I'm taking an algebra course in Galois Theory and we have now gone over the algebraic/combinatoric meaning of the discrimiannt of a polynomial $P(x) = \Sigma a_i x^i$ as:
$\Delta_P=\begin{equation*} \prod_{i<j} (r_i-r_j)^2 \end{equation*}a^{2n-2}$ where $r_i$ denotes a root of $P(x)$. This is a really neat algebraic formula but I can't help but wonder what's hiding behind it. I'm sure there is some deeper intuition to the discriminant besides knowing if a polynomial has real roots, mainly geometric interpretation hiding behind it, and its generalization to the multivariate polynomial case. Thanks!