How tangents solve algebraic equations, or a remarkable geometry of discriminant varieties

Let ${\mathcal{D}}_{d,k}$ denote the discriminant variety of degree $d$ polynomials in one variable with at least one of its roots being of multiplicity $\ge k$. We prove that the tangent cones to ${\mathcal{D}}_{d,k}$ span ${\mathcal{D}}_{d,k-1}$ thus, revealing an extreme ruled nature of these varieties. The combinatorics of the web of affine tangent spaces to $\mathcal D_{d,k}$in$\mathcal D_{d,k-1}$ is directly linked to the root multiplicities of the relevant polynomials. In fact, solving a polynomial equation $P(z)=0$ turns out to be equivalent to finding hyperplanes through a given point $P(z)\in {\mathcal{D}}_{d,1}\approx {\text{A}}^d$ which are tangent to the discriminant hypersurface ${\mathcal{D}}_{d,2}$. We also connect the geometry of the Vi`{e}te map ${\mathcal{V}}_d:{\text{A}}_{root}^d\to {\text{A}}_{coef}^d$, given by the elementary symmetric polynomials, with the tangents to the discriminant varieties $\{{\mathcal{D}}_{d,k}\}$. Various $d$-partitions $\{\mu \}$ provide a refinement ${\mathcal D_\mu^\circ}$ofthestratificationof$\A^d_{coef}$bythe$\mathcal D_{d,k}$'s. Our main result, Theorem 7.1, describes an intricate relation between the divisibility of polynomials in one variable and the families of spaces tangent to various strata $\{{\mathcal{D}}_{\mu }^{\circ }\}$.