On associated Discriminants for Polynomials in one Variable

Abstract: In this note we introduce a family $\Sigma_i, i=0,\dots ,n-2$ of discriminants in the space ${\cal P}_n$ of polynomials of degree $n$ in one variable and study some of their algebraic and topological properties following [Ar], [Va] and [GKZ]. The discriminant $\Sigma_i$ consists of all polynomials $p$ such that some nontrivial linear combination $\alpha_0p+\alpha_1p^\prime +\dots +\alpha_ip^{(i)}$ has a zero of multiplicity greater or equal $i+2$. In particular, using the inversion of differential operators with constant coefficients (which induces the nonlinear involution on ${\cal P}_n$) we obtain the algebraic isomorphism of $\Sigma_i$ and $\Sigma_{n-2-i}$ for all $i$. \item{[Ar]} Arnold, V.I.: The spaces of functions with mild singularities. Funkt. Anal. and Appl. 23 (1989). \item{[GKZ]} Gelfand, I. M.; Kapranov, M. M.;Zelevinsky, A. V.: Discriminants, resultants and multidimensional determinants. Mathematics: theory and applications, Birkhäuser 1994. \item{[Va]} Vassiliev, V. A.: Complements of discriminants of smooth maps: Topology and Applications. Transl. of Math. Monographs 98 (1992).

Keywords: discriminants; algebraic and topological properties; inversion of differential operators

Classification (MSC91): 14P05

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