# Some bounds for the construction of Gröbner bases

## Abstract

Let *R*=*K*[*X*_{1}, ..., *X*_{ n }] be a polynomial ring over a field. For any finite subset *F* of *R*, we put *m*=‖*F*‖, *d*=*max*(*deg*(*F*) : *f* ε *F*), and we let *s* be the maximal size of the coefficients of all *f* ε *F. G*=*GB*(*F*) denotes the unique reduced Gröbner basis for the ideal (*F*) (see [B3]). We show that the number *m*′=‖*G*‖ of polynomials in *G* and their maximal degree *d*^{ t } as well as the length of the computation of *G* from *F* (with unit cost operations in *K*) are bounded recursively in (*n, m, d*). The same applies to the degrees of the polynomials occuring during the computation. Moreover, for fixed (*n, m, d*), *G* can be computed from *F* in polynomial time and linear space, when the operations of *K* can be performed in polynomial time and linear space; in addition, the vector space dimension of the residue ring *R*/(*F*) is computably stable under variation of the coefficients of polynomials in F. Corresponding facts hold for polynomial rings over commutative regular rings (see [We']) and non-commutative polynomial rings of solvable type over fields (see [KRW]). Our method does not apply to polynomial rings over *Z* or other Euclidean rings; in fact, we show that over *Z*, the length of the computation of *G* from *F* with unit cost operations in *Z does* depend on *s*.

## Preview

Unable to display preview. Download preview PDF.

## References

- [B1]B. Buchberger, A note on the complexity of constructing Gröbner bases, in [EC83], pp. 137–134.Google Scholar
- [B2]B.Buchberger, A critical-pair/completion algorithm for finitely generated ideals in rings, in
*Logic and Machines: Decision Problems and Complexity*, Münster 1983, Springer LNCS vol. 171.Google Scholar - [B3]B.Buchberger, Gröbner bases: An algorithmic method in polynomial ideal theory, chap. 6 in
*Recent Trends in Multidimensional System Theory*, N. K. Bose Ed., Reidel Publ. Comp., 1985.Google Scholar - [CA]
*Computer Algebra, Symbolic and Algebraic Computation*, B.Buchberger, G. E. Collins, R. Loos Eds., Springer Verlag 1982/83.Google Scholar - [EC83]
*Computer Algebra, EUROCAL '83*, J. A. van Hulzen Ed., Springer LNCS vol. 162.Google Scholar - [EC85]
*EUROCAL '85*, B.F.Caviness Ed., Springer LNCS vol. 204.Google Scholar - [ES84]
*EUROSAM '84*, J. Fitch Ed., Springer LNCS vol. 174.Google Scholar - [Gi]M. Giusti, Some effectivity problems in polynomial ideal theory, in [ES84], pp. 159–171.Google Scholar
- [Gi']A note on the complexity of constructing standard bases, in [EC85], pp.411–412.Google Scholar
- [Hu]D. T. Huynh, The complexity of the membership problem for two subclasses of polynomial ideals, SIAM J. of Comp. 15 (1986), 581–594.Google Scholar
- [KRK]A. Kandri-Rody, D.Kapur, Algorithms for computing Gröbner bases of polynomial ideals over various Euclidean rings, in [ES84], pp. 195–205.Google Scholar
- [KRW]A. Kandri-Rody, V.Weispfenning, Non-commutative Gröbner bases in algebras of solvable type, 1986, submitted.Google Scholar
- [Ke]J. Keisler, Fundamentals of model theory, in
*Handbook of Mathematical Logic*, J. Barwise Ed., North-Holland, 1977.Google Scholar - [La]D. Lazard, Gröbner bases, Gaussian elimination and resolution of systems of algebraic equations, in [EC83], pp. 146–156.Google Scholar
- [MaMe]E.W. Mayr, A.R. Meyer, The complexity of the word problem for commutative semigroups and polynomial ideals, Adv. math. 46 (1982), 305–329.Google Scholar
- [MM]H. M. Möller, F. Mora, Upper and lower bounds for the degree of Gröbner bases, in [ES84], pp 172–183.Google Scholar
- [Ra]M. O. Rabin, Decidable theories, in
*Handbook of mathematical Logic*, J. Barwise Ed., North-Holland, 1977.Google Scholar - [We]V. Weispfenning, Model-completeness and elimination of quantifiers for subdirect products of structures, J. of Algebra, 36 (1975), 252–277.Google Scholar
- [We']V. Weispfenning, Gröbner bases in polynomial rings over commutative regular rings, to appear in Proc. EUROCAL '87, Leipzig, Springer LNCS.Google Scholar
- [We"]V. Weispfenning, Constructing universal Gröbner bases, 1987, submitted.Google Scholar
- [Wi]F. Winkler, On the complexity of the Gröbner basis algorithm over
*K*[*x, y, z*], in [ES84], pp. 184–194.Google Scholar