On Computing the Elimination Ideal Using Resultants with Applications to Gröbner Bases

We are after a generator of the elimination ideal of an ideal generated bu two polynomials in two variables. Such a generator is unique (up to multiplication by units) and can be computed via Gröbner basis. We are interested in finding the generator of the elimination ideal using the re-sultant of the generators of the ideal. All the factors of the generators are factors of the resultant but the multiplicities might be different. Geomet-rically we are looking at two algebraic curves. Factors of the resultant give us projections of all the intersection points with their multiplicities. We are investigating the difference between the multiplicities of the factors of the Gröbner basis and the resultant In the case of general ideals we can express the difference between the variety if the elimination ideal and the variety of the set of the resultants of pairs of the generators. Also we show an attempt to use resultants in Gröbner basis computations.