Free modules and vectors

As generators of finitely generated modules in Singular.jl are given as submodule of free modules over a polynomial ring $R$, Singular.jl supports creation of the free module $R^n$ and vectors of length $n$ in such a module.

The Singular.jl type for a vector is svector{T}. For the most part, these exist to help interact with the smodule{T} type provided by Singular.

The types of vectors and associated parent objects are given in the following table according to the library providing them.

Library	Element type	Parent type
Singular	`svector{T}`	`Singular.FreeMod{T}`

These types are parameterised by the type of elements in the polynomial ring $R$.

All free module types belong directly to the abstract type Module{T} and vector types belong directly to ModuleElem{T}.

Free module and vector functionality

Singular.jl modules implement standard operations one would expect on vectors and their associated parent modules.

These include:

Operations common to all AbstractAlgebra objects, such as parent, base_ring, elem_type, parent_type, parent, deepcopy, etc.
Arithmetic operations on vectors: (unary) -, +, -
Scalar multiplication of vectors by polynomials, constants and integers
Comparison operators: ==

Below, we describe all of the functionality for Singular.jl free modules that is not included in this list of basic operations.

Constructors

Given a Singular polynomial ring $R$ and a rank $n$, the following constructors are available for creating free modules.

FreeModule{T <: AbstractAlgebra.RingElem}(R::PolyRing{T}, n::Int)

Construct the free module $R^n$ over the polynomial ring $R$. Elements of the module returned by this function are vectors of length $n$, with entries in $R$.

vector{T <: AbstractAlgebra.RingElem}(R::PolyRing{T}, coords::spoly{T}...)

Construct the vector whose coordinates (which are elements of $R$) are the given parameters.

Examples

julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"])
(Singular polynomial ring (QQ),(x,y),(dp(2),C), spoly{n_Q}[x, y])

julia> M = FreeModule(R, 3)
Free module of rank 3 over Singular polynomial ring (QQ),(x,y),(dp(2),C)

julia> v2 = M([x + 1, x*y + 1, y])
x*y*gen(2)+x*gen(1)+y*gen(3)+gen(2)+gen(1)

julia> v1 = vector(R, x + 1, x*y + 1, y)
x*y*gen(2)+x*gen(1)+y*gen(3)+gen(2)+gen(1)

Basic manipulation

LinearAlgebra.rank — Method

rank(M::FreeMod)

Return the rank of the given free module.

source

AbstractAlgebra.gens — Method

gens(I::smodule)

Return the generators in the internal representation of the module $I$ as an array.

source

gens(I::sideal)

Return the generators in the internal representation of the ideal $I$ as an array.

source

gens{T <: AbstractAlgebra.RingElem}(M::FreeMod{T})

Return a Julia array whose entries are the generators of the given free module.

source

Examples

julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"])
(Singular polynomial ring (QQ),(x,y),(dp(2),C), spoly{n_Q}[x, y])

julia> M = FreeModule(R, 5)
Free module of rank 5 over Singular polynomial ring (QQ),(x,y),(dp(2),C)

julia> v = gens(M)
5-element Vector{svector{spoly{n_Q}}}:
 gen(1)
 gen(2)
 gen(3)
 gen(4)
 gen(5)

julia> r = rank(M)
5

Conversions

To convert the internal Singular representation of an svector{T} to a Julia array whose entries have type T, we have the following conversion routine.

Array{T <: Nemo.RingElem}(v::svector{spoly{T}})

Examples

julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"])
(Singular polynomial ring (QQ),(x,y),(dp(2),C), spoly{n_Q}[x, y])

julia> v1 = vector(R, x + 1, x*y + 1, y)
x*y*gen(2)+x*gen(1)+y*gen(3)+gen(2)+gen(1)

julia> V = Array(v1)
3-element Vector{spoly{n_Q}}:
 x + 1
 x*y + 1
 y

Jet of vectors

Singular.jet — Method

jet(x::svector{spoly{T}}, n::Int)

Given a vector $x$ this function truncates each entry of $x$ up to degree $n$.

source

Examples

julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"])
(Singular polynomial ring (QQ),(x,y),(dp(2),C), spoly{n_Q}[x, y])

julia> v = vector(R, x^5 + 1, 2x^3 + 3y^2, x^2)
x^5*gen(1)+2*x^3*gen(2)+x^2*gen(3)+3*y^2*gen(2)+gen(1)

julia> w = jet(v, 3)
2*x^3*gen(2)+x^2*gen(3)+3*y^2*gen(2)+gen(1)