Triangular Systems and Factorized Groebner Bases

Gräbe, Hans-Gert
Solving systems of polynomial equations in an ultimate way means to find the isolated primes of the associated variety and to present them in a way that is well suited for further computations. [5] proposes an algorithm, that uses several Gröbner basis computations for a dimension reduction argument, delaying factorization to the end of the algorithm. In this paper we investigate the opposite approach, heavily using factorization (of multivariate polynomials), delaying the computation of stable ideal quotients. At a heuristic level this is exactly the well known Gröbner algorithm with factorization and constraint inequalities, available in all major computer algebra systems. In a preceding paper [9] we reported on some experience with a new version of this algorithm, implemented in our REDUCE package CALI [8]. Here we discuss, how this approach may be refined to produce triangular systems in the sense of [12] and [13]. Such a refinement guarantees, different to the usual Gröbner factorizer, to produce a quasi prime decomposition, i.e. the resulting components are at least pure dimensional radical ideals. As in [9] our method weakens the usual restriction to lexicographic term orders.
Appeared / Erschienen in: 
Proc. AAECC-11, Paris 1995 (ed. G. Cohen, M. Giusti, T. Mora). Lecture Notes in Computer Science 948, Springer (1995), S.248 - 261) .
Pubdate / Erscheinungsdatum: 
Pages / Seitenanzahl: 
1995-9.pdf204.34 KB