On lucky primes. (English) Zbl 0787.13016

Gräbe, Hans-Gert

J. Symb. Comput. 15, No. 2, 199-209 (1993).

Let \(k\) be a given commutative Noetherian ring. Fix a term order ‘\(<\)’ on the ring \(S:= k[x_ 1,\dots,x_ n]\); any polynomial \(f\in S\) can be written as \(f=\sum_{i=0}^ r c_{a_ r} x^{a_ r}\), where \(c_{a_ i}\in k\), \(a_ 0,\dots,a_ r\in{\mathbb{N}}^ n\) and \(a_ 0>\cdots> a_ r\). Then one can define, as usual, \(\deg(f):= a_ 0\) and \(\text{lc} (f):= c_{a_ 0}\). If \(I\subseteq S\) is any ideal, \(C_ a(I)\) denotes the set \(\{\text{lc} (f)\mid f\in I,\;\deg(f)=a\}\), i.e. \(C_ a(I)\) is the ideal in \(k\) given by all the leading coefficients of polynomials in \(I\) of fixed degree \(a\).
Given an ideal \(I\subseteq S\), a prime \(p\in \text{Spec}(k)\) is called lucky if \(C_ a(I)\not\subseteq p\) for every \(a\) such that \(C_ a(I)\not=0\). In particular, one can prove that, for example, in the case \(k={\mathbb{Z}}\), there is only a finite number of unlucky primes for a given \(I\).
Let now \(k\) be a domain, \(B\subseteq S\), \(p\subseteq k\) a lucky prime; denote by \(S_ p\) the localization of \(S\) at \(p\), and by \(S^{(p)}\) the ring \(S\otimes_ k k_ p/pk_ p\). In the first part of the paper it is proved that if \(B\subseteq S\), a minimal, reduced and normalized Gröbner basis of \(B^{(p)}\) \((\subseteq S^{(p)})\) over \(S^{(p)}\) can be lifted to a minimal, reduced and normalized Gröbner basis of \(B_ p\) over \(S_ p\) and then pushed down to a minimal, reduced and normalized Gröbner basis of \(B^{(0)}\) over \(S^{(0)}\). Moreover, if \(I\) is homogeneous, then any homogeneous basis of the syzygy module of \(B^{(p)}\) over \(S^{(p)}\) can be lifted to a homogeneous basis of the syzygy module of \(B_ p\) over \(S_ p\) and then pushed down to a basis of the syzygy module of \(B^{(0)}\) over \(S^{(0)}\).
Successively, it is sketched a method to compute a Gröbner basis of an ideal \(I\subseteq {\mathbb{Z}}[x_ 1,\dots,x_ n]\) performing the computations in the ring \(k[x_ 1,\dots,x_ n]\), where \(k:=\prod_{i=1}^ N {\mathbb{Z}}/p_ i{\mathbb{Z}}\) (\(p_ 1,\dots,p_ n\) a set of given primes). In this way one can perform modular computations not only for a single prime, but for the whole set of primes \(\{p_ 1,\dots,p_ N\}\). During the computation it is possible to find the unlucky primes in the given set. Several examples are given in order to show the applicability of the algorithm. Finally, it is given a procedure which uses only linear algebra over the field \({\mathbb{Q}}\) in order to lift a Gröbner basis using the idea of the Gröbner trace introduced by C. Traverso, in Symbolic and algebraic computation. Internat. Symp. Proc. ISAAC ’88, Proc., Lect. Notes Comput. Sci. 358, 125-138 (1989).

Reviewer: A.Logar (Trieste)

MSC:

13P10	Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13D02	Syzygies, resolutions, complexes and commutative rings

Keywords:

lucky primes; unlucky primes; syzygy; Gröbner basis