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A generalization of the Gauss-Lucas theorem

J. L. Díaz-Barrero, J. J. Egozcue (2008)

Czechoslovak Mathematical Journal

Given a set of points in the complex plane, an incomplete polynomial is defined as the one which has these points as zeros except one of them. The classical result known as Gauss-Lucas theorem on the location of zeros of polynomials and their derivatives is extended to convex linear combinations of incomplete polynomials. An integral representation of convex linear combinations of incomplete polynomials is also given.

A polynomial with 2k critical values at infinity

Janusz Gwoździewicz, Maciej Sękalski (2004)

Annales Polonici Mathematici

We construct a polynomial f:ℂ² → ℂ of degree 4k+2 with no critical points in ℂ² and with 2k critical values at infinity.

A Test for the Stability of Networks

Agnieszka Rowinska-Schwarzweller, Christoph Schwarzweller (2013)

Formalized Mathematics

A complex polynomial is called a Hurwitz polynomial, if all its roots have a real part smaller than zero. This kind of polynomial plays an all-dominant role in stability checks of electrical (analog or digital) networks. In this article we prove that a polynomial p can be shown to be Hurwitz by checking whether the rational function e(p)/o(p) can be realized as a reactance of one port, that is as an electrical impedance or admittance consisting of inductors and capacitors. Here e(p) and o(p) denote...

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