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

Pauer, Franz

J. Symb. Comput. 14, No. 5, 471-482 (1992).

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.

Reviewer: A.Dickenstein (Buenos Aires)

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
13F10	Principal ideal rings

Keywords:

principal ideal ring; Gröbner bases; lucky ideals

Citations:

Zbl 0669.13009

References:

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