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Three remarks on comprehensive Gröbner and SAGBI bases. (English) Zbl 0992.13006

Ganzha, Viktor G. (ed.) et al., Computer algebra in scientific computing. CASC 2000. Proceedings of the 3rd workshop, Samarkand, Uzbekistan, October 5-9, 2000. Berlin: Springer. 191-202 (2000).
The notion of comprehensive Gröbner Basis (CGB) was introduced by V. Weispfenning [J. Symb. Comput. 14, No. 1, 1-29 (1992; Zbl 0784.13013)]. A CGB of a set \[ F \subseteq k[U_1,\dots,U_m, X_1,\dots,X_n ]= K[\mathbb U,\mathbb X] \] is a finite set \(G \subset k[\mathbb U,\mathbb X]\) which, for any specialization of the variables \({\mathbb U}\), gives a Gröbner basis for the ideal obtained specialyzing \(F\) correspondingly. The authors consider the particular case in which \(n = 1\). For this situation and the case in which \(F\) reduces to two elements, the authors give bounds for the complexity of the computation of a CGB.
Successively the authors consider the case of CGB for binomial ideals and finally they introduce the notion of comprehensive SAGBI bases for a subalgebra. For this case they give some examples obtained from rings of invariants.
For the entire collection see [Zbl 0951.00046].
Reviewer: A.Logar (Trieste)

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
68W30 Symbolic computation and algebraic computation

Citations:

Zbl 0784.13013

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