The vectors of a root-system were originally called "*roots*" because they are the zeros of a characteristic polynomial that comes from the connection of (crystallographic) root-systems to classifying semisimple Lie algbras. From an [answer by José Carlos Santos][JCS], 

> It comes from the roots of the characteristic polynomial of an endomorphism. If $\mathfrak g$ is a complex semisimple Lie algbra, $\mathfrak h$ is a Cartan subalgebra and $\alpha\in\mathfrak{h}^*$, then $\alpha$ is a root if, for every $H\in\mathfrak h$, $\alpha(H)$ is an eigenvalue of the endomorphism of $\mathfrak g$ defined by $X\mapsto[H,X]$.

Is there a way to get at this same polynomial just from the definitions of a (crystallographic) root-system without having to talk about the connection to Lie theory? 

[JCS]: https://math.stackexchange.com/a/2622161/167197