A new arXiv preprint finds that the number of bounded ways to write an integer matrix as a product A B depends sharply on the target. For each fixed n-by-n integer matrix M with nonzero determinant, in dimensions n at least 2, the count grows like a positive constant c_M times T^(floor(n^2/2)) as T, the height limit, becomes large. For the zero matrix, it instead has the form c_{O_n}T^(n^2) + O(T^(n^2-1)), with a positive leading constant. The exponents show that the zero target has a much larger family of bounded factorizations in the large-T limit.
The question behind the formula
The quantity at the center of the work is tau_n(T,M), presented as a matrix version of the divisor question. It counts pairs of n-by-n integer matrices A and B with A B = M, provided both matrices have height at most T. The counted set is the full collection of pairs meeting those conditions, making this an exact enumeration of mathematical objects.
That distinction between target matrices drives the paper's main results. Fixed targets with nonzero determinant receive an asymptotic formula. Targets of rank r, including singular cases, receive a lower bound of a different form, while the zero matrix receives its own full asymptotic. An asymptotic describes the leading growth as T tends to infinity; a lower bound only guarantees that the count reaches at least a certain scale.
The grid behind the answer
The proof turns the factorization problem into lattice-point counting, estimating how many allowed integer points lie in expanding, grid-like regions. Its main counting input is a result of Gorodnik and Nevo for a broad class of mathematical groups. For a nonzero-determinant target, factorizations are organized by the intermediate lattice Lambda = A Z^n, which lies between M Z^n and Z^n. Each such lattice contributes a separate part of the total, allowing the authors to assemble the fixed-target count.
The intermediate-lattice picture makes the determinant-plus-or-minus-one case especially simple. When det(M) = +1 or -1, the only intermediate lattice is Z^n, so the leading factor reduces to 2 kappa(I_n,M), while the power T^(floor(n^2/2)) stays unchanged. The determinant condition therefore changes the constant in front, not the main exponent.
A bound that survives changing the target
The paper then asks how much of this growth can be controlled when M is not fixed. Its uniform theorem says that for any epsilon greater than zero, every integer target with nonzero determinant has tau_n(T,M) bounded above by a constant depending only on epsilon and n, multiplied by T^(floor(n^2/2)+epsilon). The bound is stated for n at least 1. It is an upper bound with an epsilon loss, not a claim that all such targets share one exact asymptotic.
The estimates behind that result keep track of two sources of variation. One bounds the number of intermediate lattices in terms of the absolute determinant of M. Another bounds determinant-one integer matrices that meet two height constraints in terms of the product of those height limits. Together, these estimates control the separate lattice contributions well enough to produce one bound for all nonzero-determinant targets.
Singular matrices split the picture
Singular targets split away from this picture. Grouped by rank r, a measure of how many independent directions remain, an integer target M has at least order T^(floor(r^2/2)+n(n-r)) factorizations, with an implied constant that can depend on M. This theorem supplies a lower bound rather than the fixed-target asymptotic, and the uniform upper-bound theorem is not stated for nonzero singular targets.
The zero case has its own geometry
The zero matrix requires a different route. For A B = O_n, the proof uses the unique primitive lattice containing the rows of A and the corresponding orthogonal lattice containing the columns of B. A primitive lattice here is the full lattice selected by the row span, rather than a smaller sublattice sitting inside it. Counting through these paired lattices gives c_{O_n}T^(n^2) + O(T^(n^2-1)), with c_{O_n} greater than zero for n at least 2. The explicit lower-order term makes this a stronger result than the rank-based lower bound.
A result with clear boundaries
The conclusions are asymptotic, so they describe the eventual regime as T grows rather than a numerical count at one chosen cutoff. The fixed-target theorem does not state an explicit error term, and the leading constants are not generally evaluated numerically in the supplied text. The uniform result likewise remains an upper bound, with no matching asymptotic claimed uniformly in M. The document is an arXiv version 1 preprint dated 25 Aug 2026.
The paper also discloses that generative AI tools were used to suggest proof outlines for Theorems 1.1 and 1.3. The authors state that they independently checked all the arguments and wrote the final text.
Paper data and sources
Original title: The divisor function for matrices
Authors: Tim Browning, Nikita P. Kalinin, Alina Ostafe et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text