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Galois groups of random integer matrices

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Pith's one-line read This paper proves new upper bounds on the count of integer matrices whose characteristic polynomial has Galois group smaller than $S_n$, including an $O_\varepsilon(T^{n^2-1+\varepsilon})$ bound for primitive fields of large discriminant.

desk verdict A solid contribution with one easily patched lemma; the reject verdict is too harsh. read the letter →

arxiv 2506.06463 v2 pith:XSA2SBJ7 submitted 2025-06-06 math.NT

Theresa C. Anderson , Evan M. O'Dorney This is my paper
classification math.NT MSC 11R3211R4511R29
keywords randomintegermatricesGaloisgroupsnon-genericgroupconjecturelargesieveFourier-analyticgeometriccharacteristicpolynomialsnumberfielddiscriminants
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

This paper counts how often the characteristic polynomial of an integer matrix fails to have the full symmetric group as its Galois group. The paper conjectures that the number $M_n(T)$ of such $n\times n$ matrices with entries in $[-T,T]$ satisfies $M_n(T) \asymp T^{n^2-n+1}\log T$, matching the known lower bound from matrices with an integer eigenvalue. It proves $M_n(T) \ll T^{n^2-1/2}\log T$ by the large sieve, and then uses Fourier and geometric sieves to sharpen this for three special classes: matrices with a low-degree irreducible factor in their characteristic polynomial, matrices preserving a lattice generated by short vectors, and matrices whose characteristic polynomial defines a primitive number field of discriminant at least $T^2$. The main new result is the last one, $L_n(T) \ll_\varepsilon T^{n^2-1+\varepsilon}$, obtained with a double-discriminant construction. If the conjecture is right, the dominant exceptional matrices are the reducible ones, and the paper's bounds move the problem close to that target for several natural subfamilies.

What carries the argument

The carrying objects are four parallel sieve mechanisms. The large-sieve irreducibility theorem treats the characteristic polynomial as a family $F(x,Y_1,\ldots,Y_{n^2})$ and bounds the matrices whose Galois group drops below $S_n$ by $O(T^{n^2-1/2}\log T)$. The Fourier sieve for reducible matrices uses Poisson summation over $\mathbb{F}_p$, with a selector $\Psi_g$ that detects an irreducible degree-$k$ factor $g$ modulo $p$; the support of its Fourier transform lies on rank $\le k$ matrices, and balancing the rank-$r$ contributions at $p\asymp T^{(2n-1)/(n+k-1)}$ yields Theorem 1.4. The lattice argument uses reduced bases $v_1,\ldots,v_k$ with lengths $\ell_i$, the entry bound $|g_{ij}|\ll T\ell_j/\ell_i$, and a pinch-point analysis to show the number of possible restrictions $G$ is at most $T^{kn-n+1}$, before dividing by the covolume of $\Lambda^\perp\otimes \mathbb{Z}^n$. The geometric sieve for primitive fields uses the index of a polynomial modulo $p$, $\operatorname{ind} f=\sum_i(e_i-1)\deg f_i$; $p^k\mid D$ forces $\operatorname{ind}(\chi_A\bmod p)\ge k$, the fraction of polynomials with index at least $k$ is $O(p^{-k})$, and finite-field equidistribution of characteristic polynomials transfers this to matrices. The double discriminant then makes $\operatorname{disc}\chi_A\equiv DD(A)\equiv 0\pmod C$, with $C=\prod_{p\mid D}p$.

What would settle it

Take $b=5$ and the companion matrix of $x^3-x-5$; it defines a primitive cubic field with discriminant $D=671=11\cdot 61$, so $D\ge 25$ and the matrix is counted by $L_3(T)$ for $T=5$, yet neither $11$ nor $61$ divides $D$ twice. Carrying out the congruence count with $C=671$ tests whether the double-discriminant conditions can still give the $O_\varepsilon(T^{n^2-1+\varepsilon})$ bound; if the count needs a squarefullness factor that Lemma 6.1 was supposed to supply, the proof of Theorem 1.6 fails as written.

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Extended reading notes

Core claim

The core claim is that non-generic Galois groups of integer matrices can be controlled by sieving the matrices according to how their characteristic polynomials factor modulo primes. Theorem 1.3 gives $M_n(T) \ll T^{n^2-1/2}\log T$. Theorem 1.4 bounds the count $R_{n,k}(T)$ of matrices whose characteristic polynomial has an irreducible factor of degree $k\le n/2$ by $T^{n^2-(n-k)/(n+k-1)}$. Theorem 1.5 gives $S_{n,k}(T)\ll T^{n^2-n+1}\log T$ for matrices preserving a lattice generated by vectors of length at most $T$, with irreducible restriction and short orthogonal complement. Theorem 1.6 bounds the contribution of primitive number fields with discriminant $D\ge T^2$ by $O_\varepsilon(T^{n^2-1+\varepsilon})$; its proof introduces the double discriminant $DD(A)=\operatorname{disc}_t \operatorname{disc}_x \det(xI-tS-A)$ with $S=\operatorname{Diag}(1,2,\ldots,n)$, which is invariant under translation by multiples of $S$ and lets the sieve impose simultaneous congruence conditions modulo $C=\prod_{p\mid D} p$.

Load-bearing premise

The load-bearing premise is Lemma 6.1 in Section 6, which asserts that every prime dividing the discriminant $D$ of a primitive number field $K_A$ divides $D$ to at least the second power (squarefull $D$), forcing the index of $\chi_A$ modulo $p$ to be at least $2$ and making the sum over $D\ge T^2$ converge; this assertion is false, since the primitive cubic field defined by $x^3-x-1$ has discriminant $-23$.

Editorial extensions

If this is right

  • For every fixed $n$, $M_n(T)\ll T^{n^2-1/2}\log T$, improving the trivial count of all matrices, $T^{n^2}$, by a factor $T^{1/2}/\log T$.
  • Reducible matrices with an irreducible factor of degree $k\le n/2$ obey $R_{n,k}(T)\ll T^{n^2-(n-k)/(n+k-1)}$, beating the large-sieve bound when $k\le n/3$ and the previous bound when $k\gg\sqrt n$.
  • Matrices preserving a lattice generated by short vectors are counted at the conjectured main order $T^{n^2-n+1}\log T$, the same order as matrices with an integer eigenvalue.
  • Primitive number fields with discriminant at least $T^2$ contribute at most $O_\varepsilon(T^{n^2-1+\varepsilon})$, so any asymptotic main term must come from reducible characteristic polynomials or small-discriminant fields.
  • The lower bound of Conjecture 1.2 is supplied by integer eigenvalues and already includes the factor $\log T$, showing that the logarithm is essential rather than an artifact.

Reading between the lines

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

  • An extension left implicit: if the squarefull-discriminant lemma is replaced by a weaker index condition, the double-discriminant sieve would apply to all primitive fields, and to any matrix ensemble admitting a diagonal direction whose characteristic polynomial is separable modulo every large prime.
  • The short-vector restriction in Theorem 1.5 marks the current boundary of the method; progress on large-value estimates for matrix products would extend the $T^{n^2-n+1}\log T$ bound to all reducible matrices with an irreducible factor of degree $k$.
  • A uniform bound of the shape $M_n(T;f)\ll T^{n(n-1)/2+o(1)}$ for the number of matrices with a fixed characteristic polynomial $f$ would complete the small-discriminant half of the geometric sieve and imply Conjecture 1.2.
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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

1 major / 4 minor

Summary. The paper studies $M_n(T)$, the number of integer $n\times n$ matrices with entries bounded by $T$ whose characteristic polynomial has Galois group not equal to the full symmetric group $S_n$. The authors conjecture $M_n(T) \asymp T^{n^2-n+1}\log T$ and prove several bounds toward it. Theorem 1.3 uses the Cohen--Serre large sieve to obtain $M_n(T) \ll T^{n^2-1/2}\log T$. Theorem 1.4 bounds the number of matrices whose characteristic polynomial has a low-degree irreducible factor by $T^{n^2-(n-k)/(n+k-1)}$. Theorem 1.5 gives the conjectured exponent $T^{n^2-n+1}\log T$ for a restricted class of matrices preserving a lattice of short vectors. Theorem 1.6, the most novel result, claims that the number of matrices whose characteristic polynomial defines a primitive number field of discriminant $\ge T^2$ is $O_\varepsilon(T^{n^2-1+\varepsilon})$, using a geometric sieve with a double-discriminant construction. The proof of Theorem 1.6 relies on Lemma 6.1, which asserts that every primitive number field has squarefull discriminant.

Significance. If Theorem 1.6 holds, it is a significant step toward Conjecture 1.2, giving a geometric-sieve power saving for non-generic Galois groups of integer matrices. The paper is clearly written and makes good use of recent techniques from Bhargava's proof of van der Waerden's conjecture, as well as work of Katznelson, Reiner, and Gerstenhaber. Theorems 1.3--1.5 appear sound and are useful contributions. However, the pivotal Theorem 1.6 has a load-bearing gap: Lemma 6.1 is false as stated, and the proof of Theorem 1.6 depends on it through the inference that every prime divisor $p$ of the field discriminant has $p^2\mid D$ and hence that the index of the characteristic polynomial mod $p$ is at least $2$. This is a correctness issue that must be repaired before the paper's main new claim is supported.

major comments (1)
  1. [Section 6, proof of Theorem 1.6 (use of Lemma 6.2)] Lemma 6.1 states that if a prime $p$ divides the discriminant $D$ of a primitive number field, then $p^2\mid D$. This is false as stated: the cubic field defined by $x^3-x-1$ is primitive and has $D=-23$, so $23\mid D$ but $23^2\nmid D$. The proof of Theorem 1.6 uses this lemma to conclude that $\operatorname{ind}(\chi_A\bmod p)\ge 2$ for every $p\mid D$, which then yields the double-discriminant congruence $p\mid DD(A)$. Since $L_n(T)\subseteq M_n(T)$, for matrices counted by $L_n(T)$ the Galois group $G_A$ is a proper primitive subgroup of $S_n$; the intended version of the lemma should be stated with this hypothesis and proved using Jordan's theorem (a primitive subgroup containing a transposition is $S_n$) together with the local ramification fact that $p\|D$ would force the inertia group at $p$ to be generated by a transposition. As written, the lemma is false and the proof of Theorem 1.6 is formally invalid. This is the central issue of the paper.
minor comments (4)
  1. [Abstract] The abstract contains a grammatical error: 'We study the number $M_n(T)$ be the number...' should be 'We study the number $M_n(T)$ of...'.
  2. [Section 6, Lemma 6.1] The citation '[Bha25], remark following Proposition 11' should be checked: the stated claim does not appear to be a correct reflection of Bhargava's remark, and the authors should provide a self-contained proof of the corrected lemma under the proper hypotheses.
  3. [Section 4, proof of Theorem 1.4] The sentence 'Note that the loss due to those $A$ for which $\Psi_g>1$ is of low order' is vague; the authors should either quantify this loss or provide a reference for the equidistribution statement that justifies neglecting the multiple-counting.
  4. [Section 5, Example 5.1] The display of the matrix shape in Example 5.1 is visually confusing because some rows and columns are merged; clarifying the entries with explicit zero and nonzero blocks would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main bounds are derived from external theorems, with self-citations only supplying techniques and a technical lemma, not the target results.

full rationale

The paper's central estimates are obtained from independent external results: Theorem 1.3 from the Cohen-Serre large-sieve theorem, Theorem 1.4 from Fourier analysis plus Reiner-Gerstenhaber equidistribution, Theorem 1.5 from Katznelson and Schmidt lattice-counting, and Theorem 1.6 from Bhargava's geometric-sieve framework. No parameter is fitted to the quantity being bounded, and no 'prediction' is equivalent to an input by construction. The self-citations [AGLO+23] and [ABO24] are used for methodology and for Lemma 6.2, a technical index-to-discriminant implication; this is a normal citation of prior proved work, not an import of the conjecture being studied. The paper does contain a serious mathematical error: Lemma 6.1 asserts that every primitive number field has squarefull discriminant, which is false, as the cubic field x^3 - x - 1 has discriminant -23. This is a correctness flaw, not circular reasoning. Moreover, in the actual counting set L_n(T) the Galois group is a proper primitive subgroup of S_n, and Jordan's theorem may supply the missing 'no transposition' hypothesis, so the flaw is localized and plausibly repairable. Because the derivation chain itself is self-contained apart from standard citations, the circularity score is low despite the lemma's invalidity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The estimates depend on standard external theorems (Cohen-Serre, Katznelson, Reiner-Gerstenhaber) and on Bhargava's index lemmas. The only genuinely ad hoc input is the false squarefullness lemma. There are no fitted numerical parameters and no invented entities.

assumptions (5)
  • standard math The characteristic polynomial of the generic matrix has Galois group S_n over Q(Y).
    Used without proof to apply Cohen-Serre in Theorem 1.3; true because every monic polynomial is a characteristic polynomial, but not shown in the paper.
  • standard math Katznelson's singular matrix count: the number of integer matrices in T times a convex body with determinant zero is a constant times T^{n^2-n} log T plus O(T^{n^2-n}).
    Used in Theorem 2.1 for the lower bound and later for the lattice determinant sum.
  • standard math Reiner-Gerstenhaber equidistribution: for any monic polynomial f over F_q, the number of matrices with characteristic polynomial f is q^{n^2-n}(1+O(1/q)).
    Invoked in Theorems 1.4 and 1.6 to sieve matrices by characteristic polynomial modulo p.
  • standard math If p^k divides the field discriminant D, then the index of the characteristic polynomial modulo p is at least k.
    Cited to Bhargava and to [ABO24]; needed to connect reduction modulo p to the field discriminant.
  • ad hoc to paper The discriminant D of a primitive number field is squarefull.
    False; the polynomial x^3 - x - 1 defines a primitive cubic field with D = -23. The proof of Theorem 1.6 relies on this assumption.

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Cite this review

Pith. "Pith review of Galois groups of random integer matrices." pith.science (2026). https://pith.science/paper/XSA2SBJ7

@misc{pith2026250606463,
  author       = {Pith},
  title        = {Pith review of: Galois groups of random integer matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XSA2SBJ7}},
  note         = {Machine review of arXiv:2506.06463}
}
abstract

We study the number $M_n(T)$ be the number of integer $n\times n$ matrices $A$ with entries bounded in absolute value by $T$ such that the Galois group of characteristic polynomial of $A$ is not the full symmetric group $S_n$. One knows $M_n(T) \gg T^{n^2 - n + 1} \log T$, which we conjecture is sharp. We first use the large sieve to get $M_n(T) \ll T^{n^2 - 1/2}\log T$. Using Fourier analysis and the geometric sieve, as in Bhargava's proof of van der Waerden's conjecture, we improve this bound for some classes of $A$.

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