{"id":"f47566e0-5d56-4cf9-936d-85cdc256e948","arxiv_id":"2502.07100","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves upper bounds on the number of matrices with entries from a finite subset of a finite-rank multiplicative group that have a given rank, determinant, or characteristic polynomial.","lead":"This paper proves upper bounds on the number of matrices whose entries are drawn from a finite set inside a finite-rank multiplicative group, with a prescribed rank, determinant, or characteristic polynomial. The bounds improve the trivial counting estimates for large entry sets, using deep results on linear equations in multiplicative groups.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.2 proof has an arithmetic gap for even n: the singular-minor case uses a non-sharp bound on #D_{n-1}(A;0), and the simplification at (4.11) is false for even n.","rationale":"The reader's weakest-assumption pick (Lemma 3.2, the extension of [11, Corollary 16] to arbitrary characteristic-zero fields) is not the most load-bearing issue. That extension is quite defensible: for a fixed equation, the coefficients and variables lie in a finitely generated field of characteristic zero, which embeds into C, and the implied constants depend only on the number of variables and the rank of the multiplicative group. The real problem is internal to the proof of Theorem 2.2. The paper's own estimate (4.7) for #D_{n−1}(A;0) is weaker than what its Theorem 2.1 directly yields whenever n−1 is odd, i.e., whenever n is even. Using that weaker bound in the singular-minor case forces the total exponent above the theorem's stated exponent for every even n ≥ 4, and the algebraic simplification in (4.11) is simply wrong for such n. The theorem can be repaired by quoting Theorem 2.1 instead of (4.7) for the singular submatrix count, but as written the proof of a central result is incomplete. This does not affect Theorems 2.1 or 2.4, but Theorem 2.2 is part of the paper's main contribution, so the verdict should be conditional on correcting the proof rather than unconditional acceptance.","tokens_in":17397,"tokens_out":25333,"duration_ms":192098,"concrete_test":"For n = 6, recompute the exponent in the singular-minor case of Theorem 2.2, first with the paper's bound C from (4.7) (giving 23) and then with the sharp bound from Theorem 2.1, i.e., max_{1≤r≤5} δ(5,r) = 22. The total exponent with D = A^6 and E_1 = A^4 is 33 in the first case and 32 in the second, exactly matching the theorem's n^2−⌈(n+1)/2⌉ = 32. This check distinguishes a mere arithmetic slip from a genuine failure of the stated bound.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the proof of Theorem 2.2 for d ≠ 0, when at least one Laplace minor vanishes, the paper bounds the number of possible singular (n−1)×(n−1) submatrices X_{1,1} by C = #D_{n−1}(A;0) ≪ A^{(n−1)^2−⌊(n−1)/2⌋}, citing (4.7). But (4.7) is a weakened version of what Theorem 2.1 gives: for k = n−1 odd, the maximum in the derivation of (4.7) is attained at r = k−1, yielding exponent k^2−k+1+⌊(k−2)/2⌋, which is one less than the value used. Combining the paper's C with D = A^n and the t = 1 case E_1 = A^{n−2} gives total exponent n^2−1−⌊(n−1)/2⌋; for n = 6 this is 33, while the theorem claims n^2−⌈(n+1)/2⌉ = 32. The displayed equality in (4.11), reducing the t = 1 case to A^{n^2−⌈(n+1)/2⌉}, is arithmetically incorrect for even n. Replacing C by the sharper bound coming 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.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":17688,"tokens_out":9115,"duration_ms":67688,"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":[{"comment":"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.","section":"§4.2, proof of Theorem 2.2, especially (4.11)"},{"comment":"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.","section":"§3.2, Lemma 3.2"}],"minor_comments":[{"comment":"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.","section":"Abstract and §1.1"},{"comment":"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.","section":"§2.3"},{"comment":"The display of the characteristic polynomial contains 'T N-1' instead of 'T^{n-1}'.","section":"§1.3"},{"comment":"The sentence 'Once can also ask about...' contains a typo; it should read 'One can also ask about...'.","section":"§5"},{"comment":"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.","section":"§3.2, tightness discussion"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely correct in its main claims, but the proof of Theorem 2.2 as written does not handle even n, and the unproven extension of Lemma 3.2 should be addressed. These are fixable within the scope of the manuscript, but they are load-bearing enough to require a revision rather than a direct acceptance. The self-citation to [11] is not circular, since [11] is an established published result, but the missing proof of the field extension is a genuine gap. I would encourage the editor to consider a revised version favorably once these points are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The paper gets real new bounds—first statistical counts for matrices with entries from finite-rank multiplicative groups, improving trivial counts by polynomial factors in A. And the proof of Theorem 2.2 has a gap for even n that needs fixing before the result can stand as written.\n\nThe main results are genuinely new: bounds for matrices of given rank (Theorem 2.1), given determinant (Theorem 2.2), and given characteristic polynomial (Theorem 2.4, exponent α(n) ~ 3/4 n^2). The techniques adapt Mohammadi–Ostafe–Shparlinski to this setting, with the linear-equation bounds from Bourgain–Garaev–Konyagin–Shparlinski doing the heavy lifting. The paper is careful and the logical structure is mostly sound. The lower-bound constructions for tightness are nice, and the appendix with sharper characteristic-polynomial bounds is a useful addition.\n\nThe soft spot is real. In the d ≠ 0 case of Theorem 2.2, the singular-minor argument uses C = #D_{n-1}(A;0) ≤ A^{(n-1)^2 - floor((n-1)/2)} from (4.7). That bound is not tight for odd n-1; the direct application of Theorem 2.1 gives one less in the exponent. As a result, the displayed bound (4.11) for the t=1 case is arithmetically false when n is even: for n=6 it gives exponent 33, not the claimed 32. The authors seem aware—Remark 4.1 notes that the t=1 case can be eliminated using Proposition A.1, and indeed that proposition kills the bad case. But the proof as written does not use it, so the written proof does not establish the theorem for even n ≥ 4. That is a fixable gap, not a fatal one.\n\nAlso worth flagging: the extension of Lemma 3.2 from C to arbitrary characteristic-zero fields is asserted without proof, which is standard practice but still load-bearing. And there are a few typos—the abstract gives the Alon–Solymosi year as 2003, and the construction in Section 2.3 says #A_k = k when A_k has 2k elements. Neither affects the results.\n\nWho is this for? Arithmetic statisticians and Diophantine geometers who care about matrix counting over structured sets. It deserves a serious referee; the new results are interesting and the flaw is local. I would send it out, but I would insist that the proof of Theorem 2.2 be repaired (or explicitly deferred to Proposition A.1) before publication.","headline":"Solid new bounds for matrices over finite-rank multiplicative groups, but the proof of Theorem 2.2 has a fixable gap for even n.","tokens_in":18236,"tokens_out":7007,"would_cite":true,"duration_ms":52893,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11C20","15B36","60B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"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 $3n^2/4$.","keywords":["matrices over finite rank multiplicative groups","rank","determinant","characteristic polynomial","linear equations in multiplicative groups","Subspace Theorem","matrix counting","characteristic zero fields"],"falsifier":"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.","tokens_in":17162,"feed_emoji":"🔢","tokens_out":16664,"duration_ms":130700,"temperature":0.7,"pith_summary":"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 $\\Gamma$.","feed_headline":"Fixed characteristic polynomial cuts matrix counts to A^{3/4 n^2}","feed_subtitle":"New bounds for rank, determinant, and characteristic polynomial beat trivial A^{n^2} counts","key_machinery":"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.","core_discovery":"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 $\\varrho$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the absolute bound on non-degenerate solutions to linear equations in multiplicative groups, recorded as Lemma 3.1.","marker":"[7]"},{"why":"Provides Corollary 16, quoted as Lemma 3.2, the homogeneous counting bound whose characteristic-zero extension carries the main arguments.","marker":"[11]"},{"why":"Gives the earlier bound on linear equations in multiplicative groups that the paper notes is also suitable for its purposes.","marker":"[23]"},{"why":"The Subspace Theorem is the Diophantine foundation underlying the absolute non-degenerate solution bound.","marker":"[46]"},{"why":"Supplies the row and column elimination method used in the proof of the rank theorem.","marker":"[39]"},{"why":"Motivates fixing only the top two coefficients of the characteristic polynomial, the strategy behind Theorem 2.4.","marker":"[4]"}],"fun_headline_variants":["Characteristic polynomial matrix count: O(A^{3/4 n^2})","Matrix counts for char polynomial bounded by A^{3/4 n^2}","Finite-rank multiplicative groups tame matrix counts","New matrix count bounds for rank, determinant, and char polynomial","Tight matrix count bounds from finite-rank fields"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Characteristic polynomial matrix count: O(A^{3/4 n^2})","Matrix counts for char polynomial bounded by A^{3/4 n^2}","Finite-rank multiplicative groups tame matrix counts","New matrix count bounds for rank, determinant, and char polynomial","Tight matrix count bounds from finite-rank fields"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000912,"raw_usage":{"total_tokens":3920,"prompt_tokens":950,"completion_tokens":2970,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":566,"completion_tokens_details":{"reasoning_tokens":2883}},"tokens_in":566,"tokens_out":2970,"duration_ms":20000,"temperature":1.0,"reasoning_tokens":2883,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T13:50:54.954369+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Amoroso, and E","cited_arxiv_id":null,"evidence_quote":"Supplies the absolute bound on non-degenerate solutions to linear equations in multiplicative groups, recorded as Lemma 3.1."},{"cited_title":"Bourgain, M","cited_arxiv_id":null,"evidence_quote":"Provides Corollary 16, quoted as Lemma 3.2, the homogeneous counting bound whose characteristic-zero extension carries the main arguments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the earlier bound on linear equations in multiplicative groups that the paper notes is also suitable for its purposes."},{"cited_title":"Schmidt, ‘The subspace theorem in diophantine approximation s,” Compos","cited_arxiv_id":null,"evidence_quote":"The Subspace Theorem is the Diophantine foundation underlying the absolute non-degenerate solution bound."},{"cited_title":"Mohammadi, A","cited_arxiv_id":null,"evidence_quote":"Supplies the row and column elimination method used in the proof of the rank theorem."},{"cited_title":"Afifurrahman, V","cited_arxiv_id":null,"evidence_quote":"Motivates fixing only the top two coefficients of the characteristic polynomial, the strategy behind Theorem 2.4."}],"review_version":1}