TITLE: Polynomials with more than one common root
QUESTION [5 upvotes]: Given two polynomials $p_1 (x) , p_2 (x) \in \mathbb{C} [x]$, we know they have a common root iff their resultant vanishes. What is the condition ensuring $p_1$ and $p_2$ share more than one common root? Many thanks in advance.

REPLY [6 votes]: The real condition is that the greatest common divisor of the two polynomials has degree greater than one. Computationally, that's about as good as you are going to get, but if you like resultants, the vanishing of the resultant is just a way of restating that the system of linear equations on the coefficients of $a, b$ given by 
$$a p_1 + b p_2 = 1$$ has no solutions (meaning that the vector $(1, 0, \dots)$ is not in the image). If the gcd has degree greater than one, the span of $(1, 0, \dots)$ and $(0, 1, \dots)$ is not in the image, so the resultant has co-rank at least $2,$ as suggested by Fedor.