{"id":"a9a69c61-0df5-47ae-ba8d-07bbfd28eafb","arxiv_id":"2505.02723","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Root separations of random Kac polynomials, scaled by n^{5/4}, converge to a Poisson process with intensity c_* t^3 dt.","lead":"This paper proves that the distances between roots of a random Kac polynomial, scaled by the factor n^{5/4}, converge to a Poisson point process with intensity proportional to t^3. It also derives the limit law for the smallest gap between roots and shows that random Taylor series almost surely have no double zeros away from the origin.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Net event A_z(U) has the Newton-step sign reversed; the Poisson-limit proof and the intensity computation as written refer to different events.","rationale":"The reader flagged the missing real-coefficient assumption and the mismatch in the definition of X_n(U). Those are legitimate statement-level issues. However, the most load-bearing problem in the proof as written is the sign in the net event A_z(U): the displayed condition f_n/f'_n - z ∈ R°_z places the Newton step near 2z, whereas Claim 5.5, Claim 5.6, Lemma 5.7, the lower-bound argument in Proposition 5.3, and the Gaussian computation in Section 10 all require z - f_n/f'_n ∈ R°_z, i.e. f/f' near 0. This is not a cosmetic typo: with the printed sign, A^+_z ∩ G is empty for large n, so the moment computation in Section 6 and the Gaussian intensity computation in Section 10 cannot be computing probabilities of the same events. The central claim may well be true, and the issue is very likely fixable by changing the sign in (50) and (67), but the proof as written has a genuine gap at the heart of the net argument. I therefore keep the reader's conditional verdict rather than accepting the argument as it stands. I disagree with the reader's choice of weakest assumption because the more fundamental obstruction is the event mismatch, not the real-versus-complex wording; the real-coefficient issue can be resolved by adding an assumption, while the sign mismatch breaks the internal logic of Sections 5, 6, and 10.","tokens_in":66360,"tokens_out":19682,"duration_ms":246995,"concrete_test":"Verify the sign by deriving the bound in Lemma 6.2 directly from the printed definition (67): on A^+_z ∩ G, the condition f_n/f'_n - z ∈ R♯_z implies |f_n(z)| = |f_n/f'_n| · |f'_n(z)|, and with |f'_n(z)| ≥ n^{5/4}/log n and R♯_z centered at z this gives |f_n(z)| ≳ n^{5/4}/log n, contradicting both the claimed bound |f_n(z)| ≤ 2 n^{-β} log^2 n and the G bound |f_n(z)| ≤ n^{1/2} log^2 n. Then check Claim 5.6, whose hypothesis is z - f_n/f'_n ∈ R_z: if this is the intended Newton condition, the displayed event in (50) must be changed accordingly before Proposition 5.3 can connect X_n(U) to the net count.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The net event A_z(U) in (50), repeated in (67), is displayed as f_n(z)/f'_n(z) - z ∈ R°_z. Since R°_z is a rectangle centered at z (Definition 5.1), this condition says the Newton step f/f' lies near 2z. Every subsequent use of the event requires the opposite sign: Claim 5.5 states z - f_n(z)/f'_n(z) ∈ R♯_z, Claim 5.6 proves existence of a root from z - f/f' ∈ R_z, and Lemma 5.7 plus the lower bound in Proposition 5.3 use z - f/f'. With the printed sign, the event is incompatible with the 'good' event G: G gives |f_n(z)| ≤ n^{1/2} log^2 n, while the printed A_z(U) together with |f'_n(z)| ≥ n^{5/4}/log n forces |f_n(z)| = |f_n/f'_n| · |f'_n| ≈ 2|z| n^{5/4}/log n, a contradiction for large n. Moreover, the Gaussian computation in Section 10, especially (136), conditions on F_n(w)/F'_n(w) lying in a small rectangle around 0, which corresponds to f/f' near 0, i.e. to z - f/f' ∈ R°_z, not to the displayed event. Thus, as written, Proposition 5.3 and the moment computation in Section 6 establish Poisson limits for two different counting events; the reduction to the net is valid only after the sign in (50)/(67) is corrected.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Kac polynomials with i.i.d. mean-zero sub-Gaussian coefficients and proves that the set of n^{5/4}-scaled pairwise root distances converges vaguely to a non-homogeneous Poisson point process on R_{\\ge 0} with intensity c_* t^3 dt, for an explicit positive constant c_*. The proof reduces the problem to net counts in the annulus near the unit circle, uses small-ball probability estimates for (f_n, f'_n, f''_n) at 'smooth' points, compares these statistics with Gaussian ones for separated tuples, and computes the intensity by a Gaussian calculation. A separate result (Theorem 1.3) states that a random Taylor series with log-integrability has no double zeros inside the unit disk almost surely, except possibly at the origin with probability P[a_0=0]^2.","tokens_in":66644,"tokens_out":14107,"duration_ms":159360,"significance":"If the identified issues are corrected, this is a substantial universality result: root separation is one of the few genuinely pairwise quantities of random polynomials for which a full distributional limit is obtained beyond Gaussian models, and the explicit form c_* t^3 yields a limit law for the minimal separation (Corollary 1.2). The proof is a coherent chain of reductions, and Theorem 1.3 is self-contained and of independent interest. The intensity c_* is defined by an explicit positive integral rather than fitted, and the small-ball/Gaussian-comparison machinery is developed in detail. The current version, however, contains a sign inconsistency in the central event and a mismatch in the definition of the counting statistic; these are correctable but load-bearing.","major_comments":[{"comment":"The events A_z(U), A^+_z(U), and A^-_z(U) are defined with f_n(z)/f'_n(z) - z in R^\\circ_z or the analogous rectangles, but every root-locating statement in Section 5 requires the opposite sign: Claim 5.5 concludes z - f_n(z)/f'_n(z) in R^\\sharp_z, while Claim 5.6 and Lemma 5.7 use z - f_n(z)/f'_n(z) in R_z. On the good event G from (53) together with |f'_n(z)| \\ge n^{5/4}/\\log n, the printed event forces |f_n(z)| = |f_n(z)/f'_n(z)| |f'_n(z)| \\asymp 2|z| n^{5/4}/\\log n, contradicting the bound |f_n(z)| \\le n^{1/2} \\log^2 n. Moreover, the Gaussian computation in Section 10, especially (136), conditions on F_n(w)/F'_n(w) lying in a small rectangle around 0, which is the event z - f_n(z)/f'_n(z) \\in R^\\circ_z and not the displayed event. Thus, as written, Proposition 5.3 and the moment computation in Section 6 establish Poisson limits for two different counting events; the reduction to the net is valid only after the sign in (50), (55), (56), (67), and (133) is corrected.","section":"Section 5, Eqs. (50), (55), (56); see also (67) and (133)"},{"comment":"The definition of X_n(U) in (23) counts the number of roots \\alpha \\in \\Omega_K that have some root \\alpha' at distance in n^{-5/4}U, so each close pair contributes two to X_n(U). The proof of Proposition 5.3 and the proof of Theorem 1.1 treat X_n(U) as the number of unordered pairs of roots at such a distance: X^\\pm_n(U) in (57) sum over unordered pairs, and the proof of Theorem 1.1 identifies \\mu^K_n(U) with X_n(U). These two quantities differ by a factor of two. Since the intensity c_*(K) is defined and computed for the pair count (see Claim 6.6 and the factor 2^{-m} in (84)), the statement of Theorem 3.2 is inconsistent with definition (23). The definition should be corrected to count unordered pairs, or all prefactors and the limiting intensity must be rescaled consistently.","section":"Section 3, Eq. (23); proof of Theorem 1.1"},{"comment":"Theorem 1.1 does not state that the coefficients are real-valued, but the proof relies on this: Section 3 begins with the symmetry statement for real coefficients and defines X_n(U) using only roots in the upper half-plane H, and the proof of Theorem 1.1 uses only roots in H. Without the real-coefficient assumption, the symmetry argument and the identification of the limiting process with distances between H-roots fail. Either add the real-coefficient hypothesis to Theorem 1.1, the abstract, and Corollary 1.2, or provide an argument that covers complex coefficients.","section":"Theorem 1.1 and Section 3"}],"minor_comments":[{"comment":"The subscripts in the four sums in the proof are inconsistent with the definitions in (34) and (35): the sums labelled N^{(1)}_s, N^{(2)}_s, and N^{(3)}_s should be over the non-smooth parts N^{(1)}_{ns}, N^{(2)}_{ns}, and N^{(3)}_{ns}.","section":"Proof of Lemma 4.3"},{"comment":"The sentence 'Since f_n has real coefficients, f_n(z) = f_n(z)' should read f_n(z) = \\overline{f_n(\\bar z)}; as printed it is either a typographical error or an empty identity.","section":"Section 3"},{"comment":"The abstract states only that the coefficients are independent and identically distributed, omitting the mean-zero, sub-Gaussian, and real-coefficient hypotheses used in Theorem 1.1; the abstract should state the assumptions that are actually proved.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The two main technical problems are localized definitional errors rather than a broken overall strategy: the Newton-step sign in the central event and the mismatch between counting roots and counting root pairs. Both are fixable, but they affect the main theorem as stated, so the paper should not be accepted until the events, the counting statistic, and the intensity factor are made consistent and the real-coefficient assumption is stated explicitly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Core result: a Poisson limit for the n^{-5/4}-scaled distance spectrum of roots of Kac polynomials with sub-Gaussian coefficients, with explicit intensity c_* t^3 dt, plus the minimal-separation law. That settles a problem suggested in [FS17] and is new even for Gaussian coefficients. The proof is genuinely substantial: local Gaussian comparison off the unit circle, inverse Littlewood–Offord estimates, a new almost-sure no-double-zero theorem for random Taylor series, and an honest integral formula for the constant, with no fitted parameters. Credit where it is due.\n\nThe stress-test note is correct and it matters. The event A_z(U) in (50)/(67) is written as f_n(z)/f'_n(z) - z in R°_z. Every subsequent use needs the opposite sign: Claim 5.6 and Lemma 5.7 use z - f_n(z)/f'_n(z) in R_z, and the Section 10 Gaussian computation conditions on F_n(w)/F'_n(w) near 0, which is exactly z - f_n(z)/f'_n(z) in R_z. With the printed sign, the event is empty on the good event G: |f_n| ≤ n^{1/2} log^2 n and |f'_n| ≥ n^{5/4}/log n force |f_n/f'_n| ≈ |z| ≈ 1, contradicting the required near-z condition. So the net count and the intensity computation as written refer to different events. This is a typo, not a structural gap—the intended event is clear from context—but it sits at the center of the proof, and the manuscript needs the correction before a referee can verify it.\n\nMinor issues: Theorem 1.1 omits the real-coefficient assumption used in the upper-half-plane reduction; X_n(U) in (23) is formally a count of roots but the proof and prose treat it as a pair count; small typos in Section 4.3. All minor.\n\nThis paper is for specialists in random polynomials, small-ball estimates, and universality. It deserves a serious referee: fix the sign, then check the technical lemmas, especially the Gaussian comparison in Section 9 and the intensity computation in Section 10. My own reading says the main claim is true.","headline":"A substantial paper that likely resolves the n^{-5/4} root-separation problem for sub-Gaussian Kac polynomials, but the printed event A_z(U) has a sign error that makes the proof as written inconsistent until fixed.","tokens_in":67207,"tokens_out":5658,"would_cite":true,"duration_ms":59257,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G55","60F05","30C15","30B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"At scale $n^{-5/4}$, the pairwise gaps among roots of a random polynomial converge to a Poisson process with an explicit cubic intensity.","keywords":["random polynomials","Kac polynomials","root separation","Poisson point process","small ball probability","universality","double roots","Littlewood polynomials"],"falsifier":"Run the theorem's own simulation: for Rademacher or standard-Gaussian coefficients at degree $n=10^4$, sample the unnormalized pair-distance process and check that the expected number of pairs with $n^{5/4}|\\alpha_j-\\alpha_j'|\\le s$ grows like $c_*s^4/4$ and that $P[n^{5/4}m_n\\ge s]$ converges to $\\exp(-c_*s^4/4)$; any statistically clear deviation from the quartic power law refutes the $t^3$ intensity and the minimal-gap corollary. A second test: replace real coefficients with i.i.d. complex Gaussians and see whether the same Poisson limit survives, which would settle whether the real-coefficient conjugation step in Section 3 is essential.","tokens_in":66121,"feed_emoji":"📏","tokens_out":19832,"duration_ms":227947,"temperature":0.7,"pith_summary":"This paper establishes a precise limit law for how close the roots of a random polynomial can come to one another. For a Kac polynomial $f_n(z)=\\sum_{k=0}^n \\xi_k z^k$ whose coefficients are independent, identically distributed, mean-zero, and sub-Gaussian with no atom at zero, the paper proves that the collection of pairwise root distances, magnified by $n^{5/4}$, converges in distribution to a non-homogeneous Poisson process on the nonnegative reals with intensity $c_* t^3\\,dt$ for an explicit positive constant $c_*$. The exponent $5/4$ is the paper's central quantitative finding: it encodes the repulsion between roots, which pushes the natural close-pair scale from $n^{-3/2}$ (what independent points in the same annulus would produce) up to $n^{-5/4}$. A corollary gives the first limit law for the minimal distance between distinct roots, and the proof includes a standalone theorem stating that a random Taylor series with i.i.d. coefficients almost surely has no double zero away from the origin.","feed_headline":"Random root gaps turn Poisson at scale n^{-5/4}","feed_subtitle":"Pairwise root distances converge to an explicit Poisson process; the closest pair obeys a quartic limit law.","key_machinery":"The argument is carried by a net reduction in the bulk annulus $\\Omega_K=\\mathbb{H}\\cap A(1-K/n,1+K/n)$, where almost all roots lie. At scale $\\delta=n^{-5/4-\\beta}$, each net point $z$ is interrogated through the random triple $(f_n(z),f_n'(z),f_n''(z))$: the linear root prediction $z - f_n(z)/f_n'(z)$ must land in the polar cell $R_z^\\circ$, and the quadratic second-root prediction $2|f_n'(z)|/|f_n''(z)|$ must land in $n^{-5/4}U$. A close pair of roots forces both events, and conversely the quadratic approximation recovers the pair from them, giving the two-sided counting identity that reduces the Poisson limit to a method-of-moments computation over the net. To control small-ball probabilities for arbitrary coefficient laws, net points are split into 'smooth' points (angles $\\theta$ with no $p\\theta/\\pi$ close to an integer for small $p$, so that the coefficient vector is genuinely high-dimensional) where a Konyagin-Schlag / Cook-Nguyen style local Gaussian comparison (Theorem 3.12) shows asymptotic agreement with Gaussian coefficients, and 'rough' points handled by cruder arithmetic small-ball bounds. The limiting intensity emerges from an explicit Gaussian computation of the event $A_z(U)$: the function $F(x)$ of (78) integrates to give $c_*(K)=\\tfrac14\\int_{-K}^K F(x)\\,dx$ and $c_*=\\lim_{K\\to\\infty}c_*(K)$. Pairs of roots well inside the unit disk are killed by the almost-sure no-double-zero theorem for random Taylor series, so the entire limiting process is driven by the $1/n$-neighborhood of the unit circle.","core_discovery":"The central claim (Theorem 1.1) is that for $f_n(z)=\\sum_{k=0}^n \\xi_k z^k$ with i.i.d. mean-zero sub-Gaussian coefficients and $P[\\xi_0=0]=0$, the point process $\\{n^{5/4}|\\alpha_j-\\alpha_j'| : 1\\le j<j'\\le n\\}$ converges vaguely to a non-homogeneous Poisson point process on $\\mathbb{R}_{\\ge 0}$ with intensity $c_* t^3\\,dt$. The cubic shape of the intensity is universal: it depends on the coefficient law only through the positive constant $c_* = \\tfrac{1}{4}\\int_{-\\infty}^{\\infty}F(x)\\,dx$, where $F$ is given by an explicit Gaussian computation. An immediate corollary is that the minimal separation $m_n$ between roots obeys $\\lim_{n\\to\\infty} P[n^{5/4}m_n\\ge s] = \\exp(-c_*s^4/4)$, so double roots become asymptotically improbable for every coefficient law covered by the theorem. En route, Theorem 1.3 proves that a random Taylor series with i.i.d. coefficients satisfying $E\\log(1+|a_0|)<\\infty$ satisfies $P[\\exists\\alpha\\in\\mathbb{D}: F(\\alpha)=F'(\\alpha)=0] = (P[a_0=0])^2$, i.e. a double zero can occur only at the origin and only when the first two coefficients vanish.","pith_inferences":["A direct extension the paper leaves implicit: for i.i.d. complex coefficients the same $t^3$ Poisson law should hold with a possibly different constant, since the upper-half-plane reduction would be unnecessary; a complex-Gaussian simulation would show whether the conjugation symmetry in Section 3 is essential or merely organizational.","The $(n\\varepsilon)^2$ repulsion factor is a transferable template: any planar point field whose pair correlation carries such a factor at scales $\\varepsilon\\gg n^{-1}$ should exhibit the same $n^{-5/4}$ close-pair scale and $t^3$ distance intensity, and this paper supplies the first rigorous instance for random roots.","The constant $c_*=\\tfrac14\\int_{-\\infty}^{\\infty}F(x)\\,dx$, explicit in principle through (78), is never evaluated numerically; computing it to a few digits would turn Theorem 1.1 into a quantitative prediction that numerical experiments on the minimal gap could confirm to several digits."],"forward_implications":["The minimal separation $m_n$ between roots satisfies $P[n^{5/4}m_n\\ge s]\\to\\exp(-c_*s^4/4)$: a nontrivial, universal limit law for how close the two closest roots can be.","Double roots become asymptotically improbable for every mean-zero sub-Gaussian coefficient law with no atom at zero, which the paper notes is the first such statement for Kac polynomials beyond Gaussian or integer-valued coefficient distributions.","The entire pairwise-distance spectrum is universal: only the constant $c_*$ depends on the coefficient law, while the $t^3$ intensity shape is fixed by the geometry of the bulk annulus.","Roots that remain a fixed distance inside the unit disk are uniformly separated with high probability, so the limiting statistics are governed entirely by roots within distance $K/n$ of the unit circle.","As a byproduct, Theorem 1.3 states that a random Taylor series with i.i.d. coefficients has no double zero away from the origin almost surely, extending a previously known Gaussian special case to all coefficient laws with finite $E\\log(1+|a_0|)$."],"supporting_citations":[{"why":"Defines the model: the random polynomial $f_n(z)=\\sum_{k=0}^n \\xi_k z^k$ whose roots are the paper's object of study.","marker":"[Kac43]"},{"why":"Shows most roots cluster in the annulus $1\\pm K/n$, the region where the close-pair analysis is confined.","marker":"[IZ97]"},{"why":"Supplies the smooth-point small-ball method that Sections 7-8 adapt to the interior of the disk and to second derivatives.","marker":"[KS99]"},{"why":"Supplies the local Gaussian comparison for tuples of smooth points that is extended here to the annulus and to the second derivative as Theorem 3.12.","marker":"[CN21]"},{"why":"Provides the L\\'evy-Kolmogorov-Rogozin anti-concentration inequality (Lemma 2.4) that powers the proof of Theorem 1.3.","marker":"[Ess68]"},{"why":"Establishes the Gaussian special case of the no-double-zero theorem, which Theorem 1.3 extends to all i.i.d. coefficients.","marker":"[PV05]"},{"why":"Gives the algebraic double-root asymptotics for random Littlewood polynomials that Corollary 1.2 supersedes in generality.","marker":"[PSZ16]"},{"why":"Extends double-root results to integer-valued coefficient distributions and explicitly suggests the minimal-gap question answered by Corollary 1.2.","marker":"[FS17]"},{"why":"Gives the moment condition $E\\log(1+|a_0|)<\\infty$ that defines the random Taylor series in Theorem 1.3 and characterizes its analyticity.","marker":"[IZ13]"}],"fun_headline_variants":["Root gaps turn Poisson at n^{5/4} scale","Minimal root separation has quartic limit law","Double roots almost surely absent in random polynomials","Root distances converge to explicit Poisson process","n^{5/4} scaling reveals Poisson root gaps"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The coefficients must be real-valued for the proof as written: the upper-half-plane reduction and the identification of the limiting distance set both use the conjugation identity $f_n(z)=\\overline{f_n(\\bar z)}$, yet Theorem 1.1 as stated never says that the coefficients are real.","fun_headline_variants_meta":{"raw":{"variants":["Root gaps turn Poisson at n^{5/4} scale","Minimal root separation has quartic limit law","Double roots almost surely absent in random polynomials","Root distances converge to explicit Poisson process","n^{5/4} scaling reveals Poisson root gaps"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000309,"raw_usage":{"total_tokens":1785,"prompt_tokens":989,"completion_tokens":796,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":605,"completion_tokens_details":{"reasoning_tokens":723}},"tokens_in":605,"tokens_out":796,"duration_ms":10990,"temperature":1.0,"reasoning_tokens":723,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:48:49.315927+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the theorem's own simulation: for Rademacher or standard-Gaussian coefficients at degree $n=10^4$, sample the unnormalized pair-distance process and check that the expected number of pairs with $n^{5/4}|\\alpha_j-\\alpha_j'|\\le s$ grows like $c_*s^4/4$ and that $P[n^{5/4}m_n\\ge s]$ converges to $\\exp(-c_*s^4/4)$; any statistically clear deviation from the quartic power law refutes the $t^3$ intensity and the minimal-gap corollary. A second test: replace real coefficients with i.i.d. complex Gaussians and see whether the same Poisson limit survives, which would settle whether the real-coefficient conjugation step in Section 3 is essential.","supporting_citations":[],"review_version":1}