I was wondering if it is well understood under what circumstances say three univariate polynomials $f(x),g(x),h(x)$ have a common root. In this situation, I can see that the resultant of each pair must vanish but that only ensures that each pair has a common root. Is there a way to generate a finite set of polynomials in the coefficients of $f,g,h$ which tells you when all 3 share at least one common root?

Would be interested in an answer for the more general (more than 3 polynomials) case too.

EDIT: After thinking about it a tad more here is a possibly interesting observation. If you have $n$ polynomials of degree at most $n$ then you can write it as a linear system in $x,x^2,...,x^n$. Using determinants, minors, etc you will be able to get $n-1$ necessary and sufficient relations between the coefficients which tells you when the polynomials share a common root. I would suspect that means in general if you have $k$ polynomials it might be possible to give $k-1$ polynomials in the coefficients which will be necessary and sufficient conditions for having a common root.