Uniform sharp bound for matrices with a prescribed characteristic polynomial

Establish the uniform estimate N_f(T) \ll_{n,\varepsilon} T^{n(n-1)/2+\varepsilon} for every monic polynomial f\in\mathbb{Z}[X] of degree n and every \varepsilon>0, where N_f(T) counts n\times n integer matrices of height at most T with characteristic polynomial f.

Background

For a monic degree-n polynomial f\in\mathbb{Z}[X], the counting function N_f(T) enumerates n\times n integer matrices whose entries have absolute value at most T and whose characteristic polynomial equals f. The paper proves a new uniform upper bound with exponent n^2-2n+3-6/(n+2), improving earlier work for n\geq4, but this exponent remains larger than the conjectured exponent n(n-1)/2.

The conjectured estimate is consistent with known asymptotic results for matrices ordered by Euclidean norm in several settings, including fixed irreducible polynomials and squarefree polynomials that split completely over \mathbb{Q}. The paper establishes the conjecture only in previously known low-dimensional cases and does not resolve it for general n.