Creating Multivariate Rings

In this section, for the convenience of the reader, we recall from the chapters on rings and fields how to create multivariate polynomial rings and their elements, adding illustrating examples. At the same time, we introduce and illustrate a ring type for modelling multivariate polynomial rings with gradings.

Types

OSCAR provides types for dense univariate and sparse multivariate polynomials. The univariate ring types belong to the abstract type PolyRing{T}, their elements have abstract type PolyRingElem{T}. The multivariate ring types belong to the abstract type MPolyRing{T}, their elements have abstract type MPolyRingElem{T}. Here, T is the element type of the coefficient ring of the polynomial ring.

Constructors

The basic constructor below allows one to build multivariate polynomial rings:

polynomial_ring(C::Ring, xs::AbstractVector{<:VarName}; cached::Bool = true)

Given a ring C and a vector xs of Symbols, Strings, or Characters, return a tuple R, vars, say, which consists of a polynomial ring R with coefficient ring C and a vector vars of generators (variables) which print according to the entries of xs.

Note

Caching is used to ensure that a given ring constructed from given parameters is unique in the system. For example, there is only one ring of multivariate polynomials over $\mathbb{Z}$ with variables printing as x, y, z.

Examples

julia> R, (x, y, z) = polynomial_ring(ZZ, [:x, :y, :z])
(Multivariate polynomial ring in 3 variables over ZZ, ZZMPolyRingElem[x, y, z])

julia> typeof(R)
ZZMPolyRing

julia> typeof(x)
ZZMPolyRingElem

julia> S, (a, b, c) = polynomial_ring(ZZ, [:x, :y, :z])
(Multivariate polynomial ring in 3 variables over ZZ, ZZMPolyRingElem[x, y, z])

julia> T, _ = polynomial_ring(ZZ, [:x, :y, :z])
(Multivariate polynomial ring in 3 variables over ZZ, ZZMPolyRingElem[x, y, z])

julia> R === S === T
true

julia> R1, _ = polynomial_ring(ZZ, [:x, :y, :z]);

julia> R2, _ = polynomial_ring(ZZ, ["x", "y", "z"]);

julia> R3, _ = polynomial_ring(ZZ, ['x', 'y', 'z']);

julia> R1 === R2 === R3
true

julia> R1, x = polynomial_ring(QQ, [:x])
(Multivariate polynomial ring in 1 variable over QQ, QQMPolyRingElem[x])

julia> typeof(x)
Vector{QQMPolyRingElem} (alias for Array{QQMPolyRingElem, 1})

julia> R2, (x,) = polynomial_ring(QQ, [:x])
(Multivariate polynomial ring in 1 variable over QQ, QQMPolyRingElem[x])

julia> typeof(x)
QQMPolyRingElem

julia> R3, x = polynomial_ring(QQ, :x)
(Univariate polynomial ring in x over QQ, x)

julia> typeof(x)
QQPolyRingElem

julia> T, x = polynomial_ring(GF(3), ["x[1]", "x[2]"]);

julia> x
2-element Vector{FqMPolyRingElem}:
 x[1]
 x[2]

The constructor illustrated below allows for the convenient handling of variables with multi-indices:

julia> R, x, y, z = polynomial_ring(QQ, :x => (1:3, 1:4), :y => 1:2, :z => (1:1, 1:1, 1:1));

julia> x
3×4 Matrix{QQMPolyRingElem}:
 x[1, 1]  x[1, 2]  x[1, 3]  x[1, 4]
 x[2, 1]  x[2, 2]  x[2, 3]  x[2, 4]
 x[3, 1]  x[3, 2]  x[3, 3]  x[3, 4]

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

julia> z
1×1×1 Array{QQMPolyRingElem, 3}:
[:, :, 1] =
 z[1, 1, 1]

Coefficient Rings

Gröbner and standard bases are implemented for multivariate polynomial rings over the fields and rings below:

The Field of Rational Numbers $\mathbb{Q}$

julia> QQ
Rational field

Number Fields

julia> Qx, x = QQ["x"];

julia> K, a = number_field(x^2 - 5, "a")
(Number field of degree 2 over QQ, a)

julia> Kt, t = K["t"];

julia> L, b = number_field(t^3 - 3, "b")
(Relative number field of degree 3 over K, b)

julia> Qx, x = QQ["x"];

julia> julia> L, a = number_field([x^2 - 5, x^3 - 3], "a")
(Non-simple number field of degree 6 over QQ, AbsNonSimpleNumFieldElem[a1, a2])

julia> K, a = quadratic_field(5)
(Real quadratic field defined by x^2 - 5, sqrt(5))

julia> K, zeta = cyclotomic_field(3) 
(Cyclotomic field of order 3, z_3)

The Abelian Closure of $\mathbb{Q}$

julia> K, z = abelian_closure(QQ)
(Abelian closure of rational field, Generator of abelian closure of rational field)

julia> F = algebraic_closure(QQ)
Algebraic closure of rational field

julia> z(3)*z(5)
zeta(15)^7 - zeta(15)^5 + zeta(15)^4 - zeta(15)^3 + zeta(15) - 1

Finite Fields

julia> GF(3)
Prime field of characteristic 3

julia> GF(ZZ(2)^127 - 1)
Prime field of characteristic 170141183460469231731687303715884105727

julia> GF(3,2)
Finite field of degree 2 and characteristic 3

julia> GF(3,2) == GF(9)
true

julia> F, a = finite_field(2, 70, "a")
(Finite field of degree 70 and characteristic 2, a)

julia> K, a = finite_field(9, "a")
(Finite field of degree 2 and characteristic 3, a)

julia> Kx, x = K["x"];

julia> L, b = finite_field(x^3 + x^2 + x + 2, "b")
(Finite field of degree 3 over K, b)

Algebraic Closures of Prime Fields

julia> K = algebraic_closure(GF(3))
Algebraic closure of prime field of characteristic 3

julia> L = algebraic_closure(QQ)
Algebraic closure of rational field

Rational Function Fields

julia> R, (x, y) = rational_function_field(QQ, [:x, :y])
(Rational function field over QQ, AbstractAlgebra.Generic.RationalFunctionFieldElem{QQFieldElem, QQMPolyRingElem}[x, y])

Function Fields of Transcendence Degree 1

julia> K = GF(101)
Prime field of characteristic 101

julia> Kx, x = rational_function_field(K, :x)
(Rational function field over K, x)

julia> Kxy, y = polynomial_ring(Kx, :y)
(Univariate polynomial ring in y over Kx, y)

julia> F, a = function_field(x^3-y^2)
(Function Field over K with defining polynomial 100*_a^2 + x^3, _a)

The Ring of Integers $\mathbb{Z}$

julia> ZZ
Integer ring

Rings of Type $\mathbb{Z}/m\mathbb{Z}$

julia> quo(ZZ, ZZ(6))[1]
Integers modulo 6

Gradings

Given a polynomial ring $R = C[x_1, \dots, x_n]$ , we may endow $R$ with various gradings. The standard $\mathbb Z$ -grading on $R$ is the decomposition $R=\bigoplus_{d\in \mathbb Z} R_d=\bigoplus_{d\geq 0} R_d$ by the usual degree of polynomials. More general $\mathbb Z$ -gradings are obtained by assigning integer weights to the variables and considering the corresponding weighted degrees. Even more generally, we may consider multigradings: Given a finitely generated abelian group $G$ , a multigrading on $R$ by $G$ , or a $G$ -grading, or simply a grading, corresponds to a semigroup homomorphism $\phi: \mathbb N^n \to G$ : Given $\phi$ , the degree of a monomial $x^\alpha$ is the image $\deg(x^\alpha):=\phi(\alpha)\in G$ ; the induced $G$ -grading on $R$ is the decomposition $R = \bigoplus_{g\in G} R_g$ satisfying $R_g\cdot R_h\subset R_{g+h}$ , where $R_g$ is the free $C$ -module generated by the monomials of degree $g$ . This grading is determined by assigning the weights $\deg(x_i)$ to the $x_i$ . In other words, it is determined by the weight vector $W = (\deg(x_1), \dots, \deg(x_n))\in G^n.$

We refer to the textbooks [MS05] and [KR05] for details on multigradings. With respect to notation, we follow the former book.

Note

Given a $G$ -grading on $R$ , we refer to $G$ as the grading group of $R$ . Moreover, we then say that $R$ is $G$ -graded, or simply that $R$ is graded. If $R$ is a polynomial ring over a field, we say that a $G$ -grading on $R$ is positive if $G$ is free and each graded part $R_g$ , $g\in G$ , has finite dimension. We then also say that $R$ is positively graded (by $G$ ). Note that the positivity condition can be equivalently expressed by asking that $G$ is free and that the degree zero part consists of the constants only (see Theorem 8.6 in [MS05]).

Note

Given a G-grading on R in OSCAR, we say that R is $\mathbb Z^m$ -graded if is_free(G) && number_of_generators(G) == rank(G) == m evaluates to true. In this case, conversion routines allow one to switch back and forth between elements of G and integer vectors of length m. Specifically, if R is $\mathbb Z$ -graded, that is, is_free(G) && number_of_generators(G) == rank(G) == 1 evaluates to true, elements of G may be converted to integers and vice versa.

Types

Multivariate rings with gradings are modeled by objects of type MPolyDecRing{T, S} :< MPolyRing{T}, with elements of type MPolyRingElem_dec{T, S} :< MPolyRingElem{T}. Here, S is the element type of the multivariate ring, and T is the element type of its coefficient ring as above.

Note

The types MPolyDecRing{T, S} and MPolyRingElem_dec{T, S} are also meant to eventually model multivariate rings with filtrations and their elements.

The following function allows one, in particular, to distinguish between graded and filtered rings.

is_graded — Method

Constructors for Graded Rings

There are two basic ways of creating multivariate rings with gradings: While the grade function allows one to create a graded ring by assigning a grading to a polynomial ring already constructed, the graded_polynomial_ring function is meant to create a graded polynomial ring all at once.

grade — Method

grade(R::MPolyRing, W::Vector{FinGenAbGroupElem})

Given a vector W of ngens(R) elements of a finitely generated abelian group G, say, create a G-graded ring by assigning the entries of W as weights to the variables of R. Return the new ring as an object of type MPolyDecRing, together with the vector of variables.

Examples

julia> R, (t, x, y) = polynomial_ring(QQ, [:t, :x, :y])
(Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[t, x, y])

julia> typeof(R)
QQMPolyRing

julia>  typeof(x)
QQMPolyRingElem

julia> G = abelian_group([0])
Z

julia> g = gen(G, 1)
Abelian group element [1]

julia> S, (t, x, y) = grade(R, [-g, g, g])
(Graded multivariate polynomial ring in 3 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[t, x, y])

julia> S
Multivariate polynomial ring in 3 variables over QQ graded by
  t -> [-1]
  x -> [1]
  y -> [1]

julia> typeof(S)
MPolyDecRing{QQFieldElem, QQMPolyRing}

julia> S isa MPolyRing
true

julia> typeof(x)
MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}

julia> R, x = polynomial_ring(QQ, :x => 1:5)
(Multivariate polynomial ring in 5 variables over QQ, QQMPolyRingElem[x[1], x[2], x[3], x[4], x[5]])

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

julia> g = gens(G)
2-element Vector{FinGenAbGroupElem}:
 [1, 0]
 [0, 1]

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

julia> S, _ = grade(R, W)
(Graded multivariate polynomial ring in 5 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[x[1], x[2], x[3], x[4], x[5]])

julia> S
Multivariate polynomial ring in 5 variables over QQ graded by
  x[1] -> [1 0]
  x[2] -> [1 0]
  x[3] -> [0 1]
  x[4] -> [0 1]
  x[5] -> [0 1]

julia> typeof(x[1])
QQMPolyRingElem

julia> x = map(S, x)
5-element Vector{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}:
 x[1]
 x[2]
 x[3]
 x[4]
 x[5]

julia> typeof(x[1])
MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}

julia> R, x = polynomial_ring(QQ, :x => 1:5)
(Multivariate polynomial ring in 5 variables over QQ, QQMPolyRingElem[x[1], x[2], x[3], x[4], x[5]])

julia> G = abelian_group([0, 0, 2, 2])
Finitely generated abelian group
  with 4 generators and 4 relations and relation matrix
  [0   0   0   0]
  [0   0   0   0]
  [0   0   2   0]
  [0   0   0   2]

julia> g = gens(G);

julia> W = [g[1]+g[3]+g[4], g[2]+g[4], g[1]+g[3], g[2], g[1]+g[2]]
5-element Vector{FinGenAbGroupElem}:
 [1, 0, 1, 1]
 [0, 1, 0, 1]
 [1, 0, 1, 0]
 [0, 1, 0, 0]
 [1, 1, 0, 0]

julia> S, x = grade(R, W)
(Graded multivariate polynomial ring in 5 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[x[1], x[2], x[3], x[4], x[5]])

julia> S
Multivariate polynomial ring in 5 variables over QQ graded by
  x[1] -> [1 0 1 1]
  x[2] -> [0 1 0 1]
  x[3] -> [1 0 1 0]
  x[4] -> [0 1 0 0]
  x[5] -> [1 1 0 0]

source

grade — Method

grade(R::MPolyRing, W::AbstractVector{<:AbstractVector{<:IntegerUnion}})

Given a vector W of ngens(R) integer vectors of the same size m, say, create a free abelian group of type FinGenAbGroup given by m free generators, and convert the vectors in W to elements of that group. Then create a $\mathbb Z^m$ -graded ring by assigning the group elements as weights to the variables of R, and return the new ring, together with the vector of variables.

grade(R::MPolyRing, W::Union{ZZMatrix, AbstractMatrix{<:IntegerUnion}})

As above, converting the columns of W.

Examples

julia> R, x = polynomial_ring(QQ, :x => 1:5)
(Multivariate polynomial ring in 5 variables over QQ, QQMPolyRingElem[x[1], x[2], x[3], x[4], x[5]])

julia> W = [1 1 0 0 0; 0 0 1 1 1]
2×5 Matrix{Int64}:
 1  1  0  0  0
 0  0  1  1  1

julia> S, x = grade(R, W);

julia> S
Multivariate polynomial ring in 5 variables over QQ graded by
  x[1] -> [1 0]
  x[2] -> [1 0]
  x[3] -> [0 1]
  x[4] -> [0 1]
  x[5] -> [0 1]

source

grade — Method

grade(R::MPolyRing, W::AbstractVector{<:IntegerUnion})

Given a vector W of ngens(R) integers, create a free abelian group of type FinGenAbGroup given by one free generator, and convert the entries of W to elements of that group. Then create a $\mathbb Z$ -graded ring by assigning the group elements as weights to the variables of R, and return the new ring, together with the vector of variables.

grade(R::MPolyRing)

As above, where the grading is the standard $\mathbb Z$ -grading on R.

Examples

julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z])
(Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])

julia> W = [1, 2, 3];

julia> S, (x, y, z) = grade(R, W);

julia> S
Multivariate polynomial ring in 3 variables over QQ graded by
  x -> [1]
  y -> [2]
  z -> [3]

julia> T, (x, y, z) = grade(R);

julia> T
Multivariate polynomial ring in 3 variables over QQ graded by
  x -> [1]
  y -> [1]
  z -> [1]

source

graded_polynomial_ring — Method

graded_polynomial_ring(C::Ring, args...; weights = nothing, kwargs...)

Create a multivariate polynomial_ring with coefficient ring C and variables as described by args... (using the exact same syntax as for polynomial_ring), and grade this ring according to the data provided by the keyword argument weights. Return the graded ring as an object of type MPolyDecRing, together with the variables.

Note

If no weights are entered, the returned ring is standard $\mathbb Z$ -graded, i.e. all variables are graded with weight 1.

Note

kwargs allows one to set the same keywords as for polynomial_ring.

Examples

julia> G = abelian_group([0, 0, 2, 2])
Finitely generated abelian group
  with 4 generators and 4 relations and relation matrix
  [0   0   0   0]
  [0   0   0   0]
  [0   0   2   0]
  [0   0   0   2]

julia> W1 = [G[1]+G[3]+G[4], G[2]+G[4], G[1]+G[3], G[2], G[1]+G[2]];
 
julia> R1, x, y = graded_polynomial_ring(QQ, :x => 1:2, :y => 1:3, W1);

julia> R1
Multivariate polynomial ring in 5 variables over QQ graded by
  x[1] -> [1 0 1 1]
  x[2] -> [0 1 0 1]
  y[1] -> [1 0 1 0]
  y[2] -> [0 1 0 0]
  y[3] -> [1 1 0 0]

julia> W2 = [[1, 0], [0, 1], [1, 0], [4, 1]];

julia> R2, x = graded_polynomial_ring(QQ, 4, :x; weights = W2);

julia> R2
Multivariate polynomial ring in 4 variables over QQ graded by
  x1 -> [1 0]
  x2 -> [0 1]
  x3 -> [1 0]
  x4 -> [4 1]

julia> R3, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z]; weights = [1, 2, 3]);

julia> R3
Multivariate polynomial ring in 3 variables over QQ graded by
  x -> [1]
  y -> [2]
  z -> [3]

julia> R4, x = graded_polynomial_ring(QQ, :x => 1:3);

julia> R4
Multivariate polynomial ring in 3 variables over QQ graded by
  x[1] -> [1]
  x[2] -> [1]
  x[3] -> [1]

julia> R5, x, y = graded_polynomial_ring(QQ, :x => 1:3, :y => (1:2, 1:2); weights = 1:7);

julia> R5
Multivariate polynomial ring in 7 variables over QQ graded by
  x[1] -> [1]
  x[2] -> [2]
  x[3] -> [3]
  y[1, 1] -> [4]
  y[2, 1] -> [5]
  y[1, 2] -> [6]
  y[2, 2] -> [7]

source

Tests on Graded Rings

is_standard_graded — Method

is_standard_graded(R::MPolyDecRing)

Return true if R is standard $\mathbb Z$ -graded, false otherwise.

Examples

julia> S, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z]; weights = [1, 2, 3])
(Graded multivariate polynomial ring in 3 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[x, y, z])

julia> is_standard_graded(S)
false

source

is_z_graded — Method

is_z_graded(R::MPolyDecRing)

Return true if R is $\mathbb Z$ -graded, false otherwise.

Note

Writing G = grading_group(R), we say that R is $\mathbb Z$ -graded if G is free abelian of rank 1, and ngens(G) == 1.

Examples

julia> S, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z]; weights = [1, 2, 3])
(Graded multivariate polynomial ring in 3 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[x, y, z])

julia> is_z_graded(S)
true

source

is_zm_graded — Method

is_zm_graded(R::MPolyDecRing)

Return true if R is $\mathbb Z^m$ -graded for some $m$ , false otherwise.

Note

Writing G = grading_group(R), we say that R is $\mathbb Z^m$ -graded G is free abelian of rank m, and ngens(G) == m.

Examples

julia> G = abelian_group([0, 0, 2, 2])
Finitely generated abelian group
  with 4 generators and 4 relations and relation matrix
  [0   0   0   0]
  [0   0   0   0]
  [0   0   2   0]
  [0   0   0   2]

julia> W = [G[1]+G[3]+G[4], G[2]+G[4], G[1]+G[3], G[2], G[1]+G[2]];

julia> S, x = graded_polynomial_ring(QQ, :x => 1:5; weights = W)
(Graded multivariate polynomial ring in 5 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[x[1], x[2], x[3], x[4], x[5]])

julia> is_zm_graded(S)
false

julia> G = abelian_group(ZZMatrix([1 -1]))
Finitely generated abelian group
  with 2 generators and 1 relation and relation matrix
  [1   -1]

julia> g = gen(G, 1)
Abelian group element [0, 1]

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

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

julia> R
Multivariate polynomial ring in 4 variables over QQ graded by
  w -> [0 1]
  x -> [0 1]
  y -> [0 1]
  z -> [0 1]

julia> is_free(G)
true

julia> is_zm_graded(R)
false

source

is_positively_graded — Method

is_positively_graded(R::MPolyDecRing)

Return true if R is positively graded, false otherwise.

Note

We say that R is positively graded by a finitely generated abelian group $G$ if the coefficient ring of R is a field, $G$ is free, and each graded part $R_g$ , $g\in G$ , has finite dimension.

Examples

julia> S, (t, x, y) = graded_polynomial_ring(QQ, [:t, :x, :y]; weights = [-1, 1, 1])
(Graded multivariate polynomial ring in 3 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[t, x, y])

julia> grading_group(S)
Z

julia> is_positively_graded(S)
false

julia> G = abelian_group([0, 2])
Finitely generated abelian group
  with 2 generators and 2 relations and relation matrix
  [0   0]
  [0   2]

julia> is_free(G)
false

julia> W = [gen(G, 1)+gen(G, 2), gen(G, 1)]
2-element Vector{FinGenAbGroupElem}:
 [1, 1]
 [1, 0]

julia> S, (x, y) = graded_polynomial_ring(QQ, [:x, :y]; weights = W)
(Graded multivariate polynomial ring in 2 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[x, y])

julia> is_positively_graded(S)
false

source

Data Associated to Multivariate Rings

Given a multivariate polynomial ring R with coefficient ring C,

coefficient_ring(R) refers to C,
gens(R) to the generators (variables) of R,
number_of_generators(R) / ngens(R) to the number of these generators, and
gen(R, i) as well as R[i] to the i-th such generator.

Examples

julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z])
(Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])

julia> coefficient_ring(R)
Rational field

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

julia> gen(R, 2)
y

julia> R[3]
z

julia> number_of_generators(R)
3

In the graded case, we additionally have:

grading_group — Method

weights — Method

weights(R::MPolyDecRing)

Given a graded multivariate polynomial ring R, return the weights (degrees) of the variables of R.

 weights(::Type{Vector{Int}}, R::MPolyDecRing)

Given a $\mathbb Z^m$ -graded multivariate polynomial ring R, return the weights (degrees) of the variables of R, converted to vectors of integer numbers.

weights(::Type{Int}, R::MPolyDecRing)

Given a $\mathbb Z$ -graded multivariate polynomial ring R, return the weights (degrees) of R, converted to integer numbers.

Examples

julia> G = abelian_group([0, 0, 2, 2])
Finitely generated abelian group
  with 4 generators and 4 relations and relation matrix
  [0   0   0   0]
  [0   0   0   0]
  [0   0   2   0]
  [0   0   0   2]

julia> W = [G[1]+G[3]+G[4], G[2]+G[4], G[1]+G[3], G[2], G[1]+G[2]];

julia> R, x = graded_polynomial_ring(QQ, :x => 1:5; weights = W);

julia> weights(R)
5-element Vector{FinGenAbGroupElem}:
 [1, 0, 1, 1]
 [0, 1, 0, 1]
 [1, 0, 1, 0]
 [0, 1, 0, 0]
 [1, 1, 0, 0]

julia> W = [[1, 0], [0, 1], [1, 0], [4, 1]];

julia> R, x = graded_polynomial_ring(QQ, :x => 1:4, W);

julia> weights(R)
4-element Vector{FinGenAbGroupElem}:
 [1 0]
 [0 1]
 [1 0]
 [4 1]

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

julia> weights(R)
3-element Vector{FinGenAbGroupElem}:
 [1]
 [1]
 [1]

source

monomial_basis — Method

monomial_basis(R::MPolyDecRing, g::FinGenAbGroupElem)

Given a polynomial ring R over a field which is graded by a free group of type FinGenAbGroup, and given an element g of that group, return the monomials of degree g in R.

monomial_basis(R::MPolyDecRing, W::Vector{<:IntegerUnion})

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

monomial_basis(R::MPolyDecRing, d::IntegerUnion)

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

Note

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

Examples

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

julia> G = grading_group(T)
Z

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

source

homogeneous_component — Method

homogeneous_component(R::MPolyDecRing, g::FinGenAbGroupElem)

Given a polynomial ring R over a field which is graded by a free group, and given an element g of that group, return the homogeneous component of R of degree g as a standard vector space. Additionally, return the map which sends an element of that vector space to the corresponding monomial in R.

homogeneous_component(R::MPolyDecRing, W::Vector{<:IntegerUnion})

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

homogeneous_component(R::MPolyDecRing, d::IntegerUnion)

Given a $\mathbb Z$ -graded polynomial ring R over a field, and given an integer d, convert d into an element g of the grading group of R proceed as above.

Note

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

Examples

julia> W = [1 1 0 0 0; 0 0 1 1 1]
2×5 Matrix{Int64}:
 1  1  0  0  0
 0  0  1  1  1

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

julia> G = grading_group(S)
Z^2

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

julia> L[1]
S_[1 1] of dim 6

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

julia> EMB = L[2]
Map defined by a julia-function with inverse
  from S_[1 1] of dim 6
  to graded multivariate polynomial ring in 5 variables over QQ

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

source

forget_grading — Method

Elements of Multivariate Rings

Constructors

One way to create elements of a multivariate polynomial ring is to build up polynomials from the generators (variables) of the ring using basic arithmetic as shown below:

Examples

julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z])
(Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])

julia> f = 3*x^2+y*z
3*x^2 + y*z

julia> typeof(f)
QQMPolyRingElem

julia> S, (x, y, z) = grade(R)
(Graded multivariate polynomial ring in 3 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[x, y, z])

julia> g = 3*x^2+y*z
3*x^2 + y*z

julia> typeof(g)
MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}

julia> g == S(f)
true

Alternatively, there is the following constructor:

(R::MPolyRing{T})(c::Vector{T}, e::Vector{Vector{Int}}) where T <: RingElem

Its return value is the element of R whose nonzero coefficients are specified by the elements of c, with exponent vectors given by the elements of e.

Examples

julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z])
(Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])

julia> f = 3*x^2+y*z
3*x^2 + y*z

julia> g = R(QQ.([3, 1]), [[2, 0, 0], [0, 1, 1]])
3*x^2 + y*z

julia> f == g
true

An often more effective way to create polynomials is to use the MPoly build context as indicated below:

julia> R, (x, y) = polynomial_ring(QQ, [:x, :y])
(Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])

julia> B = MPolyBuildCtx(R)
Builder for an element of R

julia> for i = 1:5 push_term!(B, QQ(i), [i, i-1]) end

julia> finish(B)
5*x^5*y^4 + 4*x^4*y^3 + 3*x^3*y^2 + 2*x^2*y + x

Special Elements

Given a multivariate polynomial ring R, zero(R) and one(R) refer to the additive and multiplicative identity of R, respectively. Relevant test calls on an element f of R are iszero(f) and isone(f).

Data Associated to Elements of Multivariate Rings

Given an element f of a multivariate polynomial ring R or a graded version of such a ring,

parent(f) refers to R, and
total_degree(f) to the total degree of f.

Note

Given a set of variables $x = \{x_1, \ldots, x_n\}$ , the total degree of a monomial $x^\alpha=x_1^{\alpha_1}\cdots x_n^{\alpha_n}\in\text{Mon}_n(x)$ is the sum of the $\alpha_i$ . The total degree of a polynomial f is the maximum of the total degrees of its monomials. In particular, the notion of total degree ignores the weights given to the variables in the graded case.

For iterators which allow one to recover the monomials (terms, $\dots$ ) of f we refer to the subsection Monomials, Terms, and More of the section on Gröbner/Standard Bases.

Examples

julia> R, (x, y) = polynomial_ring(GF(5), [:x, :y])
(Multivariate polynomial ring in 2 variables over GF(5), FqMPolyRingElem[x, y])

julia> c = map(GF(5), [1, 2, 3])
3-element Vector{FqFieldElem}:
 1
 2
 3

julia> e = [[3, 2], [1, 0], [0, 1]]
3-element Vector{Vector{Int64}}:
 [3, 2]
 [1, 0]
 [0, 1]

julia> f = R(c, e)
x^3*y^2 + 2*x + 3*y

julia> parent(f)
Multivariate polynomial ring in 2 variables x, y
  over prime field of characteristic 5

julia> total_degree(f)
5

Further functionality is available in the graded case:

homogeneous_components — Method

homogeneous_component — Method

homogeneous_component(f::MPolyDecRingElem, g::FinGenAbGroupElem)

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

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

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

homogeneous_component(f::MPolyDecRingElem, g::IntegerUnion)

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

Examples

julia> G = abelian_group([0, 0, 2, 2])
Finitely generated abelian group
  with 4 generators and 4 relations and relation matrix
  [0   0   0   0]
  [0   0   0   0]
  [0   0   2   0]
  [0   0   0   2]

julia> W = [G[1]+G[3]+G[4], G[2]+G[4], G[1]+G[3], G[2], G[1]+G[2]];

julia> S, x = graded_polynomial_ring(QQ, :x => 1:5; weights = W)
(Graded multivariate polynomial ring in 5 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[x[1], x[2], x[3], x[4], x[5]])

julia> f = x[1]^2+x[3]^2+x[5]^2
x[1]^2 + x[3]^2 + x[5]^2

julia> homogeneous_component(f, 2*G[1])
x[1]^2 + x[3]^2

julia> W = [[1, 0], [0, 1], [1, 0], [4, 1]]
4-element Vector{Vector{Int64}}:
 [1, 0]
 [0, 1]
 [1, 0]
 [4, 1]

julia> R, x = graded_polynomial_ring(QQ, :x => 1:4; weights = W)
(Graded multivariate polynomial ring in 4 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[x[1], x[2], x[3], x[4]])

julia> f = x[1]^2*x[2]+x[4]
x[1]^2*x[2] + x[4]

julia> homogeneous_component(f, [2, 1])
x[1]^2*x[2]

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

julia> f = x^2+y+z
x^2 + y + z

julia> homogeneous_component(f, 1)
0

julia> homogeneous_component(f, 2)
x^2 + y

julia> homogeneous_component(f, 3)
z

source

is_homogeneous — Method

degree — Method

degree(f::MPolyDecRingElem)

Given a homogeneous element f of a graded multivariate ring, return the degree of f.

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

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

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

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

Examples

julia> G = abelian_group([0, 0, 2, 2])
Finitely generated abelian group
  with 4 generators and 4 relations and relation matrix
  [0   0   0   0]
  [0   0   0   0]
  [0   0   2   0]
  [0   0   0   2]

julia> W = [G[1]+G[3]+G[4], G[2]+G[4], G[1]+G[3], G[2], G[1]+G[2]];

julia> S, x = graded_polynomial_ring(QQ, :x => 1:5; weights = W)
(Graded multivariate polynomial ring in 5 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[x[1], x[2], x[3], x[4], x[5]])

julia> f = x[2]^2+2*x[4]^2
x[2]^2 + 2*x[4]^2

julia> degree(f)
Abelian group element [0, 2, 0, 0]

julia> W = [[1, 0], [0, 1], [1, 0], [4, 1]]
4-element Vector{Vector{Int64}}:
 [1, 0]
 [0, 1]
 [1, 0]
 [4, 1]

julia> R, x = graded_polynomial_ring(QQ, :x => 1:4, W)
(Graded multivariate polynomial ring in 4 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[x[1], x[2], x[3], x[4]])

julia> f = x[1]^4*x[2]+x[4]
x[1]^4*x[2] + x[4]

julia> degree(f)
[4 1]

julia> degree(Vector{Int}, f)
2-element Vector{Int64}:
 4
 1

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

julia> f = x^6+y^3+z^2
x^6 + y^3 + z^2

julia> degree(f)
[6]

julia> typeof(degree(f))
FinGenAbGroupElem

julia> degree(Int, f)
6

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

source

forget_grading — Method

Homomorphisms From Multivariate Rings

If $R$ is a multivariate polynomial ring, and $S$ is any ring, then a ring homomorphism $R \to S$ is determined by specifying its restriction to the coefficient ring of $R$ , and by assigning an image to each variable of $R$ . In OSCAR, such homomorphisms are created by using the following constructor:

hom — Method

hom(R::MPolyRing, S::NCRing, coeff_map, images::Vector; check::Bool = true)

hom(R::MPolyRing, S::NCRing, images::Vector; check::Bool = true)

Given a homomorphism coeff_map from C to S, where C is the coefficient ring of R, and given a vector images of nvars(R) elements of S, return the homomorphism R $\to$ S whose restriction to C is coeff_map, and which sends the i-th variable of R 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

In case check = true (default), the function 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> K, a = finite_field(2, 2, "a");

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

julia> F = hom(R, R, z -> z^2, [y, x])
Ring homomorphism
  from multivariate polynomial ring in 2 variables over K
  to multivariate polynomial ring in 2 variables over K
defined by
  x -> y
  y -> x
with map on coefficients
  #1

julia> F(a * y)
(a + 1)*x

julia> Qi, i = quadratic_field(-1)
(Imaginary quadratic field defined by x^2 + 1, sqrt(-1))

julia> S, (x, y) = polynomial_ring(Qi, [:x, :y]);

julia> G = hom(S, S, hom(Qi, Qi, -i), [x^2, y^2])
Ring homomorphism
  from multivariate polynomial ring in 2 variables over Qi
  to multivariate polynomial ring in 2 variables over Qi
defined by
  x -> x^2
  y -> y^2
with map on coefficients
  Map: Qi -> Qi

julia> G(x+i*y)
x^2 - sqrt(-1)*y^2

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

julia> f = 3*x^2+2*x+1;

julia> S, (x, y) = polynomial_ring(GF(2), [:x, :y]);

julia> H = hom(R, S, gens(S))
Ring homomorphism
  from multivariate polynomial ring in 2 variables over ZZ
  to multivariate polynomial ring in 2 variables over GF(2)
defined by
  x -> x
  y -> y

julia> H(f)
x^2 + 1

source

Given a ring homomorphism F from R to S as above, domain(F) and codomain(F) refer to R and S, respectively.

Note

Homomorphisms from quotients of multivariate rings are discussed in the section on affine algebras. In particular, in the same section, we introduce the OSCAR homomorphism type AffAlgHom which addresses 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}.