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Some Recent Results on the Geometry of Complex Polynomials: The Gauss--Lucas Theorem, Polynomial Lemniscates, Shape Analysis, and Conformal Equivalence
Trevor J. Richards
In this article, we survey the the recent literature surrounding the geometry of complex polynomials. Specific areas surveyed are i) Generalizations of the Gauss--Lucas Theorem, ii…
Rouché's Theorem and the Geometry of Rational Functions
Trevor J. Richards
In this note, we use Rouché's theorem and the pleasant properties of the arithmetic of the logarithmic derivative to establish several new results regarding the geometry of the zer…
Leaky Roots and Stable Gauss-Lucas Theorems
Trevor J. Richards, Stefan Steinerberger
Let be a polynomial. The Gauss-Lucas theorem states that its critical points, , are contained in the convex hull of its roots. A re…
On approximate Gauss-Lucas theorems
Trevor Richards
The Gauss--Lucas theorem states that any convex set which contains all zeros of a degree polynomial must also contain all cri…