Why do generic polynomials work in reality? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T07:45:48Z http://mathoverflow.net/feeds/question/28453 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/28453/why-do-generic-polynomials-work-in-reality Why do generic polynomials work in reality? Syed 2010-06-17T02:39:04Z 2010-06-17T12:28:25Z <p>I understand that a generic \$G\$-polynomial \$f(t_1,...,t_n)[X]\$ over field \$k\$ has Galois group \$G\$ over \$k(t_1,...,t_n)\$. And basically any \$G\$ extension of \$k\$ should be generated by a realization of \$f\$.(even a bit stronger but that is not the point here).</p> <p>Now as much as I understand, our motivation for hunting these polynomials is that in real (constructive) life, we would like to plug random elements of \$k\$ into \$t_1,...,t_n\$ and get a \$G\$-extension. However, it's obvious that the definition doesn't guarantee it. For example as a trivial failure, we know that \$X^n + t_1X^{n-1} + \cdots + t_n\$ is generic for \$S_n\$, but not all values for \$t_1, ..., t_n\$ (basically all polynomials) lead to an \$S_n\$-extension. </p> <p>So, basically, my question is this: what is the constructive value of the definition of generic polynomial. Is there any (although I know I'm saying nonsense) high probabilistic/statistic success rate in getting a \$G\$-extension when a random realization is chosen. Is there some kind of definition of "odd" that says those times that we don't get a \$G\$-extension are somehow odd and not normal?</p> http://mathoverflow.net/questions/28453/why-do-generic-polynomials-work-in-reality/28456#28456 Answer by Boyarsky for Why do generic polynomials work in reality? Boyarsky 2010-06-17T03:26:10Z 2010-06-17T03:26:10Z <p>Serre introduced a notion of "thin set" in the \$k\$-rational points of a \$k\$-variety (such as \$k^n\$ viewed as the \$k\$-rational points of affine \$n\$-space, or likewise for a Zariski-dense open locus in affine \$n\$-space over \$k\$, depending on denominators in the coefficients of \$f\$ in your motivating example) as a mild generalization of "nowhere Zariski-dense" precisely to quantify issues related to Hilbert irreducibility, exactly as in your question. So the answer to your question is the concept of thin sets in the \$k\$-rational points of \$k\$-varieties (with \$k\$ an infinite field). See the Wikipedia entry on "thin set" for more specific information and references to the literature. </p> http://mathoverflow.net/questions/28453/why-do-generic-polynomials-work-in-reality/28494#28494 Answer by David Speyer for Why do generic polynomials work in reality? David Speyer 2010-06-17T12:28:25Z 2010-06-17T12:28:25Z <p>Adding unto Boyarsky's answer: Stephen Cohen has given <a href="http://www.ams.org/mathscinet-getitem?mr=628276" rel="nofollow">quantative bounds</a> for how often generic polynomials work. If I've skimmed his paper correctly, when the coefficients are integers chosen from the interval \$[-N, N]\$, the probability that the Galois group comes out wrong is \$O(N^{-1/2} \log N)\$, with an explicitly computable constant which depends on the group and the precise parameterization being used.</p>