Gröbner-Bases, Gaussian elimination and resolution of systems of algebraic equations

In the past few years, two very different methods have been developed for solving systems of algebraic equations : the method of Gr6bner bases or standard bases [Buc I, Buc 2, Tri, P.Y] and the one which I presented in Eurosam 79 [Laz 2, Laz 3] based on gaussian elimination in some matrices. Although they look very different, they are, in fact, very similar, at least if we restrict ourselves to the first step of my method. On the other hand Gr6bner base algorithms are very close to the tangent cone algorithm of Mora [Mor]. All of these algorithms are related to Gaussian elimination. In the first part of this paper, we try to develop all these relations and to show that this leads to improvements in some of these algorithms. In the second part we give upper and lower bounds for the degrees of the elements of a Gr6bner base. These bounds are based on projective algebraic geometry. The choice of the ordering appears to be critical : lexicographical orderings give Gr6bner bases of high degree, while reverse lexicographical orderings lead to low degrees.