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Counting matrices over finite rank multiplicative groups

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Pith's one-line read Matrices with entries from a finite-rank multiplicative group become sharply rarer once rank, determinant, or characteristic polynomial is fixed; for characteristic polynomials the count exponent is asymptotically $3n^2/4$.

desk verdict Solid new bounds for matrices over finite-rank multiplicative groups, but the proof of Theorem 2.2 has a fixable gap for even n. read the letter →

arxiv 2502.07100 v1 pith:MGVMO5HV submitted 2025-02-10 math.NT

Aaron Manning , Alina Ostafe , Igor E. Shparlinski This is my paper · ORCID
classification math.NT MSC 11C2015B3660B20
keywords matricesoverfiniterankmultiplicativegroupsdeterminantcharacteristicpolynomiallinearequationsinSubspaceTheoremmatrixcountingzerofields
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies $n\times n$ (and more generally $m\times n$) matrices whose entries all lie in a finite set $\mathcal A$ drawn from a multiplicative subgroup $\Gamma$ of a field of characteristic zero, where $\Gamma$ has finite rank, meaning it is generated, up to finite torsion, by finitely many elements. It aims to show that imposing a fixed rank, a fixed determinant, or a fixed characteristic polynomial removes most matrices: the number remaining is substantially smaller than the trivial bounds $\mathcal A^{n^2}$, $\mathcal A^{n^2-1}$, and $\mathcal A^{n^2-2}$. The headline result is that for $n\ge 3$, at most $\mathcal A^{\alpha(n)}$ matrices can share a given characteristic polynomial, where $\alpha(n)\sim \frac34 n^2$, and the paper proves analogous savings for rank and determinant. The upshot is that multiplicative structure alone, with no additive assumptions on $\mathcal A$, already yields strong statistical rigidity for matrix ensembles. All bounds are uniform in the target determinant or polynomial, with constants depending only on the matrix dimensions and the rank of $\Gamma$.

What carries the argument

The engine is a family of counting lemmas for linear equations with variables in $\mathcal A$. Lemma 3.2, quoted from [11], bounds homogeneous equations $a_1x_1+\cdots+a_nx_n=0$ by $O(\mathcal A^{\lfloor n/2\rfloor})$; Lemma 3.3 bounds the non-homogeneous version by $O(\mathcal A^{\lfloor(n-1)/2\rfloor})$; Lemma 3.4 bounds the two-equation system $x_1+\cdots+x_n=x_1^2+\cdots+x_n^2=0$ by $O(\mathcal A^{2n/5})$. Each lemma works by decomposing an arbitrary solution into a maximal degenerate subsum, which the homogeneous lemma controls, plus a non-degenerate remainder, which the Subspace Theorem controls absolutely. The matrix theorems then reduce to these equations: rank via row and column elimination, determinant via Laplace expansion, and characteristic polynomial via fixing the trace and the trace of the square.

What would settle it

Search for a characteristic-zero field $K$ and a rank-one multiplicative group $\Gamma$ where the homogeneous equation $x_1+x_2+x_3+x_4=0$ has more than $C\mathcal A^2$ solutions for some $\mathcal A\subset\Gamma$ with $\#\mathcal A\to\infty$ and any fixed $C$. The cleanest test is $K=\mathbb Q_p$ with $\mathcal A=\{p^s:0\le s<N\}$; Lemma 3.2 predicts $O(N^2)$ solutions. If $p$-adic carries or any other field-specific mechanism produce $\omega(N^2)$ solutions, the assertion in Section 3.2 that the lemma extends to every characteristic-zero field is false, and the proofs of Theorems 2.1, 2.2, and 2.4 break at the equations where they invoke it.

Watch

Extended reading notes

Core claim

The paper's central claim is that finite-rank multiplicative structure controls matrix statistics in characteristic zero. For a finite set $\mathcal A$ in a rank-$\varrho$ multiplicative group $\Gamma$, it proves $\#\mathcal R_{m,n}(\mathcal A;r) \ll \mathcal A^{nr+m-r}$ when $2m\le n+r$ and $\mathcal A^{nr+m-r+\lfloor(r-1)/2\rfloor(2m-n-r)}$ otherwise; $\#\mathcal D_n(\mathcal A;d)\ll \mathcal A^{n^2-\lfloor n/2\rfloor}$ for $d=0$ and $\mathcal A^{n^2-\lfloor(n+1)/2\rfloor}$ for $d\neq0$; and, for $n\ge3$, $\#\mathcal P_n(\mathcal A;f)\ll \mathcal A^{\alpha(n)}$ with $\alpha(n)=n(n-1)/2+\max\{\lfloor(n-1)/2\rfloor+\lfloor n(n-1)/4\rfloor,\lfloor n/2\rfloor+\lfloor n(n-1)/4-1/2\rfloor\}$, so that $\alpha(n)/n^2\to 3/4$. For $n=2$, the characteristic-polynomial count is $O(\mathcal A)$ when exactly one of trace or determinant is zero and $O(1)$ when both are non-zero. The bounds are uniform in $d$ and $f$, with constants depending only on $m,n$ and the rank $\varrho$.

Load-bearing premise

The argument depends on Lemma 3.2, quoted from [11], asserting that $a_1x_1+\cdots+a_nx_n=0$ with $x_i\in\mathcal A$ has only $O(\mathcal A^{\lfloor n/2\rfloor})$ solutions; the paper does not prove the claimed extension of this lemma from $\mathbb C$ to arbitrary characteristic-zero fields, and if that extension fails the main theorems have no foundation.

Editorial extensions

If this is right

  • For $n\ge 3$, $\#\mathcal P_n(\mathcal A;f)=O(\mathcal A^{\alpha(n)})$ with $\alpha(n)\sim 3n^2/4$, improving on the trivial $\mathcal A^{n^2-2}$ bound and on the bound inherited from the determinant theorem.
  • The rank bound is tight in the regime $2m\le n+r$: the exponent $nr+m-r$ is attained by taking rows as scalar multiples of the first row.
  • The determinant theorem implies that singular $n\times n$ matrices over $\mathcal A$ number at most $\mathcal A^{n^2-\lfloor n/2\rfloor}$, and that matrices with a fixed nonzero determinant are even fewer.
  • For $n=2$, a characteristic polynomial with nonzero trace and nonzero determinant leaves only $O(1)$ matrices, while exactly one of them zero leaves $O(\mathcal A)$; the only case with the trivial $O(\mathcal A^2)$ count is trace and determinant both zero.
  • Taken together, the characteristic-polynomial bounds imply that matrices $X$ with entries in $\mathcal A$ and $X^k=I_n$ for some positive integer $k$ obey the same upper bounds.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the quoted extension of Lemma 3.2 holds, the same proof scheme would likely transfer to matrices over function fields such as $\mathbb Q(t)$ with a finite-rank multiplicative group; fields of positive characteristic remain open and may require new ideas.
  • Appendix B already refines the characteristic-polynomial exponent depending on whether the top two coefficients vanish, so a natural testable extension is whether fixing further coefficients of $f$, or higher power sums of the entries, forces additional savings below the $3n^2/4$ exponent.
  • For a uniformly random matrix with entries from $\mathcal A$, these counts imply that the probability of any fixed determinant or characteristic polynomial decays like $\mathcal A^{-c n^2}$; the paper does not pursue this stochastic reading, but it follows directly from the uniform bounds.
  • The linear-algebra fact in Appendix A, that a nonsingular matrix with nonzero entries has at most $n-2$ zero minors in any Laplace expansion, is independent of the multiplicative-group setting and may be useful in other determinant-counting problems.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies counting problems for matrices with entries from an arbitrary finite subset A of a finite-rank multiplicative subgroup Γ of a field K of characteristic zero. It proves upper bounds for the number of matrices of a given rank (Theorem 2.1), with a given determinant (Theorem 2.2), and with a prescribed characteristic polynomial (Theorems 2.3 and 2.4). The bounds improve on the trivial exponents A^{nr+mr-r^2}, A^{n^2-1}, and A^{n^2-2}, respectively; for characteristic polynomials the exponent is asymptotically (3/4)n^2. The arguments rely on the Subspace Theorem through bounds on linear equations in multiplicative groups (Lemmas 3.2–3.4). The paper also includes appendices with sharper bounds and a proposition on vanishing minors in Laplace expansions.

Significance. If the results are correct, they constitute a substantial advance in the arithmetic statistics of matrices with entries from multiplicative groups. The uniformity in the prescribed determinant and characteristic polynomial, and the explicit dependence only on the rank ρ, are valuable features. The main theorems give the first nontrivial upper bounds of this type for arbitrary finite subsets of finite-rank multiplicative groups in characteristic zero, complementing recent work on integer and rational matrices. The paper is clearly written and the arguments are structured around a small number of external tools, chiefly the Amoroso–Viada bound and its consequence for homogeneous linear equations. The authors also provide additional refinements in Appendix B, which indicates a careful analysis of the main counting argument.

major comments (2)
  1. [§4.2, proof of Theorem 2.2, especially (4.11)] The proof of the d ≠ 0 case has a gap for even n. In the singular-minor case, the bound C = #D_{n-1}(A;0) is taken from (4.7) as A^{(n-1)^2 - ⌊(n-1)/2⌋}. However, for odd n-1 (i.e., even n), the maximum in the derivation of (4.7) is attained at r = n-2, giving the sharper exponent (n-1)^2 - (n-1) + 1 + ⌊(n-3)/2⌋. Using the weaker (4.7) bound and combining with D = A^n and E_1 = A^{n-2} yields, for t=1, an exponent n^2 - 1 - ⌊(n-1)/2⌋, which for even n exceeds the claimed n^2 - ⌈(n+1)/2⌉ by 1; for example, n=6 gives 33 versus 32. The displayed bound (4.11) is therefore arithmetically incorrect for even n. Replacing C by the sharper bound directly from Theorem 2.1 restores the claimed exponent, so the theorem statement is likely true, but the proof as written does not establish it for even n ≥ 4.
  2. [§3.2, Lemma 3.2] Lemma 3.2 is stated as 'essentially [11, Corollary 16]' and the text asserts that the result, presented in [11] for K = C, 'extends to arbitrary fields of characteristic zero in the natural way', but no proof of this extension is given. Since Lemma 3.2 is used in the proofs of all three main theorems (Theorems 2.1, 2.2, and 2.4), this is a load-bearing input. The extension may indeed be straightforward via Lemma 3.1, but the authors should either provide a proof of the reduction or cite a reference that states the result in this full generality. Without this, the main counting arguments rely on an unproven claim.
minor comments (5)
  1. [Abstract and §1.1] The reference to Alon and Solymosi is dated 2003 in the abstract and introduction, but the bibliography entry [6] is from 2023; please correct the year.
  2. [§2.3] In the lower-bound construction for P2(A_k; T^2), the set A_k = {±2^s : 0 ≤ s < k} has cardinality 2k, not k as stated. This does not affect the exponent but should be corrected.
  3. [§1.3] The display of the characteristic polynomial contains 'T N-1' instead of 'T^{n-1}'.
  4. [§5] The sentence 'Once can also ask about...' contains a typo; it should read 'One can also ask about...'.
  5. [§3.2, tightness discussion] In the construction for odd n = 2k+1, the description of the free variables is a bit terse; the phrase 'x_{2k-1} = x_{2k} = x_{2k+1}' together with the preceding pairing could be clarified to avoid confusion about which variables are free.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the cited [11] lemma is independently published evidence, and the counting derivations do not reduce to their inputs by construction.

full rationale

The central claims of Theorems 2.1, 2.2, and 2.4 are derived from Lemmas 3.1–3.4 by explicit combinatorial counting over matrices, not by assuming the counts being estimated. The load-bearing Lemma 3.2 is quoted from [11, Corollary 16], which has an independent published proof relying on the Amoroso–Viada bound [7]; although one author of the present paper is also an author of [11], this is an external, parameter-free result that does not state or assume any of the present theorems. The paper's assertion that the result extends to arbitrary characteristic-zero fields 'in the natural way' is an unproved extension and therefore a correctness risk, but it is not a circular reduction. The possible arithmetic gap in the proof of Theorem 2.2 for even n, concerning the t=1 case in (4.11), is likewise a proof defect rather than a case of defining a prediction in terms of a fitted input. No equation in the paper is defined in terms of the target cardinality, and no fitted parameter is renamed as a prediction. Accordingly, no circularity is present.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters or invented entities. The results rest on external bounds from the Subspace Theorem literature; the self-cited Lemma 3.2 is an established published result with independent proof.

assumptions (3)
  • standard math Amoroso-Viada non-degenerate solution bound (Lemma 3.1), derived from the Subspace Theorem.
    External deep theorem; the paper quotes the bound for non-degenerate solutions to linear equations over finite-rank multiplicative groups in characteristic zero.
  • domain assumption Lemma 3.2 linear equation bound extends [11, Corollary 16] to arbitrary characteristic zero fields.
    Stated without proof; the extension is asserted as natural. The main theorems depend on it directly.
  • domain assumption Setup: K is a field of characteristic zero and Γ is a finite-rank multiplicative subgroup of K*.
    All theorems are formulated under this setup; A is an arbitrary finite subset of Γ.

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Pith. "Pith review of Counting matrices over finite rank multiplicative groups." pith.science (2026). https://pith.science/paper/MGVMO5HV

@misc{pith2026250207100,
  author       = {Pith},
  title        = {Pith review of: Counting matrices over finite rank multiplicative groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MGVMO5HV}},
  note         = {Machine review of arXiv:2502.07100}
}
abstract

Motivated by recent works on statistics of matrices over sets of number theoretic interest, we study matrices with entries from arbitrary finite subsets $\mathcal A$ of finite rank multiplicative groups infields of characteristic zero. We obtain upper bounds, in terms of the size of $\mathcal A$, on the number of such matrices of a given rank, with a given determinant and with a prescribed characteristic polynomial. In particular, in the case of ranks, our results can be viewed as a statistical version of work by Alon and Solymosi (2003).

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