We give a simpler proof of an earlier result giving an asymptotic estimate
for the number of integral matrices, in large balls, with a given monic
integral irreducible polynomial as their common characteristic polynomial. The
proof uses equidistributions of 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 estimate
