Exercise 8, § 4

[^1] Let $n$ be a natural number $\geqslant 1$.

a) For every element $x= (x_1, . . . , x_n)\in \mathbf{C}^n$, now prove that there exists a unique polynomial P with complex coefficients, monic, of degree $n+1$, such that P(0) = 0 and such that $(x_1, . . . , x_n)$ are the zeros of the derivative of P. We note $\theta (x) =$ $(P(x_1), . . . ,P(x_n))$; its coefficients are the critical values of P.

b) Now prove that $\theta$ defines a polynomial mapping from $\mathbf{C}^n$ into $\mathbf{C}^n$ such that $\overset{-1}{\theta}(0) = 0$.

c) We denote by U the open set of $\mathbf{C}^n$ whose elements are the elements $(x_1, . . . , x_n)$ whose coefficients are distinct and non-zero. Now prove that $\overset{-1}{\theta}(U)\subset U$ and that the restriction of $\theta$ to $\overset{-1}{\theta}(U)$ defines a covering $(\overset{-1}{\theta}(U), \theta_U)$ of U.

d) Exercise prove that the covering $(\overset{-1}{\theta}(U), \theta_U)$ is of degree $(n+ 1)^n$.

e) Exercise prove that the mapping $\theta$ is surjective.

[^1]: This exercise is taken from the article by A. F. Beardon, T. K. Carne and T. W. Ng, « The critical values of a polynomial », Constr. Approx. 18 (2002), 343-354.