Algebraeon

Algebraeon is a computer algebra system written purely in Rust. It implements algorithms for working with matrices, polynomials, algebraic numbers, factorizations, etc. The focus is on exact algebraic computations over approximate numerical solutions. Algebraeon is in early stages of development and the API is currently highly unstable and subject to change. Algebraeon uses Malachite for arbitrary sized integer and rational numbers.

Usage

Add

[dependencies]
algebraeon = "0.1.3"

to your cargo.toml to make Algebraeon avalible.

Factoring Polynomials

Factor the polynomials $x^2 - 5x + 6$ and $x^{15} - 1$.

use algebraeon::rings::{
    polynomial::polynomial::*,
    structure::{elements::*, structure::*},
};
use algebraeon::nzq::integer::*;

let x = &Polynomial::<Integer>::var().into_ergonomic();
let f = (x.pow(2) - 5*x + 6).into_verbose();
println!("f(λ) = {}", f.factor().unwrap());
/*
Output:
    f(λ) = 1 * ((-2)+λ) * ((-3)+λ)
*/

let f = (x.pow(15) - 1).into_verbose();
println!("f(λ) = {}", f.factor().unwrap());
/*
Output:
    f(λ) = 1 * ((-1)+λ) * (1+λ+λ^2) * (1+λ+λ^2+λ^3+λ^4) * (1+(-1)λ+λ^3+(-1)λ^4+λ^5+(-1)λ^7+λ^8)
*/

so

x^2 - 5x + 6 = (x-2)(x-3)

x^{15}-1 = (x-1)(x^2+x+1)(x^4+x^3+x^2+x+1)(x^8-x^7+x^5-x^4+x^3-x+1)

Linear Systems of Equations

Find the general solution to the linear system

a \begin{pmatrix}3 \\ 4 \\ 1\end{pmatrix} + b \begin{pmatrix}2 \\ 1 \\ 2\end{pmatrix} + c \begin{pmatrix}1 \\ 3 \\ -1\end{pmatrix} = \begin{pmatrix}5 \\ 5 \\ 3\end{pmatrix}

for integers $a$, $b$ and $c$.

use algebraeon::rings::linear::matrix::Matrix;
use algebraeon::nzq::integer::*;
let x = Matrix::<Integer>::from_rows(vec![vec![3, 4, 1], vec![2, 1, 2], vec![1, 3, -1]]);
let y = Matrix::<Integer>::from_rows(vec![vec![5, 5, 3]]);
let s = x.row_solution_lattice(y);
s.pprint();
/*
Output:
    Start Affine Lattice
    Offset
    ( 2    0    -1 )
    Start Linear Lattice
    ( 1    -1    -1 )
    End Linear Lattice
    End Affine Lattice
*/

so the general solution is all $a$, $b$, $c$ such that

\begin{pmatrix}a \\ b \\ c\end{pmatrix} = \begin{pmatrix}2 \\ 0 \\ -1\end{pmatrix} + t\begin{pmatrix}1 \\ -1 \\ -1\end{pmatrix}

for some integer $t$.

Complex Root Isolation

Find all complex roots of the polynomial $$f(x) = x^5 + x^2 - x + 1$$

use algebraeon::rings::{polynomial::polynomial::*, structure::elements::*};
use algebraeon::nzq::integer::*;

let x = &Polynomial::<Integer>::var().into_ergonomic();
let f = (x.pow(5) + x.pow(2) - x + 1).into_verbose();
// Find the complex roots of f(x)
for root in f.all_complex_roots() {
    println!("root {} of degree {}", root, root.degree());
}
/*
Output:
    root ≈-1.328 of degree 3
    root ≈0.662-0.559i of degree 3
    root ≈0.662+0.559i of degree 3
    root -i of degree 2
    root i of degree 2
*/

Despite the output, the roots found are not numerical approximations. Rather, they are stored internally as exact algebraic numbers by using isolating boxes in the complex plane.

Factoring Multivariable Polynomials

Factor the following multivariable polynomial with integer coefficients

f(x, y) = 6x^4 - 6x^3y^2 + 6xy - 6x - 6y^3 + 6y^2

use algebraeon::{nzq::integer::*, rings::{polynomial::multipoly::*, structure::{elements::*, structure::*}}};

let x = &MultiPolynomial::<Integer>::var(Variable::new("x")).into_ergonomic();
let y = &MultiPolynomial::<Integer>::var(Variable::new("y")).into_ergonomic();

let f = (6 * (x.pow(4) - x.pow(3) * y.pow(2) + x * y - x - y.pow(3) + y.pow(2))).into_verbose();
println!("f(x, y) = {}", f.factor().unwrap());

/*
Output:
    f(x, y) = 1 * ((3)1) * ((2)1) * (x+(-1)y^2) * (x^3+y+(-1)1)
*/

so the factorization of $f(x, y)$ is

f(x, y) = 2 \times 3 \times (x^3 + y - 1) \times (y^2 - x)

P-adic Root Finding

Find the $2$-adic square roots of $17$.

use algebraeon::nzq::natural::*;
use algebraeon::nzq::integer::*;
use algebraeon::rings::{polynomial::polynomial::*, structure::elements::*};
let x = Polynomial::<Integer>::var().into_ergonomic();
let f = (x.pow(2) - 17).into_verbose();
for mut root in f.all_padic_roots(&Natural::from(2u32)) {
    println!("{}", root.truncate(&20.into()).string_repr()); // Show 20 2-adic digits
}
/*
Output:
    ...00110010011011101001
    ...11001101100100010111
*/

Truncating to the last 16 bits it can be verified that, modulo $2^{16}$, the square of these values is $17$.

let a = 0b0010011011101001u16;
assert_eq!(a.wrapping_mul(a), 17u16);
let b = 0b1101100100010111u16;
assert_eq!(b.wrapping_mul(b), 17u16);

Enumerating a Finitely Generated Group

Let $G$ be the finitely generated group generated by $3$ generators $a, b, c$ subject to the relations $a^2 = b^2 = c^2 = (ab)^3 = (bc)^5 = (ac)^2 = e$.

G = \langle a, b, c : a^2 = b^2 = c^2 = (ab)^3 = (bc)^5 = (ac)^2 = e \rangle

Using Algebraeon, $G$ is found to be a finite group of order $120$.

use algebraeon::groups::free_group::todd_coxeter::*;
let mut g = FinitelyGeneratedGroupPresentation::new();
// Add the 3 generators
let a = g.add_generator();
let b = g.add_generator();
let c = g.add_generator();
// Add the relations
g.add_relation(a.pow(2));
g.add_relation(b.pow(2));
g.add_relation(c.pow(2));
g.add_relation((&a * &b).pow(3));
g.add_relation((&b * &c).pow(5));
g.add_relation((&a * &c).pow(2));
// Count elements
let (n, _) = g.enumerate_elements();
assert_eq!(n, 120);

Jordan Normal Form of a Matrix

use algebraeon::nzq::rational::*;
use algebraeon::rings::{linear::matrix::*, number::algebraic::complex::*};
use algebraeon::sets::structure::*;
// Construct a matrix
let a = Matrix::<Rational>::from_rows(vec![
    vec![5, 4, 2, 1],
    vec![0, 1, -1, -1],
    vec![-1, -1, 3, 0],
    vec![1, 1, -1, 2],
]);
// Put it into Jordan Normal Form
let j = MatrixStructure::new(ComplexAlgebraic::structure()).jordan_normal_form(&a);
j.pprint();
/*
Output:
    / 2    0    0    0 \
    | 0    1    0    0 |
    | 0    0    4    1 |
    \ 0    0    0    4 /
*/

Computing Discriminants

Algebraeon can find an expression for the discriminant of a polynomial in terms of the polynomials coefficients.

use algebraeon::rings::polynomial::{
    multipoly::{MultiPolynomial, Variable},
    polynomial::Polynomial,
};
use algebraeon::nzq::integer::*;

let a_var = Variable::new("a");
let b_var = Variable::new("b");
let c_var = Variable::new("c");
let d_var = Variable::new("d");
let e_var = Variable::new("e");

let a = MultiPolynomial::<Integer>::var(a_var);
let b = MultiPolynomial::<Integer>::var(b_var);
let c = MultiPolynomial::<Integer>::var(c_var);
let d = MultiPolynomial::<Integer>::var(d_var);
let e = MultiPolynomial::<Integer>::var(e_var);

let p =
    Polynomial::<MultiPolynomial<Integer>>::from_coeffs(vec![c.clone(), b.clone(), a.clone()]);
println!("p(λ) = {}", p);
println!("disc(p) = {}", p.discriminant().unwrap());

println!();

let p = Polynomial::<MultiPolynomial<Integer>>::from_coeffs(vec![
    d.clone(),
    c.clone(),
    b.clone(),
    a.clone(),
]);
println!("p(λ) = {}", p);
println!("disc(p) = {}", p.discriminant().unwrap());

println!();

let p = Polynomial::<MultiPolynomial<Integer>>::from_coeffs(vec![
    e.clone(),
    d.clone(),
    c.clone(),
    b.clone(),
    a.clone(),
]);
println!("p(λ) = {}", p);
println!("disc(p) = {}", p.discriminant().unwrap());

/*
Output:
    p(λ) = (c)+(b)λ+(a)λ^2
    disc(p) = (-4)ac+b^2

    p(λ) = (d)+(c)λ+(b)λ^2+(a)λ^3
    disc(p) = (-27)a^2d^2+(18)abcd+(-4)ac^3+(-4)b^3d+b^2c^2
*/

so

\mathop{\text{disc}}(ax^2 + bx + c) = b^2 - 4ac

\mathop{\text{disc}}(ax^3 + bx^2 + cx + d) = b^2c^2 - 4ac^3 - 4b^3d - 27a^2d^2 + 18abcd

Crates

Algebraeon is published to crates.io under an umbrella crate algebraeon made up of:

• algebraeon-sets
• algebraeon-nzq
• algebraeon-groups
• algebraeon-rings
• algebraeon-geometry

Algorithms

Algebraeon currently implements the following algorithms:

• Euclids algorithm for GCD and the extended version for obtaining Bezout coefficients.
• Polynomial GCD computations using subresultant pseudo-remainder sequences.
• AKS algorithm for natural number primality testing.
• Matrix algorithms including:
  • Putting a matrix into Hermite normal form. In particular putting it into echelon form.
  • Putting a matrix into Smith normal form.
  • Gram–Schmidt algorithm for orthogonalization and orthonormalization.
  • Putting a matrix into Jordan normal.
  • Finding the general solution to a linear or affine system of equations.
• Polynomial factoring algorithms including:
  • Kronecker's method for factoring polynomials over the integers (slow).
  • Berlekamp-Zassenhaus algorithm for factoring polynomials over the integers.
  • Berlekamp's algorithm for factoring polynomials over finite fields.
  • Cantor–Zassenhaus algorithm for factoring polynomials over finite fields.
  • Trager's algorithm for factoring polynomials over algebraic number fields.
• Expressing symmetric polynomials in terms of elementary symmetric polynomials.
• Computations with algebraic numbers:
  • Real root isolation and arithmetic.
  • Complex root isolation and arithmetic.
• Computations with multiplication tables for small finite groups.
• Todd-Coxeter algorithm for the enumeration of finite index cosets of a finitely generated groups.

Getting Help

If you have questions, concerns, bug reports, etc, please file an issue in this repository's Issue Tracker.

Contributing

Contributions are welcome. There are two primary ways to contribute:

Using the issue tracker

Use the issue tracker to suggest feature requests, report bugs, and ask questions.

Changing the code-base

You should fork this repository, make changes in your own fork, and then submit a pull request. New code should have associated unit tests that validate implemented features and the presence or lack of defects.

Algebraeon is organized as a cargo workspace. Run cargo test in the root directory to build and run all tests.

A suggested workflow for testing new features:

• Create a new binary in examples/src/bin, for example my_main.rs.
• To run, use cargo run --bin my_main.rs in the root directory.
• Test any changes to the codebase with unit tests and/or using my_main.rs.