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How tangents solve algebraic equations, or a remarkable geometry of discriminant varieties. (English) Zbl 1045.14021

Summary: Let \(D_{d,k}\) denote the discriminant variety of degree \(d\) polynomials in one variable with at least one of its roots being of multiplicity \(\geq k\). The author proves that the tangent cones to \(D_{d,k}\,\text{span}\,D_{d,k-1}\), thus revealing an extreme ruled nature of these varieties. The combinatorics of the web of affine tangent spaces to \(D_{d,k}\) in \(D_{d,k-1}\) 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 \(P(z)\in D_{d,l} \approx\mathbb{A}^d\) which are tangent to the discriminant hypersurface \(D_{d,2}\). The author also connects the geometry of the Viète map \(V_d:\mathbb{A}^d_{\text{root}} \to\mathbb{A}^d_{\text{coef}}\), given by the elementary symmetric polynomials, with the tangents to the discriminant varieties \(\{D_{d,k}\}\).
Various \(d\)-partitions \(\{\mu\}\) provide a refinement \(\{D_\mu^\circ\}\) of the stratification of \(\mathbb{A}^d_{\text{coef}}\) by the \(D_{d,k}\)’s. The 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 \(\{D_\mu^\circ\}\).

MSC:

14M12 Determinantal varieties
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
65H10 Numerical computation of solutions to systems of equations
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