Gröbner basis

Summary

Fix a monomial order on a polynomial ring. A Gröbner basis of an ideal is a finite generating set whose leading terms generate the ideal of all leading terms of the ideal. With a Gröbner basis, the multivariate division algorithm has a unique remainder. So one can decide whether a polynomial lies in the ideal, and one can compute in the quotient ring. Every ideal has a Gröbner basis, and Buchberger's algorithm computes one. Gröbner bases are the basic tool of computational commutative algebra and algebraic geometry, including elimination theory.