Counting integral matrices with a given characteristic polynomial

Abstract

We give a simpler proof of an earlier result giving an asymptotic estimate for the number of integral matrices in large balls whose characteristic polynomial is a given monic integral irreducible polynomial. The proof uses a result on equidistributions of multi-dimensional polynomial trajectories on SL_n(R)/SL_n(Z) which is a generalization of Ratner's theorem on equidistributions of unipotent trajectories. We also compute the exact constants appearing in the above mentioned asymptotic estimates.