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Subalgebra bases. (English) Zbl 0725.13013
Commutative algebra, Proc. Workshop, Salvador/Brazil 1988, Lect. Notes Math. 1430, 61-87 (1990).
[For the entire collection see Zbl 0699.00021.]
The authors present methods of computation about subalgebras of a polynomial ring. The theory described in the paper is an approach which does not reduce subalgebra questions to ideal questions.
The first section introduces basic notation and terminology including subduction and the definition of the Subalgebra Analog to Gröbner Bases Ideals (SAGBI) basis. - The main features of section two are the notion of tête-a-têtes and the use of tête-a-têtes to determine whether a set is a SAGBI basis. - A main feature of section three is the SAGBI basis construction method. Section four presents the graded view of SAGBI theory. Section four also presents integrality conditions which insure a finite SAGBI basis and the corollary that SAGBI theory is algorithmic for subalgebras of k[X]. Section five is a brief section which shows that homogeneous SAGBI subalgebra membership determination is algorithmic.

MSC:
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13F20 Polynomial rings and ideals; rings of integer-valued polynomials