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.
polynomial_ring — Methodpolynomial_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 R, 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 S, 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 S, 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 S
julia> g == deepcopy(g)
truePolynomial 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
leading_coefficient — Methodleading_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.
trailing_coefficient — Methodtrailing_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.
gen — Methodgen(R::NCPolyRing)Return the generator of the given polynomial ring.
is_gen — Methodis_gen(a::PolynomialElem)Return true if the given polynomial is the constant generator of its polynomial ring, otherwise return false.
is_monomial — Methodis_monomial(a::PolynomialElem)Return true if the given polynomial is a monomial.
is_term — Methodis_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 S, 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
truncate — Methodtruncate(a::PolynomialElem, n::Int)Return $a$ truncated to $n$ terms, i.e. the remainder upon division by $x^n$.
mullow — Methodmullow(a::NCPolyRingElem{T}, b::NCPolyRingElem{T}, n::Int) where T <: NCRingElemReturn $a\times b$ 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 S, 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
reverse — Methodreverse(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().
reverse — Methodreverse(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 S, 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
shift_left — Methodshift_left(f::PolynomialElem, n::Int)Return the polynomial $f$ shifted left by $n$ terms, i.e. multiplied by $x^n$.
shift_right — Methodshift_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 S, 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
evaluate — Methodevaluate(a::NCPolyRingElem, b::T) where T <: NCRingElemEvaluate the polynomial $a$ at the value $b$ and return the result.
evaluate — Methodevaluate(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 S, 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
derivative — Methodderivative(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 S, 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