Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms

  title={Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms},
  author={Igor Rivin},
  journal={Duke Mathematical Journal},
  • Igor Rivin
  • Published 23 April 2006
  • Mathematics
  • Duke Mathematical Journal
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… 
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