Computing a Gröbner basis of a polynomial ideal over a Euclidean domain (Q1111629)

The authors develop a general Gröbner basis algorithm for polynomial ideals over a Euclidean domain, based on a natural definition of reduction of polynomials using a single polynomial at a time and Euklid's algorithm. The input for the algorithm is an ideal specified by a finite set of polynomials; the output is another finite basis for the ideal, reducing every polynomial in the ideal to 0 and every polynomial in the polynomial ring to a unique normal form. The algorithm is an extension of Buchberger's algorithms for solving the same problem for polynomials over fields and \({\mathbb{Z}}\) as well as the authors' method for polynomials over \({\mathbb{Z}}\) and \({\mathbb{Z}}[i]\). The algorithm is simpler than the existing algorithms and is illustrated by computing Gröbner bases for polynomial ideals over \({\mathbb{Z}}\), \({\mathbb{Z}}[i]\), \({\mathbb{Q}}[\sqrt{-3}]\), \({\mathbb{Q}}[\sqrt{2}]\). The question of uniqueness of a reduced Gröbner basis of an ideal is discussed.