Computation of the Euler characteristic of a specific real variety - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T20:28:18Z http://mathoverflow.net/feeds/question/84076 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84076/computation-of-the-euler-characteristic-of-a-specific-real-variety Computation of the Euler characteristic of a specific real variety Lyosha 2011-12-22T07:58:09Z 2012-01-03T21:16:05Z <p>I think computation of the Euler characteristic of a real variety is not a problem in theory.</p> <p>There are some nice papers like <em><a href="http://blms.oxfordjournals.org/content/22/6/547.abstract" rel="nofollow">J.W. Bruce, Euler characteristics of real varieties</a></em>.</p> <p>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.</p> <p>(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.)</p> <p><strong>Update:</strong> 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 $b (V)$ <em>defined by one equation of degree $3$</em>. 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 $4$).</p> http://mathoverflow.net/questions/84076/computation-of-the-euler-characteristic-of-a-specific-real-variety/84095#84095 Answer by Igor Rivin for Computation of the Euler characteristic of a specific real variety Igor Rivin 2011-12-22T14:25:59Z 2011-12-22T14:25:59Z <p>This is quite nontrivial. See for example: </p> <p>On Bounding the Betti Numbers and Computing the Euler Characteristic of Semialgebraic sets, by Saugata Basu (google has full text). </p> <p>The canonical reference is a more recent book by Basu, Ricky Pollack and Marie-Francoise Roy, called "algorithms in real algebraic geometry"</p> http://mathoverflow.net/questions/84076/computation-of-the-euler-characteristic-of-a-specific-real-variety/84107#84107 Answer by Ryan Budney for Computation of the Euler characteristic of a specific real variety Ryan Budney 2011-12-22T18:41:34Z 2011-12-22T18:41:34Z <p>Presumably the least-cumbersome approach will depend on the specific variety you need to work with. </p> <p>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 $n=3$. </p> <p>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 $\chi = 5-2n$. </p> http://mathoverflow.net/questions/84076/computation-of-the-euler-characteristic-of-a-specific-real-variety/84114#84114 Answer by F. C. for Computation of the Euler characteristic of a specific real variety F. C. 2011-12-22T19:37:24Z 2011-12-22T19:37:24Z <p>One possible way to compute Euler characteristic is to use its properties:</p> <ul> <li><p>$\chi$ is additive on disjoint unions</p></li> <li><p>$\chi$ is multiplicative on fibrations</p></li> <li><p>$\chi$ of the point is $1$</p></li> </ul> <p>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.</p> <p>The same kind of procedure can be used to count points over finite fields.</p> http://mathoverflow.net/questions/84076/computation-of-the-euler-characteristic-of-a-specific-real-variety/84836#84836 Answer by Liviu Nicolaescu for Computation of the Euler characteristic of a specific real variety Liviu Nicolaescu 2012-01-03T21:16:05Z 2012-01-03T21:16:05Z <p>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). </p>