{"id":"660a0f3c-4629-4ff4-85e9-3cb4c29f2543","arxiv_id":"2507.03800","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Using a multivariate spectrahedral relaxation for Eulerian polynomials produces root bounds that strictly beat the best univariate relaxation bound, but only by an exponentially small amount.","lead":"The paper builds multivariate versions of Eulerian polynomials and applies spectrahedral relaxations to approximate their rigidly convex sets. It proves that, along the diagonal where these reduce to ordinary Eulerian polynomials, the relaxation gives lower bounds on extreme roots that improve on previous bounds.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 22.1's positivity depends on sub-leading radical asymptotics; no reproducible exact computation is provided, so an arithmetic slip in D, N, or the conjugate expansion would flip the sign of mult_v - un.","rationale":"The reader's weakest assumption and my concern coincide: the central comparison mult_v - un ~ (1/2)(3/4)^n is carried by long algebraic expressions and cancellation-heavy asymptotics that are only sketched. This is genuinely load-bearing because the final positive difference is exponentially small compared with each individual bound, so ordinary asymptotic intuition or numerical computation at low precision cannot confirm it; only exact symbolic rearrangement can. The proposed check is concrete and decisive: recompute D and N from the L-form values, evaluate the difference at several n with exact arithmetic, and redo the asymptotic expansion without discarding terms prematurely. If the check passes, the CONDITIONAL verdict can be upgraded to ACCEPT; if it fails, Proposition 18.1 is false. I see no reason to question the overall framework of applying Schweighofer's relaxation to multivariate Eulerian polynomials, and the paper has independent support in the known univariate RZ-relaxation results, but the specific improvement claim does not yet have that level of support.","tokens_in":42002,"tokens_out":5075,"duration_ms":60482,"concrete_test":"Recompute Lemma 19.1 directly: for n = 3 through 12, form v^T M_p(x) v with v = (y,0,1,-1) using the L-form values in Computation 7.17, and compare the resulting coefficients of y^0, y^1, y^2 with the stated D and N in Lemma 19.1. Then, for each n, compute with exact rational/radical arithmetic the optimal y* = (-b - sqrt(b^2 - 4ac))/(2a), the univariate bound un(n) from the determinant in Proposition 10.9, and mult_v(n) = N(y*)/D(y*); check that mult_v(n) - un(n) > 0 and that the ratio (mult_v(n) - un(n)) / (3/4)^n tends to 1/2. Finally, re-expand the numerator in Lemma 22.1 to the first nonvanishing order using full symbolic expressions before any cancellations, not discarding sub-leading terms; if any coefficient differs from the preprint, the sign of the claimed improvement is not established.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Proposition 18.1 reduces entirely to Lemma 22.1, where the improvement mult_v(n) - un(n) is shown to be positive and asymptotic to (1/2)(3/4)^n. Both bounds are of order roughly 2^{n+1}, so the claimed difference is exponentially tiny relative to the quantities being subtracted. In Fact 21.6 and Lemma 22.1, the four dominant pieces k, v, u, w all have size 2^{11n+15}3^{n+1}n^4 and are cancelled against each other; the surviving numerator has size 2^{8n+15}3^{2n+1}n^3, a factor (3/8)^n/n smaller. In such a cancellation-heavy regime, a single wrong coefficient, a misindexed summation in Lemma 19.1, or an incorrect sign in one of the conjugate steps changes the sign or the size of the difference. The paper itself warns in Warning 19.2 that the iterated sums are laborious and were done 'in a great part, manually' because current symbolic software does not recognize the closed forms. Moreover, un is defined through a square-root expression with a denominator tending to zero, as noted in the footnote to Proposition 10.9, so any independent check must reproduce the exact decomposition un = (p + r sqrt(q))/s, not merely the leading asymptotic term. Without a machine-checked or fully CAS-reproducible derivation of Lemma 19.1 and Lemma 22.1, the central claim cannot be verified from the preprint; this is not a criticism of the method but of the security of the calculation that carries the proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies spectrahedral relaxations of the rigidly convex sets defined by the multivariate Eulerian polynomials A_n(x,1) (the descents-ascents lifting of Brändén and Visontai–Williams, with ascents set to 1). The author computes the relevant L-forms of the relaxation up to degree three, observes that the univariate relaxation recovers Colucci's bound, derives an explicit univariate bound u_n from the determinant of the relaxation, and answers Mező's asymptotic question with β=0 and d=1. The central novelty is the linearization of the multivariate relaxation along the vector family (y,0,1,-1). After optimizing the parameter y, the paper claims in Proposition 18.1 and Lemma 22.1 that the resulting bound mult_v(n) satisfies mult_v(n) - u_n(n) ~ (1/2)(3/4)^n > 0 for all sufficiently large n, thereby establishing that the multivariate relaxation strictly improves the univariate relaxation along the diagonal. The proof proceeds through explicit closed forms for the numerator and denominator of the bound, an optimization step producing radical expressions, and a cancellation-heavy asymptotic comparison using conjugation.","tokens_in":42300,"tokens_out":4985,"duration_ms":62951,"significance":"If the central comparison is correct, the paper provides a concrete and nontrivial demonstration that Schweighofer's spectrahedral relaxation can exploit the growth in the number of variables of a multivariate Eulerian polynomial to improve univariate root bounds along the diagonal. This is a meaningful step toward understanding the accuracy of spectrahedral approximations to rigidly convex sets and it connects real algebraic geometry with classical Eulerian combinatorics. The paper also gives a self-contained route to Mező's question with explicit bounds, not merely asymptotic estimates. The construction is principled: the relaxation theorem from [59] is imported correctly, the choice of the vector (y,0,1,-1) is motivated by the ghost-variable structure, and the final comparison is formulated as an explicit inequality. However, the central claim rests on very long algebraic identities and on a highly cancellative asymptotic argument that are asserted rather than verified in a reproducible way; the paper itself warns in Warning 19.2 that the sums were performed \"in a great part, manually\" and that symbolic software does not recognize the closed forms.","major_comments":[{"comment":"The L-form evaluations in Computation 7.17 are load-bearing: they are used to build the relaxation, to derive the univariate bound in Proposition 10.9, and to form the expressions D and N in Lemma 19.1. These formulas are asserted without derivation, and they involve nontrivial case distinctions (for example, x_i^2 x_j depends on whether i<j or j<i). An arithmetic slip in any of these entries propagates directly to the final difference mult_v(n)-u_n(n). Please provide a machine-checkable derivation, for instance a CAS worksheet or a detailed appendix that verifies each entry of Computation 7.17 and each subsequent summation in Lemma 19.1. Without this, the central claim of Proposition 18.1 cannot be independently checked.","section":"§7, Computation 7.17"},{"comment":"Lemma 19.1 states closed forms for D and N with the proof reduced to \"performing the last sums,\" and Warning 19.2 explains that these sums were done manually because symbolic software does not recognize the closed forms. These closed forms are the exact input to the optimization in Section 20 and to the final comparison in Sections 21–22. Since the positivity and the asymptotic size of mult_v(n)-u_n(n) depend on exact coefficients of D and N, the manuscript should either (a) include a complete derivation of the iterated sums, or (b) provide a reproducible symbolic computation that verifies Lemma 19.1 for symbolic n (or, at minimum, verifies the closed forms for many concrete values of n and the asymptotic identity for the surviving terms). The current level of detail is insufficient for a proof of Proposition 18.1.","section":"§19, Lemma 19.1 and Warning 19.2"},{"comment":"The proof of Lemma 22.1 is a cancellation-heavy asymptotic argument: the four dominant pieces k, v, u, w of the numerator all have size 2^{11n+15}3^{n+1}n^4 and cancel, leaving a contribution of size 2^{8n+15}3^{2n+1}n^3. The final sign and size of mult_v-u_n come from sub-dominant terms, yet the manuscript only displays the dominant cancellations and the final surviving growth. Because u_n and mult_v are both of order 2^{n+1}, the claimed difference (1/2)(3/4)^n is exponentially tiny compared to the quantities being subtracted. A single incorrect coefficient in the next order of any of k, v, u, w, or in the conjugate expansions used in Lemma 22.1, would flip the sign or alter the size of the difference. Please provide the full expansion to the first non-cancelling order, or a CAS-verified exact expression for mult_v-u_n, before the positivity claim can be accepted.","section":"§21–22, Fact 21.6 and Lemma 22.1"},{"comment":"The optimal parameter y is defined as the left root of ay^2+by+c = N'D - ND', but the coefficients a, b, c are never displayed. The argument that the maximum appears at the left root relies on a sign analysis of the difference with the horizontal asymptote, and the resulting expression for y contains radicals that later enter Fact 21.6. Without explicit a, b, c, or an independent verification of the optimization step, the sequence y:N->R is not fully specified and the subsequent asymptotics cannot be checked. In addition, the verification of Condition (5) in Lemma 21.4 uses the asymptotic y ~ 3^{n+1}/(2^{n+1}n), which is derived after substituting the optimal y; please confirm that this step is not circular.","section":"§20, Lemma 20.6"}],"minor_comments":[{"comment":"The keyword \"Rigidily convex set\" contains a typo and should read \"Rigidly convex set\".","section":"Keywords"},{"comment":"The statement of Proposition 11.2 is incomplete: the limit expression reads \"lim_{n→∞} b_v/2^{n+1}\" with no right-hand side, and the surrounding text uses \"asymptomatic\" where \"asymptotic\" is intended.","section":"§11, Proposition 11.2"},{"comment":"In Definition 1.5, the displayed sum \"p(x):=Σ_{n>0} a_i x^i\" mixes the indices n and i; it should be \"Σ_{i>0} a_i x^i\" or equivalently \"Σ_{n>0} a_n x^n\".","section":"§1, Definition 1.5"},{"comment":"There is a typo in the sentence \"this imposes a first condition on this y... should make N >0\" and in the notation \"q_n^n\" which should be \"q^{(n)}_n\"; please proofread the notation for roots of the Eulerian polynomials throughout.","section":"§20, near Lemma 20.6"},{"comment":"The conclusion states that a sequence of vectors with the structure described in Procedure 24.5 produces a bound whose difference from u_n grows exponentially, but this is explicitly deferred to future work and no proof is given. Please label this as a conjecture or a numerical observation rather than a confirmed result, since it currently reads as an unsupported announcement.","section":"§25, Conclusion"},{"comment":"The extended quotation from Gauss is not needed for the mathematical content and makes the exposition more discursive than necessary; consider moving it to a footnote or removing it.","section":"§10, Comment 10.4"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is plausible and the overall strategy is coherent, but the manuscript currently does not provide a reproducible verification of the algebraic identities and sub-leading asymptotics that carry the proof. I would urge the editor to require the author to supply a supplementary symbolic computation (or a detailed appendix) for Lemma 19.1, Fact 21.6 and Lemma 22.1; without this, the central claim remains an unverifiable computation. The paper is also unusually long with many philosophical digressions; these do not affect the mathematics but would benefit from editorial shortening."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What should you know? The paper claims that going multivariate in Schweighofer's spectrahedral relaxation strictly improves the best univariate lower bound for extreme roots of Eulerian polynomials, with the improvement exponentially small, asymptotic to (1/2)(3/4)^n. The framework is coherent and the univariate bound was already known (Szego; the paper admits this). The genuinely new piece is the multivariate linearization with vectors (y,0,1,-1) and the proof that it beats the univariate bound. That is a legitimate extension, and the optimization over y is a real extremal construction, not a fit to the target root.\n\nWhat is good: the paper is transparent about what is old and what is new. It explicitly acknowledges that the univariate relaxation appeared in [60] and [3], and that Mezo's question was already answered by Sobolev. The recovered Colucci bound and the asymptotic tightness to order 2^{n+1} are clean. The multivariate improvement being positive, albeit vanishing, is an honest and carefully stated result.\n\nThe soft spot is exactly where the stress test points: the proof of Lemma 22.1 depends on four dominant asymptotic terms that cancel among themselves, leaving a subdominant term whose sign is the entire claim. Those cancellations are presented as a series of asserted closed forms and conjugate expansions, and the paper's own Warning 19.2 says the iterated sums were done \"in a great part, manually\" because computer algebra did not recognize the closed forms. I believe the method is sound, but I cannot certify the sign of mult_v - un without a machine-checked or fully reproduced derivation. This is not a fabricated concern; the paper itself flags the manual labor. And the footnote to Proposition 10.9 warns that software mishandles the limit due to a denominator going to zero, so any independent check must reproduce the exact decomposition of un, not just leading asymptotics.\n\nWho is this for? People working on root bounds for Eulerian polynomials or on spectrahedral relaxations of rigidly convex sets. It is a serious paper with a clear thesis and an honest account of its own provenance. The main theorem is probably true, but the evidence as written is incomplete.\n\nMy recommendation: send it to a serious referee. The referee's first job is to verify Lemma 19.1 and Lemma 22.1 or demand a supplementary CAS file. If those hold up, this is a solid contribution, modest but real.","headline":"Genuine extension of the relaxation method, but the central comparison rests on unverified algebraic cancellations; a referee should demand a reproducible derivation.","tokens_in":42861,"tokens_out":1814,"would_cite":false,"duration_ms":24081,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05A15","26C10","14P10","90C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a spectrahedral relaxation built from multivariate Eulerian polynomials bounds the extreme roots of the univariate Eulerian polynomials strictly better than the univariate relaxation, with the gap asymptotic to…","keywords":["Eulerian polynomials","real zero polynomials","rigidly convex sets","spectrahedral relaxations","linear matrix inequalities","descent tops","asymptotic root bounds","multivariate stable polynomials"],"falsifier":"Recompute Lemma 19.1 and Lemma 22.1 with exact symbolic computation: substitute the optimized $y=(-b-\\sqrt{b^2-4ac})/(2a)$ into $D$ and $N$, form $m_v(n)-u_n(n)$, and compare the exact leading term with $(1/2)(3/4)^n$ at, say, $n=10,20,30$. If at any such $n$ the difference is not positive, or its leading coefficient differs from $1/2$, then Proposition 18.1 is refuted.","tokens_in":41753,"feed_emoji":"📐","tokens_out":10429,"duration_ms":109214,"temperature":0.7,"pith_summary":"The paper claims that moving the classical univariate Eulerian polynomials into a multivariate, real-zero setting and applying a spectrahedral relaxation yields strictly sharper lower bounds for their extreme roots than the univariate relaxation, which had already matched the best bounds in the literature. The liftings used are the descents-and-ascents Eulerian polynomials, which are real stable and become real-zero once the ascent variables are set to $1$; their rigidly convex sets are outer-approximated by spectrahedra whose defining linear matrix polynomials use only the degree-three part of the polynomial. Along the diagonal $x_1=\\cdots=x_n=x$ the original polynomial $A_n$ is recovered, so the paper measures the accuracy of the global approximation by how well the relaxation bounds the extreme roots of $A_n$. Its main result is that, after optimizing a family of linearizing vectors $(y,0,1,-1)$, the multivariate bound satisfies $m_v(n)-u_n(n)\\sim (1/2)(3/4)^n>0$ for all large $n$. This establishes, in concrete asymptotic terms, that multivariability itself gives a real improvement in root bounds for Eulerian polynomials.","feed_headline":"Multivariate relaxation beats univariate bound on Eulerian roots","feed_subtitle":"Lifting Eulerian polynomials to many variables yields strictly sharper root bounds.","key_machinery":"The load-bearing object is the spectrahedral relaxation: for a real-zero polynomial $p$, one forms the linear matrix polynomial whose entries are values of the $L$-form $L_p$ on monomials of degree at most three, and the resulting spectrahedron $S(p)$ contains the rigidly convex set of $p$. The paper applies this to the dehomogenized multivariate Eulerian polynomials $A_n(x,1)$, whose real-zeroness comes from their real stability, and extracts root bounds through the vector test $v^\\top M_p(x)v\\ge 0$ rather than by solving determinants. The vector family $(y,0,1,-1)$ is what breaks the all-ones compression that would otherwise collapse the multivariate relaxation to the univariate one, and the proof of the improvement is carried by asymptotic manipulations with square roots, including conjugation after dominant-term cancellation.","core_discovery":"On the paper's own terms, the discovery is that the multivariate relaxation improves on the optimal univariate relaxation without any change in the underlying polynomial family, only by lifting to more variables. For the $n$-th univariate Eulerian polynomial, the determinant-based univariate relaxation yields the best bound $u_n(n)$; the paper constructs vectors $(y_n,0,1,-1)$ and proves that linearizing the multivariate relaxation with them gives a bound $m_v(n)$ with $m_v(n)-u_n(n)\\sim (1/2)(3/4)^n$. Since the univariate bound already has the correct leading asymptotics $2^{n+1}$, the positive difference is a strict improvement in the next order of the exponential scale. The improvement is obtained by a symbolic asymptotic analysis in which the dominant terms inside and outside square roots cancel, and conjugates are used to recover the surviving terms.","pith_inferences":["The diagonal is only a one-dimensional test: nothing in the paper measures the accuracy of the spectrahedral outer approximation far from the diagonal, so the global quality claim should not be extrapolated beyond this direction.","The same recipe of lifting a univariate real-rooted polynomial to a multivariate real-zero polynomial and linearizing its relaxation could be applied to other combinatorial families, giving a general route from stability to numerical root bounds that the paper does not develop.","The concluding claim that a more elaborate vector sequence already yields an exponentially growing difference from $u_n$ is asserted without proof in this paper; until the promised derivation appears, it is safest to treat it as a conjecture rather than an established result."],"forward_implications":["For all sufficiently large $n$, the extreme roots of $A_n$ admit a strictly stronger lower bound than the one coming from the univariate relaxation.","The multivariate bound is obtained from linear matrix inequalities of size $n+2$ whose entries are cubic functions of the coefficients of $A_n$, so the computation is cheap in the degree parameter.","The gap $(1/2)(3/4)^n$ is exponentially small next to the leading growth $2^{n+1}$, so the multivariate method refines the asymptotics without changing the dominant scale.","Since the relaxation bounds the entire rigidly convex set, the same construction supplies a global outer approximation that is highly accurate along the diagonal where Eulerian polynomials live."],"supporting_citations":[{"why":"Supplies the relaxation construction and the containment theorem for rigidly convex sets that the entire bound relies on.","marker":"[59]"},{"why":"Establishes real stability of the multivariate descents-ascents Eulerian polynomials.","marker":"[22]"},{"why":"Introduces descent tops and the stable multivariate Eulerian polynomials that become real-zero after dehomogenization.","marker":"[63]"},{"why":"Provides the Colucci root bound and the asymptotic question that define the univariate benchmark and motivate the comparison.","marker":"[45]"},{"why":"Records the earlier asymptotic answer for the extreme roots that fixes the first-order scale used in the comparison.","marker":"[14]"},{"why":"Documents the prior appearance of the univariate relaxation, establishing that the univariate bound is not new.","marker":"[60]"},{"why":"Gives the counting formulas for adequate descents used to compute the L-form values on monomials of degree up to three.","marker":"[23]"},{"why":"Provides exact cardinalities of permutations with prescribed descent top set, needed for the entries of the relaxation.","marker":"[13]"}],"fun_headline_variants":["Lifting Eulerian polynomials sharpens root bounds","Multivariate spectrahedra tighten univariate Eulerian bounds","Relaxation method yields sharper Eulerian root estimates","Better Eulerian root bounds via multivariate lift","Spectrahedral trick improves Eulerian root bounds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the long, cancellation-prone algebraic identities behind the bound—the L-form evaluations, the closed forms of $D$ and $N$, and the radical asymptotics—are all arithmetically correct, since a single sign or coefficient slip could make the claimed difference negative or change its size.","fun_headline_variants_meta":{"raw":{"variants":["Lifting Eulerian polynomials sharpens root bounds","Multivariate spectrahedra tighten univariate Eulerian bounds","Relaxation method yields sharper Eulerian root estimates","Better Eulerian root bounds via multivariate lift","Spectrahedral trick improves Eulerian root bounds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000202,"raw_usage":{"total_tokens":1403,"prompt_tokens":985,"completion_tokens":418,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":601,"completion_tokens_details":{"reasoning_tokens":345}},"tokens_in":601,"tokens_out":418,"duration_ms":4297,"temperature":1.0,"reasoning_tokens":345,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:03:29.576541+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute Lemma 19.1 and Lemma 22.1 with exact symbolic computation: substitute the optimized $y=(-b-\\sqrt{b^2-4ac})/(2a)$ into $D$ and $N$, form $m_v(n)-u_n(n)$, and compare the exact leading term with $(1/2)(3/4)^n$ at, say, $n=10,20,30$. If at any such $n$ the difference is not positive, or its leading coefficient differs from $1/2$, then Proposition 18.1 is refuted.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes real stability of the multivariate descents-ascents Eulerian polynomials."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Documents the prior appearance of the univariate relaxation, establishing that the univariate bound is not new."},{"cited_title":"Hall and Jeffrey B","cited_arxiv_id":null,"evidence_quote":"Gives the counting formulas for adequate descents used to compute the L-form values on monomials of degree up to three."},{"cited_title":"Nelson, T","cited_arxiv_id":null,"evidence_quote":"Provides exact cardinalities of permutations with prescribed descent top set, needed for the entries of the relaxation."}],"review_version":1}