Counting Reducible Matrices, Polynomials, and Surface and Free Group Automorphisms

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.