Details:
Title  An algorithm for computing a Gr\"obner basis of a polynomial ideal over a ring with zero divisors.  Author(s)  Yongyang Cai, Deepak Kapur  Type  Article in Journal  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 ℤn and ℤn[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 KandriRody 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 KandriRody–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 ℤn, the algorithm for polynomial ideals over ℤ could be used by augmenting the original ideal presented by polynomials over ℤn with n (similarly, in the case of ℤn[i], the original ideal is augmented with n and i 2 + 1).  Keywords  Gröbner basis, ring with zero divisors, polynomial ideal, ideal membership, canonical form, algebraic geometry  ISSN  16618270; 16618289/e 
URL 
http://link.springer.com/article/10.1007%2Fs117860090072z 
Language  English  Journal  Math. Comput. Sci.  Volume  2  Number  4  Pages  601634  Publisher  Springer (Birkh\"auser), Basel  Year  2009  Edition  0  Translation 
No  Refereed 
No 
