Affine Algebras

Affine Algebras

With regard to notation, we use affine algebra as a synonym for quotient ring of a multivariate polynomial ring modulo an ideal. More specifically, if $R$ is a multivariate polynomial ring with coefficient ring $C$, and $A=R/I$ is the quotient ring of $R$ modulo 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

Most of the functions discussed here rely on Gröbner basis techniques. They are implemented for affine algebras over fields (exact fields supported by OSCAR) and, if not indicated otherwise, for affine algebras over the integers.

Note

In OSCAR, elements of quotient rings are not necessarily reduced with regard to the modulus of the quotient ring. Operations involving Gröbner basis computations may lead to partial reductions. Full reductions, depending on the choice of a monomial ordering, are achieved by explicitly computing normal forms. The functions simplify and simplify! discussed in this section implements this.

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 ìs_graded, ìs_standard_graded, ìs_z_graded, ìs_zm_graded, and ìs_positively_graded which will not be discussed again here.

Types

The OSCAR type for quotient rings of multivariate polynomial rings is of parametrized form MPolyQuo{T}, with elements of type MPolyQuoElem{T}. Here, T is the element type of the polynomial ring.

Constructors

quo — Method

quo(R::MPolyRing, I::MPolyIdeal) -> MPolyQuoRing, Map

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

quo(R::MPolyRing, V::Vector{MPolyElem}) -> MPolyQuoRing, Map

As above, where $I\subset R$ is the ideal generated by the polynomials in $V$.

Examples

julia> R, (x, y) = PolynomialRing(QQ, ["x", "y"]);

julia> A, _ = quo(R, ideal(R, [x^2-y^3, x-y]))
(Quotient of Multivariate Polynomial Ring in x, y over Rational Field by ideal(x^2 - y^3, x - y), Map from
Multivariate Polynomial Ring in x, y over Rational Field to Quotient of Multivariate Polynomial Ring in x, y over Rational Field by ideal(x^2 - y^3, x - y) defined by a julia-function with inverse)

julia> typeof(A)
MPolyQuo{fmpq_mpoly}

julia> typeof(x)
fmpq_mpoly

julia> typeof(A(x))
MPolyQuoElem{fmpq_mpoly}

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

julia> p
Map from
Multivariate Polynomial Ring in x, y over Rational Field to Quotient of Multivariate Polynomial Ring in x, y over Rational Field by ideal(x^2 - y^3, x - y) defined by a julia-function with inverse

julia> p(x)
x

julia> typeof(p(x))
MPolyQuoElem{fmpq_mpoly}

julia> S, (x, y, z) = GradedPolynomialRing(QQ, ["x", "y", "z"]);

julia> B, _ = quo(S, ideal(S, [x^2*z-y^3, x-y]))
(Quotient of Multivariate Polynomial Ring in x, y, z over Rational Field graded by
  x -> [1]
  y -> [1]
  z -> [1] by ideal(x^2*z - y^3, x - y), Map from
Multivariate Polynomial Ring in x, y, z over Rational Field graded by
  x -> [1]
  y -> [1]
  z -> [1] to Quotient of Multivariate Polynomial Ring in x, y, z over Rational Field graded by
  x -> [1]
  y -> [1]
  z -> [1] by ideal(x^2*z - y^3, x - y) defined by a julia-function with inverse)

julia> typeof(B)
MPolyQuo{MPolyElem_dec{fmpq, fmpq_mpoly}}