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.



When do multiple polynomials have a common root?
Assume that $f\_n$ is monic. Then for indeterminates $u\_1,\ldots,u\_{n-1}$, all coefficients of the polynomial $Res\_x(f\_n,u\_1f\_1+\ldots+u\_{n-1}f\_{n-1})\in k[u\_1,\ldots,u\_{n-1}]$ vanish if and only if $f\_1,\ldots,f\_n$ have a common root [expanding out this resultant then gives the list of polynomials].


This is how elimination theory works - if $f\_1(x\_1,\ldots,x\_m)=0,\ldots, f\_n(x\_1,\ldots,x\_m)=0$ set-theoretically define a variety $V$, $f\_n$ is monic in $x\_m$ (which one ensure by applying a Noether normalization coordinate change), and $\pi:\mathbb{A}^m\to \mathbb{A}^{m-1}$ is the projection away from the last coordinate, then $\pi(V)$ is a variety in $\mathbb{A}^{m-1}$ defined by the coefficients of $Res\_x(f\_n,u\_1f\_1+\ldots+u\_{n-1}f\_{n-1})$. Repeatedly eliminating variables like this eventually gives you a finite morphism surjecting $V$ to $\mathbb{A}^{\dim(V)}$.


