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Resultants and Gr\"obner bases are crucial tools in studying polynomialelimination theory. We investigate relations between the variety of theresultant of two polynomials and the variety of the ideal they generate. Thenwe focus on the bivariate case, in which the elimination ideal is principal. Westudy - by means of elementary tools - the difference between the multiplicityof the factors of the generator of the elimination ideal and the multiplicityof the factors of the resultant.

On Computing the Elimination Ideal Using Resultants with Applications to Gr\"obner Bases