Details:
Title  Computational ideal theory in finitely generated extension rings  Author(s)  Joachim Apel  Type  Article in Journal  Abstract  Since Buchberger introduced the theory of Gröbner bases in 1965 it has become an important tool in constructive algebra and, nowadays, Buchberger's method is fundamental for many algorithms in the theory of polynomial ideals and algebraic geometry. Motivated by the results in polynomial rings a lot of possibilities to generalize the ideas to other types of rings have been investigated. The perhaps most general concept, though it does not cover all possible extensions, is the theory of graded structures due to Robbiano and Mora. But in order to obtain algorithmic solutions for the computation of Gröbner bases it needs additional computability assumptions. In this paper we introduce natural graded structures of finitely generated extension rings and present subclasses of such structures which allow uniform algorithmic solutions of the basic problems in the associated graded ring and, hence, of the computation of Gröbner bases with respect to the graded structure. Among the considered rings there are many of the known generalizations. But, in addition, a wide class of rings appears first time in the context of algorithmic Gröbner basis computations. Finally, we discuss which conditions could be changed in order to find further effective Gröbner structures and it will turn out that the most interesting constructive instances of graded structures are covered by our results.  Keywords  computable algebraic structures, algebraic algorithms, Groebner bases in graded structures  Length  33 
File 
 Language  English  Journal  Theoretical Computer Science  Volume  244  Pages  1  33  Publisher  Elsevier  Year  2000  Translation 
No  Refereed 
No  Organization 
Universität Leipzig 
