Exercise 1, Appendix 2

Let X be a locally connected topological space, $\mathscr{C}(X)$ the space of continuous real-valued functions on X, and $d$ an integer $\geq 0$. Let $F\in \mathscr{C}(X)[T]$ be a monic polynomial of degree $d$ with coefficients in $\mathscr{C}(X):$

Identify F with a function on $\mathbf{R}\times X$ by putting

$F(t, x) =t^d+t^{d-1}f_1(x) +\cdots +f_d(x)$ if $t\in \mathbf{R}, x\in X$.

Let $\Delta \in \mathscr{C}(X)$ be the discriminant of the polynomial F (Algebra, Chap. IV, §1, no. 10).

If U is an open subset of X, denote by $Z_U$ the set of $(t, x)$, with $t\in \mathbf{R}$ and $x\in U$, such that $F(t, x) = 0$; this is a closed subset of $\mathbf{R}\times U$.

a) Show that the projection pr$_2: Z_U\rightarrow U$ is proper (General Topology, Chap. I, §10).

b) Assume that U is connected and that $\Delta (x)\not= 0$ for all $x\in U$. Show that $Z_U\rightarrow U$ is a covering of U (General Topology, Chap. XI) of degree $\leq d$, and that the number of connected components of $\mathbf{R}\times U$-- $Z_U$ is $\leq d+ 1$.

c) Let $X'$ be the set of points of X at which $\Delta$ is $\not= 0$. Assume that $X'$ is dense in X. Denote by $\mathscr{A}$ (resp. $\mathscr{B}$) the set of connected components of $X'$ (resp. of $\mathbf{R}\times X$-- $Z_X$). Show that

Card($\mathscr{B}$)$\leq (d+$ 1)Card($\mathscr{A}$) (use $b$)$)$.

d) Assume that X is connected, and that $d\geq 1$. Show that

Card($\mathscr{B}$)$\leq 1 +d$ Card($\mathscr{A}$).