{"id":"cbe78199-e3bb-4d88-a49f-0997ea2bff3b","arxiv_id":"1908.04191","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors prove the necessity half of the Scott-Sokal conjecture for elementary symmetric polynomials and construct Riesz kernels for all sufficiently large negative powers.","lead":"Negative powers of elementary symmetric polynomials are shown to be completely monotone only when the exponent is zero or above a conjectured threshold, at least for the necessity direction. The paper builds explicit Riesz-kernel certificates and connects them to hypergeometric functions and polytope volumes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the central proofs are coherent; the main caveat is reliance on a published asymptotic-sign lemma.","rationale":"The reader correctly identifies the imported Scott-Sokal Lemma 3.1 and the m=2 base case as the least-secure assumptions in Theorem 6.6. I do not think these amount to a defect: they are published results, and the way they are used is logically valid. The sufficiency construction in Theorem 6.4 is elaborate but internally consistent; the only endpoint subtlety is that Riesz kernels may be singular at alpha = (n-2)/2, but this does not threaten an existence statement for sufficiently large alpha. I therefore see no reason to change the ACCEPT verdict. A symbolic check of Lemma 3.1 would be a worthwhile verification, though in my judgment it is not a condition for accepting the paper.","tokens_in":21427,"tokens_out":20078,"duration_ms":191358,"concrete_test":"Verify Lemma 3.1 directly for the case m=3, n=4: with a computer algebra system, compute the leading term as x1 -> infinity of the mixed derivative d_{x2}^{a2} d_{x3}^{a3} d_{x4}^{a4} E_{3,4}^{-alpha} for every multi-index with total order at most 8, and compare its sign with the same derivative of E_{2,3}^{-alpha}. A sign mismatch for any multi-index would invalidate the induction in Theorem 6.6; agreement would confirm that the imported lemma applies in the needed mixed-derivative regime.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The central necessity statement (Theorem 6.6) depends on the import of Scott-Sokal Lemma 3.1: when x1 is large, every mixed derivative in x2,...,xn of E_{m,n}^{-alpha} has the same sign as the corresponding derivative of E_{m-1,n-1}^{-alpha}. If that sign comparison failed for some high-order mixed derivative, the induction would not yield complete monotonicity of E_{m-1,n-1}^{-alpha}, and the bound alpha >= (n-m)/2 would not follow. I checked the internal logic of the reduction and it is sound: complete monotonicity of E_{m,n}^{-alpha} fixes the sign of each derivative in x2,...,xn, and Lemma 3.1 transfers that sign to E_{m-1,n-1}^{-alpha}. The m=2 base case is likewise a cited published result. I found no internal inconsistency; residual doubt is only about the exact hypotheses of the imported lemma, not about the present arguments.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops integral-representation certificates for complete monotonicity of negative powers of hyperbolic polynomials, with the main focus on elementary symmetric polynomials E_{m,n}. Its central results are Theorem 6.4, which shows that E_{m,n}^{-\\alpha} is completely monotone for all sufficiently large \\alpha and gives a constructive route to the Riesz kernel, and Theorem 6.6, which proves the necessity half of Scott and Sokal's Conjecture 4.13: for 2 \\le m < n, if E_{m,n}^{-\\alpha} is completely monotone on the positive orthant, then \\alpha = 0 or \\alpha \\ge (n-m)/2. The earlier sections set up the general framework of complete monotonicity on convex cones, the Bernstein-Hausdorff-Widder-Choquet theorem, G\\aa rding's integral representation, and the Riesz kernel. Section 3 treats products of negative powers of linear forms, relating the Riesz kernel to fiber volumes and the chamber complex. Section 5 interprets Riesz kernels inside convolution algebras and connects them to the Orlik-Terao algebra. Section 7 relates Riesz kernels to A-hypergeometric functions and to Aomoto-Gel'fand hypergeometric integrals. The paper is expository in parts but the main theorems in Section 6 are the substantive new contributions.","tokens_in":21620,"tokens_out":10307,"duration_ms":100744,"significance":"If the results hold as stated, the paper makes a genuine advance on a conjecture of Scott and Sokal. Theorem 6.6 establishes the necessity direction of Conjecture 4.13 for all 2 \\le m < n, and Theorem 6.4 gives a constructive proof that sufficiently negative powers of every E_{m,n} are completely monotone. The constructive viewpoint is valuable: explicit Riesz kernels, convolution products of measures, and the connection to A-hypergeometric functions provide tools that go beyond the particular conjecture. I found no circularity in the argument: the conjecture from [14] is used as a goal and not as an input, and the derivations are based on standard Laplace/G\\aa rding theory and on cited results of Scott and Sokal. The main caveats are that two load-bearing inductive steps are not fully formalized as written: the proof of Theorem 6.6 imports an unstated sign-comparison lemma from [19], and the proof of Theorem 6.4 applies Lemma 6.3 under an induction hypothesis that does not explicitly include the existence of a Riesz kernel. These are fixable within the manuscript's scope.","major_comments":[{"comment":"The proof of the necessity statement rests entirely on the sign-comparison lemma [19, Lemma 3.1], which is neither stated nor proved. This lemma is load-bearing: it is what transfers derivative signs from E_{m,n}^{-\\alpha} to E_{m-1,n-1}^{-\\alpha} and thereby yields complete monotonicity of E_{m-1,n-1}^{-\\alpha}. As written, the proof of Theorem 6.6 cannot be checked from the manuscript alone. Please state the precise hypotheses of [19, Lemma 3.1], verify that they apply to all mixed derivatives in x_2,\\ldots,x_n of every order, and either reproduce the proof or give a self-contained statement.","section":"Section 6, Theorem 6.6"},{"comment":"The induction in the proof of Theorem 6.4 applies Lemma 6.3, whose hypotheses require Riesz kernels for the functions f and g, i.e., absolutely continuous Riesz measures. The formal induction hypothesis of Theorem 6.4, however, only asserts complete monotonicity of E_{m-1,n-1}^{-\\alpha}; complete monotonicity alone gives a Riesz measure by Theorem 2.5, but not necessarily a density. The constructive claim of the theorem therefore needs a strengthened induction hypothesis: for \\alpha \\ge \\alpha_{m,n}, the function E_{m,n}^{-\\alpha} has an explicit nonnegative Riesz kernel q, and the application of Lemma 6.3 to g = E_{m-1,n-1}^{-\\alpha} and f = E_{2,3}^{-\\alpha} produces that kernel via equation (18). Please make this strengthened induction explicit and verify the base cases.","section":"Section 6, Theorem 6.4"}],"minor_comments":[{"comment":"The displayed numerical fraction is malformed: '- 16652440985600 / 76263809554320336/11' should be typeset as a single fraction, evidently -16652440985600 / (76263809554320336/11) or an equivalent. Please correct the typesetting.","section":"Example 2.8"},{"comment":"In the factorization (21), the identity \\sum_{i=1}^{n-1} Q_i = m E_{m,n-1} is used implicitly. Please state this identity explicitly so that the equality between the product and the displayed exponential is transparent.","section":"Section 6, equation (21)"},{"comment":"After invoking [19, Lemma 3.1], the text should state explicitly that the sign comparison holds for every mixed derivative in x_2,\\ldots,x_n of every order, since complete monotonicity of E_{m-1,n-1}^{-\\alpha} requires all infinitely many such derivative inequalities.","section":"Section 6, Theorem 6.6"},{"comment":"The proof of Theorem 7.4 relies on a nontrivial result from [9] without giving a precise reference or derivation. A theorem number in [9] or a short explanation of the integral identity would make the argument easier to verify.","section":"Section 7, Theorem 7.4"},{"comment":"There are several typographical errors: 'certifcate' in Section 1, 'Lesbesgue' in Section 2, and 'monon- tone' in the paragraph preceding Theorem 6.4. The paper should be carefully proofread before resubmission.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"I believe the mathematical strategy is sound and the results are publishable after the inductive arguments are made fully explicit. The concerns in the major comments are about completeness and formal justification rather than a fatal flaw. The reliance on [19, Lemma 3.1] is also a citation issue: since that lemma carries the necessity proof, the authors should include its statement or a proof in a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real result, not a survey. The two theorems to remember are 6.6 and 6.4. Theorem 6.6 proves the only-if direction of Scott–Sokal Conjecture 4.13 for all 2 ≤ m < n: if E_{m,n}^{-α} is completely monotone on the positive orthant, then α = 0 or α ≥ (n−m)/2. That direction was explicitly left open by Scott and Sokal. Theorem 6.4 gives an inductive construction of the Riesz kernel for sufficiently negative powers, establishing Conjecture 4.10 for elementary symmetric polynomials. Section 7's identification of the Riesz kernel with Aomoto–Gelfand hypergeometric functions is a useful bonus.\n\nWhat the paper does well: the proof strategy is transparent and honest. The reduction in Theorem 6.6 is short but clear: write E_{m,n} = x1 E_{m-1,n-1} + E_{m,n-1}, use the sign-transfer lemma of Scott–Sokal for large x1, and induction gives the bound. Lemma 6.3's convolution formula for Riesz kernels is explicit and checkable. There are no fitted parameters and no invented entities; every claim is either proved or explicitly imported from a cited source. The citation pattern is appropriate, mostly Scott–Sokal, Gårding, and the algebraic statistics literature.\n\nSoft spots, in proportion: the main one is genuine but not fatal. Theorem 6.6 leans on Scott–Sokal Lemma 3.1, a sign-comparison lemma for high-order mixed derivatives, and on the m = 2 base case, both cited without proof. I did not find the dependence circular—these are independent published results—but the central necessity statement is only as strong as that lemma's exact hypotheses. Since the lemma comes from a standard Acta Mathematica paper, this is a moderate caveat rather than a flaw. Theorem 6.4's construction is complicated; the authors walk through E_{3,5} in detail, but a referee should verify the exponents in equations (21) and (22). There is also a garbled printed fraction in Example 2.8 and a few typos; cosmetic.\n\nBottom line: for anyone working on complete monotonicity, hyperbolic polynomials, or exponential varieties, this paper is worth reading and citing. It deserves a serious referee, and the review should focus on checking the imported lemma's hypotheses and the inductive step in Theorem 6.4. I would accept after minor revisions.","headline":"Proves the necessity half of Scott–Sokal for elementary symmetric polynomials and gives a constructive Riesz-kernel sufficiency result; solid mathematics with a real but manageable caveat about imported lemmas.","tokens_in":22145,"tokens_out":2081,"would_cite":true,"duration_ms":21338,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["44A10","26B05","33C70"],"pacs":[],"model":"deepseek-v4-flash","headline":"For elementary symmetric polynomials, the paper proves that complete monotonicity of $E_{m,n}^{-\\alpha}$ forces $\\alpha=0$ or $\\alpha\\ge (n-m)/2$, and that all sufficiently negative powers are completely monotone with explicit…","keywords":["complete monotonicity","Riesz kernel","hyperbolic polynomials","elementary symmetric polynomials","Scott-Sokal conjecture","Laplace transform","convex cones","A-hypergeometric functions"],"falsifier":"Compute, at $(x_1,\\ldots,x_5)=(1000,1,1,1,1)$ and for $\\alpha=0.9$, the signed mixed derivative of $E_{3,5}^{-\\alpha}$ corresponding to the multiset used in the induction. The theorem predicts a negative value for some signed derivative; the sign-comparison lemma predicts agreement with the same derivative of $E_{2,4}^{-\\alpha}$. Finding all such derivatives nonnegative would refute the necessity claim.","tokens_in":21247,"feed_emoji":"🧮","tokens_out":11709,"duration_ms":108297,"temperature":0.7,"pith_summary":"Complete monotonicity is a very strong positivity condition: a function and all its signed partial derivatives must be nonnegative on an open convex cone. This paper studies which negative powers of the elementary symmetric polynomial $E_{m,n}(x)=\\sum_{i_1<\\cdots0}^n$, then $\\alpha=0$ or $\\alpha\\ge (n-m)/2$; conversely, for each $m,n$ there is an $\\alpha'$ such that all $\\alpha\\ge\\alpha'$ are completely monotone. The necessity statement, Theorem 6.6, is proved by induction on $m$, using the decomposition $E_{m,n}=x_1E_{m-1,n-1}+E_{m,n-1}$ and a sign-comparison lemma from Scott and Sokal that transfers complete monotonicity to $E_{m-1,n-1}^{-\\alpha}$ as $x_1\\to\\infty$. The sufficiency statement, Theorem 6.4, is constructive: it factors the relevant exponential into a product of completely monotone factors and assembles the Riesz kernel from known kernels via a convolution formula. The exact behavior at the boundary $\\alpha=(n-m)/2$ is left open here, except in the base case $m=2$.","pith_inferences":["If the missing sufficiency half of the Scott-Sokal conjecture also holds, the set of completely monotone negative powers of $E_{m,n}$ would be exactly $\\alpha\\ge(n-m)/2$ together with $\\alpha=0$; a natural test is to study the limiting behavior of the inductive Riesz kernel as $\\alpha$ approaches $(n-m)/2$ from above.","The same induction—splitting a hyperbolic polynomial as $A+yB$ and iterating the kernel-convolution formula—may yield threshold results for other recursively defined hyperbolic polynomials, such as those with interlacing factorizations.","Because the constructed kernels are hypergeometric, numerical evaluation with hypergeometric-series algorithms offers a practical, independent check of nonnegativity of candidate Riesz kernels for specific polynomials and exponents."],"forward_implications":["Scott and Sokal's Conjecture 4.13 now has a proven necessity half: no exponent below $(n-m)/2$ can work, for every $2\\le m