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

There are some nice papers like J.W. Bruce, Euler characteristics of real varieties.

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.)

Update: the original motivation is the following: the well-known results by Oleĭnik, Petrovskiĭ, Milnor, and Thom give upper bounds on $\chi (V)$ or $b(V) = \sum_i b_i (V)$ that are exponential in $n$. It is easy to see that this is unavoidable, e.g. $(X_1^2 - X_1)^2 + \cdots + (X_n^2 - X_n)^2 = 0$ is an equation of degree $4$ that defines exactly $2^n$ isolated points in $\mathbb{R}^n$. I was interested in specific families of real algebraic sets with large $\chi (V)$ or $b (V)$ defined by one equation of degree $3$. I couldn't find an appropriate reference with such examples and it seems like a proof for such example would require some computations (unlike the case of degree $4$).

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If you can compute the singular cohomology $H^*(X(\mathbb{C});\mathbb{Q})$ of the complex points and the action of conjugation $\sigma$ on it, you can compute the Euler characteristic of the real variety by the Lefschetz fixed point theorem: $\chi(X(\mathbb{R}))=\sum (-1)^i \text{tr}\big(\sigma\big|H^i(X(\mathbb(C))\big)$. – Tom Church Dec 22 '11 at 21:09
@Tom: I am guessing the first step is much easier than the second... – Igor Rivin Dec 22 '11 at 21:53

This is quite nontrivial. See for example:

On Bounding the Betti Numbers and Computing the Euler Characteristic of Semialgebraic sets, by Saugata Basu (google has full text).

The canonical reference is a more recent book by Basu, Ricky Pollack and Marie-Francoise Roy, called "algorithms in real algebraic geometry"

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I am aware of complexity, algorithms, and general upper bounds, but now I need to explore specific examples. Maybe the word "computation" is misleading, but I actually refer to some calculations by hand, not algorithmic procedures. – Lyosha Dec 22 '11 at 17:02
Well, why do you think hand computation is possible in nontrivial cases? – Igor Rivin Dec 22 '11 at 20:13
@Igor, well, it's just wishful thinking. See also the update explaining my somewhat naïve motivation. – Lyosha Dec 22 '11 at 21:21

This is tricky even in the simplest case. Suppose we are given a real polynomial in one real variable. The Euler characteristic of its zero set is equal to the number of real roots (not counted with multiplicity).

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Presumably the least-cumbersome approach will depend on the specific variety you need to work with.

In your case, I'd think of solving for $x_n$ in terms of $x_1,\cdots,x_{n-1}$. There's always at least one solution, and sometimes as many as three. This gives a fairly natural stratification of your variety and you can try to inductively compute the Euler characteristic of the variety as a union of subspaces. I think in your case the Euler characteristic is $3$ when $n=1$, $1$ for $n=2$ and $-1$ for $n=3$.

I'm just doing some quick computations by hand, so they're somewhat heuristic and not guaranteed to be accurate. I imagine a little more work and you could get the general picture, and if the pattern holds it appears that $\chi = 5-2n$.

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One possible way to compute Euler characteristic is to use its properties:

  • $\chi$ is additive on disjoint unions

  • $\chi$ is multiplicative on fibrations

  • $\chi$ of the point is $1$

So one has to either decompose the variety as a disjoint union, or prove that it fibers over some base, and then do the same for the pieces, until one reaches something with known Euler characteristic.

The same kind of procedure can be used to count points over finite fields.

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