A Gröbner basis G for a system of polynomials A is an equivalence system that possesses useful properties, for example, that another polynomial f is a combination of those in A iff the remainder of f with respect to G is 0. (Here, the division algorithm requires an order of a certain type on the monomials.) Furthermore, the set of polynomials in a Gröbner basis have the same collection of roots as the original polynomials. For linear functions in any number of variables, a Gröbner basis is equivalent to Gaussian elimination. The algorithm for computing Gröbner bases is known as Buchberger's algorithm. Calculating a Gröbner basis is typically a very time-consuming process for large polynomial systems.

Buchberger's algorithm | commutative algebra | Euclidean algorithm | Gaussian elimination | Gröbner walk | monomial | orthonormal basis