distribution of degree of minimum polynomial for eigenvalues of random matrix with elements in finite field - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T17:52:14Z http://mathoverflow.net/feeds/question/20200 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/20200/distribution-of-degree-of-minimum-polynomial-for-eigenvalues-of-random-matrix-wit distribution of degree of minimum polynomial for eigenvalues of random matrix with elements in finite field John Jiang 2010-04-02T23:32:07Z 2010-04-03T13:30:19Z <p>This is an attempt to extend the current full fledged random matrix theory to fields of positive characteristics. So here is a possible setup for the problem: Let $A_{n,p}$ be an $n \times n$ matrix with entries iid taking values uniformly in $F_p$. Then one should be able to find its eigenvalues together with multiplicities, which might lie in some finite extension of the field $F_p$. To ensure diagonalizability, one might even take $A_{n,p}$ to be symmetric or antisymmetric (I am not so sure if that guarantees diagonalizability in $F_p$ but I have no counterexamples either). Now the question is if we associate to each eigenvalue $\lambda$ the degree of its minimal polynomial $d(\lambda)$, then does the distribution of $d(\lambda)$ as $n$ goes to infinite converge to some law upon normalization (say maybe Gaussian)? I am very curious whether others have studied this problem before. Maybe it's completely trivial.</p> http://mathoverflow.net/questions/20200/distribution-of-degree-of-minimum-polynomial-for-eigenvalues-of-random-matrix-wit/20237#20237 Answer by Bjorn Poonen for distribution of degree of minimum polynomial for eigenvalues of random matrix with elements in finite field Bjorn Poonen 2010-04-03T13:30:19Z 2010-04-03T13:30:19Z <p>The survey article</p> <p><a href="http://www.ams.org/bull/2002-39-01/S0273-0979-01-00920-X/S0273-0979-01-00920-X.pdf" rel="nofollow">Jason Fulman, Random matrix theory over finite fields, <em>Bulletin of the AMS</em> <strong>39</strong> (2002), 51-85</a></p> <p>and the references therein should answer your questions to the extent that the answers are currently known. See in particular Example 3 in Section 2.2. Roughly, the distribution of the degrees of the factors of the characteristic polynomial of a random matrix over a finite field is close to the distribution of the degrees of the factors of a random polynomial over the same finite field, which is close to the distribution of the cycle lengths of a random element of a symmetric group.</p>