A note on the elimination theory

We consider certain generalisation of the resultant of two polynomials in one variable. Using the Schur symmetricfunctions we describe the ideal of all polynomials in the coefficients of two equations, which vanish if these equations haver+1 roots in common, where r≥0. We discuss also related (classical)criterions giving the conditions when two equations have r+1 roots in common, where r≥0.