# Introduction

The commutative algebra part of OSCAR provides functionality for dealing with

• multivariate polynomial rings and their ideals,
• quotients of multivariate polynomial rings modulo ideals and ideals of such quotients,
• localizations of the above rings and ideals of such localizations, and
• modules over all rings above.

We use affine algebra as a synonym for quotient of a multivariate polynomial ring modulo an ideal.

Fundamental to computational commutative algebra is the concept of standard bases. Each such basis is defined relative to a monomial ordering. If this ordering is a well-ordering, a standard basis is also called a Gröbner basis. We refer to the corresponding section in this chapter for details.

Note

Each multivariate polynomial ring in OSCAR comes equipped with a monomial ordering according to which the polynomials are stored and displayed. Independently of this ordering, standard bases can be computed with respect to any monomial ordering: The groebner_basis and standard_basis functions provided by OSCAR allow one to specify the desired monomial ordering as a key word argument. Typically, however, the user does not have to worry about Gröbner (standard) bases: The functions discussed in this chapter compute such bases behind the scenes when needed. Once computed, each such basis is cached for later reuse.

Note

In Oscar, it is possible to equip multivariate polynomial rings with gradings by finitely presented groups. Most functions discussed in this chapter apply to both ungraded and graded polynomial rings. However, for simplicity of the presentation, in this documentation, the functions are often only illustrated by examples with focus on the former case, but work similarly for homogeneous ideals and graded modules in the latter case.

Note

Our main focus in this chapter is on multivariate polynomial rings over fields (exact fields supported by OSCAR). Where not indicated otherwise, the presented functions also apply to polynomial rings over $\mathbb Z$.

General textbooks offering details on theory and algorithms include: