# Gröbner Basis and Its Applications

## Abstract

Computer Algebra is a field of mathematics and computer science that studies algorithms for symbolic computation. A fundamental tool in computer algebra to study polynomial ideals is the theory of Geöbner basis. The notion of the Gröbner basis and the Buchberger’s algorithm for computing it was proposed by Bruno Buchberger in 1965. Gröbner bases have numerous applications in commutative algebra, algebraic geometry, combinatorics, coding theory, cryptography, theorem proving, etc. The Buchberger’s algorithm is implemented in many computer algebra systems, such as *Risa/Asir*, *Macaulay2*, *Singular*, *CoCoa*, *Maple*, and *Mathematica*. In this chapter, we will give a short introduction on Gröbner basis theory, and then we will present some applications of Gröbner bases.

### Keywords

Gröbner basis Toric ideal### References

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