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

TL;DR

The paper bounds the sparsity of $n\times n$ integer matrices with entries bounded by $H$ whose characteristic polynomial discriminant is fixed, focusing on $R_n(d,H)$. It establishes a sharp, general upper bound for the zero-discriminant case $d=0$ and uniform bounds for $d\neq 0$, with $H^{(n^2+n-2)/2} \ll N_n(H) \le R_n(H) \ll H^{n^2-\Delta_n}\log H$ and $R_n(d,H) \ll H^{n^2-2+o(1)}$ respectively; special improvements occur for $n=2,5,7,8$. The methods combine modular reductions and finite-field counting for $d=0$ with absolute irreducibility of discriminant shifts and Salberger-type point-count bounds for $d\neq 0$, complemented by a simple lower-bound construction. The results advance understanding of non-diagonalisable matrices and discriminant constraints, with implications for matrix lifting, singular matrix polynomials, and conditioned-number analyses of integral matrix ensembles.

Abstract

We consider the set $\mathcal M_n(\mathbb Z; H)$ of $n\times n$-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 $d\ne 0$, 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).

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Remark 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof