The Buchberger Algorithm as a Tool for Ideal Theory of Polynomial Rings in Constructive Mathematics

Abstract

One of the aims of Constructive Mathematics is to provide effective methods (algorithms) to compute objects whose existence is asserted by Classical Mathematics. Moreover, all proofs should be constructive, i.e., have an underlying effective content. E.g. the classical proof of the correctness of Buchberger algorithm, based on noetherianity, is non constructive : the closest consequence is that we know that the algorithm ends, but we don’t know when. In this paper we explain how the Buchberger algorithm can be used in order to give a constructive approach to the Hilbert basis theorem and more generally to the constructive content of ideal theory in polynomial rings over “discrete” fields. Mines, Richman and Ruitenburg in 1988 ([5]) (following Richman [6] and Seidenberg [7]) attained this aim without using Buchberger algorithm and Grobner bases, through a general theory of “coherent noetherian rings” with a constructive meaning of these words (see [5], chap. VIII, th. I.5). Moreover, the results in [5] are more general than in our paper and the Seidenberg version gives a slightly different result. Here, we get the Richman version when dealing with a discrete field as coefficient ring (“discrete” means the equality is decidable in k). In classical texts (cf. Cox, Little and O’Shea [2]) about Grobner bases, the correctness of the Buchberger algorithm and the Hilbert basis theorem are both proved by using a non constructive version of Dickson’s Lemma. So, from a constructive point of view, the classical approach gives a constructive tool with a gap in the proof. E.g., it is impossible to give bounds for the Buchberger algorithm by a detailed inspection of the classical proof. Moreover, the classical formulation of the Hilbert basis theorem is nonconstructive. Here we give a constructive version of Dickson’s Lemma, we deduce constructively the correctness of Buchberger algorithm and from this result we get the Hilbert basis theorem in a constructive form. In our opinion Grobner bases are a very good tool, the more natural one in the present time, for understanding the constructive content of ideal theory in polynomial rings over a discrete field. ∗UMR CNRS 6623. Univ. de Franche-Comte, 25030 Besancon cedex, France. henri.lombardi@univ-fcomte.fr †UMR CNRS 6623. Univ. de Franche-Comte, 25030 Besancon cedex, France. perdry@univ-fcomte.fr