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

Abstract

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 $\geq k$. 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 $P(z)\in \mathcal D_{d,1} \approx \A^d$ which are tangent to the discriminant hypersurface $\mathcal D_{d,2}$. We also connect the geometry of the Vi\`{e}te map $\mathcal V_d: \A^d_{root} \to \A^d_{coef}$, given by the elementary symmetric polynomials, with the tangents to the discriminant varieties $\{\mathcal D_{d,k}\}$. Various $d$-partitions $\{\mu\}$ provide a refinement $\{\mathcal D_\mu^\circ\}$ of the stratification of $\A^d_{coef}$ by the $\mathcal D_{d,k}$'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 $\{\mathcal D_\mu^\circ\}$.