2014/05/04 by Andreas Rüdinger, Rüdinger, Andreas
Engineering · Mathematics · Physics and Astronomy · #30C15 #Advanced Mathematical Theories and Applications #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Mathematics and Applications #math.CA #math.CV #msc:30C15
paper · pdf · doi:10.48550/arxiv.1405.0689
6 pages, 4 figures
arxiv created 2014/05/04 · openalex publication_date 2014/05/04 · arxiv updated 2014/05/06 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
According to the classical Gauss-Lucas theorem all zeros of the derivative of a complex non-constant polynomial p lie in the convex hull of the zeros of p. It is proved that for a polynomial p of degree four with four different zeros forming a concave quadrilateral, the zeros of the derivative lie in two of the three triangles formed by the zeros of p. Thus a strengthening of the classical Gauss-Lucas theorem is established for this case, which can be extended to the case of a polynomial of degree n for which the zeros do not form a convex n-polygon.