On lucky ideals for Gröbner basis computations. (English) Zbl 0776.13014

Let \(R\) be a principal ideal ring, \(R[x]\) the polynomial ring in \(n\) variables over \(R\) and \(I\) an ideal in \(R[x]\). Intuitively, an ideal \(P\) of \(R\) is lucky for \(I\) if we do not loose too much information on Gröbner bases of \(I\), when we project \(I\) to \((R/P)[x]\). One is led to this concept when trying to apply modular or \(p\)-adic methods in order to control the possibly enormous growth of coefficients during the computation of a Gröbner basis of a given ideal in the polynomial ring in \(n\) variables over the field of rational numbers.
Let \(F\) be a Gröbner basis of \(I\). The author shows that \(F\) gives direct and full information about lucky ideals for \(I\), and gets “projection” and “reconstruction” results of a paper. As an application, a short proof of the main result of a paper by F. Winkler [J. Symb. Comput. 6, No. 2/3, 287-304 (1988; Zbl 0669.13009)] is given. Complexity aspects are not considered.


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
13F10 Principal ideal rings


Zbl 0669.13009
Full Text: DOI


[1] Buchberger, B., Some properties of Gröbner bases of polynomial ideas, ACM SIGSAM Bull., 10, 4, 19-24 (1976)
[2] Buchberger, B., A critical-pair / completion algorithm for finitely generated ideals in rings, (Börger, E.; etal., Logic and Machines: Decision Problems and Complexity. Logic and Machines: Decision Problems and Complexity, Springer Lecture Notes in Computer Science, 171 (1984)), 137-155 · Zbl 0546.68021
[3] Ebert, G., Some comments on the modular approach to Gröbner bases, ACM SIGSAM Bull., 17, 2, 28-32 (1983) · Zbl 0527.13001
[4] Davenport, J.; Siret, Y.; Tournier, E., Calcul formel (1987), Masson: Masson Paris
[5] Furukawa, A.; Sasaki, T.; Kobayashi, H., Gröbner bases of a module over \(K[x_1\),…,\(x_n]\) and polynomial solutions of a system of linear equations, (Char, B., Proc. SYMSAC’86 ACM. Proc. SYMSAC’86 ACM, 1986 (1986)), 222-224
[6] Gianni, P.; Trager, B.; Zacharias, G., Gröbner Bases and Primary Decomposition of Polynomial Ideals, J. Symbolic Computation, 6, 149-167 (1988) · Zbl 0667.13008
[7] Kandri-Rody, A.; Kapur, D., Computing a Gröbner Basis of a Polynomial Ideal over a Euclidean Domain, J. Symbolic Computation, 6, 37-58 (1988) · Zbl 0658.13016
[8] Kornerup, P.; Gregory, R., Mapping integers and Hensel codes onto Farey fractions, Bit, 23, 9-20 (1983) · Zbl 0521.10007
[9] Möller, H. M., On the Construction of Gröbner bases Using Syzgies, J. Symbolic Computation, 6, 345-359 (1988) · Zbl 0695.13003
[10] Möller, H. M.; Mora, T., New Constructive Methods in Classical Ideal Theory, J. Algebra, 100, 138-178 (1986) · Zbl 0621.13007
[11] Pan, L., On the D-bases of Polynomial Ideals Over Principal Ideal Domains, J. Symbolic Computation, 7, 55-69 (1988) · Zbl 0668.68034
[12] Pauer, F.; Pfeifhofer, M., The Theory of Gröbner Bases, L’Enseignement Math, 34, 215-232 (1988) · Zbl 0702.13019
[13] Pauer, F., Gröbner Basen und ihre Anwendungen, (Chatterji, S.; etal., Jahrbuch Überblicke Mathematik (1991), Vieweg Verlag: Vieweg Verlag Braunschweig), 127-149 · Zbl 0725.13012
[14] Traverso, G., Gröbner Trace Algorithms, (Proc. ISSAC’88. Proc. ISSAC’88, Springer Lecture Notes in Computer Science, 358 (1988)), 125-138
[15] Trinks, W., Über B. Buchbergers Verfahren, Systeme algebraischer Gleichungen zu lösen, J. Number Theory, 10, 475-488 (1978) · Zbl 0404.13004
[16] Winkler, F., A p-adic Approach to the Computation of Gröbner Bases, J. Symbolic Computation, 6, 287-304 (1988) · Zbl 0669.13009
[17] Zacharias, G., Generalized Gröbner bases in commutative polynomial rings, (Bachelor Thesis (1978), MIT, Dept. Comp. Sci.)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.