caa a22 4500 477118267 CHVBK 20180405111628.0 cr unu---uuuuu 170330e19960101xx s 000 0 eng 10.1007/BF02259625 doi (NATIONALLICENCE)springer-10.1007/BF02259625 Generalized Rolle theorem in ℝ n and ℂ [Elektronische Daten] [A. Khovanskii, S. Yakovenko] The Rolle theorem for functions of one real variable asserts that the number of zeros off on a real connected interval can be at most that off′ plus 1. The following inequality is a multidimensional generalization of the Rolle theorem: if ℓ[0,1] → ℝ n ,t→x(t), is a closed smooth spatial curve and L(ℓ) is the length of its spherical projection on a unit sphere, then for thederived curve ℓ′ [0,1], → ℝ n $$t \mapsto \dot x(t)$$ , the following inequality holds: L(ℓ) ⩽ L(ℓ′). For the analytic functionF(z) defined in a neighborhood of a closed plane curve Г ⊂ ℂ ≃ ℝ2 this inequality implies that $$\tilde V$$ Γ(F) ⩽ $$\tilde V$$ Γ(F′) + ϰ(Γ), where $$\tilde V$$ Γ(F) is the total variation of the argument ofF along Γ, and ϰ(Γ) is the integral absolute curvature of Γ. As an application of this inequality, we find an upper bound for the number of complex isolated zeros of quasipolynomials. We also establish a two-sided inequality between the variation index $$\tilde V$$ Γ(F) and another quantity, called theBernstein index, which is expressed in terms of the modulus growth of an analytic function. Plenum Publishing Corporation, 1996 Smooth spatial curve nationallicence integral absolute curvature nationallicence quasipolynomials nationallicence Bernstein index nationallicence Khovanskii A. Department of Mathematics, University of Toronto, 100 St. George Str., M5S 1A1, Toronto, ON, Canada aut Yakovenko S. Department of Theoretical Mathematics, The Weizmann Institute of Science, 76100, Rehovot, Israel aut Journal of Dynamical and Control Systems Kluwer Academic Publishers-Plenum Publishers 2/1(1996-01-01), 103-123 1079-2724 2:1<103 1996 2 10883 https://doi.org/10.1007/BF02259625 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF02259625 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Khovanskii A. Department of Mathematics, University of Toronto, 100 St. George Str., M5S 1A1, Toronto, ON, Canada aut NATIONALLICENCE 700 1- Yakovenko S. Department of Theoretical Mathematics, The Weizmann Institute of Science, 76100, Rehovot, Israel aut NATIONALLICENCE 773 0- Journal of Dynamical and Control Systems Kluwer Academic Publishers-Plenum Publishers 2/1(1996-01-01), 103-123 1079-2724 2:1<103 1996 2 10883 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer