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Algebra Homomorphisms

Singular.jl allows the creation of algebra homomorphisms of Singular polynomial rings over Nemo/Singular coefficient rings.

The default algebra homomorphism type in Singular.jl is the Singular SAlgHom type.

Additionally, a special type for the identity homomorphism has been implemented. The type in Singular.jl for the latter is SIdAlgHom.

All algebra homomorphism types belong directly to the abstract type AbstractAlgebraHomomorphism{T}.

Algebra Homomorphism functionality

Constructors

Given two Singular polynomial rings $R$ and $S$ over the same base ring, the following constructors are available for creating algebra homomorphisms.

Singular.AlgebraHomomorphism — Method
AlgebraHomomorphism(D::PolyRing, C::PolyRing, V::Vector)

Constructs an algebra homomorphism $f: D \to C$, where the $i$-th variable of $D$ is mapped to the $i$-th entry of $V$. $D$ and $C$ must be polynomial rings over the same base ring.

source
Singular.IdentityAlgebraHomomorphism — Method
IdentityAlgebraHomomorphism(R::PolyRing)

Constructs the canonical identity algebra homomorphism $id: D \to D$, where the $i$-th variable of $D$ is mapped to itself.

source

Examples

julia> L = FiniteField(3, 2, "a")
(Finite Field of Characteristic 3 and degree 2, a)

julia> R, (x, y, z, w) = polynomial_ring(L[1], ["x", "y", "z", "w"];
                                    ordering=:negdegrevlex)
(Singular Polynomial Ring (9,a),(x,y,z,w),(ds(4),C), spoly{n_GF}[x, y, z, w])

julia> S, (a, b, c) = polynomial_ring(L[1], ["a", "b", "c"];
                                    ordering=:degrevlex)
(Singular Polynomial Ring (9,a),(a@1,b,c),(dp(3),C), spoly{n_GF}[a, b, c])

julia> V = [a, a + b^2, b - c, c + b]
4-element Vector{spoly{n_GF}}:
 a
 b^2 + a
 b + a^4*c
 b + c

julia> f = AlgebraHomomorphism(R, S, V)
Algebra Homomorphism with

Domain: Singular Polynomial Ring (9,a),(x,y,z,w),(ds(4),C)

Codomain: Singular Polynomial Ring (9,a),(a@1,b,c),(dp(3),C)

Defining Equations: spoly{n_GF}[a, b^2 + a, b + a^4*c, b + c]

Operating on objects

It is possible to act on polynomials and ideals via algebra homomorphisms.

Examples

R, (x, y, z, w) = polynomial_ring(Nemo.ZZ, ["x", "y", "z", "w"];
                             ordering=:negdegrevlex)

S, (a, b, c) = polynomial_ring(Nemo.ZZ, ["a", "b", "c"];
                             ordering=:degrevlex)

V = [a, a + b^2, b - c, c + b]

f = AlgebraHomomorphism(R, S, V)

id  = IdentityAlgebraHomomorphism(S)


J = Ideal(R, [x, y^3])

p = x + y^3 + z*w

K = f(J)

q = f(p)

Composition

AbstractAlgebra.compose — Method
compose(f::AbstractAlgebra.Map(Singular.SAlgHom),
                     g::AbstractAlgebra.Map(Singular.SAlgHom))

Returns an algebra homomorphism $h: domain(f) \to codomain(g)$, where $h = g(f)$.

source

A short command for the composition of $f$ and $g$ is f*g, which is the same as compose(f, g).

Examples

julia> R, (x, y, z, w) = polynomial_ring(QQ, ["x", "y", "z", "w"];
                                    ordering=:negdegrevlex)
(Singular Polynomial Ring (QQ),(x,y,z,w),(ds(4),C), spoly{n_Q}[x, y, z, w])

julia> S, (a, b, c) = polynomial_ring(QQ, ["a", "b", "c"];
                                    ordering=:degrevlex)
(Singular Polynomial Ring (QQ),(a,b,c),(dp(3),C), spoly{n_Q}[a, b, c])

julia> V = [a, a + b^2, b - c, c + b]
4-element Vector{spoly{n_Q}}:
 a
 b^2 + a
 b - c
 b + c

julia> W = [x^2, x + y + z, z*y]
3-element Vector{spoly{n_Q}}:
 x^2
 x + y + z
 y*z

julia> f = AlgebraHomomorphism(R, S, V)
Algebra Homomorphism with

Domain: Singular Polynomial Ring (QQ),(x,y,z,w),(ds(4),C)

Codomain: Singular Polynomial Ring (QQ),(a,b,c),(dp(3),C)

Defining Equations: spoly{n_Q}[a, b^2 + a, b - c, b + c]


julia> g = AlgebraHomomorphism(S, R, W)
Algebra Homomorphism with

Domain: Singular Polynomial Ring (QQ),(a,b,c),(dp(3),C)

Codomain: Singular Polynomial Ring (QQ),(x,y,z,w),(ds(4),C)

Defining Equations: spoly{n_Q}[x^2, x + y + z, y*z]


julia> idR  = IdentityAlgebraHomomorphism(R)
Identity Algebra Homomorphism with

Domain: Singular Polynomial Ring (QQ),(x,y,z,w),(ds(4),C)

Defining Equations: spoly{n_Q}[x, y, z, w]


julia> h1 = f*g
Algebra Homomorphism with

Domain: Singular Polynomial Ring (QQ),(x,y,z,w),(ds(4),C)

Codomain: Singular Polynomial Ring (QQ),(x,y,z,w),(ds(4),C)

Defining Equations: spoly[x^2, 2*x^2 + 2*x*y + y^2 + 2*x*z + 2*y*z + z^2, x + y + z - y*z, x + y + z + y*z]


julia> h2 = idR*f
Algebra Homomorphism with

Domain: Singular Polynomial Ring (QQ),(x,y,z,w),(ds(4),C)

Codomain: Singular Polynomial Ring (QQ),(a,b,c),(dp(3),C)

Defining Equations: spoly{n_Q}[a, b^2 + a, b - c, b + c]


julia> h3 = g*idR
Algebra Homomorphism with

Domain: Singular Polynomial Ring (QQ),(a,b,c),(dp(3),C)

Codomain: Singular Polynomial Ring (QQ),(x,y,z,w),(ds(4),C)

Defining Equations: spoly{n_Q}[x^2, x + y + z, y*z]


julia> h4 = idR*idR
Identity Algebra Homomorphism with

Domain: Singular Polynomial Ring (QQ),(x,y,z,w),(ds(4),C)

Defining Equations: spoly{n_Q}[x, y, z, w]

Preimages

AbstractAlgebra.preimage — Method
preimage(f::AbstractAlgebra.Map(SAlgHom), I::sideal)

Returns the preimage of the ideal $I$ under the algebra homomorphism $f$.

source
AbstractAlgebra.preimage — Method
preimage(f::AbstractAlgebra.Map(SIdAlgHom), I::sideal)

Returns the preimage of the ideal $I$ under the identity algebra homomorphism.

source

Examples

julia> R, (x, y, z, w) = polynomial_ring(QQ, ["x", "y", "z", "w"];
                                    ordering=:negdegrevlex)
(Singular Polynomial Ring (QQ),(x,y,z,w),(ds(4),C), spoly{n_Q}[x, y, z, w])

julia> S, (a, b, c) = polynomial_ring(QQ, ["a", "b", "c"];
                                    ordering=:degrevlex)
(Singular Polynomial Ring (QQ),(a,b,c),(dp(3),C), spoly{n_Q}[a, b, c])

julia> I = Ideal(S, [a, a + b^2, b - c, c + b])
Singular ideal over Singular Polynomial Ring (QQ),(a,b,c),(dp(3),C) with generators (a, b^2 + a, b - c, b + c)

julia> f = AlgebraHomomorphism(R, S, gens(I))
Algebra Homomorphism with

Domain: Singular Polynomial Ring (QQ),(x,y,z,w),(ds(4),C)

Codomain: Singular Polynomial Ring (QQ),(a,b,c),(dp(3),C)

Defining Equations: spoly{n_Q}[a, b^2 + a, b - c, b + c]


julia> idS  = IdentityAlgebraHomomorphism(S)
Identity Algebra Homomorphism with

Domain: Singular Polynomial Ring (QQ),(a,b,c),(dp(3),C)

Defining Equations: spoly{n_Q}[a, b, c]


julia> P1 = preimage(f, I)
Singular ideal over Singular Polynomial Ring (QQ),(x,y,z,w),(ds(4),C) with generators (x, y, z, w)

julia> P2 = preimage(idS, I)
Singular ideal over Singular Polynomial Ring (QQ),(a,b,c),(dp(3),C) with generators (a, b^2 + a, b - c, b + c)

julia> K1 = kernel(f)
Singular ideal over Singular Polynomial Ring (QQ),(x,y,z,w),(ds(4),C) with generators (4*x - 4*y + z^2 + 2*z*w + w^2)

julia> K2 = kernel(idS)
Singular ideal over Singular Polynomial Ring (QQ),(a,b,c),(dp(3),C) with generators (0)