Exercise 2, Appendix 2

Let V be a real vector space of finite dimension $n$, and F a polynomial function on V of degree $d$. Let $V'$ be the set of points of V at which $F\not= 0$. Show that the number of connected components of $V'$ is finite and bounded by a constant depending only on $n$ and $d$. (Argue by induction on $n$. Reduce to the case in which F has no multiple factors, and show that V can be decomposed as $\mathbf{R}\times X$ in such a way that the results of Exerc. 1 are applicable to $F.$)$^{10}$

$^{10}$For other results in the same direction, cf. J. MILNOR, On the Betti numbers of

real varieties, Proc. Amer. Math. Soc., Vol. XV (1964), pp. 275-280.