**This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.**

### Abstract

This formalization is concerned with the theory of Gröbner bases in
(commutative) multivariate polynomial rings over fields, originally
developed by Buchberger in his 1965 PhD thesis. Apart from the
statement and proof of the main theorem of the theory, the
formalization also implements Buchberger's algorithm for actually
computing Gröbner bases as a tail-recursive function, thus allowing to
effectively decide ideal membership in finitely generated polynomial
ideals. Furthermore, all functions can be executed on a concrete
representation of multivariate polynomials as association lists.

### License

### History

- April 18, 2019
- Specialized Gröbner bases to less abstract representation of polynomials, where
power-products are represented as polynomial mappings.

### Topics

### Session Groebner_Bases

- General
- Confluence
- Reduction
- Groebner_Bases
- Algorithm_Schema
- Buchberger
- Benchmarks
- Algorithm_Schema_Impl
- Code_Target_Rat
- Buchberger_Examples
- More_MPoly_Type_Class
- Auto_Reduction
- Reduced_GB
- Reduced_GB_Examples
- Macaulay_Matrix
- F4
- F4_Examples
- Syzygy
- Syzygy_Examples
- Groebner_PM