On the sparsity of non-diagonalisable integer matrices and matrices with a given discriminant

2023/12/19 by Alina Ostafe, Ostafe, Alina, Igor E. Shparlinski +1
Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #Graph theory and applications #Number Theory (math.NT) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2312.12626

openalex publication_date 2023/12/19 · openalex created_date 2023/12/22 · openalex updated_date 2026/07/28

Abstract

We consider the set \mathcal M_n(\mathbb Z; H) of n× n-matrices with integer elements of size at most H and obtain upper bounds on the number of matrices from \mathcal M_n(\mathbb Z; H), for which the characteristic polynomial has a fixed discriminant d. When d=0, this corresponds to counting matrices with a repeated eigenvalue, and thus is related to counting non-diagonalisable matrices. For d≠ 0, this problem seems not to have been studied previously, while for d=0, both our approach and the final result improve on those of A. J. Hetzel, J. S. Liew and K. Morrison (2007).

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