I think computation of the Euler characteristic of a real variety is not a problem in theory.

There are some nice papers like <em><a href="http://blms.oxfordjournals.org/content/22/6/547.abstract">J.W. Bruce, Euler characteristics of real varieties</a></em>.

But suppose we have, say, a very specific real nonsingular hypersurface, given by a polynomial, or a nice family of such hypersurfaces. What is the least cumbersome approach to computation of $\chi(V)$? One can surely count the critical points of an appropriate Morse function, but I hope it's not the only possible way.

(Since I am talking about dealing with specific examples, here's one:
$f (X_1,\ldots,X_n) = X_1^3 - X_1 + \cdots + X_n^3 - X_n = 0$, where $n$ is odd.)