# Getting Started

Nemo is a computer algebra package for the Julia programming language, maintained by William Hart, Tommy Hofmann, Claus Fieker, Fredrik Johansson with additional code by Oleksandr Motsak, Marek Kaluba and other contributors.

- https://nemocas.org (Website)
- https://github.com/Nemocas/Nemo.jl (Source code)
- https://nemocas.github.io/Nemo.jl/latest/ (Online documentation)

The features of Nemo so far include:

- Multiprecision integers and rationals
- Integers modulo n
- p-adic numbers
- Finite fields (prime and non-prime order)
- Number field arithmetic
- Algebraic numbers
- Exact real and complex numbers
- Arbitrary precision real and complex balls
- Univariate and multivariate polynomials and matrices over the above

Nemo depends on AbstractAlgebra.jl which provides Nemo with generic routines for:

- Univariate and multivariate polynomials
- Absolute and relative power series
- Laurent series
- Fraction fields
- Residue rings
- Matrices and linear algebra
- Young Tableaux
- Permutation groups
- Characters

## Installation

To use Nemo we require Julia 1.6 or higher. Please see https://julialang.org/downloads/ for instructions on how to obtain julia for your system.

At the Julia prompt simply type

`julia> using Pkg; Pkg.add("Nemo")`

## Quick start

Here are some examples of using Nemo.

This example computes recursive univariate polynomials.

```
julia> using Nemo
julia> R, x = PolynomialRing(ZZ, "x")
(Univariate Polynomial Ring in x over Integer Ring,x)
julia> S, y = PolynomialRing(R, "y")
(Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Integer Ring,y)
julia> T, z = PolynomialRing(S, "z")
(Univariate Polynomial Ring in z over Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Integer Ring,z)
julia> f = x + y + z + 1
z+(y+(x+1))
julia> p = f^30; # semicolon supresses output
julia> @time q = p*(p+1);
0.325521 seconds (140.64 k allocations: 3.313 MB)
```

Here is an example using generic recursive ring constructions.

```
julia> using Nemo
julia> R, x = FiniteField(7, 11, "x")
(Finite field of degree 11 over F_7,x)
julia> S, y = PolynomialRing(R, "y")
(Univariate Polynomial Ring in y over Finite field of degree 11 over F_7,y)
julia> T = ResidueRing(S, y^3 + 3x*y + 1)
Residue ring of Univariate Polynomial Ring in y over Finite field of degree 11 over F_7 modulo y^3+(3*x)*y+(1)
julia> U, z = PolynomialRing(T, "z")
(Univariate Polynomial Ring in z over Residue ring of Univariate Polynomial Ring in y over Finite field of degree 11 over F_7 modulo y^3+(3*x)*y+(1),z)
julia> f = (3y^2 + y + x)*z^2 + ((x + 2)*y^2 + x + 1)*z + 4x*y + 3;
julia> g = (7y^2 - y + 2x + 7)*z^2 + (3y^2 + 4x + 1)*z + (2x + 1)*y + 1;
julia> s = f^12;
julia> t = (s + g)^12;
julia> @time resultant(s, t)
0.426612 seconds (705.88 k allocations: 52.346 MB, 2.79% gc time)
(x^10+4*x^8+6*x^7+3*x^6+4*x^5+x^4+6*x^3+5*x^2+x)*y^2+(5*x^10+x^8+4*x^7+3*x^5+5*x^4+3*x^3+x^2+x+6)*y+(2*x^10+6*x^9+5*x^8+5*x^7+x^6+6*x^5+5*x^4+4*x^3+x+3)
```

Here is an example using matrices.

```
julia> using Nemo
julia> R, x = PolynomialRing(ZZ, "x")
(Univariate Polynomial Ring in x over Integer Ring,x)
julia> S = MatrixSpace(R, 40, 40)
Matrix Space of 40 rows and 40 columns over Univariate Polynomial Ring in x over Integer Ring
julia> M = rand(S, 2:2, -20:20)
julia> @time det(M);
0.131212 seconds (1.12 M allocations: 39.331 MiB, 4.77% gc time)
```

And here is an example with power series.

```
julia> using Nemo
julia> R, x = QQ["x"]
(Univariate Polynomial Ring in x over Rational Field,x)
julia> S, t = PowerSeriesRing(R, 100, "t")
(Univariate power series ring in t over Univariate Polynomial Ring in x over Rational Field,t+O(t^101))
julia> u = t + O(t^100)
t+O(t^100)
julia> @time divexact((u*exp(x*u)), (exp(u)-1));
0.042663 seconds (64.01 k allocations: 1.999 MB, 15.40% gc time)
```

## Building dependencies from source

Nemo depends on various C libraries which are installed using binaries by default. With julia version >= 1.3, the use of these binaries can be overridden by putting the following into the file `~/.julia/artifacts/Overrides.toml`

:

```
[e134572f-a0d5-539d-bddf-3cad8db41a82]
FLINT = "/prefix/for/libflint"
[d9960996-1013-53c9-9ba4-74a4155039c3]
Arb = "/prefix/for/libarb"
[e21ec000-9f72-519e-ba6d-10061e575a27]
Antic = "/prefix/for/libantic"
```

(If only a specific library should be overridden, only the specific entry should be added.)

## Experimental threading support for flint

Enabling a threaded version of flint can be done by setting the evironment variable `NEMO_THREADED=1`

. To set the actual number of threads, use `Nemo.flint_set_num_threads($numberofthreads)`

.