1.3 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.

## Examples of usage

```
julia> using Oscar
...
----- ----- ----- - -----
| | | | | | | | | |
| | | | | | | |
| | ----- | | | |-----
| | | | |-----| | |
| | | | | | | | | |
----- ----- ----- - - - -
...combining (and extending) GAP, Hecke, Nemo, Polymake and Singular
Version 0.5.1-DEV ...
... which comes with absolutely no warranty whatsoever
Type: '?Oscar' for more information
(c) 2019-2021 by The Oscar Development Team
julia> k, a = quadratic_field(-5)
(Number field over Rational Field with defining polynomial x^2+5, sqrt(-5))
julia> zk = maximal_order(k)
Maximal order of Number field over Rational Field with defining polynomial x^2+5
with basis nf_elem[1, sqrt(-5)]
julia> factorisations(zk(6))
2-element Array{Fac{NfAbsOrdElem{AnticNumberField,nf_elem}},1}:
-1 * (2) * (-3)
-1 * (sqrt(-5)+1) * (sqrt(-5)-1)
julia> Qx, x = PolynomialRing(QQ, :x=>1:2)
(Multivariate Polynomial Ring in x1, x2 over Rational Field, fmpq_mpoly[x1, x2])
julia> R = grade(Qx, [1,2])
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 = FreeModule(R, 1)
Free module of rank 1 over R, graded as R^1([0])
julia> s = sub(F, [f*F[1]])
Subquotient by Array of length 1
1 -> (x1^2 + x2)*e[1]
a> mH(H[1])
Map with following data
Domain:
=======
s
Codomain:
=========
Subquotient of Array of length 1
1 -> (1)*e[1]
by Array of length 1
1 -> (x1^2 + x2)*e[1]
defined on the Singular side
julia> H, mH = hom(s, quo(F, s))
(hom of (s, Subquotient of Array of length 1
1 -> (1)*e[1]
by Array of length 1
1 -> (x1^2 + x2)*e[1]
defined on the Singular side
), Map from
H to Set of all homomorphisms from Subquotient by Array of length 1
1 -> (x1^2 + x2)*e[1]
defined on the Singular side
to Subquotient of Array of length 1
1 -> (1)*e[1]
by Array of length 1
1 -> (x1^2 + x2)*e[1]
defined on the Singular side
defined by a julia-function with inverse
)
julia> D = decoration(H)
GrpAb: Z
julia> homogenous_component(H, D[0])
(H_[0] of dim 2, Map from
H_[0] of dim 2 to H defined by a julia-function with inverse
)
```

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)
```

## Funding

The development of this Julia package is supported by the Deutsche Forschungsgemeinschaft DFG within the Collaborative Research Center TRR 195.