Computational ideal theory in finitely generated extension rings

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...

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Vydané v:Theoretical computer science Ročník 244; číslo 1; s. 1 - 33
Hlavný autor: Apel, Joachim
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Amsterdam Elsevier B.V 06.08.2000
Elsevier
Predmet:
Algebra
Algebraic algorithms
Algebraic geometry
Algorithmics. Computability. Computer arithmetics
Applied sciences
Computable algebraic structures
Computer science; control theory; systems
Exact sciences and technology
Groebner bases in graded structures
Mathematics
Nonassociative rings and algebras
Sciences and techniques of general use
Theoretical computing
F.u.1
I.1.2
Algebraic algorithms
Computable algebraic structures
Groebner bases in graded structures
F.2.2
Annihilation
Gröbner basis
Noether theorem
Ideal
Algebraic algorithm
Algebraic struture
Group theory
Buchberger algorithm
Graded structure
Algorithm
Computability
ISSN:0304-3975, 1879-2294
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Popis
Shrnutí: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.
ISSN:0304-3975
1879-2294
DOI:10.1016/S0304-3975(00)00172-9