# Nemo

## Introduction

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 and other contributors.

The features of Nemo so far include:

• Multiprecision integers and rationals
• Integers modulo n
• Finite fields (prime and non-prime order)
• Number field arithmetic
• Maximal orders of number fields
• Arithmetic of ideals in maximal orders
• Arbitrary precision real and complex balls
• Univariate polynomials and matrices over the above
• Generic polynomials, power series, fraction fields, residue rings and matrices

## Installation

At the Julia prompt simply type

julia> Pkg.add("Nemo")
julia> Pkg.build("Nemo")


Alternatively, if you don't want to set Julia up yourself, Julia and Nemo are available on https://cloud.sagemath.com/.

## 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> M = MatrixSpace(R, 40, 40)();

julia> R, x = PolynomialRing(ZZ, "x")
(Univariate Polynomial Ring in x over Integer Ring,x)

julia> M = MatrixSpace(R, 40, 40)();

julia> for i in 1:40
for j in 1:40
M[i, j] = R(map(fmpz, rand(-20:20, 3)))
end
end

julia> @time det(M);
0.174888 seconds (268.40 k allocations: 26.537 MB, 4.47% 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)