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

Let Dd,k denote the discriminant variety of degree d polynomials in one variable with at least one of its roots being of multiplicity ≥ k. We prove that the tangent cones to Dd,k span Dd,k − 1 thus, revealing an extreme ruled nature of these varieties. The combinatorics of the web of affine tangent spaces to Dd,k in Dd,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 which are tangent to the discriminant hypersurface Dd,2. We also connect the geometry of the Viète map Vd: , given by the elementary symmetric polynomials, with the tangents to the discriminant varieties {Dd,k}. Various d-partitions {μ} provide a refinement {Doμ} of the stratification of by the Dd,kʹs. 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 {Doμ}.