Positivity Certificates via Integral Representations

Bernd Sturmfels; Khazhgali Kozhasov; Mateusz Micha{\l}ek

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Positivity Certificates via Integral Representations

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Pith's one-line read For elementary symmetric polynomials, the paper proves that complete monotonicity of $E_{m,n}^{-\alpha}$ forces $\alpha=0$ or $\alpha\ge (n-m)/2$, and that all sufficiently negative powers are completely monotone with explicit…

desk verdict Proves the necessity half of Scott–Sokal for elementary symmetric polynomials and gives a constructive Riesz-kernel sufficiency result; solid mathematics with a real but manageable caveat about imported lemmas. read the letter →

arxiv 1908.04191 v1 pith:Q5GH4KRI submitted 2019-08-12 math.FA math.AGmath.COmath.OC

Khazhgali Kozhasov , Mateusz Micha{l}ek , Bernd Sturmfels This is my paper · ORCID

classification math.FAmath.AGmath.COmath.OC MSC 44A1026B0533C70

keywords completemonotonicityRieszkernelhyperbolicpolynomialselementarysymmetricScott-SokalconjectureLaplacetransformconvexconesA-hypergeometricfunctions

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

Complete monotonicity is a very strong positivity condition: a function and all its signed partial derivatives must be nonnegative on an open convex cone. This paper studies which negative powers of the elementary symmetric polynomial $E_{m,n}(x)=\sum_{i_1<\cdots

What carries the argument

The Riesz kernel is the nonnegative function $q(y)$ on the dual cone whose Laplace transform reproduces the function being certified; by the Bernstein-Hausdorff-Widder-Choquet theorem, the existence of such a kernel is equivalent to complete monotonicity. For hyperbolic polynomials $p$, Gårding's integral representation defines $q(y)$ as an oscillatory integral over $\mathbb{R}^n$. The proof for elementary symmetric polynomials combines the splitting $E_{m,n}=E_{m,n-1}+x_nE_{m-1,n-1}$, Scott-Sokal's sign-comparison lemma for large $x_n$, and a generalized kernel-convolution formula that writes the Riesz kernel of $B^{-\alpha}f(x,A/B)$ as an integral of the two individual kernels.

What would settle it

Compute, at $(x_1,\ldots,x_5)=(1000,1,1,1,1)$ and for $\alpha=0.9$, the signed mixed derivative of $E_{3,5}^{-\alpha}$ corresponding to the multiset used in the induction. The theorem predicts a negative value for some signed derivative; the sign-comparison lemma predicts agreement with the same derivative of $E_{2,4}^{-\alpha}$. Finding all such derivatives nonnegative would refute the necessity claim.

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Extended reading notes

Core claim

The central claim is the dichotomy for $E_{m,n}^{-\alpha}$: if $2\le m<n$ and the function is completely monotone on $\mathbb{R}_{>0}^n$, then $\alpha=0$ or $\alpha\ge (n-m)/2$; conversely, for each $m,n$ there is an $\alpha'$ such that all $\alpha\ge\alpha'$ are completely monotone. The necessity statement, Theorem 6.6, is proved by induction on $m$, using the decomposition $E_{m,n}=x_1E_{m-1,n-1}+E_{m,n-1}$ and a sign-comparison lemma from Scott and Sokal that transfers complete monotonicity to $E_{m-1,n-1}^{-\alpha}$ as $x_1\to\infty$. The sufficiency statement, Theorem 6.4, is constructive: it factors the relevant exponential into a product of completely monotone factors and assembles the Riesz kernel from known kernels via a convolution formula. The exact behavior at the boundary $\alpha=(n-m)/2$ is left open here, except in the base case $m=2$.

Load-bearing premise

The necessity induction rests on Scott and Sokal's sign-comparison lemma, which is cited rather than proved in this paper: for very large $x_1$, the signs of derivatives in $x_2,\ldots,x_n$ of $E_{m,n}^{-\alpha}$ and of $E_{m-1,n-1}^{-\alpha}$ are asserted to coincide; if that comparison fails for some high-order mixed derivative, the threshold conclusion collapses.

Editorial extensions

If this is right

Scott and Sokal's Conjecture 4.13 now has a proven necessity half: no exponent below $(n-m)/2$ can work, for every $2\le m<n$.
Every elementary symmetric polynomial has a range of sufficiently negative exponents for which complete monotonicity holds, so each such power admits an explicit Riesz-kernel certificate and a Laplace-transform representation.
For products of negative powers of linear forms, the Riesz kernel is a piecewise-polynomial volume function on chambers of the dual cone, connecting these positivity certificates to polytope volumes.
Riesz kernels of hyperbolic polynomials are $A$-hypergeometric in the polynomial coefficients, so hypergeometric-system methods can be used to derive and simplify positivity certificates.

Reading between the lines

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

If the missing sufficiency half of the Scott-Sokal conjecture also holds, the set of completely monotone negative powers of $E_{m,n}$ would be exactly $\alpha\ge(n-m)/2$ together with $\alpha=0$; a natural test is to study the limiting behavior of the inductive Riesz kernel as $\alpha$ approaches $(n-m)/2$ from above.
The same induction—splitting a hyperbolic polynomial as $A+yB$ and iterating the kernel-convolution formula—may yield threshold results for other recursively defined hyperbolic polynomials, such as those with interlacing factorizations.
Because the constructed kernels are hypergeometric, numerical evaluation with hypergeometric-series algorithms offers a practical, independent check of nonnegativity of candidate Riesz kernels for specific polynomials and exponents.

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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

Short version: this is a real result, not a survey. The two theorems to remember are 6.6 and 6.4. Theorem 6.6 proves the only-if direction of Scott–Sokal Conjecture 4.13 for all 2 ≤ m < n: if E_{m,n}^{-α} is completely monotone on the positive orthant, then α = 0 or α ≥ (n−m)/2. That direction was explicitly left open by Scott and Sokal. Theorem 6.4 gives an inductive construction of the Riesz kernel for sufficiently negative powers, establishing Conjecture 4.10 for elementary symmetric polynomials. Section 7's identification of the Riesz kernel with Aomoto–Gelfand hypergeometric functions is a useful bonus.

What the paper does well: the proof strategy is transparent and honest. The reduction in Theorem 6.6 is short but clear: write E_{m,n} = x1 E_{m-1,n-1} + E_{m,n-1}, use the sign-transfer lemma of Scott–Sokal for large x1, and induction gives the bound. Lemma 6.3's convolution formula for Riesz kernels is explicit and checkable. There are no fitted parameters and no invented entities; every claim is either proved or explicitly imported from a cited source. The citation pattern is appropriate, mostly Scott–Sokal, Gårding, and the algebraic statistics literature.

Soft spots, in proportion: the main one is genuine but not fatal. Theorem 6.6 leans on Scott–Sokal Lemma 3.1, a sign-comparison lemma for high-order mixed derivatives, and on the m = 2 base case, both cited without proof. I did not find the dependence circular—these are independent published results—but the central necessity statement is only as strong as that lemma's exact hypotheses. Since the lemma comes from a standard Acta Mathematica paper, this is a moderate caveat rather than a flaw. Theorem 6.4's construction is complicated; the authors walk through E_{3,5} in detail, but a referee should verify the exponents in equations (21) and (22). There is also a garbled printed fraction in Example 2.8 and a few typos; cosmetic.

Bottom line: for anyone working on complete monotonicity, hyperbolic polynomials, or exponential varieties, this paper is worth reading and citing. It deserves a serious referee, and the review should focus on checking the imported lemma's hypotheses and the inductive step in Theorem 6.4. I would accept after minor revisions.

Referee Report

2 major / 5 minor

Summary. The paper develops integral-representation certificates for complete monotonicity of negative powers of hyperbolic polynomials, with the main focus on elementary symmetric polynomials E_{m,n}. Its central results are Theorem 6.4, which shows that E_{m,n}^{-\alpha} is completely monotone for all sufficiently large \alpha and gives a constructive route to the Riesz kernel, and Theorem 6.6, which proves the necessity half of Scott and Sokal's Conjecture 4.13: for 2 \le m < n, if E_{m,n}^{-\alpha} is completely monotone on the positive orthant, then \alpha = 0 or \alpha \ge (n-m)/2. The earlier sections set up the general framework of complete monotonicity on convex cones, the Bernstein-Hausdorff-Widder-Choquet theorem, G\aa rding's integral representation, and the Riesz kernel. Section 3 treats products of negative powers of linear forms, relating the Riesz kernel to fiber volumes and the chamber complex. Section 5 interprets Riesz kernels inside convolution algebras and connects them to the Orlik-Terao algebra. Section 7 relates Riesz kernels to A-hypergeometric functions and to Aomoto-Gel'fand hypergeometric integrals. The paper is expository in parts but the main theorems in Section 6 are the substantive new contributions.

Significance. If the results hold as stated, the paper makes a genuine advance on a conjecture of Scott and Sokal. Theorem 6.6 establishes the necessity direction of Conjecture 4.13 for all 2 \le m < n, and Theorem 6.4 gives a constructive proof that sufficiently negative powers of every E_{m,n} are completely monotone. The constructive viewpoint is valuable: explicit Riesz kernels, convolution products of measures, and the connection to A-hypergeometric functions provide tools that go beyond the particular conjecture. I found no circularity in the argument: the conjecture from [14] is used as a goal and not as an input, and the derivations are based on standard Laplace/G\aa rding theory and on cited results of Scott and Sokal. The main caveats are that two load-bearing inductive steps are not fully formalized as written: the proof of Theorem 6.6 imports an unstated sign-comparison lemma from [19], and the proof of Theorem 6.4 applies Lemma 6.3 under an induction hypothesis that does not explicitly include the existence of a Riesz kernel. These are fixable within the manuscript's scope.

major comments (2)

[Section 6, Theorem 6.6] The proof of the necessity statement rests entirely on the sign-comparison lemma [19, Lemma 3.1], which is neither stated nor proved. This lemma is load-bearing: it is what transfers derivative signs from E_{m,n}^{-\alpha} to E_{m-1,n-1}^{-\alpha} and thereby yields complete monotonicity of E_{m-1,n-1}^{-\alpha}. As written, the proof of Theorem 6.6 cannot be checked from the manuscript alone. Please state the precise hypotheses of [19, Lemma 3.1], verify that they apply to all mixed derivatives in x_2,\ldots,x_n of every order, and either reproduce the proof or give a self-contained statement.
[Section 6, Theorem 6.4] The induction in the proof of Theorem 6.4 applies Lemma 6.3, whose hypotheses require Riesz kernels for the functions f and g, i.e., absolutely continuous Riesz measures. The formal induction hypothesis of Theorem 6.4, however, only asserts complete monotonicity of E_{m-1,n-1}^{-\alpha}; complete monotonicity alone gives a Riesz measure by Theorem 2.5, but not necessarily a density. The constructive claim of the theorem therefore needs a strengthened induction hypothesis: for \alpha \ge \alpha_{m,n}, the function E_{m,n}^{-\alpha} has an explicit nonnegative Riesz kernel q, and the application of Lemma 6.3 to g = E_{m-1,n-1}^{-\alpha} and f = E_{2,3}^{-\alpha} produces that kernel via equation (18). Please make this strengthened induction explicit and verify the base cases.

minor comments (5)

[Example 2.8] The displayed numerical fraction is malformed: '- 16652440985600 / 76263809554320336/11' should be typeset as a single fraction, evidently -16652440985600 / (76263809554320336/11) or an equivalent. Please correct the typesetting.
[Section 6, equation (21)] In the factorization (21), the identity \sum_{i=1}^{n-1} Q_i = m E_{m,n-1} is used implicitly. Please state this identity explicitly so that the equality between the product and the displayed exponential is transparent.
[Section 6, Theorem 6.6] After invoking [19, Lemma 3.1], the text should state explicitly that the sign comparison holds for every mixed derivative in x_2,\ldots,x_n of every order, since complete monotonicity of E_{m-1,n-1}^{-\alpha} requires all infinitely many such derivative inequalities.
[Section 7, Theorem 7.4] The proof of Theorem 7.4 relies on a nontrivial result from [9] without giving a precise reference or derivation. A theorem number in [9] or a short explanation of the integral identity would make the argument easier to verify.
[General] There are several typographical errors: 'certifcate' in Section 1, 'Lesbesgue' in Section 2, and 'monon- tone' in the paragraph preceding Theorem 6.4. The paper should be carefully proofread before resubmission.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems reduce to external Scott-Sokal lemmas and Garding theory, not to their own conclusions.

full rationale

The paper's central claims are Theorems 6.4 and 6.6. Theorem 6.4 proves existence of a sufficiently negative exponent making E_{m,n}^{-alpha} completely monotone by induction; its base case is [19, Corollary 1.10] and its induction step uses Lemma 6.3, which is proved in the paper from the Bernstein-Hausdorff-Widder-Choquet theorem and injectivity of the Laplace transform. Theorem 6.6 proves necessity of alpha >= (n-m)/2 by induction, importing [19, Lemma 3.1] to transfer derivative signs from E_{m,n}^{-alpha} to E_{m-1,n-1}^{-alpha}; the m=2 base is again [19]. Neither target statement is used as an input: Conjecture 4.13 is the goal of Theorem 6.6, not an assumption, and Conjecture 4.10 from the authors' own [14] is proved rather than presupposed. The only in-paper citation to [14] in the load-bearing part (Theorem 4.7) points back to external Scott-Sokal results. The imported lemmas are published, parameter-free, and do not include the target conclusion, so they count as independent support even though some are cited rather than reproved. Residual risk concerns the correctness or hypotheses of the imported Lemma 3.1, which is a dependency, not a circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data; the exponents alpha are variables, and the threshold alpha' is constructed recursively. The paper's central theorems rely on standard results in Laplace analysis, Garding's integral representation, and Scott-Sokal's lemmas rather than on new postulated entities.

assumptions (4)

standard math Bernstein-Hausdorff-Widder-Choquet theorem: a C-infinity function on an open convex cone is completely monotone iff it is the Laplace transform of a unique Borel measure on the dual cone.
Used throughout as the equivalence certifying complete monotonicity and defining Riesz measures; cited to [23] and [4].
standard math Garding's integral representation: for a complete hyperbolic polynomial p with hyperbolicity cone C, q_alpha(y) in (13) is independent of e in C, vanishes outside C*, and p(x)^{-alpha} = integral over C* e^{-<y,x>} q_alpha(y) dy for Re(alpha) > n.
Basis for computing Riesz kernels in Sections 4 through 7; cited to Garding [7].
standard math Scott-Sokal base cases and lemmas: E_{2,n}^{-alpha} is completely monotone for alpha >= (n-2)/2, Lemma 3.1 on sign comparison of derivatives as x1 tends to infinity, Theorem 1.3 for the determinant case, and Corollary 5.8 for Riesz kernels of E_{2,k}.
Supplies the induction base and the sign-comparison step in Theorems 6.4 and 6.6; accepted from [19] without reproduction.
standard math Aomoto-Gelfand evaluation used in Theorem 7.4: Phi(alpha; y,-y1,...,-y_{m-n}) equals |L|^{-1} times the integral of product x_i^{alpha_i-1} over the fiber L^{-1}(y).
Connects the Riesz kernel for monomials in linear forms to hypergeometric functions; cited to [9].

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Pith. "Pith review of Positivity Certificates via Integral Representations." pith.science (2026). https://pith.science/paper/Q5GH4KRI

@misc{pith2026190804191,
  author       = {Pith},
  title        = {Pith review of: Positivity Certificates via Integral Representations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q5GH4KRI}},
  note         = {Machine review of arXiv:1908.04191}
}

read the original abstract

Complete monotonicity is a strong positivity property for real-valued functions on convex cones. It is certified by the kernel of the inverse Laplace transform. We study this for negative powers of hyperbolic polynomials. Here the certificate is the Riesz kernel in Garding's integral representation. The Riesz kernel is a hypergeometric function in the coefficients of the given polynomial. For monomials in linear forms, it is a Gel'fand-Aomoto hypergeometric function, related to volumes of polytopes. We establish complete monotonicity for sufficiently negative powers of elementary symmetric functions. We also show that small negative powers of these polynomials are not completely monotone, proving one direction of a conjecture by Scott and Sokal.

Figures

Figures reproduced from arXiv: 1908.04191 by the authors.

**Figure 1.** Four linear forms in two variables. They are positive in the shaded region. Integration over the unit circle reduces to integration over the displayed circular arc. We change the integration contour to the horizontal segment. Integrating along the segment between (0, 1) and (− c a , 1), we obtain the formula Φ(α1, α3, α4; a, b, c, d) = Z − c a 0 x α1−1 1 (ax1 + c) α3−1 (bx1 + d) α4−1 dx1. This integral can be expres… view at source ↗

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Pith tools

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

[1] [14]

Micha lek, B

M. Micha lek, B. Sturmfels, C. Uhler, and P. Zwiernik: Exponential varieties, Proceedings of the London Mathematical Society 112 (2016) 27–56

work page 2016

[2] [19]

Scott and A

A. Scott and A. Sokal: Complete monotonicity for inverse powers of some combinatorially deﬁned polynomials, Acta Mathematica 212 (2014) 323–392

work page 2014

[3] [1]

Aomoto and M

K. Aomoto and M. Kita: Theory of Hypergeometric Functions, Springer, New York, 1994

work page 1994

[4] [2]

Atiyah, R

M. Atiyah, R. Bott, and L. G˚ arding: Lacunas for hyperbolic diﬀerential operators with constant coeﬃcients. I, Acta Math. 131 (1973) 145–206

work page 1973

[5] [3]

Blekherman, P

G. Blekherman, P. Parrilo and R. Thomas: Semideﬁnite Optimization and Convex Algebraic Ge- ometry, MOS-SIAM Series on Optimization 13, 2012

work page 2012

[6] [4]

Choquet: Deux exemples classiques de repr´ esentation int´ egrale, L’Enseignement Math´ ematique15 (1969) 63–75

G. Choquet: Deux exemples classiques de repr´ esentation int´ egrale, L’Enseignement Math´ ematique15 (1969) 63–75

work page 1969

[7] [5]

De Concini and C

C. De Concini and C. Procesi: Topics in Hyperplane Arrangements, Polytopes and Box-Splines , Universitext, Springer, New York, 2010

work page 2010

[8] [6]

Dressler, S

M. Dressler, S. Iliman and T. de Wolﬀ: A positivstellensatz for sums of nonnegative circuit poly- nomials, SIAM Journal on Applied Algebra and Geometry 1 (2017) 536–555

work page 2017