An Algorithm for Computing a Gröbner Basis of a Polynomial Ideal over a Ring with Zero Divisors

Abstract.

An algorithm for computing a Gröbner basis of an ideal of polynomials whose coefficients are taken from a ring with zero divisors, is presented; such rings include $\mathbb {Z}_n$ and $\mathbb {Z}_n[i]$, where n is not a prime number. The algorithm is patterned after (1) Buchberger’s algorithm for computing a Gröbner basis of a polynomial ideal whose coefficients are from a field and (2) its extension developed by Kandri-Rody and Kapur when the coefficients appearing in the polynomials are from a Euclidean domain. The algorithm works as Buchberger’s algorithm when a polynomial ideal is over a field and as Kandri-Rody–Kapur’s algorithm when a polynomial ideal is over a Euclidean domain. The proposed algorithm and the related technical development are quite different from a general framework of reduction rings proposed by Buchberger in 1984 and generalized later by Stifter to handle reduction rings with zero divisors. These different approaches are contrasted along with the obvious approach where for instance, in the case of ${\mathbb {Z}}_n$, the algorithm for polynomial ideals over ${\mathbb {Z}}$ could be used by augmenting the original ideal presented by polynomials over ${\mathbb {Z}}_n$ with n (similarly, in the case of ${\mathbb {Z}}_n[i]$, the original ideal is augmented with n and i² + 1).