2001/06/12 by Gudlaugur Thorbergsson, Thorbergsson, Gudlaugur, Masaaki Umehara +1
Mathematics · #51L15 #53C75 #Advanced Differential Equations and Dynamical Systems #Differential Geometry (math.DG) #FOS: Mathematics #Mathematical Dynamics and Fractals #math.DG #msc:51L15 #msc:53C75
paper · pdf · doi:10.48550/arxiv.math/0106088
39 pages, 6 figures
arxiv created 2001/06/12 · openalex publication_date 2001/06/12 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a real valued periodic smooth function u on R, n≥ 0, one defines the osculating polynomial ϕs (of order 2n+1) at a point s∈ R to be the unique trigonometric polynomial of degree n, whose value and first 2n derivatives at s coincide with those of u at s. We will say that a point s is a clean maximal flex (resp. clean minimal flex) of the function u on S1 if and only if ϕs≥ u (resp. ϕs≤ u) and the preimage (ϕ-u)-1(0) is connected. We prove that any smooth periodic function u has at least n+1 clean maximal flexes of order 2n+1 and at least n+1 clean minimal flexes of order 2n+1. The assertion is clearly reminiscent of Morse theory and generalizes the classical four vertex theorem for convex plane curves.