If your goal is to check if a sequence of polynomial equations has solutions, results are not what you want. There is a number of extremely sophisticated algorithms to do this (from the variations on subgresultant algorithms (due initially to Collins) to Grobner bases. Grobner bases work particularly well when the variety you are trying to determine is zero-dimensional (I have no way of knowing if this is your situation). All the computer algebra systems (Mathematica, Maple, Singular) have these implemented. In the case of mathematica the function Solve[] is what you want.

If your goal is to check if a set of polynomial equations has solutions, resultants are not what you want. There is a number of extremely sophisticated algorithms to do this (from the variations on subgresultant algorithms (due initially to Collins) to Grobner bases. Grobner bases work particularly well when the variety you are trying to determine is zero-dimensional (I have no way of knowing if this is your situation). All the computer algebra systems (Mathematica, Maple, Singular) have these implemented. In the case of mathematica the function Solve[] is what you want.