# Univariate polynomials over a noncommutative ring

AbstractAlgebra.jl provides a module, implemented in `src/NCPoly.jl`

for univariate polynomials over any noncommutative ring in the AbstractAlgebra type hierarchy.

## Generic type for univariate polynomials over a noncommutative ring

AbstractAlgebra.jl implements a generic univariate polynomial type over noncommutative rings in `src/generic/NCPoly.jl`

.

These generic polynomials have type `Generic.NCPoly{T}`

where `T`

is the type of elements of the coefficient ring. Internally they consist of a Julia array of coefficients and some additional fields for length and a parent object, etc. See the file `src/generic/GenericTypes.jl`

for details.

Parent objects of such polynomials have type `Generic.NCPolyRing{T}`

.

The string representation of the variable of the polynomial ring and the base/coefficient ring $R$ is stored in the parent object.

## Abstract types

The polynomial element types belong to the abstract type `NCPolyRingElem{T}`

and the polynomial ring types belong to the abstract type `NCPolyRing{T}`

. This enables one to write generic functions that can accept any AbstractAlgebra polynomial type.

Note that both the generic polynomial ring type `Generic.NCPolyRing{T}`

and the abstract type it belongs to, `NCPolyRing{T}`

are both called `NCPolyRing`

. The former is a (parameterised) concrete type for a polynomial ring over a given base ring whose elements have type `T`

. The latter is an abstract type representing all polynomial ring types in AbstractAlgebra.jl, whether generic or very specialised (e.g. supplied by a C library).

## Polynomial ring constructors

In order to construct polynomials in AbstractAlgebra.jl, one must first construct the polynomial ring itself. This is accomplished with the following constructor.

`AbstractAlgebra.polynomial_ring`

— Method`polynomial_ring(R::NCRing, s::VarName = :x; cached::Bool = true)`

Given a base ring `R`

and symbol/string `s`

specifying how the generator (variable) should be printed, return a tuple `S, x`

representing the new polynomial ring $S = R[x]$ and the generator $x$ of the ring.

By default the parent object `S`

depends only on `R`

and `x`

and will be cached. Setting the optional argument `cached`

to `false`

will prevent the parent object `S`

from being cached.

**Examples**

```
julia> R, x = polynomial_ring(ZZ, :x)
(Univariate polynomial ring in x over integers, x)
julia> S, y = polynomial_ring(R, :y)
(Univariate polynomial ring in y over univariate polynomial ring, y)
```

A shorthand version of this function is provided: given a base ring `R`

, we abbreviate the constructor as follows.

`R["x"]`

Here are some examples of creating polynomial rings and making use of the resulting parent objects to coerce various elements into the polynomial ring.

**Examples**

```
julia> R = matrix_ring(ZZ, 2)
Matrix ring of degree 2
over integers
julia> S, x = polynomial_ring(R, "x")
(Univariate polynomial ring in x over matrix ring, x)
julia> T, y = polynomial_ring(S, "y")
(Univariate polynomial ring in y over univariate polynomial ring in x over matrix ring, y)
julia> U, z = R["z"]
(Univariate polynomial ring in z over matrix ring, z)
julia> f = S()
0
julia> g = S(123)
[123 0; 0 123]
julia> h = T(BigInt(1234))
[1234 0; 0 1234]
julia> k = T(x + 1)
x + 1
julia> m = U(z + 1)
z + 1
```

All of the examples here are generic polynomial rings, but specialised implementations of polynomial rings provided by external modules will also usually provide a `polynomial_ring`

constructor to allow creation of their polynomial rings.

## Basic ring functionality

Once a polynomial ring is constructed, there are various ways to construct polynomials in that ring.

The easiest way is simply using the generator returned by the `polynomial_ring`

constructor and build up the polynomial using basic arithmetic, as described in the Ring interface.

The Julia language also has special syntax for the construction of polynomials in terms of a generator, e.g. we can write `2x`

instead of `2*x`

.

The polynomial rings in AbstractAlgebra.jl implement the full Ring interface. Of course the entire Univariate Polynomial Ring interface is also implemented.

We give some examples of such functionality.

**Examples**

```
julia> R = matrix_ring(ZZ, 2)
Matrix ring of degree 2
over integers
julia> S, x = polynomial_ring(R, "x")
(Univariate polynomial ring in x over matrix ring, x)
julia> T, y = polynomial_ring(S, "y")
(Univariate polynomial ring in y over univariate polynomial ring in x over matrix ring, y)
julia> f = x^3 + 3x + 21
x^3 + [3 0; 0 3]*x + [21 0; 0 21]
julia> g = (x + 1)*y^2 + 2x + 1
(x + 1)*y^2 + [2 0; 0 2]*x + 1
julia> h = zero(T)
0
julia> k = one(S)
1
julia> isone(k)
true
julia> iszero(f)
false
julia> n = length(g)
3
julia> U = base_ring(T)
Univariate polynomial ring in x over matrix ring
julia> V = base_ring(y + 1)
Univariate polynomial ring in x over matrix ring
julia> v = var(T)
:y
julia> U = parent(y + 1)
Univariate polynomial ring in y over univariate polynomial ring in x over matrix ring
julia> g == deepcopy(g)
true
```

## Polynomial functionality provided by AbstractAlgebra.jl

The functionality listed below is automatically provided by AbstractAlgebra.jl for any polynomial module that implements the full Univariate Polynomial Ring interface over a noncommutative ring. This includes AbstractAlgebra.jl's own generic polynomial rings.

But if a C library provides all the functionality documented in the Univariate Polynomial Ring interface over a noncommutative ring, then all the functions described here will also be automatically supplied by AbstractAlgebra.jl for that polynomial type.

Of course, modules are free to provide specific implementations of the functions described here, that override the generic implementation.

### Basic functionality

`AbstractAlgebra.leading_coefficient`

— Method`leading_coefficient(a::PolynomialElem)`

Return the leading coefficient of the given polynomial. This will be the nonzero coefficient of the term with highest degree unless the polynomial in the zero polynomial, in which case a zero coefficient is returned.

`AbstractAlgebra.trailing_coefficient`

— Method`trailing_coefficient(a::PolynomialElem)`

Return the trailing coefficient of the given polynomial. This will be the nonzero coefficient of the term with lowest degree unless the polynomial is the zero polynomial, in which case a zero coefficient is returned.

`AbstractAlgebra.gen`

— Method`gen(R::NCPolyRing)`

Return the generator of the given polynomial ring.

`AbstractAlgebra.is_gen`

— Method`is_gen(a::PolynomialElem)`

Return `true`

if the given polynomial is the constant generator of its polynomial ring, otherwise return `false`

.

`AbstractAlgebra.is_monomial`

— Method`is_monomial(a::PolynomialElem)`

Return `true`

if the given polynomial is a monomial.

`AbstractAlgebra.is_term`

— Method`is_term(a::PolynomialElem)`

Return `true`

if the given polynomial has one term.

**Examples**

```
julia> R = matrix_ring(ZZ, 2)
Matrix ring of degree 2
over integers
julia> S, x = polynomial_ring(R, "x")
(Univariate polynomial ring in x over matrix ring, x)
julia> T, y = polynomial_ring(S, "y")
(Univariate polynomial ring in y over univariate polynomial ring in x over matrix ring, y)
julia> a = zero(T)
0
julia> b = one(T)
1
julia> c = BigInt(1)*y^2 + BigInt(1)
y^2 + 1
julia> d = x*y^2 + (x + 1)*y + 3
x*y^2 + (x + 1)*y + [3 0; 0 3]
julia> f = leading_coefficient(d)
x
julia> y = gen(T)
y
julia> g = is_gen(y)
true
julia> m = is_unit(b)
true
julia> n = degree(d)
2
julia> is_term(2y^2)
true
julia> is_monomial(y^2)
true
```

### Truncation

`Base.truncate`

— Method`truncate(a::PolynomialElem, n::Int)`

Return $a$ truncated to $n$ terms, i.e. the remainder upon division by $x^n$.

`AbstractAlgebra.mullow`

— Method`mullow(a::NCPolyRingElem{T}, b::NCPolyRingElem{T}, n::Int) where T <: NCRingElem`

Return $a\times b$ truncated to $n$ terms.

**Examples**

```
julia> R = matrix_ring(ZZ, 2)
Matrix ring of degree 2
over integers
julia> S, x = polynomial_ring(R, "x")
(Univariate polynomial ring in x over matrix ring, x)
julia> T, y = polynomial_ring(S, "y")
(Univariate polynomial ring in y over univariate polynomial ring in x over matrix ring, y)
julia> f = x*y^2 + (x + 1)*y + 3
x*y^2 + (x + 1)*y + [3 0; 0 3]
julia> g = (x + 1)*y + (x^3 + 2x + 2)
(x + 1)*y + x^3 + [2 0; 0 2]*x + [2 0; 0 2]
julia> h = truncate(f, 1)
[3 0; 0 3]
julia> k = mullow(f, g, 4)
(x^2 + x)*y^3 + (x^4 + [3 0; 0 3]*x^2 + [4 0; 0 4]*x + 1)*y^2 + (x^4 + x^3 + [2 0; 0 2]*x^2 + [7 0; 0 7]*x + [5 0; 0 5])*y + [3 0; 0 3]*x^3 + [6 0; 0 6]*x + [6 0; 0 6]
```

### Reversal

`Base.reverse`

— Method`reverse(x::PolynomialElem, len::Int)`

Return the reverse of the polynomial $x$, thought of as a polynomial of the given length (the polynomial will be notionally truncated or padded with zeroes before the leading term if necessary to match the specified length). The resulting polynomial is normalised. If `len`

is negative we throw a `DomainError()`

.

`Base.reverse`

— Method`reverse(x::PolynomialElem)`

Return the reverse of the polynomial $x$, i.e. the leading coefficient of $x$ becomes the constant coefficient of the result, etc. The resulting polynomial is normalised.

**Examples**

```
julia> R = matrix_ring(ZZ, 2)
Matrix ring of degree 2
over integers
julia> S, x = polynomial_ring(R, "x")
(Univariate polynomial ring in x over matrix ring, x)
julia> T, y = polynomial_ring(S, "y")
(Univariate polynomial ring in y over univariate polynomial ring in x over matrix ring, y)
julia> f = x*y^2 + (x + 1)*y + 3
x*y^2 + (x + 1)*y + [3 0; 0 3]
julia> g = reverse(f, 7)
[3 0; 0 3]*y^6 + (x + 1)*y^5 + x*y^4
julia> h = reverse(f)
[3 0; 0 3]*y^2 + (x + 1)*y + x
```

### Shifting

`AbstractAlgebra.shift_left`

— Method`shift_left(f::PolynomialElem, n::Int)`

Return the polynomial $f$ shifted left by $n$ terms, i.e. multiplied by $x^n$.

`AbstractAlgebra.shift_right`

— Method`shift_right(f::PolynomialElem, n::Int)`

Return the polynomial $f$ shifted right by $n$ terms, i.e. divided by $x^n$.

**Examples**

```
julia> R = matrix_ring(ZZ, 2)
Matrix ring of degree 2
over integers
julia> S, x = polynomial_ring(R, "x")
(Univariate polynomial ring in x over matrix ring, x)
julia> T, y = polynomial_ring(S, "y")
(Univariate polynomial ring in y over univariate polynomial ring in x over matrix ring, y)
julia> f = x*y^2 + (x + 1)*y + 3
x*y^2 + (x + 1)*y + [3 0; 0 3]
julia> g = shift_left(f, 7)
x*y^9 + (x + 1)*y^8 + [3 0; 0 3]*y^7
julia> h = shift_right(f, 2)
x
```

### Evaluation

`AbstractAlgebra.evaluate`

— Method`evaluate(a::NCPolyRingElem, b::T) where T <: NCRingElem`

Evaluate the polynomial $a$ at the value $b$ and return the result.

`AbstractAlgebra.evaluate`

— Method`evaluate(a::NCPolyRingElem, b::Union{Integer, Rational, AbstractFloat})`

Evaluate the polynomial $a$ at the value $b$ and return the result.

We also overload the functional notation so that the polynomial $f$ can be evaluated at $a$ by writing $f(a)$.

**Examples**

```
julia> R = matrix_ring(ZZ, 2)
Matrix ring of degree 2
over integers
julia> S, x = polynomial_ring(R, "x")
(Univariate polynomial ring in x over matrix ring, x)
julia> T, y = polynomial_ring(S, "y")
(Univariate polynomial ring in y over univariate polynomial ring in x over matrix ring, y)
julia> f = x*y^2 + (x + 1)*y + 3
x*y^2 + (x + 1)*y + [3 0; 0 3]
julia> k = evaluate(f, 3)
[12 0; 0 12]*x + [6 0; 0 6]
julia> m = evaluate(f, x^2 + 2x + 1)
x^5 + [4 0; 0 4]*x^4 + [7 0; 0 7]*x^3 + [7 0; 0 7]*x^2 + [4 0; 0 4]*x + [4 0; 0 4]
julia> r = f(23)
[552 0; 0 552]*x + [26 0; 0 26]
```

### Derivative

`AbstractAlgebra.derivative`

— Method`derivative(a::PolynomialElem)`

Return the derivative of the polynomial $a$.

**Examples**

```
julia> R = matrix_ring(ZZ, 2)
Matrix ring of degree 2
over integers
julia> S, x = polynomial_ring(R, "x")
(Univariate polynomial ring in x over matrix ring, x)
julia> T, y = polynomial_ring(S, "y")
(Univariate polynomial ring in y over univariate polynomial ring in x over matrix ring, y)
julia> f = x*y^2 + (x + 1)*y + 3
x*y^2 + (x + 1)*y + [3 0; 0 3]
julia> h = derivative(f)
[2 0; 0 2]*x*y + x + 1
```