User lyosha - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T10:42:50Z http://mathoverflow.net/feeds/user/20101 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86099/projections-of-real-algebraic-curves/86155#86155 Answer by Lyosha for Projections of real algebraic curves Lyosha 2012-01-20T00:52:05Z 2012-01-20T00:52:05Z <p>As already mentioned, the answer is negative for varieties.</p> <p>However, if you are interested in a more general setting when the answer is positive (i.e. a more general class of sets that <em>is</em> closed under projections), you can look at</p> <ul> <li>semialgebraic sets in the case of real closed fields (e.g. $\mathbb{R}$),</li> <li>constructible sets (boolean combinations of zero sets) in the case of algebraically closed fields (e.g. $\mathbb{C}$).</li> </ul> <p>This is related to quantifier elimination. The theory of real closed fields (RCF) and the theory of algebraically closed fields (ACF) admit quantifier elimination (see e.g. <em>David Marker, Model Theory, §3.2, 3.3.</em>), which means that if you take a definable set in RCF or ACF, then its projection is also a definable set. And the definable sets are semialgebraic sets and constructible sets respectively.</p> 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/86494/software-for-combinatorial-algebra-sought Comment by Lyosha Lyosha 2012-01-24T05:36:55Z 2012-01-24T05:36:55Z I think GAP can be useful here. <a href="http://www.gap-system.org/" rel="nofollow">gap-system.org</a> http://mathoverflow.net/questions/84076/computation-of-the-euler-characteristic-of-a-specific-real-variety/84095#84095 Comment by Lyosha Lyosha 2011-12-22T21:21:05Z 2011-12-22T21:21:05Z @Igor, well, it's just wishful thinking. See also the update explaining my somewhat na&#239;ve motivation. http://mathoverflow.net/questions/84076/computation-of-the-euler-characteristic-of-a-specific-real-variety/84095#84095 Comment by Lyosha Lyosha 2011-12-22T17:02:28Z 2011-12-22T17:02:28Z I am aware of complexity, algorithms, and general upper bounds, but now I need to explore specific examples. Maybe the word &quot;computation&quot; is misleading, but I actually refer to some calculations by hand, not algorithmic procedures.