Postive Semidefinite and Sum of Squares Biquadratic Polynomials

Chunfeng Cui; Liqun Qi; Yi Xu

REVIEW 3 major objections 7 minor 14 references

Postive Semidefinite and Sum of Squares Biquadratic Polynomials

T0 review · 3 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that every 2×2 PSD biquadratic polynomial—a homogeneous quartic in four variables that is nonnegative everywhere—is a sum of squares of three quadratic polynomials, with the proof running through a reduction to a…

desk verdict New 3-square certificate for 2x2 PSD biquadratic forms via a tripartite reduction; the proof has a repairable gap and Section 4 needs corrections. read the letter →

arxiv 2506.17260 v4 pith:4HUS2MUY submitted 2025-06-09 math.OC

Chunfeng Cui , Liqun Qi , Yi Xu This is my paper

classification math.OC MSC 11E2512D1514P1015A6990C23

keywords biquadraticpolynomialssumofsquarespositivesemidefinitetripartitequarticSOSrankternaryquarticsNewtonpolytopetensors

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

This paper's central claim is that every $2\times 2$ biquadratic polynomial—a homogeneous quartic in four variables, two $x$-variables and two $y$-variables, of the form $f(x,y)=\sum a_{ijkl}x_ix_jy_ky_l$—that is nonnegative on $\mathbb{R}^4$ can be written as a sum of squares of three quadratic polynomials. The route is a reduction: any $m\times n$ biquadratic is shown to be equivalent, for both nonnegativity and sum-of-squares status, to a tripartite quartic in $m+n-1$ variables, and for $m=n=2$ that reduced object is exactly a ternary quartic, where a classical 1888 theorem guarantees three squares. If this is right, the known certificate size for $2\times2$ biquadratics drops from nine squares to three. The paper also gives a necessary and sufficient condition for a general $m\times n$ PSD biquadratic to be SOS, with SOS rank at most $mn$, and constructs explicit three-square decompositions in three $2\times2$ coefficient regimes.

What carries the argument

The central object is the tripartite quartic polynomial $h(x,y,z)=h_0z^4+h_1(x,y)z^3+h_2(x,y)z^2+h_3(x,y)z+h_4(x,y)$, obtained by setting $x_m=y_n=1$ and homogenizing with a single flexible variable $z$. It carries the argument because the paper proves that $f$ is PSD if and only if $h$ is PSD, and $f$ is SOS if and only if $h$ is SOS; the 1888 ternary-quartic theorem then applies exactly when $m=n=2$. The second piece of machinery is the parameterized flattening $M(\Gamma)=B+P(\Gamma)$, with $z=x\otimes y$, where $\Gamma$ encodes the non-unique split of full-cross terms; existence of $\Gamma$ with $M(\Gamma)\succeq0$ characterizes SOS biquadratics.

What would settle it

For a dense set of $2\times2$ PSD biquadratic polynomials with coefficients satisfying the paper's inequalities (10), solve the semidefinite program that minimizes the rank of $M(\Gamma)$ subject to $M(\Gamma)\succeq0$; if any such polynomial has minimum rank 4, Theorem 2.1's three-square conclusion is false, because a three-square certificate would give rank at most 3.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that biquadratic structure compensates for having four variables: although a general PSD quartic in four variables need not be SOS, a $2\times2$ biquadratic always is. The proof places the object in a tripartite quartic $h(x,y,z)=h_0z^4+h_1(x,y)z^3+h_2(x,y)z^2+h_3(x,y)z+h_4(x,y)$ in three variables, preserving PSD and SOS equivalence. The 1888 theorem for ternary quartics then gives three quadratic summands for $h$, and the paper asserts these convert back to three quadratic summands for the original biquadratic. In the general $m\times n$ setting, the paper characterizes SOS by the existence of a parameter matrix $\Gamma$ such that the flattened matrix $M(\Gamma)=B+P(\Gamma)$ is positive semidefinite, which forces SOS rank at most $mn$ whenever $f$ is SOS.

Load-bearing premise

The argument's weakest point is the unproved assertion that after setting one $x$ and one $y$ variable to 1, a sum-of-squares decomposition of the smaller polynomial can always be lifted back to a sum of squares of bilinear forms in the original variables.

Editorial extensions

If this is right

For $2\times2$ PSD biquadratics, the SOS certificate uses three squares, improving the previous nine-square bound from 1973.
If an $m\times n$ PSD biquadratic is SOS, it admits an SOS decomposition with at most $mn$ squares whose coefficient vectors can be taken orthogonal; by the paper's equivalence, this amounts to choosing $\Gamma$ with $M(\Gamma)\succeq0$.
The PSD-SOS problem for biquadratics is equivalently a problem about tripartite quartics in one fewer variable, so any algorithmic or theoretical progress on tripartite quartics transfers directly.
When the tripartite quartic is degenerated, meaning $h_0=0$, PSD is equivalent to $h_1\equiv0$ and three explicit PSD conditions involving $h_2$, $h_4$, and $4h_2h_4-h_3^2$, giving a checkable certificate in that regime.
For the three $2\times2$ cases treated constructively, the coefficient inequalities given are necessary and sufficient for PSD and simultaneously yield an explicit SOS decomposition.

Reading between the lines

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

The rank bound $mn$ from the general theorem is likely not tight: for $m=n=2$ it gives 4 while the paper's main theorem gives 3, so a natural extension is to determine the maximal SOS rank for $m\times2$ and $3\times3$ biquadratics, which the paper leaves open.
Because the reduction identifies biquadratics with tripartite quartics, a numerical SDP that solves the ternary-quartic SOS problem could serve as a practical SOS test for $2\times2$ biquadratics without expanding to the full four-variable certificate.
The three explicit cases cover no half-cross terms, one half-cross term, and two neighbor half-cross terms without a full-cross term; the omitted case with two half-cross terms plus a full-cross term appears to be the next testbed for a fully constructive classification of all $2\times2$ SOS biquadratics.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear Colleague,

The headline is that this paper has a real new idea—the tripartite reduction of an m×n biquadratic form to a homogeneous quartic in m+n−1 variables—and it uses that reduction with Hilbert's theorem to claim that every 2×2 PSD biquadratic form is a sum of three squares of quadratics, improving Calderon's nine-square certificate. I believe the result is likely true, but the written proof leaves a load-bearing gap, and Section 4 has inconsistencies.

What's good: Theorem 3.1 gives a clean characterization: a PSD biquadratic is SOS iff some matrix M(Γ) in the affine Gram-matrix set is PSD, and then the SOS rank is at most mn. The proof of that theorem is standard but correct. The main idea—reducing to the ternary quartic case where Hilbert's theorem applies—is elegant, and the 3-square conclusion is a genuine improvement over Calderon.

The soft spot is Theorem 2.1. The equivalence 'f is PSD iff g is PSD iff h is PSD' is fine. The SOS direction is not. When you dehomogenize and then homogenize with a single z, an SOS of h may have square factors that are not in the image of the substitution x_m = y_n = z. A factor containing x_i x_j or y_k y_l would produce monomials in its square that don't occur in h, so such factors can't appear; but this must be argued. The missing tool is Reznick's Newton-polytope lemma, which the paper actually cites later. With it, every square factor's support lies in half the Newton polytope, so each factor is affine-bilinear in (x,y,z) and lifts uniquely to a bilinear form. As written, the proof simply asserts 'f is SOS iff h is SOS' and the 2×2 result does not follow from Hilbert's theorem without this step. It's repairable, but it needs to be written down.

Section 4 also needs care. The threshold formulas in Theorem 4.8 don't match the expressions derived in the proof; the case with c_x = c_y and a11 between c_y^2 and 5/4 c_y^2 derives γ*_3 = 1/(a11 − c_y^2) but the statement and Example 4.9 use a different, more elaborate expression. The numerical value 4.8431 in the example is inconsistent with the displayed formula. These are fixable mistakes, but they make the constructive part hard to trust as it stands.

Who should read this: anyone working on nonnegative biquadratic forms or low-dimensional SOS theory. The main claim is new and likely correct, and the gap is fillable. I would not cite it in its current form, but I would send it to a serious referee and ask the authors to supply the lifting proof and correct Section 4. If those are done, the paper will be a solid contribution.

Referee Report

3 major / 7 minor

Summary. The paper studies the relationship between positive semidefinite (PSD) biquadratic polynomials and sum-of-squares (SOS) decompositions. The main theorem (Thm 2.1) claims that an m×n biquadratic f is PSD (resp. SOS) if and only if an associated tripartite homogeneous quartic h in m+n−1 variables is PSD (resp. SOS); for m=n=2 this gives, via Hilbert's theorem on ternary quartics, that every PSD 2×2 biquadratic is an SOS of three quadratic polynomials, improving Calderón's nine-square certificate. Theorem 3.1 reformulates SOS of a PSD biquadratic as existence of a PSD matrix M(Γ) in a gauge family, with SOS rank at most mn; Theorem 4.8 and the surrounding propositions provide explicit SOS constructions for three restricted families of 2×2 PSD biquadratics.

Significance. If the main theorem is fully established, the paper gives a clean and elegant route to a known but nontrivial result: every 2×2 PSD biquadratic is SOS, with the sharp number three of squares. The gauge-matrix characterization in Theorem 3.1 is a useful reformulation with a finite-dimensional certificate, and the constructive cases in Section 4 provide concrete decompositions. The use of Reznick's Newton-polytope lemma, though only cited, is appropriate. However, the central SOS lifting step is not proved in the manuscript, and there are local errors in Theorem 3.1 and Theorem 4.8, so the present version is not yet publication-ready.

major comments (3)

[Theorem 2.1 and 2.2 (proof of SOS equivalence)] The proof asserts 'f is SOS if and only if g is SOS' after setting x_m=y_n=1 and then 'g is SOS if and only if h is SOS' after homogenizing. The only nontrivial direction is lifting an SOS representation of g (or h) to an SOS representation of f as squares of bilinear forms. This requires the Newton-polytope support lemma of Reznick [11]: since every variable has exponent at most two in New(f) (resp. New(g)), half the Newton polytope forces each square factor to have exponent at most one in each x- and y-variable. Without this argument, the chain h SOS ⇒ g SOS ⇒ f SOS does not follow, and Hilbert's theorem cannot be applied. The fix is straightforward, but as written Theorem 2.1 and the paper's main claim are not proved.
[Theorem 3.1, proof of (ii)⇒(i)] The proof states that Cholesky decomposition M(Γ)=CC^T gives f=Σ(c_t^T z)^2 with orthogonal coefficients. For C with columns c_t, M=CC^T=Σ c_t c_t^T, and the c_t are not orthogonal in general. Orthogonality is instead obtained from an eigendecomposition M=QΛQ^T, taking coefficient vectors √λ_t q_t. The statement of (i) is true, but the supplied proof is incorrect.
[Theorem 4.8, condition (c), second case] In the subcase cx=cy and c_y^2 ≤ a11 ≤ 5/4 c_y^2, the proof of Case (i2) derives the threshold a22 ≥ 1/(a11−c_y^2)=1/(a11−c_x^2), but the displayed formula in condition (c) for this subcase contains an additional term −4/(3c_y−√(9c_y^2−4a11))^2. This extra term is not justified and contradicts the proof; Example 4.9 uses the erroneous numerical value 4.8431 instead of 5. Please correct the theorem and the example.

minor comments (7)

[Title] The title misspells 'Positive' as 'Postive'.
[Abstract and Section 1] The abstract credits the first explicit PNS quartic in four variables to Choi (1975) [3], while the introduction credits Choi and Lam (1977) [4]; these statements should be harmonized.
[Section 3, definition of B] The tensor symmetry is stated as a_{ijkl}=a_{klij}, while the polynomial representation (2) also requires symmetry under i↔j and k↔l; please state the full symmetry or explain the convention.
[Lemmas 4.4 and 4.5, Theorem 4.5] Expressions such as √a11a22 and √a12a21 are ambiguous; use √(a11a22) and √(a12a21).
[Theorem 4.8, proof of Case (i1)] The displayed γ3* is written as c_y/(α*)^4 −1/(α*)^2; the preceding equations give c_y/(α*)^3 −1/(α*)^2, which matches the denominator (3c_y−√(9c_y^2−4a11))^3 in the theorem statement.
[Section 4, after Proposition 4.2] The phrase 'the 2 × 2 bipartite polynomial' should be 'the 2 × 2 biquadratic polynomial'; also, the sentence 'Some final remarks and open questions are made and raised' is awkward and should be rewritten.
[Theorem 2.1 proof] The PSD equivalence after setting x_m=y_n=1 should be justified explicitly using the two-variable homogeneity f(λx,μy)=λ^2μ^2f(x,y), including continuity at points where the normalization variables vanish.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 2.1 reduces to Hilbert's theorem and Reznick's external Newton-polytope lemma, not to the paper's own inputs.

full rationale

The central derivation is not circular. Theorem 2.1 maps an m×n biquadratic f to a tripartite quartic h and invokes Hilbert's 1888 theorem (external). The PSD equivalences are valid by homogeneity: evaluating at x_m=y_n=1 and rehomogenizing with z preserves nonnegativity. The nontrivial SOS-lifting direction (g SOS or h SOS implies f SOS) is asserted in the proof without proof; it requires Reznick's Newton-polytope support lemma, which is external and is in fact cited later in Theorem 3.1 ('By Theorem 1 in [11]'). This is a proof-completeness gap, not a circular reduction, because the missing ingredient is not an input of the paper and does not rest on the conclusion. Theorem 3.1 is an exact matrix reformulation of SOS (existence of Γ with M(Γ) PSD and Cholesky decomposition), not a fitted parameter renamed as a prediction. The constructive sections 4.1–4.3 posit explicit SOS templates and verify both the SOS direction and the failure/PNS direction by coefficient comparison, so the ansatz is exhibited and checked rather than smuggled in. Self-citations [9] and [10] appear only for background and for one illustrative example; no load-bearing claim is justified solely by a self-citation. Accordingly no circular step can be exhibited, and the score is 0.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The paper adds no fitted constants and postulates no new physical entities. It relies on two external theorems, Hilbert and Reznick, and on an implicit homogenization fact. The main analytical device is the tripartite reduction with a flexible variable z, which is a coordinate transformation rather than a new entity.

free parameters (1)

Full-cross coefficient gauge Gamma = gamma = -3 in Example 3.2
Theorem 3.1 introduces a free gauge Gamma in the affine matrix M(Gamma). The example picks gamma = -3 to make M(Gamma) PSD. This is an existential certificate variable, not a fitted physical constant.

assumptions (3)

standard math Hilbert's 1888 theorem: every PSD homogeneous quartic in three variables is SOS of three quadratic forms.
Used in Theorem 2.1 to conclude the 2x2 result from the reduction to a ternary quartic.
standard math Reznick's Newton polytope lemma: in any SOS decomposition, each square's Newton polytope is contained in half the Newton polytope of the sum.
Used in the proof of Theorem 3.1 to force each square to be a bilinear form in z = x tensor y.
domain assumption Separate homogeneity of biquadratic forms: f(lambda x, mu y) = lambda^2 mu^2 f(x,y), so scaling the last x and last y to 1 preserves PSD on the dense open set.
Implicit in Theorem 2.1's substitution x_m = y_n = 1; boundary cases are handled by continuity, but the paper does not spell this out.

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Cite this review

Pith. "Pith review of Postive Semidefinite and Sum of Squares Biquadratic Polynomials." pith.science (2026). https://pith.science/paper/4HUS2MUY

@misc{pith2026250617260,
  author       = {Pith},
  title        = {Pith review of: Postive Semidefinite and Sum of Squares Biquadratic Polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4HUS2MUY}},
  note         = {Machine review of arXiv:2506.17260}
}

abstract

Hilbert proved in 1888 that a positive semi-definite (PSD) homogeneous quartic polynomial of three variables always can be expressed as the sum of squares (SOS) of three quadratic polynomials, and a psd homogeneous quartic polynomial of four variables may not be sos. Only after 87 years, in 1975, Choi gave the explicit expression of such a psd-not-sos (PNS) homogeneous quartic polynomial of four variables. An $m \times n$ biquadratic polynomial is a homogeneous quartic polynomial of $m+n$ variables. In this paper, we show that an $m \times n$ biquadratic polynomial can be expressed as a tripartite homogeneous quartic polynomial of $m+n-1$ variables. Therefore, {by Hilbert's theorem}, a $2 \times 2$ PSD biquadratic polynomial can be expressed as the sum of squares of three quadratic polynomials. This improves the result of Calder\'{o}n in 1973, who proved that a $2 \times 2$ biquadratic polynomial can be expressed as the sum of squares of nine quadratic polynomials. Furthermore, we present a necessary and sufficient condition for an $m \times n$ psd biquadratic polynomial to be sos, and show that if such a polynomial is sos, then its sos rank is at most $mn$. Then we give a constructive proof of the sos form of a $2 \times 2$ psd biquadratic polynomial in three cases.

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Works this paper leans on

14 extracted references · 14 canonical work pages

[11]

An elementary proof of Hilbert’s theorem on ternary quartics

A. Pfister and C. Scheiderer, “An elementary proof of Hilbert’s theorem on ternary quartics”, Journal of Algebra 371 (2012) 1-25

work page 2012
[1]

Calder\' o n, ``A note on biquadratic forms'', Linear Algebra and Its Applications 7 (1973) 175-177

A.P. Calder\' o n, ``A note on biquadratic forms'', Linear Algebra and Its Applications 7 (1973) 175-177

work page 1973
[2]

Chesi, ``On the gap between positive polynomials and SOS of polynomials'', IEEE Trans

G. Chesi, ``On the gap between positive polynomials and SOS of polynomials'', IEEE Trans. Automat. Control 52 (2007) 1066-1072.\

work page 2007
[3]

Choi, ``Positive semidefinite biquadratic forms'', Linear Algebra and Its Applications 12 (1975) 95-100

M.-D. Choi, ``Positive semidefinite biquadratic forms'', Linear Algebra and Its Applications 12 (1975) 95-100

work page 1975
[4]

Choi and T.-Y

M.-D. Choi and T.-Y. Lam, ``Extremal positive semi-definite forms'', Mathematische Annalen 231 (1977) 1-18

work page 1977
[5]

Nonnegative Biquadratic Tensors

C. Cui and L. Qi, ``Nonnegative biquadratic tensors'', March 2025, arXiv:2503.16176v2

work page Pith review arXiv 2025
[6]

Fidalgo and A

C. Fidalgo and A. Kovacec, ``Positive semi-definite diagonal minus tail forms are sums of squares'', Mathematische Zeitschrift 269 (2011) 629-645

work page 2011
[7]

C. Goel, S. Kuhlmann and B. Reznick, ``The analogue of Hilbert’s 1888 theorem for even symmetric forms'', Journal of Pure and Applied Algebra 221 (2017) 1438–1448

work page 2017

Show all 14 references

[8]

Hilbert, ``\" U ber die darstellung definiter formen als summe von formenquadraten'', Mathematische Annalen 32 (1888) 342-350

D. Hilbert, ``\" U ber die darstellung definiter formen als summe von formenquadraten'', Mathematische Annalen 32 (1888) 342-350
[9]

C. Ling, J. Nie, L. Qi and Y. Ye, ``Biquadratic optimization over unit spheres and semidefinite programming relaxations'', SIAM Journal on Optimization 20 (2010) 1286-1310

2010
[10]

Motzkin, ``The arithmetic geometry inequality'', in: Proceedings of the Symposium , Wright-Patterson Air Force Base, Ohio, 1965, pp.205-224

T.S. Motzkin, ``The arithmetic geometry inequality'', in: Proceedings of the Symposium , Wright-Patterson Air Force Base, Ohio, 1965, pp.205-224

1965
[12]

Qi and C

L. Qi and C. Cui, ``Biquadratic tensors: Eigenvalues and structured tensors'', March 2025, arXiv:2502.06447v2

2025 arXiv
[13]

L. Qi, H.H. Dai and D. Han, ``Conditions for strong ellipticity and M-eigenvalues'', Frontiers of Mathematics in China 4 (2009) 349-364

2009
[14]

Reznick, ``Extremal PSD forms with few terms'', Duke Mathematical Journal 45 (1978) 363-374

B. Reznick, ``Extremal PSD forms with few terms'', Duke Mathematical Journal 45 (1978) 363-374

1978

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [11]

An elementary proof of Hilbert’s theorem on ternary quartics

A. Pfister and C. Scheiderer, “An elementary proof of Hilbert’s theorem on ternary quartics”, Journal of Algebra 371 (2012) 1-25

work page 2012

[2] [1]

Calder\' o n, ``A note on biquadratic forms'', Linear Algebra and Its Applications 7 (1973) 175-177

A.P. Calder\' o n, ``A note on biquadratic forms'', Linear Algebra and Its Applications 7 (1973) 175-177

work page 1973

[3] [2]

Chesi, ``On the gap between positive polynomials and SOS of polynomials'', IEEE Trans

G. Chesi, ``On the gap between positive polynomials and SOS of polynomials'', IEEE Trans. Automat. Control 52 (2007) 1066-1072.\

work page 2007

[4] [3]

Choi, ``Positive semidefinite biquadratic forms'', Linear Algebra and Its Applications 12 (1975) 95-100

M.-D. Choi, ``Positive semidefinite biquadratic forms'', Linear Algebra and Its Applications 12 (1975) 95-100

work page 1975

[5] [4]

Choi and T.-Y

M.-D. Choi and T.-Y. Lam, ``Extremal positive semi-definite forms'', Mathematische Annalen 231 (1977) 1-18

work page 1977

[6] [5]

Nonnegative Biquadratic Tensors

C. Cui and L. Qi, ``Nonnegative biquadratic tensors'', March 2025, arXiv:2503.16176v2

work page Pith review arXiv 2025

[7] [6]

Fidalgo and A

C. Fidalgo and A. Kovacec, ``Positive semi-definite diagonal minus tail forms are sums of squares'', Mathematische Zeitschrift 269 (2011) 629-645

work page 2011

[8] [7]

C. Goel, S. Kuhlmann and B. Reznick, ``The analogue of Hilbert’s 1888 theorem for even symmetric forms'', Journal of Pure and Applied Algebra 221 (2017) 1438–1448

work page 2017