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 SLn(R)/SLn(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.

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