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}}