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 P (z) ∈ Dd,1 ≈ A d which are tangent to the discriminant hypersurface Dd,2. We also connect the geometry of the Viète map Vd : A d root → A d coef , given by the elementary symmetric polynomials, with the tangents to the discriminant varieties {Dd,k}. Various d-partitions {μ} provide a refinement {D μ} of the stratification of A coef by the Dd,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 {D μ}.