Resolving algebraic equations by contour integrals (Q5959493)

Roots of some non-linear equations can be found explicitly by making use of simple complex analysis [see, e.g., \textit{P. Kravanja} and \textit{M. Van Barel}, Computing the zeros of analytic functions. Lecture Notes in Mathematics. 1727. Berlin: Springer (2000; Zbl 0945.65018), or the reviewer's note, Zesz. Nauk. Uniw. Jagiell., Univ. Iagell. 1245, Acta Math. 38, 299--305 (2000; Zbl 1001.30008)] and the references therein. In the paper under review the author derives explicit formulas for all the roots \(z_k\) of a polynomial of degree \(n\) as contour integrals \[ z_k = \frac{1}{2\pi i} \int _{\gamma _k} \frac{z P_k'(z) dz}{P_k (z)}, k=1,2, \ldots ,n \] where the contours \(\gamma _k\) and the polynomials \(P_k\) of degrees not greater than \(n\) are given in the paper.