Stability and Buchberger criterion for standard bases in power series rings. (English) Zbl 0707.13008

Becker, Thomas

J. Pure Appl. Algebra 66, No. 3, 219-227 (1990).

One considers a formal power series ring \(k[[X_ 1,...,X_ n]]\) over a field k. One puts an admissible order on the set of all terms T (which is a well-ordering of T!, following from Dickson’s lemma). Let LT(F) denote the least element of a finite set F. The Hironaka theorem summarizes the most important facts, known about standard bases. As one passes from formal power series to polynomials, then the Hironaka theorem becomes, mutatis mutandis, a summary of the basic facts of Gröbner bases.
This paper adds some results, known for Gröbner bases, to the theory of standard bases. - First of all, the stability theorem for Gröbner bases (i.e. the property of a finite set of polynomials to be a Gröbner basis is locally stable in the topological space of admissible term orders), is “translated” into an analogous statement for standard bases. - Then the critical-pair-criterion, known for Gröbner bases, is carried over into a criterion for a finite set of power series to be a standard basis of the ideal it generates.
As all these results are non-algorithmic, it is natural to question the computability. The author concludes by discussing this point.

Reviewer: G.Molenbergh

MSC:

13P10	Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13F25	Formal power series rings

Keywords:

Buchsberger algorithm; power series ring; standard bases; Gröbner bases; computability

References:

[1]	Alonso, M.E.; Mora, T.; Raimondo, M., Computing with algebraic series, (), 101-111
[2]	Bayer, D.A., The division algorithm and the Hilbert scheme, ()
[3]	T. Becker, Standard bases and some computations in rings of power series, J. Symbolic Comput., to appear. · Zbl 0709.13013
[4]	Buchberger, B., An algorithm for finding a basis for the residue class ring of a zero-dimensional polynomial ideal, (), (in German) · Zbl 1158.01306
[5]	Buchberger, B., Gröbner bases: a criterion for the solvability of algebraic systems of equations, Aequationes math., 4, 3, 374-383, (1970), (in German)
[6]	Galligo, A., A propos du théorème de préparation de Weierstrass, (), 543-579, (Oct. 1970-Dec. 1973)
[7]	Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of math., 79, 109-326, (1964) · Zbl 0122.38603
[8]	Mora, T.; Robbiano, L., The Groebner Fan of an ideal, J. symbolic comput., 6, 183-208, (1988) · Zbl 0668.13017
[9]	Mora, T., (), Preprint
[10]	Schwartz, N., Stability of Groebner bases, J. pure appl. algebra, 53, 171-186, (1988) · Zbl 0664.13006
[11]	Weispfenning, V., Constructing universal Groebner bases, Proc. AAECC, (1987), Menorca
[12]	Weispfenning, V., Admissible orders and linear forms, ACM SIGSAM bull., 21, 16-18, (1987) · Zbl 0655.13017