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  "slug": "ddvv-sum-of-squares-thresholds",
  "title": "Exact sum-of-squares thresholds for the DDVV inequality on real symmetric and Hermitian matrices",
  "shortTitle": "The DDVV commutator inequality holds with constant 1, but its simplest sum-of-squares proof needs 9/8 for 3 × 3 matrices and up to 3 in general",
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  "oneLine": "For every matrix size n and number of matrices m, the exact least constant for which the DDVV commutator inequality has a quadratic sum-of-squares proof: the true constant 1 only when n or m is 2, 9/8 for 3 × 3 real symmetric matrices, approaching but never reaching 3; for Hermitian matrices the true constant (4/3, or 1 for two matrices) only when n = 2, m = 2, or m = 3 with n ≤ 7, up to 4 in general.",
  "abstract": "The DDVV inequality, which began as a conjecture in submanifold geometry and is now a theorem, says that for real symmetric n × n matrices B_1, …, B_m the sum of the squared Frobenius norms of all commutators [B_p, B_q] is at most (Σ_p ‖B_p‖_F²)². Ge, Li and Zhou asked whether inequalities of this type have sum-of-squares proofs. This candidate answers the question for quadratic sum-of-squares certificates, that is, sums of squares of quadratic forms in the matrix entries. For all n, m ≥ 2 it computes the least coefficient c for which c(Σ_p ‖B_p‖_F²)² − Σ_{p,q} ‖[B_p, B_q]‖_F² is such a sum of squares. For real symmetric matrices this least coefficient σ(n, m) equals the true constant 1 exactly when min(n, m) = 2. Otherwise it is an explicit three-branch rational function of n and m (Theorem 1.1): it equals 9/8 for 3 × 3 matrices whenever m ≥ 3, it is nondecreasing in both parameters, and its supremum is 3, not attained. For Hermitian matrices, whose sharp constant is 1 for m = 2 and 4/3 for m ≥ 3, an analogous four-branch formula σ_H(n, m) is proved (Theorem 1.2). The sharp Hermitian inequality has a quadratic sum-of-squares proof exactly when n = 2, m = 2, or m = 3 and n ≤ 7, and the supremum of σ_H is 4. Both proofs average Gram matrices over the symmetry group, which reduces every certificate to a cone with ten explicit generators, and then match explicit sum-of-squares identities with explicit dual functionals; the thresholds are also the exact values of the degree-four moment relaxation. Scripts in exact arithmetic replay the identities and several independent finite certificates. AI-assisted, unrefereed candidate; no human peer review, independent reproduction or proof-assistant formalisation is claimed.",
  "datePublished": "2026-11-03",
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  "keywords": [
    "DDVV inequality",
    "commutator",
    "sum of squares",
    "semidefinite programming",
    "moment relaxation",
    "symmetry reduction",
    "Hermitian matrices",
    "normal scalar curvature"
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    "Theorem 1.1 (proof): for real symmetric tuples, σ(n, m) = 1 iff min(n, m) = 2; for n, m ≥ 3, σ = 3n²(m−1)/(mn²+7mn−6m−12n+12) if m(n+12) ≥ 6(n+4), 4n(n+8)/((n+2)(3n+14)) if m = 3 and n ∈ {5, 6}, and 4(m−1)(n+2)/(mn+7m+2n−2) otherwise.",
    "Corollary 1.3 (proof): the sharp defect is a sum of squares iff min(n, m) = 2 (for n, m ≥ 3 it is nonnegative but not a sum of squares of polynomials of any degree); σ(3, m) = 9/8 for every m ≥ 3; σ is nondecreasing in n and m; sup σ = 3, not attained.",
    "Theorem 1.2 (proof): for Hermitian tuples, σ_H(n, 2) = 1, σ_H = 4/3 iff n = 2 (m ≥ 3) or m = 3 and n ≤ 7, and an explicit four-branch formula otherwise; sup σ_H = 4, not attained; σ_H ≥ σ.",
    "Sections 7–8 (proof): the skew-Hermitian threshold equals σ_H; restriction bounds for complex symmetric, general real and general complex tuples; σ and σ_H are the exact values of the degree-four moment relaxation, with optimal pseudo-moments given explicitly.",
    "Section 9 (exact certificates): an independent zero-set obstruction at (3, 3) and exact PSD moment matrices at (3, 3) and (4, 3), regenerated byte-identically."
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  "relatedWorks": [
    {
      "citation": "Ge, Li and Zhou (2021). Some generalizations of the DDVV and BW inequalities. Transactions of the American Mathematical Society. Asked whether DDVV-type inequalities have sum-of-squares proofs; the source question answered here for real symmetric and Hermitian matrices, and the Hermitian constant 4/3.",
      "url": "https://doi.org/10.1090/tran/8360",
      "doi": "10.1090/tran/8360"
    },
    {
      "citation": "Ge and Tang (2008). A proof of the DDVV conjecture and its equality case. Pacific Journal of Mathematics. The DDVV inequality with constant 1, proved; nonnegativity at the sharp constant is prior work.",
      "url": "https://doi.org/10.2140/pjm.2008.237.87",
      "doi": "10.2140/pjm.2008.237.87"
    },
    {
      "citation": "Lu (2011). Normal Scalar Curvature Conjecture and its applications. Journal of Functional Analysis. The independent proof of the DDVV conjecture.",
      "url": "https://doi.org/10.1016/j.jfa.2011.05.002",
      "doi": "10.1016/j.jfa.2011.05.002"
    },
    {
      "citation": "Ge, Xu, You and Zhou (2017). DDVV-type inequality for Hermitian matrices. Linear Algebra and its Applications. The sharp Hermitian constant 4/3 used in Theorem 1.2.",
      "url": "https://doi.org/10.1016/j.laa.2017.04.028",
      "doi": "10.1016/j.laa.2017.04.028"
    },
    {
      "citation": "Dillen, Fastenakels and Van der Veken (2007). A pinching theorem for the normal scalar curvature of invariant submanifolds. Journal of Geometry and Physics. Geometric background of the conjecture.",
      "url": "https://doi.org/10.1016/j.geomphys.2006.06.006",
      "doi": "10.1016/j.geomphys.2006.06.006"
    },
    {
      "citation": "Chern, do Carmo and Kobayashi (1970). Minimal Submanifolds of a Sphere with Second Fundamental Form of Constant Length. Functional Analysis and Related Fields. The two-matrix commutator estimate behind the m = 2 case.",
      "url": "https://doi.org/10.1007/978-3-642-48272-4_2",
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    },
    {
      "citation": "Böttcher and Wenzel (2008). The Frobenius norm and the commutator. Linear Algebra and its Applications. The Böttcher–Wenzel commutator inequality, a related matrix inequality.",
      "url": "https://doi.org/10.1016/j.laa.2008.05.020",
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    },
    {
      "citation": "Hilbert (1888). Ueber die Darstellung definiter Formen als Summe von Formenquadraten. Mathematische Annalen. Nonnegative forms need not be sums of squares, the distinction this paper quantifies.",
      "url": "https://doi.org/10.1007/bf01443605",
      "doi": "10.1007/bf01443605"
    },
    {
      "citation": "Choi and Lam (1977). Extremal positive semidefinite forms. Mathematische Annalen. The zero-set obstruction used for the independent certificate at (3, 3).",
      "url": "https://doi.org/10.1007/bf01360024",
      "doi": "10.1007/bf01360024"
    },
    {
      "citation": "Gatermann and Parrilo (2004). Symmetry groups, semidefinite programs, and sums of squares. Journal of Pure and Applied Algebra. The general symmetry-reduction method the proofs specialise.",
      "url": "https://doi.org/10.1016/j.jpaa.2003.12.011",
      "doi": "10.1016/j.jpaa.2003.12.011"
    },
    {
      "citation": "Riener, Theobald, Andrén and Lasserre (2013). Exploiting symmetries in SDP-relaxations for polynomial optimization. Mathematics of Operations Research. Symmetry reduction of moment relaxations (Section 8).",
      "url": "https://doi.org/10.1287/moor.1120.0558",
      "doi": "10.1287/moor.1120.0558"
    },
    {
      "citation": "Ge and Tang (2022). Isoparametric Polynomials and Sums of Squares. International Mathematics Research Notices. Related non-SOS results for isoparametric quartics.",
      "url": "https://doi.org/10.1093/imrn/rnac297",
      "doi": "10.1093/imrn/rnac297"
    },
    {
      "citation": "Ge (2014). DDVV-type inequality for skew-symmetric matrices and Simons-type inequality for Riemannian submersions. Advances in Mathematics. Skew-symmetric DDVV, a matrix type not classified here.",
      "url": "https://doi.org/10.1016/j.aim.2013.10.010",
      "doi": "10.1016/j.aim.2013.10.010"
    },
    {
      "citation": "Klep, Magron and Volčič (2021). Optimization Over Trace Polynomials. Annales Henri Poincaré. The dimension-free trace-polynomial setting discussed in Section 8.",
      "url": "https://doi.org/10.1007/s00023-021-01095-4",
      "doi": "10.1007/s00023-021-01095-4"
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