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Siebeck curves and two refinements of the Gauss-Lucas theorem. (English) Zbl 1281.14047

By an old result of F. H. Siebeck [J. Reine Angew. Math. 64, 175–182 (1865; Zbl 02750115)], the zeros of the derivative \(f'\) of a complex cubic polynomial \(f\) are the focal points of the Steiner ellipse to the triangle of roots of \(f\). The Steiner ellipse is tangent to the sides of a triangle at their respective midpoints. More generally, if \(f\) is of arbitrary degree, the roots of \(f'\) are the foci of the Siebeck curve which is defined by a number of similar tangent conditions that involve the roots of \(f\) [B.-Z. Linfield, American M. S. Bull. 27, 17–21 (1920; JFM 47.0073.01)].
In this article, the author provides a precise and general formulation of Linfield’s result. He also refines the Gauss-Lucas Theorem which states that the zeros of \(f'\) are contained in the convex hull of the zeros of \(f\). The zeros of \(f'\) lie in the interior of the convex hull of the Siebeck curve (which is strictly contained in the convex hull of the zeros of \(f\) unless all zeros are aligned) and only zeros of \(f'\) which are also zeros of \(f\) can lie on the closure of its exterior.
The mere formulation of above results requires a careful and detailed investigation of the Siebeck curve. It is, for example, not a priori clear that a well-defined exterior exists. The scrutiny of the author’s exposition certainly gives credibility to his statement that some of Linfield’s original arguments are based on dubious assumptions.

MSC:

14Q05 Computational aspects of algebraic curves
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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