2006/04/23 by Igor Rivin, Rivin, Igor
Mathematics · #11G99 #20E05 #57M60 #FOS: Mathematics #Geometric Topology (math.GT) #Number Theory (math.NT) #math.GT #math.NT #msc:11G99 #msc:20E05 #msc:57M60
paper · pdf · doi:10.48550/arxiv.math/0604489
14 pages; fixed some egregious typos and improved notation
arxiv created 2006/04/27 · arxiv updated 2016/09/07
We give upper bounds on the numbers of various classes of polynomials reducible over the integers and over integers modulo a prime and on the number of matrices in SL(n), GL(n) and Sp(2n) with reducible characteristic polynomials, and on polynomials with non-generic Galois groups. We use our result to show that a random (in the appropriate sense) element of the mapping class group of a closed surface is pseudo-Anosov, and that a random automorphism of a free group is strongly irreducible (aka irreducible with irreducible powers). We also give a necessary condition for all powers of an algebraic integers to be of the same degree, and give a simple proof (in the Appendix) that the distribution of cycle structures modulo a prime p for polynomials with a restricted coefficient is the same as that for general polynomials.