Univariate Polynomial Ring Interface

# Univariate Polynomial Ring Interface

Univariate polynomial rings are supported in AbstractAlgebra, and in addition to the standard Ring interface, numerous additional functions are required to be present for univariate polynomial rings.

Univariate polynomial rings can be built over both commutative and noncommutative rings.

Univariate polynomial rings over a field are also Euclidean and therefore such rings must implement the Euclidean interface.

Since a sparse distributed multivariate format can generally also handle sparse univariate polynomials, the univariate polynomial interface is designed around the assumption that they are dense. This is not a requirement, but it may be easier to use the multivariate interface for sparse univariate types.

## Types and parents

AbstractAlgebra provides two abstract types for polynomial rings and their elements over a commutative ring:

• `PolyRing{T}` is the abstract type for univariate polynomial ring parent types
• `PolyElem{T}` is the abstract type for univariate polynomial types

Similarly there are two abstract types for polynomial rings and their elements over a noncommutative ring:

• `NCPolyRing{T}` is the abstract type for univariate polynomial ring parent types
• `NCPolyElem{T}` is the abstract type for univariate polynomial types

We have that `PolyRing{T} <: AbstractAlgebra.Ring` and `PolyElem{T} <: AbstractAlgebra.RingElem`. Similarly we have that `NCPolyRing{T} <: AbstractAlgebra.NCRing` and `NCPolyElem{T} <: AbstractAlgebra.NCRingElem`.

Note that the abstract types are parameterised. The type `T` should usually be the type of elements of the coefficient ring of the polynomial ring. For example, in the case of \$\mathbb{Z}[x]\$ the type `T` would be the type of an integer, e.g. `BigInt`.

If the parent object for such a ring has type `MyZX` and polynomials in that ring have type `MyZXPoly` then one would have:

• `MyZX <: PolyRing{BigInt}`
• `MyZXPoly <: PolyElem{BigInt}`

Polynomial rings should be made unique on the system by caching parent objects (unless an optional `cache` parameter is set to `false`). Polynomial rings should at least be distinguished based on their base (coefficient) ring. But if they have the same base ring and symbol (for their variable/generator), they should certainly have the same parent object.

See `src/generic/GenericTypes.jl` for an example of how to implement such a cache (which usually makes use of a dictionary).

## Required functionality for univariate polynomials

In addition to the required functionality for the Ring/NCRing interface (and in the case of polynomials over a field, the Euclidean Ring interface), the Polynomial Ring interface has the following required functions.

We suppose that `R` is a fictitious base ring (coefficient ring) and that `S` is a univariate polynomial ring over `R` (i.e. \$S = R[x]\$) with parent object `S` of type `MyPolyRing{T}`. We also assume the polynomials in the ring have type `MyPoly{T}`, where `T` is the type of elements of the base (coefficient) ring.

Of course, in practice these types may not be parameterised, but we use parameterised types here to make the interface clearer.

Note that the type `T` must (transitively) belong to the abstract type `RingElem` or `NCRingElem`.

We describe the functionality below for polynomials over commutative rings, i.e. with element type belonging to `RingElem`, however similar constructors should be available for element types belonging to `NCRingElem` instead, if the coefficient ring is noncommutative.

### Constructors

In addition to the standard constructors, the following constructors, taking an array of coefficients, must be available.

``(S::MyPolyRing{T})(A::Array{T, 1}) where T <: AbstractAlgebra.RingElem``

Create the polynomial in the given ring whose degree \$i\$ coefficient is given by `A[i]`.

``(S::MyPolyRing{T})(A::Array{U, 1}) where T <: AbstractAlgebra.RingElem, U <: AbstractAlgebra.RingElem``

Create the polynomial in the given ring whose degree \$i\$ coefficient is given by `A[i]`. The elements of the array are assumed to be able to be coerced into the base ring `R`.

``(S::MyPolyRing{T})(A::Array{U, 1}) where T <: AbstractAlgebra.RingElem, U <: Integer``

Create the polynomial in the given ring whose degree \$i\$ coefficient is given by `A[i]`.

It may be desirable to have a additional version of the function that accepts an array of Julia `Int` values if this can be done more efficiently.

Examples

``````julia> S, x = PolynomialRing(QQ, "x")
(Univariate Polynomial Ring in x over Rationals, x)

julia> f = S(Rational{BigInt}[2, 3, 1])
x^2+3//1*x+2//1

julia> g = S(BigInt[1, 0, 4])
4//1*x^2+1//1

julia> h = S([4, 7, 2, 9])
9//1*x^3+2//1*x^2+7//1*x+4//1
``````

### Data type and parent object methods

``var(S::MyPolyRing{T}) where T <: AbstractAlgebra.RingElem``

Return a `Symbol` representing the variable (generator) of the polynomial ring. Note that this is a `Symbol` not a `String`, though its string value will usually be used when printing polynomials.

``symbols(S::MyPolyRing{T}) where T <: AbstractAlgebra.RingElem``

Return the array `[s]` where `s` is a `Symbol` representing the variable of the given polynomial ring. This is provided for uniformity with the multivariate interface, where there is more than one variable and hence an array of symbols.

Examples

``````julia> S, x = PolynomialRing(QQ, "x")
(Univariate Polynomial Ring in x over Rationals, x)

julia> vsym = var(S)
:x

julia> V = symbols(S)
1-element Array{Symbol,1}:
:x
``````

### Basic manipulation of rings and elements

``length(f::MyPoly{T}) where T <: AbstractAlgebra.RingElem``

Return the length of the given polynomial. The length of the zero polynomial is defined to be \$0\$, otherwise the length is the degree plus \$1\$. The return value should be of type `Int`.

``set_length!(f::MyPoly{T}, n::Int) where T <: AbstractAlgebra.RingElem``

This function must zero any coefficients beyond the requested length \$n\$ and then set the length of the polynomial to \$n\$. This function does not need to normalise the polynomial and is not useful to the user, but is used extensively by the AbstractAlgebra generic functionality.

This function mutates the existing polynomial in-place, but does not return the polynomial.

``coeff(f::MyPoly{T}, n::Int) where T <: AbstractAlgebra.RingElem``

Return the coefficient of the polynomial `f` of degree `n`. If `n` is larger than the degree of the polynomial, it should return zero in the coefficient ring.

``setcoeff!(f::MyPoly{T}, n::Int, a::T) where T <: AbstractAlgebra.RingElem``

Set the degree \$n\$ coefficient of \$f\$ to \$a\$. This mutates the polynomial in-place if possible and returns the mutated polynomial (so that immutable types can also be supported). The function must not assume that the polynomial already has space for \$n + 1\$ coefficients. The polynomial must be resized if this is not the case.

Note that this function is not required to normalise the polynomial and is not necessarily useful to the user, but is used extensively by the generic functionality in AbstractAlgebra.jl. It is for setting raw coefficients in the representation.

``normalise(f::MyPoly{T}, n::Int) where T <: AbstractAlgebra.RingElem``

Given a polynomial whose length is currently \$n\$, including any leading zero coefficients, return the length of the normalised polynomial (either zero or the length of the polynomial with nonzero leading coefficient). Note that the function does not actually perform the normalisation.

``fit!(f::MyPoly{T}, n::Int) where T <: AbstractAlgebra.RingElem``

Ensure that the polynomial \$f\$ internally has space for \$n\$ coefficients. This function must mutate the function in-place if it is mutable. It does not return the mutated polynomial. Immutable types can still be supported by defining this function to do nothing.

Some interfaces for C polynomial types automatically manage the internal allocation of polynomials in every function that can be called on them. Explicit adjustment by the generic code in AbstractAlgebra.jl is not required. In such cases, this function can also be defined to do nothing.

Examples

``````julia> S, x = PolynomialRing(ZZ, "x")
(Univariate Polynomial Ring in x over Integers, x)

julia> f = x^3 + 3x + 1
x^3+3*x+1

julia> g = S(BigInt[1, 2, 0, 1, 0, 0, 0]);

julia> n = length(f)
4

julia> c = coeff(f, 1)
3

julia> set_length!(g, normalise(g, 7))
4

julia> g = setcoeff!(g, 2, BigInt(11))
x^3+11*x^2+2*x+1

julia> fit!(g, 8)

julia> g = setcoeff!(g, 7, BigInt(4))
4*x^7+x^3+11*x^2+2*x+1``````

## Optional functionality for polynomial rings

Sometimes parts of the Euclidean Ring interface can and should be implemented for polynomials over a ring that is not necessarily a field.

When divisibility testing can be implemented for a polynomial ring over a field, it should be possible to implement the following functions from the Euclidean Ring interface:

• divides
• remove
• valuation

When the given polynomial ring is a GCD domain, with an effective GCD algorithm, it may be possible to implement the following functions:

• gcd
• lcm

Polynomial rings can optionally implement any part of the generic univariate polynomial functionality provided by AbstractAlgebra.jl, using the same interface.

Obviously additional functionality can also be added to that provided by AbstractAlgebra.jl on an ad hoc basis.