On the Bézout Construction of the Resultant

This paper deals with some ideas of Bézout and his successors Poisson, Netto and Laurent for solving polynomial systems. We treat them from the determinantal and from the Gröbner basis point of view. This results in effective algorithms for constructing the multivariate resultant. Other problems of Elimination Theory are discussed: how to find an eliminant for a polynomial system, how to represent its zeroes as the rational functions of the roots of this eliminant and how to separate zeroes.