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

TL;DR

This paper establishes bounds on the reducibility of polynomials and matrices over various fields, and applies these results to properties of random surface and free group automorphisms, revealing their typical pseudo-Anosov and irreducible nature.

Contribution

It provides new upper bounds on reducible polynomials and matrices, and connects these bounds to probabilistic properties of automorphisms in surface and free groups.

Findings

Random surface automorphisms are pseudo-Anosov with high probability.

Random free group automorphisms are strongly irreducible.

Distribution of cycle structures modulo prime matches that of general polynomials.

Abstract

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.