REVIEW 2 major objections 5 minor 25 references
An SOS counterexample to an inequality of symmetric functions
T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read The paper refutes the 2011 converse conjecture for homogeneous symmetric functions by giving a sum-of-squares certificate that $H_{44}-H_{521}$ is nonnegative despite incomparable dominance.
desk verdict A real SOS certificate and a clever method, but the advertised disproof covers only n=3 and the universal CGS conjecture remains open. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing mechanism is the Gram-matrix characterization of sums of squares: a homogeneous polynomial $h$ of degree $2d$ is a sum of squares iff $h = m^T A m$ for some positive semidefinite matrix $A$ indexed by monomials of degree $d$. The paper turns this into a semidefinite program whose solution must exactly reproduce $h = (H_{44}-H_{521})(x_1^2,x_2^2,x_3^2)$, then uses three structural reductions: symmetrization with respect to the variable-permutation group, linear constraints forced by real zeros of $h$, and rational rounding to convert floating-point output into an exact rational matrix. Factoring this $45\times 45$ matrix yields the 41 squares.
What would settle it
Expand the 41 listed square summands symbolically and compare with the displayed polynomial: exact equality verifies the three-variable certificate. To test the universal claim, run the same semidefinite search for $(H_{44}-H_{521})(x_1^2,\ldots,x_n^2)$ with $n=4$; a negative value or an infeasible SOS program at any point would show the three-variable proof does not extend to the definition's 'any number of variables'.
Extended reading notes
Core claim
The central claim, Theorem 2, is that $H_{44}-H_{521} \geq 0$ provides a degree-minimal counterexample to the proposed converse: the partitions $44$ and $521$ are incomparable in dominance order, yet the normalized homogeneous symmetric function difference is nonnegative after the substitution $x_i \mapsto x_i^2$, which certifies nonnegativity on the nonnegative orthant. The nonnegativity is not asserted abstractly: the paper gives a rational sum-of-squares decomposition into 41 squares, produced by solving a semidefinite program, imposing $S_3$-symmetry and real-zero constraints, rounding the numerical Gram matrix to exact rational entries, and factoring it. Because the certificate is explicit and checkable by squaring and summing, the counterexample is a proof rather than a numerical heuristic. The paper also reports many further numerical counterexamples in degrees 8, 9, and 10 and organizes them in a poset showing how dominance order would need to be modified.
Load-bearing premise
The proof certifies nonnegativity only in three variables, while the statement being refuted requires the inequality to hold for every number of variables $n$, and no argument in the paper bridges that gap.
Editorial extensions
If this is right
- The 2011 converse conjecture is false as stated for the homogeneous basis, so dominance order cannot fully classify normalized homogeneous symmetric function inequalities.
- The explicit certificate upgrades the counterexample from a numerical find to a checkable algebraic proof, and it shows the existence of rational SOS certificates at degree 16 despite known obstructions at degree 4.
- The paper's poset of numerically certified differences suggests that many more incomparable pairs satisfy the inequality; converting those to exact certificates would map the true nonnegativity order for fixed small $n$.
- For the paper's stated definition, which demands the inequality for any number of variables, the proof covers only $n=3$; whether the counterexample survives as $n$ grows is explicitly left open.
Reading between the lines
- If the same symmetry-plus-real-zeros rounding pipeline were run in four or more variables, it would either extend the counterexample or show that the three-variable certificate is an artifact of low dimension, directly addressing the paper's open large-$n$ question.
- The appearance of a rational SOS certificate at degree 16 with large denominators suggests that exact rational certificates may be common in the interior of the SOS cone, so symmetry reduction may be more important than raw degree for finding them.
- The modified poset in Section 3 could be tested against candidate partial orders generated by weighted cumulative sums; the blue arrows would provide immediate falsification data for any proposed order.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper addresses the Cuttler-Greene-Skandera conjecture on term-normalized homogeneous symmetric functions. It claims that H44−H521 is nonnegative on the nonnegative orthant even though the partitions (4,4) and (5,2,1) are incomparable in dominance order, and that this gives a degree-minimal counterexample to the conjecture. The proof is computational: an SDP is run in three variables, symmetrized, constrained using real zeros, and then rounded to exact rational data, yielding a claimed representation of (H44−H521)(x1^2,x2^2,x3^2) as a sum of 41 rational squares. The full certificate is not printed; the paper refers to an external GitHub repository and a Max Planck mathrepo page. Section 3 reports additional numerical SOS certificates for incomparable pairs in degrees 8, 9, and 10.
Significance. If the three-variable certificate is correct, it is a nontrivial computational result: it is a clean example of exact rational SOS certification for a degree-16 polynomial, and it demonstrates the practical value of symmetry reduction and real-zero constraints in semidefinite programming. However, the advertised significance as a disproof of the CGS conjecture depends on a quantifier that the proof does not deliver. As written, the paper establishes a fixed-n=3 SOS statement, not the universal inequality required by the paper's own definition in Section 1. The computational certificate, if fully supplied with a permanent location, could still be a useful benchmark contribution, but the main claim needs substantial revision.
major comments (2)
- [Section 1 and Section 4] The paper defines Gλ ≥ Gμ as the inequality holding for any number of variables n, and the abstract and Theorem 2 claim to disprove the CGS conjecture. The proof in Section 4 only certifies nonnegativity on R^3_{\ge 0}, via the SOS decomposition of (H44−H521)(x1^2,x2^2,x3^2). No argument is given that this three-variable certificate extends to n>3; setting extra variables to zero does not reduce the n-variable inequality to the certified one because the normalization denominators hλ(1^n) and hμ(1^n) depend on n. Remark 6 explicitly leaves open whether counterexamples persist for large n. Thus the stated universal counterexample is not established; the proven statement is a fixed-n=3 SOS result.
- [Section 4 and reference [14]] The claimed explicit SOS certificate is not self-contained. The proof prints the first row of the Gram matrix and the first square, and states that the remaining squares are in a .txt file on a GitHub repository and a mathrepo page. Since Theorem 2 rests entirely on this certificate, the full list of 41 squares, or the verifying code, should be included as supplementary/ancillary material with a permanent identifier. As submitted, the reader cannot verify Theorem 2 from the text alone.
minor comments (5)
- [Abstract and Section 4] The abstract says the paper provides a sums-of-squares decomposition of H44−H521, but the certificate is for (H44−H521)(x1^2,x2^2,x3^2); this wording should be corrected.
- [Section 4] The phrase "nonnegative octant" should be "nonnegative orthant".
- [Theorem 2] The notation in Theorem 2, "Hµ−Hλ≥ 0 ... provided by H44−H521," should specify explicitly which of the two partitions is λ and which is μ.
- [Section 3] The text states that a poset of SOS certifications is provided below, but the figure or diagram is not present in this version; it should be included.
- [Section 1] The definition of term-normalized symmetric functions should state the condition on the number of variables n needed so that g(1) is nonzero, since for l(λ)>n the normalized polynomial is not defined.
Circularity Check
No circularity: the SOS certificate is independently checkable and does not reduce to its inputs.
full rationale
The central result is an explicit rational sum-of-squares decomposition of (H_44 - H_521)(x1^2, x2^2, x3^2), and the paper supplies the first squared polynomial in the text, refers to a file with the remaining squares, and provides open-source code that squares and sums them. This is a self-contained, externally verifiable certificate: nonnegativity on R^3_{>=0} follows by Lemma 1 from a literal SOS identity, not from any fitted parameter renamed as a prediction or from any equation that defines the target in terms of the certificate. The SDP search is only heuristic for finding the certificate; the proof is the exact rational identity. The cited symmetry and rational-rounding tools (Gatermann-Parrilo, RealCertify) are used as computational aids, not as load-bearing self-citations, and the authors' own website is data/code rather than an imported uniqueness theorem. The remaining gap between the 3-variable certificate and the all-n definition of G_lambda >= G_mu is a correctness/scope issue, not a circularity: Remark 6 openly questions the large-n behavior, and no equation in the paper makes the all-n claim equal to the 3-variable SOS by construction. Hence no circular step is exhibited.
Assumptions & free parameters
assumptions (2)
- domain assumption The relation Gλ ≥ Gμ is defined as holding for any number of variables n on the nonnegative orthant.
- standard math The known equivalence results for monomial, elementary, power-sum, and Schur bases in Theorem 1 (from [10]) are correct.
Cite this review
Pith. "Pith review of An SOS counterexample to an inequality of symmetric functions." pith.science (2026). https://pith.science/paper/QB4U3UK6
@misc{pith2026190900081,
author = {Pith},
title = {Pith review of: An SOS counterexample to an inequality of symmetric functions},
year = {2026},
howpublished = {\url{https://pith.science/paper/QB4U3UK6}},
note = {Machine review of arXiv:1909.00081}
}
abstract
It is known that differences of symmetric functions corresponding to various bases are nonnegative on the nonnegative orthant exactly when the partitions defining them are comparable in dominance order. The only exception is the case of homogeneous symmetric functions where it is only known that dominance of the partitions implies nonnegativity of the corresponding difference of symmetric functions. It was conjectured by Cuttler, Greene, and Skandera in 2011 that the converse also holds, as in the cases of the monomial, elementary, power-sum, and Schur bases. In this paper we provide a counterexample, showing that homogeneous symmetric functions break the pattern. We use semidefinite programming to find an explicit sums of squares decomposition of the polynomial $H_{44} - H_{521}$ as a sum of 41 squares. This rational certificate of nonnegativity disproves the conjecture, since a polynomial which is a sum of squares cannot be negative, and since the partitions 44 and 521 are incomparable in dominance order.
Reference graph
Works this paper leans on
-
[1]
There are significantly more nonnegative polynomials than sums of squares
Grigoriy Blekherman. There are significantly more nonnegative polynomials than sums of squares. Israel J. Math., 153:355–380, 2006
work page 2006
-
[2]
Nonnegative polynomials and sums of squares
Grigoriy Blekherman. Nonnegative polynomials and sums of squares. In Semidefinite optimization and convex algebraic geometry, volume 13 of MOS-SIAM Ser. Optim. , pages 159–202. SIAM, Philadelphia, PA, 2013
work page 2013
-
[3]
Algebraic boundaries of Hilbert’s SOS cones
Grigoriy Blekherman, Jonathan Hauenstein, John Christian Ottem, Kristian Ranestad, and Bernd Sturmfels. Algebraic boundaries of Hilbert’s SOS cones. Compos. Math., 148(6):1717–1735, 2012. 8
work page 2012
-
[4]
Dimensional differences between faces of the cones of nonnegative polynomials and sums of squares
Grigoriy Blekherman, Sadik Iliman, and Martina Juhnke-Kubitzke. Dimensional differences between faces of the cones of nonnegative polynomials and sums of squares. InCombinatorial methods in topology and algebra, volume 12 of Springer INdAM Ser. , pages 69–77. Springer, Cham, 2015
work page 2015
-
[5]
Grigoriy Blekherman, Pablo A. Parrilo, and Rekha R. Thomas, editors. Semidefinite optimization and convex algebraic geometry, volume 13 of MOS-SIAM Series on Optimization . Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA, 2013
work page 2013
-
[6]
Symmetric non-negative forms and sums of squares
Grigoriy Blekherman and Cordian Riener. Symmetric non-negative forms and sums of squares. Discrete and Computational Geometry , 2012
work page 2012
-
[7]
M. D. Choi, T. Y. Lam, and Bruce Reznick. Even symmetric sextics. Math. Z., 195(4):559–580, 1987
work page 1987
-
[8]
Man Duen Choi and Tsit Yuen Lam. An old question of Hilbert. In Conference on Quadratic Forms— 1976 (Proc. Conf., Queen’s Univ., Kingston, Ont., 1976) , pages 385–405. Queen’s Papers in Pure and Appl. Math., No. 46, 1977
work page 1976
Show all 25 references
-
[9]
Sum of squares decomposition for symmetric polynomial inequalities
Logan Coe. Sum of squares decomposition for symmetric polynomial inequalities. San Francisco State University Master’s Thesis, 2017
2017
-
[10]
Inequalities for symmetric means
Allison Cuttler, Curtis Greene, and Mark Skandera. Inequalities for symmetric means. European J. Combin., 32(6):745–761, 2011
2011
-
[11]
Karin Gatermann and Pablo A. Parrilo. Symmetry groups, semidefinite programs, and sums of squares. J. Pure Appl. Algebra , 192(1-3):95–128, 2004
2004
-
[12]
The analogue of Hilbert’s 1888 theorem for even symmetric forms
Charu Goel, Salma Kuhlmann, and Bruce Reznick. The analogue of Hilbert’s 1888 theorem for even symmetric forms. J. Pure Appl. Algebra , 221(6):1438–1448, 2017
2017
-
[13]
A general formula for the algebraic degree in semidefinite programming
Hans-Christian Graf von Bothmer and Kristian Ranestad. A general formula for the algebraic degree in semidefinite programming. Bull. Lond. Math. Soc. , 41(2):193–197, 2009
2009
-
[14]
Sos counterexample
Alexander Heaton and Isabelle Shankar. Sos counterexample. https://github.com/alexheaton2/ SOS-counterexample. Accessed: 2019-09-18
2019
-
[15]
Exact algorithms for linear matrix inequalities
Didier Henrion, Simone Naldi, and Mohab Safey El Din. Exact algorithms for linear matrix inequalities. SIAM J. Optim. , 26(4):2512–2539, 2016
2016
-
[16]
¨Uber den Vergleich des arithmetischen und des geometrischen Mittels
Adolf Hurwitz. ¨Uber den Vergleich des arithmetischen und des geometrischen Mittels. Journal f¨ ur die reine und angewandte Mathematik , 108:266–268, 1891
-
[17]
On exact Polya and Putinar’s representations
Victor Magron and Mohab Safey El Din. On exact Polya and Putinar’s representations. In ISSAC’18— Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation , pages 279–286. ACM, New York, 2018
2018
-
[18]
Realcertify: A maple package for certifying non-negativity
Victor Magron and Mohab Safey El Din. Realcertify: A maple package for certifying non-negativity. arXiv: https://arxiv.org/abs/1805.02201 , 2018
2018 arXiv
-
[19]
Gromov-Witten theory and Noether-Lefschetz theory
Davesh Maulik and Rahul Pandharipande. Gromov-Witten theory and Noether-Lefschetz theory. In A celebration of algebraic geometry , volume 18 of Clay Math. Proc., pages 469–507. Amer. Math. Soc., Providence, RI, 2013
2013
-
[20]
R. F. Muirhead. Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters. Proceedings of the Edinburgh Mathematical Society, 21:144–162, 1902
1902
-
[21]
The algebraic degree of semidefinite program- ming
Jiawang Nie, Kristian Ranestad, and Bernd Sturmfels. The algebraic degree of semidefinite program- ming. Math. Program., 122(2, Ser. A):379–405, 2010
2010
-
[22]
A quantitative version of hurwitz’ theorem on the arithmetic-geometric inequality
Bruce Reznick. A quantitative version of hurwitz’ theorem on the arithmetic-geometric inequality. Journal fur die Reine und Angewandte Mathematik , 1987(377):108–112, January 1987. 9
1987
-
[23]
Sums of squares of polynomials with rational coefficients
Claus Scheiderer. Sums of squares of polynomials with rational coefficients. J. Eur. Math. Soc. (JEMS) , 18(7):1495–1513, 2016
2016
-
[24]
On inequalities for normalized Schur functions
Suvrit Sra. On inequalities for normalized Schur functions. European J. Combin., 51:492–494, 2016
2016
-
[25]
Richard P. Stanley. Enumerative combinatorics. Vol. 2 , volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. 10
1999
Reviewed August 14, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.