A refinement of Reznick's Positivstellensatz with applications to quantum information theory

Alexander M\"uller-Hermes; David Reeb; Ion Nechita

REVIEW 2 major objections 4 minor 30 references

A refinement of Reznick's Positivstellensatz with applications to quantum information theory

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims explicit, improved exponents for Reznick's Positivstellensatz by inverting the Chiribella identity from quantum cloning.

desk verdict Explicit Chiribella inversion yields a solid complex Positivstellensatz and de Finetti bounds; the real section has a concrete arithmetic error in Lemma 4.1 and its claimed improvement over Reznick is currently unsupported. read the letter →

arxiv 1909.01705 v4 pith:2CRPQRBM submitted 2019-09-04 quant-ph math-phmath.MP

Alexander Müller-Hermes , Ion Nechita , David Reeb This is my paper · ORCID

classification quant-phmath-phmath.MP

keywords PositivstellensatzsumofsquaresReznick'stheoremChiribellaidentitymeasure-and-preparemapexponentialdeFinettisphericaldesignsquantumcloning

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 claims that Reznick's Positivstellensatz---a strictly positive homogeneous polynomial becomes a sum of squares once multiplied by a large enough power of the squared norm---can be reproved and sharpened by translating it into the language of quantum information theory. The translation represents polynomials by Hermitian operators on symmetric tensor powers, so that multiplication by $\|x\|^2$ and Laplacians become partial traces and their adjoints. The main result is an explicit bound on the required exponent $n$ in both the complex and the real case, together with a constructive sum-of-squares decomposition based on finite spherical designs. If the bounds are correct, they improve previously known constants and yield sharper exponential de Finetti theorems for quantum states.

What carries the argument

The load-bearing mechanism is the Chiribella identity, which expresses the measure-and-prepare map $\mathrm{MP}_{n\to k}$ as a weighted sum of partial traces followed by their adjoints, i.e. by approximate-cloning maps. The paper's main technical move is an explicit inverse $\Psi^{(n)}_{k\to k}$ of the map $\Phi^{(n)}_{k\to k}$ appearing in that identity, with coefficients $q(n,k,t)$ given in closed form. Applying this inverse inside the adjoint of the measure-and-prepare map rewrites $p_W$ as an integral of a new form $p_{\widetilde W}$; positivity of $p_{\widetilde W}$ is then controlled by Bernstein-type inequalities for the Laplacian. Finite complex spherical designs convert the continuous integral into an explicit finite sum, yielding the sum-of-squares certificate.

What would settle it

Apply Lemma 4.1 to $p(x) = (v_1 x_1 + \cdots + v_d x_d)^{2n}$ with $\|v\|=1$. The proof's own computation gives $\Delta p = 2n(2n-1)\|v\|^2\langle x|v\rangle^{2n-2}$, while the claimed partial trace is $\|v\|^2 v^{\otimes(2n-2)}$, so each Laplacian step contributes the factor $2n(2n-1)$, not $(2n)^2$. Iterating, the ratio $\Delta^{n-k} p / \mathrm{tr}_{n\to k}(p)$ is $(2n(2n-1))^{n-k}$; if this calculation stands, the real-case bound (19) does not follow from the proof as written.

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Extended reading notes

Core claim

The paper's central claim is Theorem 3.1: for a Hermitian operator $W$ on $\vee^k \mathbb{C}^d \otimes \mathbb{C}^D$ with $m(W)>0$, whenever $n \ge d k(2k-1)/\ln(1 + m(W)/M(W)) - d - k + 1$, the identity $\|x\|^{2(n-k)} p_W(x,y) = \int p_{\widetilde{W}}(\varphi,y) |\langle\varphi|x\rangle|^{2n} d\varphi$ holds with $p_{\widetilde{W}} \ge 0$, so the left-hand side is a sum of squares. For $k=1$ the bound improves to $n \ge d M(W)/m(W) - d$. In the real case, Theorem 4.2 claims the analogous bound $2n \ge d k(2k-1)/\ln(1 + m(v)/M(v)) + 2 - 2k - d$, with $2n \ge d M(v)/m(v) - d$ when $k=1$; the authors state that this improves Reznick's bound by shrinking the leading constant. The same inversion of the Chiribella identity gives Theorem 5.1, an exponential de Finetti theorem with error at most $\delta^{r+1}/(1-3\delta)$.

Load-bearing premise

The real-case bound in Theorem 4.2 rests on Lemma 4.1, which asserts that the iterated real Laplacian equals $(2n)^{2(n-k)}$ times the partial trace; the displayed calculation in the lemma's proof appears to yield $2n(2n-1)$ per iteration instead, so unless that constant is corrected or the calculation reconciled, the claimed real-case improvement over Reznick is not established.

Editorial extensions

If this is right

For $k=1$, a strictly positive complex bi-homogeneous form admits the representation as soon as $n \ge d M(W)/m(W) - d$, a bound the paper compares against earlier complex-case results.
For general $k$, the required exponent grows like $d k(2k-1)/\ln(1 + m(W)/M(W))$, giving a logarithmic rather than linear dependence on the ratio $M/m$.
Using finite complex spherical designs, the representation becomes an explicit sum of squares with at most $(n+k+1)^{2d}$ terms of the form $|\langle\varphi|x\rangle|^{2n}|\langle w_\varphi^{(i)}|y\rangle|^2$.
In the real case, the paper claims the leading constant in Reznick's bound is reduced; the same operator inversion yields an exponential de Finetti bound with error $\delta^{r+1}/(1-3\delta)$ and real parameter $\delta_R = k(2k+d-2)/(2n+2k+d-2)$.

Reading between the lines

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

The explicit inverse of the Chiribella identity is a standalone operator identity: it expands partial traces into approximate-cloning channels, and de Finetti-type bounds are only the first place it is likely to be useful.
If the real-case constant in Lemma 4.1 is corrected, the qualitative structure of Theorem 4.2 should survive, but the exponent may grow by a constant factor; the complex theorem does not depend on that lemma.
A numerical scan over random low-dimensional Hermitian $W$ could test how close the analytic bounds (7) and (9) are to the minimal $n$ for which (8) holds, extending the Motzkin example the paper computes in the real case.

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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

Things you should know: this paper gives an explicit inversion of Chiribella's identity and uses it to prove a complex Positivstellensatz with better constants than To–Yeung, plus an exponential de Finetti bound. That part is real, new, and carefully derived. But the real-case extension is not sound as written: Lemma 4.1 mis-states the constant connecting the real Laplacian to the partial trace, and every real-case bound that relies on it (Theorem 4.2, Remark 4.4, Example 4.5, Remark 5.2) is unsupported until the factor is corrected and the omitted computation supplied.

The technical heart is Lemma 3.2. The coefficients q(n,k,t) are explicit, and the proof that Φ∘Ψ = id is clean. The derivation of Theorem 3.1 via measure-and-prepare maps and the Bernstein inequality is a nice unification of Reznick and To–Yeung; the k=1 bound is genuinely better than TY06. The de Finetti application is a canonical use of the inverse, and the error bound with δ is a clear improvement over Harrow's unspecified constants. The Laguerre-root construction of complex spherical designs in Appendix B is elementary and clearly presented.

Now the soft spot. Lemma 4.1 states (2n)^{2(n−k)} tr_{n→k}(p) = Δ^{n−k}p, but on the spanning set v^{⊗2n} the proof itself computes Δp = 2n(2n−1)||v||² p and tr = ||v||²p, so the constant per step is 2n(2n−1), not (2n)². Iterating gives a product of 2j(2j−1) from j=k+1 to n. The d=1, n=2, k=1 example of x⁴ makes the failure concrete: 12x² versus 16x². Theorem 4.2's proof then uses the wrong constant and leaves the final computation to the reader, so the claimed improvement over Reznick (Remark 4.4) is not established. The complex results do not depend on this lemma, so a corrected real section could plausibly restore the claims; the product is just a different constant. Still, the paper as submitted should not claim the real improvement, and the real de Finetti remark should be checked as well.

Citation pattern looks fine: the external benchmarks are genuinely prior work and are credited. The appendices are review material but are useful and carefully done.

Who should read this: anyone working with effective Positivstellensätze, symmetric subspace methods, or explicit de Finetti rates. The complex part alone deserves a serious referee. For the real part, the authors need to fix Lemma 4.1 and carry out the calculation.

My recommendation: send it to peer review. The complex part is substantive and correct; the real error is local and probably repairable. A referee can check whether the corrected constants still improve on Reznick. Desk rejection would be wrong.

Referee Report

2 major / 4 minor

Summary. The paper develops quantum-information-theoretic proofs of Reznick-type Positivstellensätze. In the complex case (Theorem 3.1) it proves an explicit integral representation of ||x||^{2(n-k)} p_W(x,y) as ∫ p_{\widetilde W}(φ,y)|⟨φ|x⟩|^{2n} dφ with p_{\widetilde W} ≥ 0, for n satisfying (7), and shows how this representation yields sum-of-squares decompositions supported on complex spherical designs. In the real case (Theorem 4.2) it claims an analogous bound that improves on Reznick's classical bound. Section 5 applies the inverse of the Chiribella identity to derive an exponential quantum de Finetti theorem in diamond norm. The appendices construct Gaussian-based Hilbert identities and an elementary family of complex spherical designs.

Significance. If the real-case claims were repaired, the paper would be a solid contribution: Theorem 3.1 is a complete, self-contained proof with explicit constants and constructively computable SOS decompositions, and the de Finetti application in Theorem 5.1 gives an explicit exponential error rate. The complex spherical design construction in Appendix B is elementary and potentially useful. The main additional advertised value, however, is the improvement over Reznick in the real setting, and that part is currently unsupported because of the error in Lemma 4.1. The complex results do not rely on the faulty lemma, so the paper's central complex contribution remains intact.

major comments (2)

[Section 4, Lemma 4.1] Lemma 4.1 asserts (2n)^{2(n-k)} tr_{n→k}(p) = Δ_R^{n-k} p for p ∈ H^{2n}(R^d). The proof's own computation gives Δ_R p_{v^{⊗2n}}(x) = 2n(2n-1)||v||²⟨x|v⟩^{2n-2} and tr_{n→n-1}(v^{⊗2n}) = ||v||² v^{⊗(2n-2)}, so the correct comparison is 2n(2n-1) tr_{n→n-1} = Δ_R, not (2n)² tr = Δ_R. Iterating gives Δ_R^{n-k} p_v = [∏_{j=k+1}^{n} 2j(2j-1)] ||v||^{2(n-k)} ⟨x|v⟩^{2k}, not (2n)^{2(n-k)} tr_{n→k}(p). For a concrete check in d=1, n=2, k=1, p(x)=x⁴ satisfies Δ_R p = 12x² and tr_{2→1}(p) = x², so the constant is 12, not 16. The lemma as stated is false.
[Section 4, Theorem 4.2, Eqs. (19), (22), (24); Example 4.5] Theorem 4.2's proof uses the incorrect identity p_{(tr_{k→t}⊗id)(v)} = ((2k)^{2(k-t)})^{-1} Δ_R^{k-t} p_v, which is exactly the relation that Lemma 4.1 was supposed to establish. Since the estimates leading to (19), (22), (24) and the numerical comparison in Example 4.5 all depend on this coefficient, the claimed improvement over Reznick in Remark 4.4 is not established. The proof also explicitly leaves the final computation to the reader, so the corrected constants have not been propagated; a revised version must supply this calculation and verify that the advertised improvement survives. The complex Theorem 3.1 and the complex de Finetti theorem in Theorem 5.1 do not depend on Lemma 4.1.

minor comments (4)

[Abstract] The printed abstract says the denominator can be chosen as an 'N-th power of a linear form', but Reznick's denominator is an N-th power of the squared norm, ||x||^{2N}; the arXiv metadata abstract states this correctly, so the printed abstract should be corrected.
[Example 4.5] In the paragraph after (26), 'Not that this latter bound is necessarily better' should read 'Note that this latter bound is necessarily better'.
[Remark 3.4] There is a typo in the phrase 'even by roughly a factor of 2 for the case k = 1 in Eq. (9)': the comparison is with To--Yeung's bound, and the wording should make clear which bound is being compared; also 'Renzick' is misspelled in Remark 4.4.
[Section 4, Figure captions] The captions of Figures 3 and 4 are verbose and would be clearer if they identified which curves correspond to which equations in the text rather than only in prose; the typo 'Not' in the right-panel description of Figure 3 also occurs there.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main derivations are self-contained or use external, independently established ingredients.

full rationale

The paper does not fit parameters and then rename them as predictions. Theorem 3.1 derives an explicit SOS decomposition using the Chiribella identity, which is proved in Section 2.3, the explicit inverse in Lemma 3.2, and the Bernstein inequality Lemma 2.6, which is proved by reduction to Reznick's real inequality. No equation in the proof is assumed to hold by construction; the coefficients q(n,k,t) are computed and verified. The real-case Section 4 likewise derives a bound from Reznick's Bernstein inequality and its own Lemma 4.1. The reader's concern about Lemma 4.1 (the constant 2n(2n-1) versus (2n)^2, and the omitted computation behind (19), (22), (24)) is a potential mathematical error in the real-case improvement, not a circularity: the claim would depend on an incorrect lemma, not on its own conclusion. There are no load-bearing self-citations: the cited works by Reznick, To-Yeung, Harrow, Chiribella, and Hobson are external, and the key identities are re-proved in the text. The paper's benchmarks are genuinely prior independent results. The 'small improvements' claims are comparisons, not fitted predictions. Overall, no circular step is present.

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

The complex-case derivation is essentially self-contained: the Chiribella identity and the Hilbert identities are derived in the text, and the only unproved inputs are standard theorems. There are no data-fitted parameters and no new physical entities.

assumptions (3)

standard math Real Bernstein inequality: for W in H(∨^k R^d) and ||x||=1, |(∆_R^t p_W)(x)| <= d^t (2k)^{2t} M(W) (Lemma 2.5, from [Rez95]).
Invoked to prove the complex Bernstein inequality (Lemma 2.6) and to bound the Laplacian terms in the real Positivstellensatz proof (Theorem 4.2). The paper cites rather than proves it.
standard math The set {x^{⊗n} : x in C^d} spans the symmetric subspace ∨^n C^d, and similarly over R.
Used to verify operator identities on the spanning set in Lemmas 2.1, 4.1 and Theorems 2.4, 3.2. Cited to Bhatia and Harrow.
standard math Wick's (Isserlis) formula for Gaussian moments.
Used in Appendix A to derive the real and complex spherical Hilbert identities.

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Pith. "Pith review of A refinement of Reznick's Positivstellensatz with applications to quantum information theory." pith.science (2026). https://pith.science/paper/2CRPQRBM

@misc{pith2026190901705,
  author       = {Pith},
  title        = {Pith review of: A refinement of Reznick's Positivstellensatz with applications to quantum information theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2CRPQRBM}},
  note         = {Machine review of arXiv:1909.01705}
}

abstract

In his solution of Hilbert's 17th problem Artin showed that any positive definite polynomial in several variables can be written as the quotient of two sums of squares. Later Reznick showed that the denominator in Artin's result can always be chosen as an $N$-th power of the squared norm of the variables and gave explicit bounds on $N$. By using concepts from quantum information theory (such as partial traces, optimal cloning maps, and an identity due to Chiribella) we give simpler proofs and minor improvements of both real and complex versions of this result. Moreover, we discuss constructions of Hilbert identities using Gaussian integrals and we review an elementary method to construct complex spherical designs. Finally, we apply our results to give improved bounds for exponential quantum de Finetti theorems in the real and in the complex setting.

Figures

Figures reproduced from arXiv: 1909.01705 by the authors.

**Figure 1.** Graphical representation of the correspondence between self-adjoint operators W acting on the symmetric subspace, and polynomials. From left to right, we have depicted the diagrams for pW (z), kzk 2kpW (z), and ((n + k)k) −2∆kpW (z) respectively, where ∆ is the complex Laplacian (1). This emphasized in particular that multiplying with the norm and the iterated complex Laplacian are, up to constants, dual operation… view at source ↗

**Figure 2.** Graphical representation of the correspondence between symmetric tensors v and homogeneous polynomials. From left to right, we have depicted the diagrams for pv(x), kxk 2pv(x), and (n + 2)2∆pv(x) respectively, where ∆ is the real Laplacian. Note that multiplying with the norm and the Laplacian are, up to constants, dual operations. On the other hand trn→(n−1)(v ⊗2n ) = kvk 2 v ⊗(2n−2) . Direct comparison of the two… view at source ↗

**Figure 3.** Comparing lower bounds for n needed for the real Positivstellensatz to hold for the shifted Motzkin polynomial from (26), as a function of ε. In the left panel, Reznick’s bound (25) (black, dotted curve), the bounds from (19) (red, dashed curve) and (24) (blue curve) are plotted. On the right, the bound obtained from equation (22) (black curve) is plotted against the bound from (24) (blue, dashed curve). be non-nega… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Lower and upper bounds for the minimal n such that the decomposition (20) holds for the shifted Motzkin polynomial (26), as functions of ε. The upper bound (filled blue region) comes from equation (22). The lower bounds (red horizontal bars, one for each value of n) c…

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Works this paper leans on

30 extracted references · 24 canonical work pages

[1]

U ber die Z erlegung definiter F unktionen in Q uadrate. In Abhandlungen aus dem mathematischen S eminar der U niversit \

Emil Artin. \"U ber die Z erlegung definiter F unktionen in Q uadrate. In Abhandlungen aus dem mathematischen S eminar der U niversit \"a t H amburg , volume 5, pages 100--115. Springer, 1927

work page 1927
[2]

Real algebraic geometry , volume 36

Jacek Bochnak, Michel Coste, and Marie-Fran c oise Roy. Real algebraic geometry , volume 36. Springer Science & Business Media, 2013

2013
[3]

Matrix analysis , volume 169

Rajendra Bhatia. Matrix analysis , volume 169. Springer Science & Business Media, 1997

1997
[4]

On quantum estimation, quantum cloning and finite quantum de finetti theorems

Giulio Chiribella. On quantum estimation, quantum cloning and finite quantum de finetti theorems. In Conference on Quantum Computation, Communication, and Cryptography , pages 9--25. Springer, 2010

work page 2010
[5]

Spherical codes and designs

Philippe Delsarte, Jean-Marie Goethals, and Johan Jacob Seidel. Spherical codes and designs. In Geometry and Combinatorics , pages 68--93. Elsevier, 1991

work page 1991
[6]

Waring's problem

William J Ellison. Waring's problem. The American Mathematical Monthly , 78(1):10--36, 1971

work page 1971
[7]

The sum-of-squares hierarchy on the sphere, and applications in quantum information theory

Kun Fang and Hamza Fawzi. The sum-of-squares hierarchy on the sphere, and applications in quantum information theory. arXiv preprint arXiv:1908.05155 , 2019

arXiv 1908
[8]

Concrete Mathematics, 2nd Edition

Ronald L Graham, Donald E Knuth, and Oren Patashnik. Concrete Mathematics, 2nd Edition . Massachusetts: Addison-Wesley, 1994

work page 1994

Show all 30 references

[9]

The church of the symmetric subspace

Aram W Harrow. The church of the symmetric subspace. arXiv preprint arXiv:1308.6595 , 2013

2013 arXiv
[10]

Zur Hilbertschen L \"o sung des Waringschen Problems

Felix Hausdorff. Zur Hilbertschen L \"o sung des Waringschen Problems . Mathematische Annalen , 67(3):301--305, 1909

1909
[11]

Beweis f \"u r die D arstellbarkeit der ganzen Z ahlen durch eine feste A nzahl n-ter P otenzen ( W aringsches P roblem)

David Hilbert. Beweis f \"u r die D arstellbarkeit der ganzen Z ahlen durch eine feste A nzahl n-ter P otenzen ( W aringsches P roblem). Mathematische Annalen , 67(3):281--300, 1909

1909
[12]

The theory of spherical and ellipsoidal harmonics

Ernest W Hobson. The theory of spherical and ellipsoidal harmonics . CUP Archive, 1931

1931
[13]

Some formal properties of the density matrix

K \^o di Husimi. Some formal properties of the density matrix. Proceedings of the Physico-Mathematical Society of Japan. 3rd Series , 22(4):264--314, 1940

1940
[14]

On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables

Leon Isserlis. On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables. Biometrika , 12(1/2):134--139, 1918

1918
[15]

A most compendious and facile quantum de finetti theorem

Robert K \"o nig and Graeme Mitchison. A most compendious and facile quantum de finetti theorem. Journal of Mathematical Physics , 50(1):012105, 2009

2009
[16]

Anneaux pr \'e ordonn \'e s

Jean-Louis Krivine. Anneaux pr \'e ordonn \'e s. Journal d'analyse math \'e matique , 12(1):307--326, 1964

1964
[17]

Optimal cloning of pure states, testing single clones

Michael Keyl and Reinhard F Werner. Optimal cloning of pure states, testing single clones. Journal of Mathematical Physics , 40(7):3283--3299, 1999

1999
[18]

Positive polynomials and sums of squares

Murray Marshall. Positive polynomials and sums of squares . Number 146. American Mathematical Soc., 2008

2008
[19]

Effective aspects of positive semi-definite real and complex polynomials

Mok Hoi Nam. Effective aspects of positive semi-definite real and complex polynomials. Master's thesis, Department of Mathematics, National University of Singapore, 2008

2008
[20]

On Waring's problem (elementary methods)

Yu V Nesterenko. On Waring's problem (elementary methods) . Journal of Mathematical Sciences , 137(2):4699--4715, 2006

2006
[21]

Positive polynomials on compact semi-algebraic sets

Mihai Putinar. Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal , 42(3):969--984, 1993

1993
[22]

On the representation of hermitian forms as sums of squares

Daniel G Quillen. On the representation of hermitian forms as sums of squares. Inventiones mathematicae , 5(4):237--242, 1968

1968
[23]

Symmetry of large physical systems implies independence of subsystems

Renato Renner. Symmetry of large physical systems implies independence of subsystems. Nature Physics , 3(9):645, 2007

2007
[24]

Uniform denominators in H ilbert's seventeenth problem

Bruce Reznick. Uniform denominators in H ilbert's seventeenth problem. Math. Z. , 220(1):75--97, 1995

1995
[25]

The K -moment problem for compact semi-algebraic sets

Konrad Schm \"u dgen. The K -moment problem for compact semi-algebraic sets. Mathematische Annalen , 289(1):203--206, 1991

1991
[26]

Tight informationally complete quantum measurements

Andrew J Scott. Tight informationally complete quantum measurements. Journal of Physics A: Mathematical and General , 39(43):13507, 2006

2006
[27]

A Nullstellensatz and a Positivstellensatz in semialgebraic geometry

Gilbert Stengle. A Nullstellensatz and a Positivstellensatz in semialgebraic geometry . Mathematische Annalen , 207(2):87--97, 1974

1974
[28]

Orthogonal polynomials , volume 23

Gabor Szeg \"o . Orthogonal polynomials , volume 23. American Mathematical Soc., 1939

1939
[29]

Effective isometric embeddings for certain H ermitian holomorphic line bundles

Wing-Keung To and Sai-Kee Yeung. Effective isometric embeddings for certain H ermitian holomorphic line bundles. Journal of the London Mathematical Society , 73(3):607--624, 2006

2006
[30]

Quantum information theory

Mark M Wilde. Quantum information theory . Cambridge University Press, 2017

2017

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

U ber die Z erlegung definiter F unktionen in Q uadrate. In Abhandlungen aus dem mathematischen S eminar der U niversit \

Emil Artin. \"U ber die Z erlegung definiter F unktionen in Q uadrate. In Abhandlungen aus dem mathematischen S eminar der U niversit \"a t H amburg , volume 5, pages 100--115. Springer, 1927

work page 1927

[2] [2]

Real algebraic geometry , volume 36

Jacek Bochnak, Michel Coste, and Marie-Fran c oise Roy. Real algebraic geometry , volume 36. Springer Science & Business Media, 2013

2013

[3] [3]

Matrix analysis , volume 169

Rajendra Bhatia. Matrix analysis , volume 169. Springer Science & Business Media, 1997

1997

[4] [4]

On quantum estimation, quantum cloning and finite quantum de finetti theorems

Giulio Chiribella. On quantum estimation, quantum cloning and finite quantum de finetti theorems. In Conference on Quantum Computation, Communication, and Cryptography , pages 9--25. Springer, 2010

work page 2010

[5] [5]

Spherical codes and designs

Philippe Delsarte, Jean-Marie Goethals, and Johan Jacob Seidel. Spherical codes and designs. In Geometry and Combinatorics , pages 68--93. Elsevier, 1991

work page 1991

[6] [6]

Waring's problem

William J Ellison. Waring's problem. The American Mathematical Monthly , 78(1):10--36, 1971

work page 1971

[7] [7]

The sum-of-squares hierarchy on the sphere, and applications in quantum information theory

Kun Fang and Hamza Fawzi. The sum-of-squares hierarchy on the sphere, and applications in quantum information theory. arXiv preprint arXiv:1908.05155 , 2019

arXiv 1908

[8] [8]

Concrete Mathematics, 2nd Edition

Ronald L Graham, Donald E Knuth, and Oren Patashnik. Concrete Mathematics, 2nd Edition . Massachusetts: Addison-Wesley, 1994

work page 1994