Strengthening the Gauss-Lucas theorem for polynomials with Zeros in the interior of the convex hull

Andreas Rüdinger

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arXiv · 1405.0689

Strengthening the Gauss-Lucas theorem for polynomials with Zeros in the interior of the convex hull

Andreas Rüdinger

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Abstract

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.

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BibTeX RIS

Andreas Rüdinger. 2014-05-04. Strengthening the Gauss-Lucas theorem for polynomials with Zeros in the interior of the convex hull. https://arxiv.org/abs/1405.0689

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