Affine Algebras and Their Ideals

With regard to notation, we use affine algebra as a synonym for quotient of a multivariate polynomial ring by an ideal. More specifically, if $R$ is a multivariate polynomial ring with coefficient ring $C$ , and $A=R/I$ is the quotient of $R$ by an ideal $I$ of $R$ , we refer to $A$ as an affine algebra over $C$ , or an affine $C$ -algebra. In this section, we discuss functionality for handling such algebras in OSCAR.

Note

We emphasize: In this section, we view $R/I$ together with its ring structure. Realizing $R/I$ as an $R$ -module means to implement it as the quotient of a free $R$ -module of rank 1. See the section on modules.

Note

Most functions discussed here rely on Gröbner basis techniques. In particular, they typically make use of a Gröbner basis for the modulus of the quotient. Nevertheless, the construction of quotients is lazy in the sense that the computation of such a Gröbner basis is delayed until the user performs an operation that indeed requires it. The Gröbner basis is then computed with respect to the monomial ordering entered by the user when creating the quotient; if no ordering is entered, OSCAR will use the default_ordering on the underlying polynomial ring. See the section on Gröbner/Standard Bases for default orderings in OSCAR. Once computed, the Gröbner basis is cached for later reuse.

Note

Recall that Gröbner basis methods are implemented for multivariate polynomial rings over fields (exact fields supported by OSCAR) and, where not indicated otherwise, for multivariate polynomial rings over the integers.

Note

In OSCAR, elements of a quotient $A = R/I$ are not necessarily represented by polynomials which are reduced with regard to $I$ . That is, if $f\in R$ is the internal polynomial representative of an element of $A$ , then $f$ may not be the normal form mod $I$ with respect to the default ordering on $R$ (see the section on Gröbner/Standard Bases for normal forms). Operations involving Gröbner basis computations may lead to (partial) reductions. The function simplify discussed in this section computes fully reduced representatives.

Note

Each grading on a multivariate polynomial ring R in OSCAR descends to a grading on the affine algebra A = R/I (recall that OSCAR ideals of graded polynomial rings are required to be homogeneous). Functionality for dealing with such gradings and our notation for describing this functionality descend accordingly. This applies, in particular, to the functions is_graded, is_standard_graded, is_z_graded, is_zm_graded, and is_positively_graded which will not be discussed again here.

Types

The OSCAR type for quotients of multivariate polynomial rings is of parametrized form MPolyQuoRing{T}, with elements of type MPolyQuoRingElem{T}. Here, T is the element type of the polynomial ring.

Constructors

quo — Method

quo(R::MPolyRing, I::MPolyIdeal; ordering::MonomialOrdering = default_ordering(R)) -> MPolyQuoRing, Map

Create the quotient ring R/I and return the new ring as well as the projection map R $\to$ R/I.

quo(R::MPolyRing, V::Vector{MPolyRingElem}; ordering::MonomialOrdering = default_ordering(R)) -> MPolyQuoRing, Map

As above, where I is the ideal of R generated by the polynomials in V.

Note

Once R/I is created, all computations within R/I relying on division with remainder and/or Gröbner bases are done with respect to ordering.

Examples

julia> R, (x, y) = polynomial_ring(QQ, [:x, :y]);

julia> A, p = quo(R, ideal(R, [x^2-y^3, x-y]));

julia> A
Quotient
  of multivariate polynomial ring in 2 variables x, y
    over rational field
  by ideal (x^2 - y^3, x - y)

julia> typeof(A)
MPolyQuoRing{QQMPolyRingElem}

julia> typeof(x)
QQMPolyRingElem

julia> p
Map defined by a julia-function with inverse
  from multivariate polynomial ring in 2 variables over QQ
  to quotient of multivariate polynomial ring by ideal (x^2 - y^3, x - y)

julia> p(x)
x

julia> typeof(p(x))
MPolyQuoRingElem{QQMPolyRingElem}

julia> S, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z]);

julia> B, _ = quo(S, ideal(S, [x^2*z-y^3, x-y]))
(Quotient of multivariate polynomial ring by ideal (x^2*z - y^3, x - y), Map: S -> B)

julia> typeof(B)
MPolyQuoRing{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}

source

Data Associated to Affine Algebras

Basic Data

If A=R/I is the quotient of a multivariate polynomial ring R modulo an ideal I of R, then

base_ring(A) refers to R,
modulus(A) to I,
gens(A) to the generators of A,
number_of_generators(A) / ngens(A) to the number of these generators,
gen(A, i) as well as A[i] to the i-th such generator, and
ordering(A) to the monomial ordering used in the construction of A.

Examples

julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z]);

julia> A, _ = quo(R, ideal(R, [y-x^2, z-x^3]));

julia> base_ring(A)
Multivariate polynomial ring in 3 variables x, y, z
  over rational field

julia> modulus(A)
Ideal generated by
  -x^2 + y
  -x^3 + z

julia> gens(A)
3-element Vector{MPolyQuoRingElem{QQMPolyRingElem}}:
 x
 y
 z

julia> number_of_generators(A)
3

julia> gen(A, 2)
y

julia> ordering(A)
degrevlex([x, y, z])

In the graded case, we additionally have:

grading_group — Method

monomial_basis — Method

monomial_basis(A::MPolyQuoRing, g::FinGenAbGroupElem)

Given an affine algebra A over a field which is graded by a free group of type FinGenAbGroup, and given an element g of that group, return a vector of monomials of R such that the residue classes of these monomials form a K-basis of the graded part of A of degree g.

monomial_basis(A::MPolyQuoRing, W::Vector{<:IntegerUnion})

Given a $\mathbb Z^m$ -graded affine algebra A over a field and a vector W of $m$ integers, convert W into an element g of the grading group of A and proceed as above.

monomial_basis(A::MPolyQuoRing, d::IntegerUnion)

Given a $\mathbb Z$ -graded affine algebra A over a field and an integer d, convert d into an element g of the grading group of A and proceed as above.

Note

If the component of the given degree is not finite dimensional, an error message will be thrown.

Examples

julia> R, (x, y) = graded_polynomial_ring(QQ, [:x, :y]);

julia> I = ideal(R, [x^2])
Ideal generated by
  x^2

julia> A, _ = quo(R, I)
(Quotient of multivariate polynomial ring by ideal (x^2), Map: R -> A)

julia> L = monomial_basis(A, 3)
2-element Vector{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}:
 y^3
 x*y^2

source

homogeneous_component — Method

homogeneous_component(A::MPolyQuoRing{<:MPolyDecRingElem}, g::FinGenAbGroupElem)

Given a graded quotient A of a multivariate polynomial ring over a field, where the grading group is free of type FinGenAbGroup, and given an element g of that group, return the homogeneous component of A of degree g. Additionally, return the embedding of the component into A.

homogeneous_component(A::MPolyQuoRing{<:MPolyDecRingElem}, g::Vector{<:IntegerUnion})

Given a $\mathbb Z^m$ -graded quotient A of a multivariate polynomial ring over a field, and given a vector g of $m$ integers, convert g into an element of the grading group of A, and return the homogeneous component of A whose degree is that element. Additionally, return the embedding of the component into A.

homogeneous_component(A::MPolyQuoRing{<:MPolyDecRingElem}, g::IntegerUnion)

Given a $\mathbb Z$ -graded quotient A of a multivariate polynomial ring over a field, and given an integer g, convert g into an element of the grading group of A, and return the homogeneous component of A whose degree is that element. Additionally, return the embedding of the component into A.

Note

If the component is not finite dimensional, an error message will be thrown.

Examples

julia> R, (w, x, y, z) = graded_polynomial_ring(QQ, [:w, :x, :y, :z])
(Graded multivariate polynomial ring in 4 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[w, x, y, z])

julia> L = homogeneous_component(R, 2);

julia> HC = gens(L[1]);

julia> EMB = L[2]
Map defined by a julia-function with inverse
  from R_[2] of dim 10
  to graded multivariate polynomial ring in 4 variables over QQ

julia> for i in 1:length(HC) println(EMB(HC[i])) end
z^2
y*z
y^2
x*z
x*y
x^2
w*z
w*y
w*x
w^2

julia> PTC = ideal(R, [-x*z + y^2, -w*z + x*y, -w*y + x^2]);

julia> A, _ = quo(R, PTC);

julia> L = homogeneous_component(A, 2);

julia> HC = gens(L[1]);

julia> EMB = L[2]
Map defined by a julia-function with inverse
  from quotient space over QQ with 7 generators and no relations
  to quotient of multivariate polynomial ring by ideal (-x*z + y^2, -w*z + x*y, -w*y + x^2)

julia> for i in 1:length(HC) println(EMB(HC[i])) end
z^2
y*z
x*z
w*z
w*y
w*x
w^2

julia> G = abelian_group([0, 0])
Z^2

julia> W = [G[1], G[1], G[2], G[2], G[2]];

julia> S, x, y = graded_polynomial_ring(QQ, :x => 1:2, :y => 1:3; weights = W);

julia> L = homogeneous_component(S, [2,1]);

julia> HC = gens(L[1]);

julia> EMB = L[2]
Map defined by a julia-function with inverse
  from S_[2 1] of dim 9
  to graded multivariate polynomial ring in 5 variables over QQ

julia> for i in 1:length(HC) println(EMB(HC[i])) end
x[2]^2*y[3]
x[2]^2*y[2]
x[2]^2*y[1]
x[1]*x[2]*y[3]
x[1]*x[2]*y[2]
x[1]*x[2]*y[1]
x[1]^2*y[3]
x[1]^2*y[2]
x[1]^2*y[1]

julia> I = ideal(S, [x[1]*y[1]-x[2]*y[2]]);

julia> A, = quo(S, I);

julia> L = homogeneous_component(A, [2,1]);

julia> HC = gens(L[1]);

julia> EMB = L[2]
Map defined by a julia-function with inverse
  from quotient space over QQ with 7 generators and no relations
  to quotient of multivariate polynomial ring by ideal (x[1]*y[1] - x[2]*y[2])

julia> for i in 1:length(HC) println(EMB(HC[i])) end
x[2]^2*y[3]
x[2]^2*y[2]
x[2]^2*y[1]
x[1]*x[2]*y[3]
x[1]*x[2]*y[2]
x[1]^2*y[3]
x[1]^2*y[2]

source

Dimension

dim — Method

is_finite_dimensional_vector_space — Method

vector_space_dimension — Method

monomial_basis — Method

Elements of Affine Algebras

Types

The OSCAR type for elements of quotients of multivariate polynomial rings is of parametrized form MPolyQuoRing{T}, where T is the element type of the polynomial ring.

Creating Elements of Affine Algebras

Elements of an affine algebra A=R/I are created as images of elements of R under the projection map or by directly coercing elements of R into A.

Examples

julia> R, (x, y) = polynomial_ring(QQ, [:x, :y]);

julia> A, p = quo(R, ideal(R, [x^3*y^2-y^3*x^2, x*y^4-x*y^2]));

julia> f = p(x^3*y^2-y^3*x^2+x*y)
x^3*y^2 - x^2*y^3 + x*y

julia> typeof(f)
MPolyQuoRingElem{QQMPolyRingElem}

julia> g = A(x^3*y^2-y^3*x^2+x*y)
x^3*y^2 - x^2*y^3 + x*y

julia> f == g
true

Reducing Polynomial Representatives

simplify — Method

Tests on Elements of Affine Algebras

== — Method

is_invertible_with_inverse — Method

In the graded case, we additionally have:

is_homogeneous — Method

Data associated to Elements of Affine Algebras

Given an element f of an affine algebra A,

parent(f) refers to A.

In the graded case, we also have:

homogeneous_components — Method

homogeneous_component — Method

homogeneous_component(f::MPolyQuoRingElem{<:MPolyDecRingElem}, g::FinGenAbGroupElem)

Given an element f of a graded affine algebra, and given an element g of the grading group of that algebra, return the homogeneous component of f of degree g.

homogeneous_component(f::MPolyQuoRingElem{<:MPolyDecRingElem}, g::Vector{<:IntegerUnion})

Given an element f of a $\mathbb Z^m$ -graded affine algebra A, say, and given a vector g of $m$ integers, convert g into an element of the grading group of A, and return the homogeneous component of f whose degree is that element.

homogeneous_component(f::MPolyQuoRingElem{<:MPolyDecRingElem}, g::IntegerUnion)

Given an element f of a $\mathbb Z$ -graded affine algebra A, say, and given an integer g, convert g into an element of the grading group of A, and return the homogeneous component of f whose degree is that element.

Examples

julia> R, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z]);

julia> A, p = quo(R, ideal(R, [y-x, z^3-x^3]));

julia> f = p(y^2-x^2+x*y*z+z^4)
-x^2 + x*y*z + y^2 + z^4

julia> homogeneous_component(f, 4)
z^4

source

degree — Method

degree(f::MPolyQuoRingElem{<:MPolyDecRingElem})

Given a homogeneous element f of a graded affine algebra, return the degree of f.

degree(::Type{Vector{Int}}, f::MPolyQuoRingElem{<:MPolyDecRingElem})

Given a homogeneous element f of a $\mathbb Z^m$ -graded affine algebra, return the degree of f, converted to a vector of integer numbers.

degree(::Type{Int}, f::MPolyQuoRingElem{<:MPolyDecRingElem})

Given a homogeneous element f of a $\mathbb Z$ -graded affine algebra, return the degree of f, converted to an integer number.

Examples

julia> R, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z] );

julia> A, p = quo(R, ideal(R, [y-x, z^3-x^3]))
(Quotient of multivariate polynomial ring by ideal (-x + y, -x^3 + z^3), Map: R -> A)

julia> f = p(y^2-x^2+z^4)
-x^2 + y^2 + z^4

julia> degree(f)
[4]

julia> typeof(degree(f))
FinGenAbGroupElem

julia> degree(Int, f)
4

julia> typeof(degree(Int, f))
Int64

source

Ideals in Affine Algebras

Constructors

ideal — Method

Reducing Polynomial Representatives of Generators

simplify — Method

Data Associated to Ideals in Affine Algebras

Basic Data

If a is an ideal of the affine algebra A, then

base_ring(a) refers to A,
gens(a) to the generators of a,
number_of_generators(a) / ngens(a) to the number of these generators, and
gen(a, i) as well as a[i] to the i-th such generator.

Examples

julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z]);

julia> A, _ = quo(R, ideal(R, [y-x^2, z-x^3]));

julia> a = ideal(A, [x-y, z^4])
Ideal generated by
  x - y
  z^4

julia> base_ring(a)
Quotient
  of multivariate polynomial ring in 3 variables x, y, z
    over rational field
  by ideal (-x^2 + y, -x^3 + z)

julia> gens(a)
2-element Vector{MPolyQuoRingElem{QQMPolyRingElem}}:
 x - y
 z^4

julia> number_of_generators(a)
2

julia> gen(a, 2)
z^4

Dimension of Ideals in Affine Algebras

dim — Method

Minimal Sets of Generators

In the graded case, we have:

minimal_generating_set — Method

Operations on Ideals in Affine Algebras

Simple Ideal Operations in Affine Algebras

Powers of Ideal

^ — Method

Sum of Ideals

+ — Method

Product of Ideals

* — Method

Intersection of Ideals

intersect — Method

Ideal Quotients

quotient — Method

Tests on Ideals in Affine Algebras

Basic Tests

is_zero — Method

Containment of Ideals in Affine Algebras

is_subset — Method

Equality of Ideals in Affine Algebras

== — Method

Ideal Membership

ideal_membership — Method

Homomorphisms From Affine Algebras

If $A=R/I$ is an affine $C$ -algebra, and $S$ is any ring, then defining a ring homomorphism $\overline{\phi}: A \to S$ means to define a ring homomorphism $\phi: R \to S$ such that $I\subset \ker(\phi)$ . Thus, $\overline{\phi}$ is determined by specifying its restriction to $C$ , and by assigning an image to each generator of $A$ . In OSCAR, such homomorphisms are created as follows:

hom — Method

hom(A::MPolyQuoRing, S::NCRing, coeff_map, images::Vector; check::Bool = true)

hom(A::MPolyQuoRing, S::NCRing, images::Vector; check::Bool = true)

Given a homomorphism coeff_map from C to S, where C is the coefficient ring of the base ring of A, and given a vector images of ngens(A) elements of S, return the homomorphism A $\to$ S whose restriction to C is coeff_map, and which sends the i-th generator of A to the i-th entry of images.

If no coefficient map is entered, invoke a canonical homomorphism of C to S, if such a homomorphism exists, and throw an error, otherwise.

Note

The function returns a well-defined homomorphism A $\to$ S iff the given data defines a homomorphism base_ring(A) $\to$ S whose kernel contains the modulus of A. This condition is checked by the function in case check = true (default).

Note

In case check = true (default), the function also checks the conditions below:

If S is graded, the assigned images must be homogeneous with respect to the given grading.
If S is noncommutative, the assigned images must pairwise commute.

Examples

julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z]);

julia> A, _ = quo(R, ideal(R, [y-x^2, z-x^3]));

julia> S, (s, t) = polynomial_ring(QQ, [:s, :t]);

julia> F = hom(A, S, [s, s^2, s^3])
Ring homomorphism
  from quotient of multivariate polynomial ring by ideal (-x^2 + y, -x^3 + z)
  to multivariate polynomial ring in 2 variables over QQ
defined by
  x -> s
  y -> s^2
  z -> s^3

source

Given a ring homomorphism F : A $\to$ S as above, domain(F) and codomain(F) refer to A and S, respectively. Given ring homomorphisms F : A $\to$ B and G : B $\to$ T as above, compose(F, G) refers to their composition.

Homomorphisms of Affine Algebras

The OSCAR homomorphism type AffAlgHom models ring homomorphisms R $\to$ S such that the types of R and S are subtypes of Union{MPolyRing{T}, MPolyQuoRing{U1}} and Union{MPolyRing{T}, MPolyQuoRing{U2}}, respectively. Here, T <: FieldElem and U1 <: MPolyRingElem{T}, U2 <: MPolyRingElem{T}. Functionality for these homomorphism is discussed in what follows.

Data Associated to Homomorphisms of Affine Algebras

preimage — Method

kernel — Method

Examples

julia> D1, (w, x, y, z) = graded_polynomial_ring(QQ, [:w, :x, :y, :z]);

julia> C1, (s,t) = graded_polynomial_ring(QQ, [:s, :t]);

julia> V1 = [s^3, s^2*t, s*t^2, t^3];

julia> para = hom(D1, C1, V1)
Ring homomorphism
  from graded multivariate polynomial ring in 4 variables over QQ
  to graded multivariate polynomial ring in 2 variables over QQ
defined by
  w -> s^3
  x -> s^2*t
  y -> s*t^2
  z -> t^3

julia> twistedCubic = kernel(para)
Ideal generated by
  -x*z + y^2
  -w*z + x*y
  -w*y + x^2

julia> C2, p2 = quo(D1, twistedCubic);

julia> D2, (a, b, c) = graded_polynomial_ring(QQ, [:a, :b, :c]);

julia> V2 = [p2(w-y), p2(x), p2(z)];

julia> proj = hom(D2, C2, V2)
Ring homomorphism
  from graded multivariate polynomial ring in 3 variables over QQ
  to quotient of multivariate polynomial ring by ideal (-x*z + y^2, -w*z + x*y, -w*y + x^2)
defined by
  a -> w - y
  b -> x
  c -> z

julia> nodalCubic = kernel(proj)
Ideal generated by
  -a^2*c + b^3 - 2*b^2*c + b*c^2

julia> D3,y = polynomial_ring(QQ, :y => 1:3);

julia> C3, x = polynomial_ring(QQ, :x => 1:3);

julia> V3 = [x[1]*x[2], x[1]*x[3], x[2]*x[3]];

julia> F3 = hom(D3, C3, V3)
Ring homomorphism
  from multivariate polynomial ring in 3 variables over QQ
  to multivariate polynomial ring in 3 variables over QQ
defined by
  y[1] -> x[1]*x[2]
  y[2] -> x[1]*x[3]
  y[3] -> x[2]*x[3]

julia> sphere = ideal(C3, [x[1]^3 + x[2]^3  + x[3]^3 - 1])
Ideal generated by
  x[1]^3 + x[2]^3 + x[3]^3 - 1

julia> steinerRomanSurface = preimage(F3, sphere)
Ideal generated by
  y[1]^6*y[2]^6 + 2*y[1]^6*y[2]^3*y[3]^3 + y[1]^6*y[3]^6 + 2*y[1]^3*y[2]^6*y[3]^3 + 2*y[1]^3*y[2]^3*y[3]^6 - y[1]^3*y[2]^3*y[3]^3 + y[2]^6*y[3]^6

Tests on Homomorphisms of Affine Algebras

is_injective — Method

is_surjective — Method

is_bijective — Method

is_finite — Method

Examples

julia> D, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z]);

julia> S, (a, b, c) = polynomial_ring(QQ, [:a, :b, :c]);

julia> C, p = quo(S, ideal(S, [c-b^3]));

julia> V = [p(2*a + b^6), p(7*b - a^2), p(c^2)];

julia> F = hom(D, C, V)
Ring homomorphism
  from multivariate polynomial ring in 3 variables over QQ
  to quotient of multivariate polynomial ring by ideal (-b^3 + c)
defined by
  x -> 2*a + c^2
  y -> -a^2 + 7*b
  z -> c^2

julia> is_surjective(F)
true

julia> D1, _ = quo(D, kernel(F));

julia> F1 = hom(D1, C, V);

julia> is_bijective(F1)
true

julia> R, (x, y, z) = polynomial_ring(QQ, [ :x, :y, :z]);

julia> C, (s, t) = polynomial_ring(QQ, [:s, :t]);

julia> V = [s*t, t, s^2];

julia> paraWhitneyUmbrella = hom(R, C, V)
Ring homomorphism
  from multivariate polynomial ring in 3 variables over QQ
  to multivariate polynomial ring in 2 variables over QQ
defined by
  x -> s*t
  y -> t
  z -> s^2

julia> D, _ = quo(R, kernel(paraWhitneyUmbrella));

julia> is_finite(hom(D, C, V))
true

Inverting Homomorphisms of Affine Algebras

inverse — Method

Presenting Finite Extension Rings

present_finite_extension_ring — Method

  present_finite_extension_ring(F::Oscar.AffAlgHom)

Given a finite homomorphism F $:$ A $\rightarrow$ B of algebras of type <: Union{MPolyRing, MPolyQuoRing} over a field, return a presentation

$A^r \rightarrow A^s\rightarrow B \rightarrow 0$

of B as an A-module.

More precisely, return a tuple (gs, PM, sect), say, where

gs is a vector of polynomials representing generators for B as an A-module,
PM is an r $\times$ s-matrix of polynomials defining the map $A^r \rightarrow A^s$ , and
sect is a function which gives rise to a section of the augmentation map $A^s\rightarrow B$ .

Note

The finiteness condition on F is checked by the function.

Note

The function is implemented so that the last element of gs is one(B).

Examples

julia> RA, (h,) = polynomial_ring(QQ, [:h]);

julia> A, _ = quo(RA, ideal(RA, [h^9]));

julia> RB, (k, l) = polynomial_ring(QQ, [:k, :l]);

julia> B, _ = quo(RB, ideal(RB, [k^3, l^3]));

julia> F = hom(A, B, [k+l])
Ring homomorphism
  from quotient of multivariate polynomial ring by ideal (h^9)
  to quotient of multivariate polynomial ring by ideal (k^3, l^3)
defined by
  h -> k + l

julia> gs, PM, sect = present_finite_extension_ring(F);

julia> gs
3-element Vector{QQMPolyRingElem}:
 l^2
 l
 1

julia> PM
3×3 Matrix{QQMPolyRingElem}:
 h^3     0       0
 -3*h^2  h^3     0
 3*h     -3*h^2  h^3

julia> sect(k*l)
3-element Vector{QQMPolyRingElem}:
 -1
 h
 0

julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z]);

julia> I = ideal(R, [z^2-y^2*(y+1)]);

julia> A, _ = quo(R, I);

julia> B, (s,t) =  polynomial_ring(QQ, [:s, :t]);

julia> F = hom(A,B, [s, t^2-1, t*(t^2-1)])
Ring homomorphism
  from quotient of multivariate polynomial ring by ideal (-y^3 - y^2 + z^2)
  to multivariate polynomial ring in 2 variables over QQ
defined by
  x -> s
  y -> t^2 - 1
  z -> t^3 - t

julia> gs, PM, sect = present_finite_extension_ring(F);

julia> gs
2-element Vector{QQMPolyRingElem}:
 t
 1

julia> PM
2×2 Matrix{QQMPolyRingElem}:
 y   -z
 -z  y^2 + y

julia> sect(t)
2-element Vector{QQMPolyRingElem}:
 1
 0

julia> sect(one(B))
2-element Vector{QQMPolyRingElem}:
 0
 1

julia> sect(s)
2-element Vector{QQMPolyRingElem}:
 0
 x

julia> A, (a, b, c) = polynomial_ring(QQ, [:a, :b, :c]);

julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z]);

julia> I = ideal(R, [x*y]);

julia> B, _ = quo(R, I);

julia> (x, y, z) = gens(B);

julia> F = hom(A, B, [x^2+z, y^2-1, z^3])
Ring homomorphism
  from multivariate polynomial ring in 3 variables over QQ
  to quotient of multivariate polynomial ring by ideal (x*y)
defined by
  a -> x^2 + z
  b -> y^2 - 1
  c -> z^3

julia> gs, PM, sect = present_finite_extension_ring(F);

julia> gs
2-element Vector{QQMPolyRingElem}:
 y
 1

julia> PM
2×2 Matrix{QQMPolyRingElem}:
 a^3 - c  0
 0        a^3*b + a^3 - b*c - c

julia> sect(y)
2-element Vector{QQMPolyRingElem}:
 1
 0

julia> sect(one(B))
2-element Vector{QQMPolyRingElem}:
 0
 1

source

Algebraic Independence

is_algebraically_independent — Method

is_algebraically_independent_with_relations — Method

Subalgebras

Subalgebra Membership

subalgebra_membership — Method

Minimal Subalgebra Generators

minimal_subalgebra_generators — Method

Noether Normalization

noether_normalization — Method

noether_normalization(A::MPolyQuoRing)

Given an affine algebra $A=R/I$ over a field $K$ , return a triple $(V,F,G)$ such that:

$V$ is a vector of $d=\dim A$ elements of $A$ , represented by linear forms $l_i\in R$ , and such that $K[V]\hookrightarrow A$ is a Noether normalization for $A$ ;
$F: A=R/I \to B = R/\phi(I)$ is an isomorphism, induced by a linear change $\phi$ of coordinates of $R$ which maps the $l_i$ to the the last $d$ variables of $R$ ;
$G = F^{-1}.$

Warning

The algorithm may not terminate over a small finite field. If it terminates, the result is correct.

source

Examples

julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z]);

julia> A, _ = quo(R, ideal(R, [x*y, x*z]));

julia> L = noether_normalization(A);

julia> L[1]
2-element Vector{MPolyQuoRingElem{QQMPolyRingElem}}:
 -2*x + y
 -5*y + z

julia> L[2]
Ring homomorphism
  from quotient of multivariate polynomial ring by ideal (x*y, x*z)
  to quotient of multivariate polynomial ring by ideal (2*x^2 + x*y, 10*x^2 + 5*x*y + x*z)
defined by
  x -> x
  y -> 2*x + y
  z -> 10*x + 5*y + z

julia> L[3]
Ring homomorphism
  from quotient of multivariate polynomial ring by ideal (2*x^2 + x*y, 10*x^2 + 5*x*y + x*z)
  to quotient of multivariate polynomial ring by ideal (x*y, x*z)
defined by
  x -> x
  y -> -2*x + y
  z -> -5*y + z

Normalization

normalization — Method

normalization(A::MPolyQuoRing; algorithm = :equidimDec)

Find the normalization of a reduced affine algebra over a perfect field $K$ . That is, given the quotient $A=R/I$ of a multivariate polynomial ring $R$ over $K$ modulo a radical ideal $I$ , compute the integral closure $\overline{A}$ of $A$ in its total ring of fractions $Q(A)$ , together with the embedding $f: A \to \overline{A}$ .

Implemented Algorithms and how to Read the Output

The function relies on the algorithm of Greuel, Laplagne, and Seelisch which proceeds by finding a suitable decomposition $I=I_1\cap\dots\cap I_r$ into radical ideals $I_k$ , together with maps $A = R/I \to A_k=\overline{R/I_k}$ which give rise to the normalization map of $A$ :

$A\hookrightarrow A_1\times \dots\times A_r=\overline{A}$

For each $k$ , the function specifies two representations of $A_k$ : It returns an array of triples $(A_k, f_k, \mathfrak a_k)$ , where $A_k$ is represented as an affine $K$ -algebra, and $f_k$ as a map of affine $K$ -algebras. The third entry $\mathfrak a_k$ is a tuple $(d_k, J_k)$ , consisting of an element $d_k\in A$ and an ideal $J_k\subset A$ , such that $\frac{1}{d_k}J_k = A_k$ as $A$ -submodules of the total ring of fractions of $A$ .

By default (algorithm = :equidimDec), as a first step on its way to find the decomposition $I=I_1\cap\dots\cap I_r$ , the algorithm computes an equidimensional decomposition of the radical ideal $I$ . Alternatively, if specified by algorithm = :primeDec, the algorithm computes $I=I_1\cap\dots\cap I_r$ as the prime decomposition of the radical ideal $I$ . If specified by algorithm = :isPrime, assume that $I$ is prime.

See [GLS10].

Warning

The function does not check whether $A$ is reduced. Use is_reduced(A) in case you are unsure (this may take some time).

Examples

julia> R, (x, y) = polynomial_ring(QQ, [:x, :y]);

julia> A, _ = quo(R, ideal(R, [(x^2-y^3)*(x^2+y^2)*x]));

julia> L = normalization(A);

julia> size(L)
(2,)

julia> LL = normalization(A, algorithm = :primeDec);

julia> size(LL)
(3,)

julia> LL[1][1]
Quotient
  of multivariate polynomial ring in 3 variables T(1), x, y
    over rational field
  by ideal (-T(1)*y + x, -T(1)*x + y^2, T(1)^2 - y, -x^2 + y^3)

julia> LL[1][2]
Ring homomorphism
  from quotient of multivariate polynomial ring by ideal (x^5 - x^3*y^3 + x^3*y^2 - x*y^5)
  to quotient of multivariate polynomial ring by ideal (-T(1)*y + x, -T(1)*x + y^2, T(1)^2 - y, -x^2 + y^3)
defined by
  x -> x
  y -> y

julia> LL[1][3]
(y, Ideal (x, y))

source

normalization_with_delta — Method

normalization_with_delta(A::MPolyQuoRing; algorithm::Symbol = :equidimDec)

Compute the normalization

$A\hookrightarrow A_1\times \dots\times A_r=\overline{A}$

of $A$ as does normalize(A), but return additionally the delta invariant of $A$ , that is, the dimension

$\dim_K(\overline{A}/A).$

How to Read the Output

The return value is a tuple whose first element is normalize(A), whose second element is an array containing the delta invariants of the $A_k$ , and whose third element is the (total) delta invariant of $A$ . The return value -1 in the third element indicates that the delta invariant is infinite.

Examples

julia> R, (x, y) = polynomial_ring(QQ, [:x, :y]);

julia> A, _ = quo(R, ideal(R, [(x^2-y^3)*(x^2+y^2)*x]));

julia> L = normalization_with_delta(A);

julia> L[2]
3-element Vector{Int64}:
 1
 1
 0

julia> L[3]
13

julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z]);

julia> A, _ = quo(R, ideal(R, [z^3-x*y^4]));

julia> L = normalization_with_delta(A);

julia> L[3]
-1

source

Integral Bases

integral_basis — Method

integral_basis(f::MPolyRingElem, i::Int; algorithm::Symbol = :normal_local)

Given a polynomial $f$ in two variables with coefficients in a perfect field $K$ , and given an integer $i\in\{1,2\}$ specifying one of the variables, $f$ must be irreducible and monic in the specified variable: Say, $f\in\mathbb K[x,y]$ is monic in $y$ . Then the normalization of $A = K[x,y]/\langle f \rangle$ , that is, the integral closure $\overline{A}$ of $A$ in its quotient field, is a free module over $K[x]$ of finite rank, and any set of free generators for $\overline{A}$ over $K[x]$ is called an integral basis for $\overline{A}$ over $K[x]$ . The function returns a pair $(d, V)$ , where $d$ is an element of $A$ , and $V$ is a vector of elements in $A$ , such that the fractions $v/d, v\in V$ , form an integral basis for $\overline{A}$ over $K[x]$ .

By default (algorithm = :normal_local), the function relies on the local-to-global approach to normalization presented in [BDLPSS13]. Alternatively, if specified by algorithm = :normal_global, the global normalization algorithm in [GLS10] is used. If $K = \mathbb Q$ , it is recommended to apply the algorithm in [BDLP19], which makes use of Puiseux expansions and Hensel lifting (algorithm = :hensel).

Note

The conditions on $f$ are automatically checked.

Examples

julia> R, (x, y) = polynomial_ring(QQ, [:x, :y]);

julia> f = (y^2-2)^2 + x^5
x^5 + y^4 - 4*y^2 + 4

julia> integral_basis(f, 2)
(x^2, MPolyQuoRingElem{QQMPolyRingElem}[x^2, x^2*y, y^2 - 2, y^3 - 2*y])

source

Tests on Affine Algebras

Reducedness Test

is_reduced — Method

Normality Test

is_normal — Method

Cohen-Macaulayness Test

is_cohen_macaulay — Method

is_cohen_macaulay(A::MPolyQuoRing)

Given a $\mathbb Z$ -graded affine algebra A = R/I over a field, say, K, where the grading is inherited from the standard $\mathbb Z$ -grading on the polynomial ring R, return true if A is a Cohen-Macaulay ring, false otherwise.

Examples

julia> R, (w, x, y, z) = graded_polynomial_ring(QQ, [:w, :x, :y, :z]);

julia> I = ideal(R, [x*z-y^2, w*z-x*y, w*y-x^2]);

julia> A, _ = quo(R, I);

julia> is_cohen_macaulay(A)
true

julia> R, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z]);

julia> I = ideal(R, [x*z, y*z]);

julia> A, _ = quo(R, I);

julia> is_cohen_macaulay(A)
false

source

Hilbert Series and Hilbert Polynomial

Given a multivariate polynomial ring $R$ over a field $K$ together with a (multi)grading on $R$ by a finitely generated abelian group $G$ , let $I$ be an ideal of $R$ which is homogeneous with respect to this grading. Then the affine $K-$ algebra $A=R/I$ inherits the grading: $A = \bigoplus_{g\in G} A_g$ . Suppose now that $R$ is positively graded by $G$ . That is, $G$ is free and each graded piece $R_g$ has finite dimension. Then also $A_g$ is a finite dimensional $K$ -vector space for each $g$ , and we have the well-defined Hilbert function of $A$ ,

$H(A, \underline{\phantom{d}}): G \to \mathbb{N}, \; g\mapsto \dim_K(A_g).$

The Hilbert series of $A$ is the generating function

$H_A(\mathbb t)=\sum_{g\in G} H(A, g) \mathbb t^g$

(see Section 8.2 in [MS05] for a formal discussion extending the classical case of $\mathbb Z$ -gradings with positive weights to the more general case of multigradings). As in the classical case, the infinitely many values of the Hilbert function can be expressed in finite terms by representing the Hilbert series as a rational function (see Theorem 8.20 in [MS05] for a precise statement).

By a result of Macaulay, if $A = R/I$ is an affine algebra, and $L_{>}(I)$ is the leading ideal of $I$ with respect to a global monomial ordering $>$ , then the Hilbert function of $A$ equals that of $R/L_{>}(I)$ (see Theorem 15.26 in [Eis95]). Thus, using Gröbner bases, the computation of Hilbert series can be reduced to the case where the modulus of the affine algebra is a monomial ideal. In the latter case, we face a problem of combinatorial nature, and there are various strategies of how to proceed (see [KR05]). The functions hilbert_series, hilbert_series_reduced, hilbert_series_expanded, hilbert_function, hilbert_polynomial, and degree address the case of $\mathbb Z$ -gradings with positive weights, relying on corresponding Singular functionality. The functions multi_hilbert_series, multi_hilbert_series_reduced, and multi_hilbert_function offer a variety of different strategies and allow one to handle positive gradings in general.

$\mathbb Z$ -Gradings With Positive Weights

Let $R=K[x_1, \dots x_n]$ be a polynomial ring in $n$ variables over a field $K$ . Assign positive integer weights $w_i$ to the variables $x_i$ , and grade $R=\bigoplus_{d\in \mathbb Z} R_d=\bigoplus_{d\geq 0} R_d$ according to the corresponding weighted degree. Let $I$ be an ideal of $R$ which is homogeneous with respect to this grading. Then the affine $K$ -algebra $A=R/I$ inherits the grading: $A = \bigoplus_{d\geq 0} A_d$ , where each graded piece $A_d$ is a finite dimensional $K$ -vector space. In this situation, the Hilbert function of $A$ is of type

$H(A, \underline{\phantom{d}}): \mathbb{N} \to \mathbb{N}, \;d \mapsto \dim_K(d),$

and the Hilbert series of $A$ is the formal power series

$H_A(t)=\sum_{d\geq 0} H(A, d) t^d\in\mathbb Z[[t]].$

The Hilbert series can be written as a rational function $p(t)/q(t)$ , with denominator

$q(t) = (1-t^{w_1})\cdots (1-t^{w_n}).$

In the standard $\mathbb Z$ -graded case, where the weights on the variables are all 1, the Hilbert function is of polynomial nature: There exists a unique polynomial $P_A(t)\in\mathbb{Q}[t]$ , the Hilbert polynomial, which satisfies $H(M,d)=P_M(d)$ for all $d \gg 0$ . Furthermore, the degree of $A$ is defined as the dimension of $A$ over $K$ if this dimension is finite, and as the integer $d$ such that the leading term of the Hilbert polynomial has the form $d t^e/e!$ , otherwise.

hilbert_series — Method

hilbert_series(A::MPolyQuoRing; backend::Symbol=:Singular, algorithm::Symbol=:BayerStillmanA)

Given a $\mathbb Z$ -graded affine algebra $A = R/I$ over a field $K$ , where the grading is inherited from a $\mathbb Z$ -grading on the polynomial ring $R$ defined by assigning positive integer weights to the variables, return a pair $(p,q)$ , say, of univariate polynomials $p, q\in\mathbb Z[t]$ such that $p/q$ represents the Hilbert series of $A$ as a rational function with denominator

$q = (1-t^{w_1})\cdots (1-t^{w_n}),$

where $n$ is the number of variables of $R$ , and $w_1, \dots, w_n$ are the assigned weights.

Positive Gradings in General

multi_hilbert_series — Method

multi_hilbert_series_reduced — Method

multi_hilbert_function — Method

multi_hilbert_function(A::MPolyQuoRing, g::FinGenAbGroupElem)

Given a positively graded affine algebra $A$ over a field $K$ with grading group $G$ , say, and given an element $g$ of $G$ , return the value $H(A, g)$ of the Hilbert function

$H(A, \underline{\phantom{d}}): G \to \mathbb{N}, \; g\mapsto \dim_K(A_g).$

multi_hilbert_function(A::MPolyQuoRing, g::Vector{<:IntegerUnion})

Given a positively $\mathbb Z^m$ -graded affine algebra $A$ over a field $K$ , and given a vector $g$ of $m$ integers, convert $g$ into an element of the grading group of $A$ , and return the value $H(A, g)$ as above.

multi_hilbert_function(A::MPolyQuoRing, g::IntegerUnion)

Given a positively $\mathbb Z$ -graded affine algebra $A$ over a field $K$ , and given an integer $g$ , convert $g$ into an element of the grading group of $A$ , and return the value $H(A, g)$ as above.

Examples

julia> W = [1 1 1; 0 0 -1];

julia> R, x = graded_polynomial_ring(QQ, :x => 1:3, W)
(Graded multivariate polynomial ring in 3 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[x[1], x[2], x[3]])

julia> I = ideal(R, [x[1]^3*x[2], x[2]*x[3]^2, x[2]^2*x[3], x[3]^4]);

julia> A, _ = quo(R, I);

julia> multi_hilbert_function(A::MPolyQuoRing, [1, 0])
2

julia> R, (w, x, y, z) = graded_polynomial_ring(QQ, [:w, :x, :y, :z], [-1, -1, -1, -1]);

julia> A, _ = quo(R, ideal(R, [w*y-x^2, w*z-x*y, x*z-y^2]));

julia> multi_hilbert_function(A, -7)
22

julia> G = abelian_group(ZZMatrix([1 -1]));

julia> g = gen(G, 1);

julia> W = [g, g, g, g];

julia> R, (w, x, y, z) = graded_polynomial_ring(QQ, [:w, :x, :y, :z], W);

julia> A, _ = quo(R, ideal(R, [w*y-x^2, w*z-x*y, x*z-y^2]));

julia> multi_hilbert_function(A, 7*g)
22

source