Sankhya: The Indian Journal of Statistics

2000, Volume 62, Series A, Pt. 3, pp. 386--412

COUNTING INTEGRAL MATRICES WITH A GIVEN CHARACTERISTIC POLYNOMIAL

By

NIMISH A. SHAH, *Tata Institute of
Fundamental Research, Mumbai*

*SUMMARY. *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.

*AMS (1991) subject classification. * Primary 22E40;
secondary 11G99, 11P21.

*Key words and phrases. *Lattice
points counting, unipotent flows, uniform distribution, integral
matrices, characteristic polynomial.