When do multiple polynomials have a common root?

Question

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.

Answer 1

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 $\mathbb{A}^{\dim(V)}$.

Answer 2

Well, the "naïve" answer (already implied in @Somnium's comment) is that a general method for finding the solutions of a system of simultaneous polynomial equations $$ f_1(x)=0, \ \ f_2(x)=0, \ \ \cdots \ \ f_n(x)=0 $$ is finding the $\gcd\bigl(f_1, f_2, \cdots, f_n\bigr)=g(x)$, of these polynomials:

Assuming that we are speaking about polynomials with complex coefficients and since $\mathbb{C}[x]$ is a PID, the ideal of $\mathbb{C}[x]$ generated by the polynomials $f_1, f_2, \dotsc, f_n$ is the ideal generated by their greatest common divisor: $$ \langle f_1, f_2, \dotsc, f_n\rangle=\langle\gcd\bigl(f_1, f_2, \dotsc, f_n\bigr)\rangle=\langle g\rangle. $$

Thus, the solution of the initial system, is equivalent to the solution of the single equation $$ g(x)=0. $$

Maybe there is no explicit formula for the gcd of a number of univariate polynomials (in terms of their coefficients) but there is certainly an algorithm: The $\gcd\bigl(f_1, f_2, \dotsc, f_n\bigr)$, can be found using the Euclidean division algorithm for finding the $\gcd$ of two univariate polynomials in $\mathbb{C}[x]$ together with the fact that for $n\geq 3$ $$ \gcd(f_1, f_2, \dotsc, f_n)=\gcd\bigl(f_1,\gcd(f_2, \dotsc, f_n)\bigr). $$

P.S.: I am not sure if this satisfies you as an answer but I was led to this description driven by the way the question has been posed and the subsequent discussion in the comments.