On the Sparsity of Nondiagonalizable Integer Matrices and Matrices with a Given Discriminant

We consider the set [Formula: see text] of [Formula: see text]-matrices with integer elements of size at most [Formula: see text] and obtain upper bounds on the number of matrices from [Formula: see text], for which the characteristic polynomial has a fixed discriminant [Formula: see text]. When [Formula: see text], this corresponds to counting matrices with a repeated eigenvalue and thus is related to counting nondiagonalizable matrices. For [Formula: see text], this problem seems not to have been studied previously, while for [Formula: see text], both our approach and the final result improve on those of A. J. Hetzel, J. S. Liew, and K. Morrison [ Amer. Math. Monthly, 114 (2007), pp. 491–499].