Manifolds and Polynomials - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T06:40:12Z http://mathoverflow.net/feeds/question/71415 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71415/manifolds-and-polynomials Manifolds and Polynomials Markus Ulke 2011-07-27T15:54:41Z 2011-07-27T17:33:50Z <p>Given a compact smooth manifold $M \subset R^k$ there is a Polynom $f\in R[x_1,..x_n]$ such that the zero set of $f$ is diffeomorphic to $M$. Can the coefficients of $f$ be pertubated slightly to a Polynomial $g \in Q[x_1,..x_n]$ such that the zero set of $g$ is diffeotopic to $M$? Are their conditions on the homology or homotopy on $M$ such that such a pertubation process is possible / not possible? What happens if Q is replaced by an arbitrary number field K?</p> http://mathoverflow.net/questions/71415/manifolds-and-polynomials/71419#71419 Answer by Ben McKay for Manifolds and Polynomials Ben McKay 2011-07-27T17:02:44Z 2011-07-27T17:02:44Z <p>Yes: proven in Ballico, E., Tognoli, A., <em>Algebraic models deﬁned over $\mathbb{Q}$ of diﬀerential manifolds.</em> <strong>Geom. Dedicata</strong> 42 (1992), no. 2, 155–161. In fact, you can get the zero set to be diffeomorphic to $M$, not just diffeotopic.</p>