STRUCTURE OF GR ¨ OBNER BASES WITH RESPECT TO ELIMINATION ORDERS M’HAMMED EL KAHOUI AND SAID RAKRAK Abstract. In this paper we study the structure of Gr¨ obner bases with respect to elimination orders. We give a structure theorem for such Gr¨ obner bases, which turns out to be an extension of Lazard’s theorem established in the case of two variables. We then give a more precise structure theorem in the case of radical zero- dimensional ideals. As an application of this we give the basic theoretical ingredients of an algorithm to compute a lexicographic Gr¨ obner basis of a finite intersection of maximal ideals. 1. Introduction The concept of Gr¨obner bases, introduced by Buchberger [5] in 1965, is nowadays one of the main tools for studying algebraic systems and various related problems in computational algebra, see [6] and the stan- dard reference books [4, 2, 10, 25] for basic facts and applications of such a concept. Many computational aspects of Gr¨obner bases theory have been studied so far. For example, the way to get the solutions of an algebraic system from a Gr¨obner basis with respect to the lexicographic order is studied in [26]. The same question, but with respect to other monomial orders, is treated for example in [16]. The complexity aspect of Gr¨obner basis computation is studied in [8, 9, 14]. We are concerned in this work with the question of understanding the structure of lexicographic Gr¨obner bases. In the case of two vari- ables, Lazard’s theorem [15] gives a complete structural understanding of lexicographic Gr¨obner bases. This result is extended to the case of univariate polynomial rings over Dedekind domains in [3]. For higher dimensions, a structure theorem is given in [13] for the case of radical zero-dimensional ideals. This paper is aimed toward a better understanding of the structure, in high dimensions, of Gr¨obner bases with respect to elimination orders. It is organized as follows. After setting up the necessary notation in 2000 Mathematics Subject Classification. 13P10, 12Y05. Key words and phrases. Elimination order, Gr¨ obner basis, Triangular system. 1