{"id":"70a158ea-0af4-4a04-ba6b-697ddbb6df33","arxiv_id":"2608.06576","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For broad classes of random p-adic polynomials and matrices, limiting root and eigenvalue-factor statistics match the Haar coefficient model.","lead":"Random p-adic polynomials with independent, spread-out coefficients have the same limiting root statistics as the uniform Haar model, and the same idea also works for random p-adic matrices. The paper introduces a resultant-based method that may become a standard tool for p-adic universality questions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Matrix theorem's proof depends on unstated hypotheses of the external Cheong-Yu cokernel theorem; if those require i.i.d. entries or restrict the test polynomial, Theorem 1.7 is not established.","rationale":"The central claim of the paper is twofold: polynomial universality (Theorem 1.2) and matrix universality (Theorem 1.7). The polynomial part is self-contained after Caruso's published formula: Theorem 4.1 proves the Fourier-based resultant convergence and Proposition 4.3 proves moment tightness, and I found no internal gap in these arguments. The matrix part, however, is not self-contained. Proposition 5.2 reduces resultant convergence to a weak convergence statement for Cok(Z(A_n)) taken verbatim from [4, Theorem 1.3], and Proposition 5.3 reduces degree tightness to [4, Theorem 1.12] plus [21, Lemma 6.3]. Since the manuscript never states the hypotheses of these external theorems, the proof as written cannot be checked for the exact setting of Theorem 1.7. The most likely point of failure is the entry assumption: Theorem 1.7 says 'independent and epsilon-balanced', which does not require identical distributions, whereas many cokernel universality theorems are stated for i.i.d. entries. A second possible failure is the class of polynomials Z: [4] might require unit constant term or some regularity, while the method needs all Z in P^S including Z=x for S containing 1. The reader's weakest_assumption is exactly this external dependence, so I agree. If the hypotheses of [4] are as broad as needed, Theorem 1.7 holds and the reader's ACCEPT is justified; if not, the matrix half of the paper is unproved. Because the manuscript does not provide the verification, the appropriate verdict conditional on the outcome of the concrete test is CONDITIONAL. I have no independent objection to the polynomial theorem.","tokens_in":34614,"tokens_out":28531,"duration_ms":255921,"concrete_test":"Read [4] and verify: (a) Theorem 1.3 applies to every monic Z in Z_p[x] of positive degree with A_n having independent, possibly non-identically distributed, epsilon-balanced Z_p entries, and yields an almost surely finite limiting module; (b) Theorem 1.12 gives lim_{n->infinity} E[#Sur_{F_p[t]}(Cok(tI_n - B), G)] = 1 for every finite F_p[t]-module G with B having independent epsilon-balanced entries. If either fails, re-prove Proposition 5.2 for the minimal case Z=x under the paper's hypotheses; without [4] the convergence of val(det A_n) is not established, so Theorem 1.7 lacks proof. If both pass, the concern is resolved.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 1.7 and its proof rest entirely on two black-box results from [4]: Proposition 5.2 invokes [4, Theorem 1.3] to assert Cok(Z(A_n)) converges for every monic Z in Z_p[x] when A_n has independent epsilon-balanced entries, and the appendix invokes [4, Theorem 1.12] to get surjection-moment limits for A_n mod p. The paper nowhere states the hypotheses of either theorem. Two concrete mismatches would be fatal: (i) if [4] requires entries to be identically distributed (Theorem 1.7 only assumes independent epsilon-balanced entries, not i.i.d.), then Proposition 5.2 and Proposition 5.3 are not justified; (ii) if [4] restricts the polynomial (e.g., to Z with Z(0) not in pZ_p or to squarefree reductions), then the resultant convergence for arbitrary Z in P^S, including Z=x when S contains 1, is unproved, and with it the uniqueness step via Theorem 3.1. The polynomial theorem has no such dependence; its resultant estimate is proved internally. But the matrix universality claim is exactly the part that would collapse, and no internal check in the manuscript can rule this out because the external theorem is used as an unstated-hypotheses black box.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a 'resultant distribution method' for universality of zeros of random p-adic polynomials. For P_n(x)=ξ_n x^n+...+ξ_0 with independent ε-balanced coefficients in Z_p, it proves (Theorem 1.2) that for every finite étale algebra E=K_1×...×K_m and every clopen U⊂O^{×,new}_E, the expected number of tuples of pairwise non-conjugate roots in U converges to the integral over U of the stabilized Caruso Haar correlation function ρ^{(∞)}_{K_1,...,K_m}. The method consists of three structural theorems: resultant valuations against all test polynomials in P^S determine the law (Theorem 3.1); convergence of all resultant distributions plus degree tightness yields weak convergence of the random polynomials (Theorem 3.2); and, with moment tightness and an almost-surely squarefree limit, one obtains convergence of expected root statistics (Theorem 3.3). The required resultant-distribution convergence (Theorem 4.1) and the moment tightness of the distinguished factors (Proposition 4.3) are proved internally via Fourier analysis and exponential tail bounds. Corollaries give the limiting second moment of the number of roots in Z_p^× and the expected number of roots generating a fixed quadratic extension. The paper also states an application to random matrices (Theorem 1.7): the S-distinguished factors of characteristic polynomials of matrices with independent ε-balanced entries converge weakly to a universal limit, using the cokernel distribution theorem of Cheong–Yu [4].","tokens_in":34831,"tokens_out":20949,"duration_ms":170648,"significance":"The polynomial part of the paper is a substantial and largely self-contained advance: it extends Shmueli's first-order universality for Z_p^× to arbitrary finite extensions of Q_p and to joint root statistics, and it introduces a genuinely new proof technique. Theorems 3.1–3.3 are clean and appear correct, the Fourier estimate (4.2) and the exponential tail bound (4.9) are effective, and the explicit formulas in Corollaries 1.4 and 1.5 provide concrete, checkable predictions. If Theorem 1.7 holds, it would be the first universality result for the eigenvalue structure of p-adic random matrices and would strengthen the heuristic of Ellenberg–Jain–Venkatesh. However, the proof of Theorem 1.7 rests entirely on two external theorems from [4] whose hypotheses are never stated, so the matrix theorem cannot currently be verified from the manuscript. The polynomial theorem, by contrast, does not have this dependency and appears to be solidly established.","major_comments":[{"comment":"The proof of Proposition 5.2 applies [4, Theorem 1.3] as a black box to conclude that Cok(Z(A_n)) converges weakly to a finite Z_p[t]/(Z)-module G_Z, but the hypotheses of that theorem are not stated anywhere in the paper. Theorem 1.7 assumes only that the entries of A_n are independent and ε-balanced, which is a substantially weaker condition than i.i.d. entries; in particular, the definition of ε-balanced in Definition 1.1 constrains only the mod-p residues and not the laws of the entries themselves. If [4, Theorem 1.3] requires i.i.d. entries or imposes additional restrictions on the test polynomial Z (for example Z(0)∉pZ_p or squarefree reduction), then the asserted convergence of val(Res(P_A,Z)) for all monic Z∈Z_p[x] is not established, and with it the resultant-convergence hypothesis in the proof of Theorem 1.7 collapses. The manuscript must either restate the precise statement of [4, Theorem 1.3] and verify each hypothesis for independent ε-balanced entries, or replace this step with a proof that works under the stated hypotheses.","section":"Section 5, Proposition 5.2"},{"comment":"The degree-tightness step for the matrix theorem is proved in the appendix by invoking [4, Theorem 1.12] to assert that lim_{n→∞} E[#Sur_{F_p[t]}(Cok(tI_n - B),G)] = 1 for every fixed finite F_p[t]-module G, where B is the reduction of A_n modulo p. As with Proposition 5.2, the hypotheses of [4, Theorem 1.12] are never stated. The entries of B are independent ε-balanced F_p-valued variables, not necessarily uniform or i.i.d., so the applicability of the theorem cannot be checked from the manuscript. This step is directly responsible for the convergence of the joint degrees of the distinguished factors (Proposition 5.3), which is the tightness input for Theorem 3.2 in the proof of Theorem 1.7. The appendix should state the exact theorem used and either prove the surjection-moment convergence under the ε-balanced hypothesis or verify that the theorem is indeed applicable.","section":"Appendix A, Proposition 5.3"}],"minor_comments":[{"comment":"In the typeset version several words are run together without spaces (for example 'randomp-adic', 'independentϵ-balanced', and 'PolynomialsVia' in the title block); please correct the formatting in the final version.","section":"General typography"},{"comment":"The object d_S is called a 'random variable' but is an element of Z^s_{≥0}; consider calling it a random vector for clarity.","section":"Proposition 5.3"},{"comment":"The reference [23] is cited as 'In preparation'; if its results are used only for motivation, consider stating explicitly in the text that no result of [23] is used in the proofs.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The polynomial theorem (Theorem 1.2 and its corollaries) is, in my reading, sound and well within the scope of the journal; the proofs of the resultant distribution method are self-contained and the explicit corollaries are valuable. The matrix theorem (Theorem 1.7) is plausible but currently unverifiable because the hypotheses of the two Cheong–Yu results are not stated. I recommend major revision rather than rejection because the gap may be fixable by restating and checking the external theorems; however, if the authors find that [4] requires stronger entry assumptions than ε-balanced, then Theorem 1.7 as stated cannot be repaired without additional proof, and the paper would need to be restructured around the polynomial result alone."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe headline: this is a genuinely strong paper on p-adic random polynomials. The resultant distribution method is new and the proof of the polynomial universality theorem is essentially self-contained: Fourier decay for ε-balanced coefficients, resultant convergence to the Haar limit, and exponential moment tightness of the distinguished factor. I checked the main lines and found no internal errors. Theorem 1.2 is a real advance over Shmueli's first-order count and Caruso's Haar-only formulas; the corollaries (second moment in Z_p^×, quadratic extension counts) are correctly derived.\n\nThe matrix part is the soft spot. Theorem 1.7 rests entirely on two black-box invocations of Cheong-Yu's cokernel distribution theorem ([4, Thm 1.3] in Proposition 5.2 and [4, Thm 1.12] in the appendix). The paper never states the hypotheses of either theorem. If Cheong-Yu requires i.i.d. entries, or restricts the test polynomial, Theorem 1.7 as stated is not established. This is not a minor drafting issue: the degree tightness and the resultant convergence for arbitrary monic Z in P^S both depend on exactly those hypotheses. I could not verify them from the manuscript. The polynomial theorem does not share this problem; its key estimates are proved in the text.\n\nThe rest of the paper is careful. The limitation to unit roots is explicit and argued, the open-vs-clopen counterexample is instructive, and the arithmetic in the corollaries checks out. Citation pattern is fine; self-citations are to relevant work.\n\nWho this is for: anyone working on p-adic random polynomials, p-adic random matrices, or cokernel universality. It deserves a serious referee. For the matrix theorem, the referee should ask the author to either state and verify the external theorem's hypotheses, or prove the needed cokernel convergence directly for ε-balanced independent entries.\n\nMy recommendation: send to peer review. The polynomial theorem alone is publishable; the matrix theorem is likely fixable but needs the black box opened.","headline":"The polynomial universality theorem is strong and self-contained via a genuinely new resultant distribution method; the matrix theorem is a real advance but has a black-box gap over the Cheong-Yu hypotheses that the paper never states.","tokens_in":35355,"tokens_out":4782,"would_cite":true,"duration_ms":39633,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","11S15","11S05","15B52"],"pacs":[],"model":"deepseek-v4-flash","headline":"Random $p$-adic polynomials have universal root statistics under a mild non-concentration condition.","keywords":["p-adic random polynomial","universality","resultant distribution method","Haar measure","finite étale algebra","p-adic random matrix","cokernel distribution","characteristic polynomial"],"falsifier":"Simulate or compute, for a fixed prime $p$ and a concrete $\\epsilon$-balanced coefficient distribution that is not Haar (for example uniform on $\\{1,-1\\}$ when $p$ is odd), the limiting expected number of unit roots or of roots generating a fixed quadratic extension. If either limit differs from the formula in Theorem 2.3 or Corollary 1.5 -- equivalently, if the resultant-valuation distribution of $P_n$ against some unit-constant monic test polynomial fails to match the Haar-model distribution -- the main theorem is false. The boundary is also testable: a coefficient distribution supported on a single residue class modulo $p$ violates $\\epsilon$-balancedness and should produce different root statistics, as the paper's Remark 1.3 explains.","tokens_in":34383,"feed_emoji":"🎲","tokens_out":9319,"duration_ms":73697,"temperature":0.7,"pith_summary":"This paper claims that the fine root statistics of random $p$-adic polynomials are universal. For polynomials with independent, sufficiently non-concentrated ('$\\epsilon$-balanced') coefficients taking values in $\\mathbb{Z}_p$, the limiting expected number of root tuples in any clopen region of units inside a finite étale algebra over $\\mathbb{Q}_p$ equals the corresponding statistic of the Haar coefficient model, whose stabilized correlation functions were derived in [3]. The proof rests on a new resultant distribution method: instead of tracking roots directly, one studies the $p$-adic valuations of the resultants of the random polynomial against fixed monic test polynomials; the paper shows these distributions determine the law of the polynomial and, together with degree tightness, force convergence of expected root statistics. The same machinery is then applied to characteristic polynomials of random $p$-adic matrices with independent $\\epsilon$-balanced entries, proving that the distinguished factors of the characteristic polynomial converge to a universal limit independent of the entry distribution. If the theorems are correct, explicit formulas previously available only for Haar-distributed coefficients -- including the second moment of the number of unit roots and the expected number of roots generating a prescribed quadratic extension -- hold for all $\\epsilon$-balanced independent coefficient distributions.","feed_headline":"Random p-adic roots obey Haar statistics for balanced coefficients","feed_subtitle":"New resultant-distribution method makes joint root statistics and matrix eigenvalue factors universal.","key_machinery":"The resultant distribution method. For a finite set $S$ of lifted subspaces of $\\mathbb{Z}_p$, the paper proves (Theorem 3.1) that the law of a random monic polynomial whose roots lie in those subspaces is determined by the distributions of $\\mathrm{val}(\\mathrm{Res}(P,Z))$ for every test polynomial $Z$ from the same space. Theorem 3.2 upgrades this to weak convergence: if resultant valuations converge for every fixed $Z$ and the degrees are tight, the polynomials converge weakly; Theorem 3.3 adds a moment-tightness condition and squarefreeness of the limit to obtain convergence of expected root statistics. The resultant valuations themselves are accessed by a Fourier argument: on the quotient ring $\\mathbb{Z}_p[x]/(Z,p^k)$, $\\epsilon$-balancedness forces a uniform character decay bound (equation (4.2)), so $P_n$ modulo $(Z,p^k)$ becomes asymptotically uniform. For matrices the same valuations are logarithms of cokernel sizes through the identity $\\mathrm{Res}(P_A,Z)=\\det(Z(A))$, so the convergence of resultant distributions follows from the cokernel distribution theorem of [4], while degree tightness in the matrix case is handled via rational canonical forms over $\\mathbb{F}_p$.","core_discovery":"The central discovery is that the limiting joint root statistics of a random $p$-adic polynomial are insensitive to the coefficient distribution beyond a mild non-concentration condition. Theorem 1.2 states that for $P_n(x)=\\xi_n x^n+\\cdots+\\xi_0$ with independent $\\epsilon$-balanced coefficients, for every finite étale algebra $E=K_1\\times\\cdots\\times K_m$ and every clopen $U\\subset \\mathcal{O}^{\\times,\\mathrm{new}}_E$, one has $\\lim_{n\\to\\infty} \\mathbb{E}[Z_U(P_n)] = \\int_U \\rho^{(\\infty)}_{K_1,\\dots,K_m}(x_1,\\dots,x_m)\\,dx_1\\cdots dx_m$, where $\\rho^{(\\infty)}$ is the stabilized correlation function of the Haar coefficient model from [3]. This includes joint statistics over arbitrary finite extensions of $\\mathbb{Q}_p$ and recovers the first-order universality result of [25] when $m=1$, $K_1=\\mathbb{Q}_p$, $U=\\mathbb{Z}_p^{\\times}$. Separately, Theorem 1.7 shows that the $S$-distinguished factors of characteristic polynomials of random $p$-adic matrices with independent $\\epsilon$-balanced entries converge weakly to a universal random polynomial depending only on $p$ and on the finite set $S$ of lifted subspaces, so the eigenvalue structure of such matrices is robust to the entry distribution.","pith_inferences":["The same recipe should extend beyond i.i.d. coefficients: any coefficient sequence whose $p$-adic Fourier characters decay as in (4.2) -- including weakly dependent or exchangeable arrays -- should enjoy the same limiting root statistics; the paper's proofs only use the factorized expectation (4.1) and the character bound.","A natural next step is a quantitative version: explicit rates for resultant convergence (via the contraction factor $\\rho_k$ in (4.2)) and for the moment-tightness bound would convert the qualitative Theorem 1.2 into rates, paralleling the known rate for the first moment in [25].","The paper explicitly leaves open a stronger moment-tightness estimate for random matrices; proving such an estimate would upgrade Theorem 1.7 from weak convergence of distinguished factors to universality of expected eigenvalue statistics such as $\\mathbb{E}[Z_{\\mathbb{Z}_p}(P_A)] = 1$ from the Haar matrix model.","The clopen restriction on $U$ is tight: the paper's own counterexample (Remark 4.5) shows that for open-only or closed-only regions even the Rademacher coefficient model separates from the Haar limit, so universality for root statistics is fundamentally a property of compact-open counting windows."],"forward_implications":["Every clopen-root-statistic formula derived for the Haar coefficient model -- including the second moment of unit roots (Corollary 1.4) and the expected number of roots generating unramified or ramified quadratic extensions (Corollary 1.5) -- holds verbatim for all independent $\\epsilon$-balanced coefficient distributions.","The limiting correlation functions of Haar random polynomials over finite étale algebras, once evaluated explicitly, transfer automatically to the whole $\\epsilon$-balanced class; this includes general finite configurations of distinct unit roots.","For random $p$-adic matrices with independent $\\epsilon$-balanced entries, the part of the characteristic polynomial attached to any fixed finite collection of lifted subspaces has a universal weak limit, so the heuristic random-matrix model of $p$-adic L-function distinguished factors is insensitive to the choice of entry distribution.","The resultant distribution method gives a reusable, model-independent route to $p$-adic root universality: establish resultant-valuation convergence plus (moment) degree tightness, and root-statistic convergence follows from the three general theorems of Section 3."],"supporting_citations":[{"why":"Provides the Fourier-analysis background used to prove the $\\epsilon$-balanced character decay estimate (4.2).","marker":"[2]"},{"why":"Supplies the stabilized Haar-model correlation functions $\\rho^{(\\infty)}$ that are the universal limits in Theorem 1.2.","marker":"[3]"},{"why":"The cokernel distribution theorem that yields convergence of resultant valuations for random matrix characteristic polynomials.","marker":"[4]"},{"why":"The surjection-moment framework used in the appendix for degree tightness in the matrix setting; also the uniqueness/robustness philosophy the resultant method parallels.","marker":"[21]"},{"why":"The author's earlier rational-canonical-form universality over $\\mathbb{F}_p$, which provides the weak convergence of degrees used in Proposition 5.3.","marker":"[22]"},{"why":"The first-order universality result for the expected number of unit roots that Theorem 1.2 extends to joint statistics over finite extensions.","marker":"[25]"}],"fun_headline_variants":["Random p-adic roots follow Haar statistics regardless of coefficients","Resultant distribution method yields universal p-adic root statistics","p-adic matrices: eigenvalue factors universal, entry distribution irrelevant","Coefficient-balanced p-adic polynomials share universal root laws"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"For the matrix theorem, the load-bearing premise is that the external cokernel distribution theorem of [4] applies to polynomial evaluations of matrices whose entries are independent and $\\epsilon$-balanced; the paper does not restate the precise hypotheses of that theorem, and if they are stricter the proof of Theorem 1.7 would not go through. For the polynomial theorem alone, the load-bearing estimate is the $\\epsilon$-balanced Fourier decay bound (4.2), which the paper proves in full.","fun_headline_variants_meta":{"raw":{"variants":["Random p-adic roots follow Haar statistics regardless of coefficients","Resultant distribution method yields universal p-adic root statistics","p-adic matrices: eigenvalue factors universal, entry distribution irrelevant","Coefficient-balanced p-adic polynomials share universal root laws"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000274,"raw_usage":{"total_tokens":1694,"prompt_tokens":1058,"completion_tokens":636,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":674,"completion_tokens_details":{"reasoning_tokens":568}},"tokens_in":674,"tokens_out":636,"duration_ms":5862,"temperature":1.0,"reasoning_tokens":568,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:32:11.243008+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate or compute, for a fixed prime $p$ and a concrete $\\epsilon$-balanced coefficient distribution that is not Haar (for example uniform on $\\{1,-1\\}$ when $p$ is odd), the limiting expected number of unit roots or of roots generating a fixed quadratic extension. If either limit differs from the formula in Theorem 2.3 or Corollary 1.5 -- equivalently, if the resultant-valuation distribution of $P_n$ against some unit-constant monic test polynomial fails to match the Haar-model distribution -- the main theorem is false. The boundary is also testable: a coefficient distribution supported on a single residue class modulo $p$ violates $\\epsilon$-balancedness and should produce different root statistics, as the paper's Remark 1.3 explains.","supporting_citations":[{"cited_title":"Irreducibility of random polynomials of large degree.Acta Mathe- matica, 2019","cited_arxiv_id":null,"evidence_quote":"Provides the Fourier-analysis background used to prove the $\\epsilon$-balanced character decay estimate (4.2)."},{"cited_title":"Where are the zeroes of a randomp-adic polynomial?Forum Math","cited_arxiv_id":null,"evidence_quote":"Supplies the stabilized Haar-model correlation functions $\\rho^{(\\infty)}$ that are the universal limits in Theorem 1.2."},{"cited_title":"Universality of rational canonical form for random matrices over a finite field.arXiv preprint arXiv:2510.16225, 2025","cited_arxiv_id":null,"evidence_quote":"The author's earlier rational-canonical-form universality over $\\mathbb{F}_p$, which provides the weak convergence of degrees used in Proposition 5.3."},{"cited_title":"The expected number of roots over the field ofp-adic numbers.International Mathematics Research Notices, 2023(3):2543–2571, 2023","cited_arxiv_id":null,"evidence_quote":"The first-order universality result for the expected number of unit roots that Theorem 1.2 extends to joint statistics over finite extensions."}],"review_version":2}