-Polynomials are mathematical algebraic structures that play an important role in science and engineering by capturing physical phenomena and expressing any outcome as a function of some unknown variables. Formally speaking, a polynomial $p(x)$ of degree $n$ having $n$ coefficients in the complex plane $\mathbb{C}$ is: \begin{equation}p(x)=\sum_{i=0}^{n}{a_ix^i}.\end{equation}
-\LZK{Dans ce cas le polynôme est de degré $n-1$!}
-
-The root-finding problem consists in finding the $n$ different values of the unknown variable $x$ for which $p(x)=0$. Such values are called zeros of $p$ (\textit{i.e.} roots). If zeros are $\alpha_{i}$, $i=1,\ldots,n$, then $p(x)$ can be written as :
+%Polynomials are mathematical algebraic structures that play an important role in science and engineering by capturing physical phenomena and expressing any outcome as a function of some unknown variables. Formally speaking, a polynomial $p(x)$ of degree $n$ having $n$ coefficients in the complex plane $\mathbb{C}$ is:
+%\begin{equation}
+%p(x)=\sum_{i=0}^{n}{a_ix^i}.
+%\end{equation}
+%\LZK{Dans ce cas le polynôme a $n+1$ coefficients et non pas $n$!}
+The issue of finding the roots of polynomials of very high
+degrees arises in many complex problems in various fields,
+such as algebra, biology, finance, physics or climatology [1].
+In algebra for example, finding eigenvalues or eigenvectors of
+any real/complex matrix amounts to that of finding the roots
+of the so-called characteristic polynomial.
+
+The root-finding problem consists in finding the $n$ different values of the unknown variable $x$ for which $p(x)=0$. Such values are called zeros or roots of $p$. If zeros are $\{\alpha_{i}\}_{1\leq i\leq n}$, then $p(x)$ can be written as :
