### 2-D polynomial equations

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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.

Let $P$ and $Q$ be polynomials in one variable with complex coefficients and let $n$ be a natural number. Suppose that $Q$ is not constant and has only simple roots. Then there is a rational function $\varphi $ with ${\varphi}^{\text{'}}=P/{Q}^{n+1}$ if and only if the Wronskian of the functions ${Q}^{\text{'}},{\left({Q}^{2}\right)}^{\text{'}},...,{\left({Q}^{n}\right)}^{\text{'}},P$ is divisible by $Q$.

It is shown that if A is a bounded linear operator on a complex Hilbert space, then $w\left(A\right)\le 1/2\left(\right|\left|A\right||+|\left|A\xb2\right|{|}^{1/2})$, where w(A) and ||A|| are the numerical radius and the usual operator norm of A, respectively. An application of this inequality is given to obtain a new estimate for the numerical radius of the Frobenius companion matrix. Bounds for the zeros of polynomials are also given.

In the paper an elementary and simple proof of the Fundamental Theorem of Algebra is given.