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Spectrahedral relaxations of Eulerian rigidly convex sets

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Pith's one-line read 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…

desk verdict Genuine extension of the relaxation method, but the central comparison rests on unverified algebraic cancellations; a referee should demand a reproducible derivation. read the letter →

arxiv 2507.03800 v1 pith:5WEEY747 submitted 2025-07-04 math.CO cs.NAmath.AGmath.NAmath.OC

Alejandro González Nevado This is my paper · ORCID
classification math.COcs.NAmath.AGmath.NAmath.OC MSC 05A1526C1014P1090C22
keywords Eulerianpolynomialsrealzerorigidlyconvexsetsspectrahedralrelaxationslinearmatrixinequalitiesdescenttopsasymptoticrootboundsmultivariatestable
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

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.

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 (4)
  1. [§7, Computation 7.17] 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.
  2. [§19, Lemma 19.1 and Warning 19.2] 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.
  3. [§21–22, Fact 21.6 and Lemma 22.1] 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.
  4. [§20, Lemma 20.6] 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.
minor comments (6)
  1. [Keywords] The keyword "Rigidily convex set" contains a typo and should read "Rigidly convex set".
  2. [§11, Proposition 11.2] 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.
  3. [§1, Definition 1.5] 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".
  4. [§20, near Lemma 20.6] 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.
  5. [§25, Conclusion] 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.
  6. [§10, Comment 10.4] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

Circularity not found: mult_v and un are independent symbolic bounds, the y-optimization is a legitimate extremal construction, and the positivity of mult_v − un is derived by a separate asymptotic computation; remaining risks are unverified manual algebra (Warning 19.2) and reliance on the advisor's external relaxation theorem [59].

full rationale

The derivation chain is self-contained from combinatorial inputs to the final comparison. un(n) (Definition 10.8) is the exact root of the determinant of the 2x2 LMP obtained by applying Schweighofer's relaxation to the univariate Eulerian polynomial, computed directly from the L-form values of Computation 10.7. mult_v(n) is the quotient N/D of Lemma 19.1, obtained by evaluating the L-form values of Computation 7.17 on the linearizing vector (y,0,1,-1). The two bounds are independently defined symbolic expressions; neither is defined in terms of the other or in terms of the target root q_1^(n). The parameter y is chosen as the maximizer of N/D over R (Computation 20.5, Lemma 20.6), a legitimate extremal construction over a family each of whose members is a valid relaxation bound; nothing about the positivity of mult_v - un is forced by this optimization, which is why the asymptotic comparison in Lemma 22.1 must be (and is) carried out separately, with the difference shown to be ~ (1/2)(3/4)^n > 0. The enabling relaxation theorem (Theorem 4.14) is cited to [59] by the author's advisor; this is load-bearing in the sense that the whole method rests on it, but it is a published theorem with an independent proof (the paper notes it relies on the Helton-Vinnikov theorem, Observation 17.3), with stated assumptions that do not include the target result, and with external consistency checks provided by the paper's own recovery of the Colucci and Szego bounds. The vector ansatz (y,0,1,-1) is disclosed as numerically motivated (Objective 16.1), not smuggled in via citation, and no uniqueness claim is imported from prior work. The genuine weakness is reproducibility, not circularity: Warning 19.2 states that the iterated sums behind Lemma 19.1 were performed 'in a great part, manually' because current symbolic software 'does not recognize' the closed forms, and Fact 21.6/Lemma 22.1 rest on cancellation-heavy radical asymptotics (four dominant terms of size 2^(11n+15)3^(n+1)n^4 annihilate, leaving a survivor smaller by a factor (3/8)^n/n) with no machine verification; also the Proposition 10.9 footnote reports that even Mathematica fails on the un-limit unless manually conjugated. An arithmetic slip in D, N, or the conjugate expansions would be a correctness failure and could flip the sign of the central difference, but under the given rules that is a correctness risk, not circularity.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The method introduces one optimized free parameter y in the linearizing vector. It imports the relaxation theorem, real stability theorems, and Eulerian number asymptotics from prior work. No new particles, forces, or empirical entities are introduced.

free parameters (1)
  • y(n) optimization parameter in the linearizing vector (y,0,1,-1) = y(n) ~ 3^(n+1) / (2^(n+1) n), chosen as the maximizing root of a quadratic
    The vector family and the value of y are selected through numerical experiments and optimization to maximize the lower bound mult_v(n). The resulting bound is valid for any admissible y, but this is still a free parameter tuned by the method.
assumptions (6)
  • domain assumption For every real zero polynomial p, the rigidly convex set rcs(p) is contained in the spectrahedron S(p) built from L-forms (Schweighofer's relaxation theorem).
    Imported from [59, Theorem 3.35] without proof; all bounds in the paper inherit this containment.
  • domain assumption The multivariate Eulerian polynomial An(x,y) is real stable.
    Quoted from [22, Theorem 3.2] and [63, Theorem 3.3]; it is the route to proving An(x,1) is a real zero polynomial.
  • domain assumption Setting the ascent variables y to 1 preserves real stability, as does identifying them with a single variable y.
    Used in Observation 6.8 and Corollary 6.9, relying on [64, Lemma 2.4].
  • standard math The Eulerian numbers satisfy E(n+1,n-k) / (k+1)^(n+1) -> 1 for fixed k.
    Proposition 8.4, used in the asymptotic comparison of the bounds.
  • domain assumption The passage from real stability in positive directions to hyperbolicity in a coordinate direction is valid for the dehomogenized polynomial.
    Based on [53, Proposition 1.3] and [59, Proposition 6.7]; needed to establish that An(x,1) is RZ.
  • ad hoc to paper The asymptotic expansions of the radical expressions in the comparison are governed by the listed dominant terms after conjugation.
    This is the fragile computational premise in Fact 21.6, Remark 21.7, and Lemma 22.1. The paper asserts these cancellations without a fully detailed verification.

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Cite this review

Pith. "Pith review of Spectrahedral relaxations of Eulerian rigidly convex sets." pith.science (2026). https://pith.science/paper/5WEEY747

@misc{pith2026250703800,
  author       = {Pith},
  title        = {Pith review of: Spectrahedral relaxations of Eulerian rigidly convex sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5WEEY747}},
  note         = {Machine review of arXiv:2507.03800}
}
read the original abstract

We study a generalization of Eulerian polynomials to the multivariate setting introduced by Br\"and\'en. Although initially these polynomials were introduced using the language of hyperbolic and stable polynomials, we manage to translate some restrictions of these polynomials to our real zero setting. Once we are in this setting, we focus our attention on the rigidly convex sets (RCSs) defined by these polynomials. In particular, we study the corresponding rigidly convex sets looking at spectrahedral relaxations constructed through the use of monic symmetric linear matrix polynomials (MSLMPs) of small size and depending polynomially (actually just cubically) on the coefficients of the corresponding polynomials. We analyze how good are the obtained spectrahedral approximations to these rigidly convex sets. We do this analysis by measuring the behavior along the diagonal, where we precisely recover the original univariate Eulerian polynomials. Thus we conclude that, measuring through the diagonal, our relaxation-based spectrahedral method for approximation of the rigidly convex sets defined by multivariate Eulerian polynomials is highly accurate. In particular, we see that this relaxation-based spectrahedral method for approximation of the rigidly convex sets defined by multivariate Eulerian polynomials provides bounds for the extreme roots of the corresponding univariate Eulerian polynomials that are better than these already found in the literature. All in all, this tells us that, at least close to the diagonal, the global outer approximation to the rigidly convex sets provided by this relaxation-based spectrahedral method is itself highly accurate.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Guessing sequences of eigenvectors for LMPs defining spectrahedral relaxations of Eulerian rigidly convex sets

    math.CO 2025-07 conditional novelty 5.0 of 10

    For even n, a carefully chosen sequence of vectors makes the spectrahedral relaxation bound for Eulerian polynomial roots exceed the univariate bound by asymptotically (3/8)(9/8)^{n/2}.

Reference graph

Works this paper leans on

69 extracted references · 58 canonical work pages · cited by 1 Pith paper

  1. [60]

    G´ abor Szeg˝ o,Orthogonal polynomials, American Mathematical Society, Volume 23, 1939

    work page 1939

  2. [3]

    Grigoriy Blekherman, Mario Kummer, Cordian Riener, Markus Schweighofer, and Cynthia Vinzant, Generalized eigenvalue methods for Gaussian quadrature rules , Annales Henri Lebesgue, Volume 3, Pages 1327–1341, 2020

    work page 2020

  3. [59]

    Markus Schweighofer, Spectrahedral relaxations of hyperbolicity cones , arXiv preprint arXiv:1907.13611, 2023

    arXiv 1907

  4. [1]

    Anari, S

    N. Anari, S. O. Gharan, A. Saberi, N. Srivastava, Approximating the largest root and applications to interlacing families , Proceedings of the 68 Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms (2018), 1015–1028

    work page 2018

  5. [2]

    Alexander Belton, Dominique Guillot, Apoorva Khare, and Mihai Puti- nar, Matrix compression along isogenic blocks , Acta Scientiarum Math- ematicarum, Volume 88, Number 1, Pages 417–448, Springer, 2022

    work page 2022

  6. [4]

    Boese and Wolfram J

    Fritz G. Boese and Wolfram J. Luther, Accurate enclosure of the zero set of multivariate polynomials , BIT Numerical Mathematics, Volume 43, Pages 245–261, Springer, 2003

    work page 2003

  7. [5]

    Mikl´ os B´ ona,Combinatorics of Permutations , CRC Press, 2022

    work page 2022

  8. [6]

    Wag- ner, Proof of the monotone column permanent conjecture , in *Notions of Positivity and the Geometry of Polynomials*, Pages 63–78, Springer, 2011

    Petter Br¨ and´ en, James Haglund, Mirk´ o Visontai, and David G. Wag- ner, Proof of the monotone column permanent conjecture , in *Notions of Positivity and the Geometry of Polynomials*, Pages 63–78, Springer, 2011

    2011

Show all 69 references
  1. [7]

    Br¨ and´ en, M

    P. Br¨ and´ en, M. Leander, M. Visontai,Multivariate Eulerian polynomials and exclusion processes, Combinatorics, Probability and Computing, 25 (2016), 4, 486–499

    2016

  2. [8]

    Br¨ and´ en, M

    P. Br¨ and´ en, M. Leander,Multivariate p-Eulerian polynomials , arXiv preprint arXiv:1604.04140, 2016

    2016 arXiv

  3. [9]

    Caviness and Richard J

    Bob F. Caviness and Richard J. Fateman, Simplification of radical ex- pressions, Proceedings of the Third ACM Symposium on Symbolic and Algebraic Computation, Pages 329–338, 1976

    1976

  4. [10]

    Hongwei Cheng, Zydrunas Gimbutas, Per-Gunnar Martinsson, and Vladimir Rokhlin, On the compression of low rank matrices, SIAM Jour- nal on Scientific Computing, Volume 26, Number 4, Pages 1389–1404, SIAM, 2005

    2005

  5. [11]

    Marc Chevalier and J´ erˆ ome Feret,Sharing ghost variables in a collec- tion of abstract domains, In *Verification, Model Checking, and Abstract 69 Interpretation: 21st International Conference, VMCAI 2020, New Or- leans, LA, USA, January 16–21, 2020, Proceedings 21*, Pages 1...

    2020

  6. [12]

    J. H. Davenport, M. Mignotte, On finding the largest root of a poly- nomial, ESAIM: Mathematical Modelling and Numerical Analysis 24 (1990), no. 6, 693–696

    1990

  7. [13]

    Nelson, T

    Robert Davis, Sarah A. Nelson, T. Kyle Petersen, and Bridget E. Ten- ner, The pinnacle set of a permutation , Discrete Mathematics, Volume 341, Number 11, Pages 3249–3270, Elsevier, 2018

    2018

  8. [14]

    Demidenko and Vladimir L

    Gennadii V. Demidenko and Vladimir L. Vaskevich, Selected works of S.L. Sobolev, Springer, 2006

    2006

  9. [15]

    Peter Denton, Stephen Parke, Terence Tao, and Xining Zhang, Eigen- vectors from eigenvalues: A survey of a basic identity in linear algebra , Bulletin of the American Mathematical Society, Volume 59, Number 1, Pages 31–58, 2022

    2022

  10. [16]

    Pardalos, and Weili Wu, Mathematical Theory of Optimization , Volume 56, Springer Science & Business Media, 2001

    Ding-Zhu Du, Panos M. Pardalos, and Weili Wu, Mathematical Theory of Optimization , Volume 56, Springer Science & Business Media, 2001

    2001

  11. [17]

    Haas, Frederick R

    Ahmed Elgohary, Matthias Boehm, Peter J. Haas, Frederick R. Reiss, and Berthold Reinwald, Compressed linear algebra for large-scale ma- chine learning, The VLDB Journal, Volume 27, Number 5, Pages 719– 744, Springer, 2018

    2018

  12. [18]

    Sebastian Falkensteiner and J. Rafael Sendra, Transforming Radical Dif- ferential Equations to Algebraic Differential Equations , Mediterranean Journal of Mathematics, Volume 21, Number 3, Article 87, Springer, 2024

    2024

  13. [19]

    Juan Carlos Garcia-Escartin, Finding eigenvectors with a quantum vari- ational algorithm , Quantum Information Processing, Volume 23, Num- ber 7, Page 254, Springer, 2024

    2024

  14. [20]

    Letter to Wolfgang Bolyai , G¨ ottingen, Septem- ber 2, 1808

    Carl Friedrich Gauß. Letter to Wolfgang Bolyai , G¨ ottingen, Septem- ber 2, 1808. Archive: G¨ ottingen, Nieders¨ achsische Staats- und Uni- versit¨ atsbibliothek, Signature: Cod. Ms. Gauß Briefe B: Bolyai, Brief 70 Nr. 19, 4 S.; deutsch. Available online: https://gauss.adw-go...

  15. [21]

    A. V. Gnedin and G. I. Olshanskii, The boundary of the Eulerian number triangle, Mosc. Math. J., Volume 6, Issue 3, 461–475, 2006

    2006

  16. [22]

    James Haglund and Mirk´ o Visontai,Stable multivariate Eulerian polyno- mials and generalized Stirling permutations , European Journal of Com- binatorics, Volume 33, Number 4, Pages 477–487, Elsevier, 2012

    2012

  17. [23]

    Hall and Jeffrey B

    John T. Hall and Jeffrey B. Remmel, Counting descent pairs with pre- scribed tops and bottoms , Journal of Combinatorial Theory, Series A, Volume 115, Number 5, Pages 693–725, Elsevier, 2008

    2008

  18. [24]

    J. William Helton and Victor Vinnikov, Linear matrix inequality repre- sentation of sets , Communications on Pure and Applied Mathematics, Volume 60, Number 5, Pages 654–674, Wiley Online Library, 2007

    2007

  19. [25]

    Didier Henrion, Milan Korda, and Jean-Bernard Lasserre, Polynomial argmin for recovery and approximation of multivariate discontinuous functions, arXiv preprint arXiv:2302.06945, 2023

    2023 arXiv

  20. [26]

    Masaru Ito and Bruno F. Louren¸ co,Automorphisms of rank-one gener- ated hyperbolicity cones and their derivative relaxations , SIAM Journal on Applied Algebra and Geometry, Volume 7, Number 1, Pages 236–263, SIAM, 2023

    2023

  21. [27]

    The Laguerre–Samuelson inequality with exten- sions and applications in statistics and matrix theory

    Jensen, Shane Tyler. “The Laguerre–Samuelson inequality with exten- sions and applications in statistics and matrix theory.” 1999. McGill University

    1999

  22. [28]

    Gaurav Jindal, Neelam Sharma, Harshita Chadha, and Nitish Pathak, Compression using Matrix Folding Algorithm , In *Proceedings of the 2021 9th International Conference on Reliability, Infocom Technologies and Optimization (Trends and Future Directions) (ICRITO)*, Pages 1–4, IEEE, 2021

    2021

  23. [29]

    Thorsten J¨ orgens and Thorsten Theobald,Hyperbolicity cones and imag- inary projections, Proceedings of the American Mathematical Society, Volume 146, Number 10, Pages 4105–4116, 2018. 71

    2018

  24. [30]

    Tadashi Kadowaki and Mitsuru Ambai, Lossy compression of matrices by black box optimisation of mixed integer nonlinear programming , Sci- entific Reports, Volume 12, Number 1, Page 15482, Nature Publishing Group UK London, 2022

    2022

  25. [31]

    Christian Karpfinger, Numerical Calculation of Eigenvalues and Eigen- vectors, in *Calculus and Linear Algebra in Recipes: Terms, Phrases and Numerous Examples in Short Learning Units*, Pages 445–456, Springer, 2022

    2022

  26. [32]

    Eric Katz and Max Kutler, Matroidal mixed Eulerian numbers , Alge- braic Combinatorics, Volume 7, Number 5, Pages 1479–1506, 2024

    2024

  27. [33]

    Sergey Kitaev, Patterns in permutations and words , Springer, Volume 1, 2011

    2011

  28. [34]

    Greg Knese, Determinantal representations of semihyperbolic polyno- mials, Michigan Mathematical Journal, Volume 65, Number 3, Pages 473–487, University of Michigan, Department of Mathematics, 2016

    2016

  29. [35]

    Mario Kummer, Daniel Plaumann, and Cynthia Vinzant, Hyperbolic polynomials, interlacers, and sums of squares , Mathematical Program- ming, Volume 153, Pages 223–245, Springer, 2015

    2015

  30. [36]

    James Levitt and Per-Gunnar Martinsson, Linear-complexity black- box randomized compression of rank-structured matrices, SIAM Journal on Scientific Computing, Volume 46, Number 3, Pages A1747–A1763, SIAM, 2024

    2024

  31. [37]

    James Levitt and Per-Gunnar Martinsson, Randomized compression of rank-structured matrices accelerated with graph coloring , Journal of Computational and Applied Mathematics, Volume 451, Page 116044, Elsevier, 2024

    2024

  32. [38]

    Louren¸ co, Vera Roshchina, and James Saunderson, Hyper- bolicity cones are amenable , Mathematical Programming, Volume 204, Number 1, Pages 753–764, Springer, 2024

    Bruno F. Louren¸ co, Vera Roshchina, and James Saunderson, Hyper- bolicity cones are amenable , Mathematical Programming, Volume 204, Number 1, Pages 753–764, Springer, 2024

    2024

  33. [39]

    Maier, Triangular recurrences, generalized Eulerian numbers, and related number triangles, Advances in Applied Mathematics, Volume 146, Article 102485, 2023

    Robert S. Maier, Triangular recurrences, generalized Eulerian numbers, and related number triangles, Advances in Applied Mathematics, Volume 146, Article 102485, 2023. 72

    2023

  34. [40]

    John Michael MacNamee and Victor Yakovlevich Pan, Numerical Meth- ods for Roots of Polynomials – Parts I and II , Studies in Computational Mathematics, Volumes 14 and 16, Elsevier

  35. [41]

    Simon J. A. Malham, An introduction to asymptotic analysis , Heriot- Watt University, 2005

    2005

  36. [42]

    Matthieu Martel, Compressed matrix computations, In *Proceedings of the 2022 IEEE/ACM International Conference on Big Data Computing, Applications and Technologies (BDCAT)*, Pages 68–76, IEEE, 2022

    2022

  37. [43]

    McSorley and Philip Feinsilver, Multivariate matching polyno- mials of cyclically labelled graphs , Discrete Mathematics, Volume 309, Number 10, Pages 3205–3218, Elsevier, 2009

    John P. McSorley and Philip Feinsilver, Multivariate matching polyno- mials of cyclically labelled graphs , Discrete Mathematics, Volume 309, Number 10, Pages 3205–3218, Elsevier, 2009

    2009

  38. [44]

    Stephen Melczer, Multivariate Series and Diagonals , in *An Invitation to Analytic Combinatorics: From One to Several Variables*, Pages 93– 141, Springer, 2021

    2021

  39. [45]

    Istv´ an Mez˝ o,Combinatorics and number theory of counting sequences , Chapman and Hall/CRC, 2019

    2019

  40. [46]

    Mourrain, V

    B. Mourrain, V. Y. Pan, Multivariate polynomials, duality, and struc- tured matrices, Journal of Complexity, 16 (1) (2000), 110–180

    2000

  41. [47]

    Arik Nemtsov, Amir Averbuch, and Alon Schclar, Matrix compression using the Nystr¨ om method, Intelligent Data Analysis, Volume 20, Num- ber 5, Pages 997–1019, SAGE Publications, 2016

    2016

  42. [48]

    Laguerre–Samuelson type inequalities

    Niezgoda, Marek. “Laguerre–Samuelson type inequalities.” Linear Alge- bra and its Applications , vol. 422, no. 2–3, 2007, pp. 574–581. Elsevier

    2007

  43. [49]

    Rafael Oliveira, Conditional lower bounds on the spectrahedral repre- sentation of explicit hyperbolicity cones , Proceedings of the 45th Inter- national Symposium on Symbolic and Algebraic Computation, Pages 396–401, 2020

    2020

  44. [50]

    Tomas Oppelstrup, Matrix compression by common subexpression elim- ination, Journal of Computational Physics, Volume 247, Pages 100–108, Elsevier, 2013. 73

    2013

  45. [51]

    Paixao and Fl´ avio Code¸ co Coelho, Matrix compression methods, Technical Report, PeerJ PrePrints, 2015

    Crysttian A. Paixao and Fl´ avio Code¸ co Coelho, Matrix compression methods, Technical Report, PeerJ PrePrints, 2015

    2015

  46. [52]

    Wilson, Twenty combinatorial examples of asymptotics derived from multivariate generating functions , SIAM Review, Volume 50, Number 2, Pages 199–272, SIAM, 2008

    Robin Pemantle and Mark C. Wilson, Twenty combinatorial examples of asymptotics derived from multivariate generating functions , SIAM Review, Volume 50, Number 2, Pages 199–272, SIAM, 2008

    2008

  47. [53]

    Robin Pemantle, Hyperbolicity and stable polynomials in combinatorics and probability, arXiv preprint arXiv:1210.3231, 2012

    2012 arXiv

  48. [54]

    Kyle Petersen, Eulerian numbers, Springer, 2015

    T. Kyle Petersen, Eulerian numbers, Springer, 2015

    2015

  49. [55]

    Goldman, Some geometric results in semidefinite programming, Journal of Global Optimization, Volume 7, Number 1, Pages 33–50, Citeseer, 1995

    Motakuri Ramana and Alan J. Goldman, Some geometric results in semidefinite programming, Journal of Global Optimization, Volume 7, Number 1, Pages 33–50, Citeseer, 1995

    1995

  50. [56]

    Rump, Computational error bounds for multiple or nearly multiple eigenvalues , Linear Algebra and its Applications, Volume 324, Numbers 1–3, Pages 209–226, Elsevier, 2001

    Siegfried M. Rump, Computational error bounds for multiple or nearly multiple eigenvalues , Linear Algebra and its Applications, Volume 324, Numbers 1–3, Pages 209–226, Elsevier, 2001

    2001

  51. [57]

    Rajarshi Saha, Varun Srivastava, and Mert Pilanci, Matrix compression via randomized low rank and low precision factorization , Advances in Neural Information Processing Systems, Volume 36, 2023

    2023

  52. [58]

    How deviant can you be?

    Samuelson, Paul A. “How deviant can you be?” Journal of the American Statistical Association, vol. 63, no. 324, 1968, pp. 1522–1525. Taylor & Francis

    1968

  53. [61]

    Dominique Unruh, Quantum Hoare logic with ghost variables , In *Pro- ceedings of the 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)*, Pages 1–13, IEEE, 2019

    2019

  54. [62]

    William Helton*, Pages 325–349, Springer, 2012

    Victor Vinnikov, LMI representations of convex semialgebraic sets and determinantal representations of algebraic hypersurfaces: past, present, and future , in *Mathematical Methods in Systems, Optimization, and 74 Control: Festschrift in Honor of J. William Helton*, Pages 325–...

    2012

  55. [63]

    Mirk´ o Visontai and Nathan Williams, Stable multivariate W-Eulerian polynomials, Journal of Combinatorial Theory, Series A, Volume 120, Number 7, Pages 1929–1945, Elsevier, 2013

    1929

  56. [64]

    53–84, 2011

    David Wagner, Multivariate stable polynomials: theory and applications , Bulletin of the American Mathematical Society, Volume 48, Number 1, pp. 53–84, 2011

    2011

  57. [65]

    Wilf, generatingfunctionology, CRC Press, 2005

    Herbert S. Wilf, generatingfunctionology, CRC Press, 2005

    2005

  58. [66]

    P. B. Zhang, X. Zhang, Multivariate stable Eulerian polynomials on segmented permutations, European J. Combin., 78 (2019), 155–162

    2019

  59. [67]

    Bao-Xuan Zhu, A generalized Eulerian triangle from staircase tableaux and tree-like tableaux , Journal of Combinatorial Theory, Series A, Vol- ume 172, Article 105206, 2020

    2020

  60. [68]

    The finite element method: its basis and fundamentals

    Zienkiewicz, Olgierd Cecil, Taylor, Robert Leroy, and Zhu, Jian Z. The finite element method: its basis and fundamentals . Elsevier, 2005

    2005

  61. [69]

    Richard Zippel, Simplification of expressions involving radicals , Journal of Symbolic Computation, Volume 1, Number 2, Pages 189–210, Else- vier, 1985. 75

    1985

Pith tools

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