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

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dc.creatorKatz, Gabriel
dc.date2002-11-18
dc.date.accessioned2026-07-07T04:53:03Z
dc.date.available2026-07-07T04:53:03Z
dc.descriptionLet $\mathcal D_{d,k}$ 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 Viète 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 $\{μ\}$ provide a refinement $\{\mathcal D_μ^\circ\}$ 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 $\{\mathcal D_μ^\circ\}$.
dc.description43 pages, 12 figures
dc.identifierhttps://arxiv.org/abs/math/0211281
dc.identifierhttp://arxiv.org/abs/math/0211281
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/65701
dc.subjectAlgebraic Geometry
dc.subject14N10
dc.titleHow tangents solve algebraic equations, or a remarkable geometry of discriminant varieties
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