Construct resultant for 3 polynomials

[pyfgcr]

For 2 polynomials f,g, resultant Res(f,g) vanish if and only if f and g has at least a common root.
However, is there any way to construct a coefficients polynomial of 3 polynomials f,g,h [Res(f,g,h)] that vanish if and only if f,g,h has at least a common root?

 

[fzero]

I'm not familiar with the subject, but google turned up http://arxiv.org/abs/math/0007036, which seems helpful.