Algebraic relations between moments of plane polygons

Boris Shapiro; Ralf Fr\"oberg; Yurii Burman

REVIEW 3 major objections 6 minor 21 references

Algebraic relations between moments of plane polygons

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read All harmonic and anti-harmonic moments of an n-vertex polygon are rational functions of the separately symmetric vertex data plus the single quadratic moment $\nu_2$.

desk verdict Solid structural result on moment relations for polygons; the main proofs are mostly complete but one unproven determinant nonvanishing assumption needs a patch. read the letter →

arxiv 1908.07621 v1 pith:Y2XOL2DU submitted 2019-08-20 math.CV math.CA

Yurii Burman , Ralf Fröberg , Boris Shapiro This is my paper · ORCID

classification math.CVmath.CA MSC 44A6031B20

keywords harmonicmomentsanti-harmonicpolygonalmeasuresalgebraicrelationslogarithmicpotentialGaloisgroupsymmetricfunctionsSegrevariety

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

The paper asks which algebraic relations hold among the infinitely many harmonic moments $m_j=\int z^j\,d\mu$ and anti-harmonic moments $\bar m_j=\int \bar z^j\,d\mu$ of a planar polygon with $n$ vertices. It answers by complexifying the vertices, treating $z$ and $\bar z$ as independent variables, and proving that the field generated by all these moments is an algebraic extension of the field of rational functions symmetric in the $z$'s and symmetric in the $\bar z$'s separately, generated by a single element $\nu_2$, the normalized quadratic moment that for real polygons is the signed area. The degree of this extension is $n!(n-1)!$ for odd $n$ and $2((n-1)!)^2$ for even $n$. For harmonic moments alone, the paper proves that the field is generated rationally by the first $2n-2$ moments, so a finite list of $2n-2$ numbers carries all information. A reader should care because this reduces an infinite, a priori highly redundant system of moment relations to one algebraic equation with an explicit Galois symmetry.

What carries the argument

The load-bearing object is $\nu_2(z,\bar z)$, the normalized quadratic moment, a bilinear form giving the signed area of the polygonal fan; it is the unique moment that generates everything in the geometric approach. Its orbit under the action of $S_n\times S_n$ has a stabilizer $G$ described in Proposition 3.2, and the polynomial $P(t)=\prod_{(\$\sigma$,\tau)\in(S_n\times S_n)/G}(t-\$nu_2^{{(\sigma,\tau)}}$)$ is shown to be the minimal polynomial of $\nu_2$ over $H$, with the exact degree above. The technical engine on the harmonic side is the normalized generating function $\Psi_\mu(w)=\sum_{j\ge 2}\nu_j $w^{{j-2}}$=A_D(z,w)/\prod_{j=1}^n(1-z_j w)$, whose denominator produces a constant-coefficient recurrence relating consecutive moments to the elementary symmetric functions $e_1(z),\dots,e_n(z)$; writing the first $n$ recurrences gives the Toeplitz system $U\cdot E=V$ (equation (2.5)), which solves for the $e_j$ in terms of $\nu_2,\dots,\nu_{2n-1}$ with common denominator $D_n=\det U$. The algebra generated by all orbit copies of $\nu_2$ is isomorphic to the Segre coordinate ring $\mathbb{C}[d_{ij}]/\langle I_2\rangle$, the ring of a product of two projective spaces cut out by all $2\times 2$ minors, and from that presentation the Galois group of the Galois closure is described.

What would settle it

Compute the symbolic determinant $D_n$ of the Toeplitz matrix (2.6) from the explicit moment formulas (1.1), starting with $n=4$. If $D_n$ is identically zero as a polynomial in $z_1,\dots,z_n,\bar z_1,\dots,\bar z_n$, then the recurrence (2.4)-(2.10) cannot express the elementary symmetric functions $e_j$ in terms of $\nu_2,\dots,\nu_{2n-1}$, and Theorem 1.7(i) would need a different argument.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is Theorem 1.9: with independent complex variables $z_1,\dots,z_n,\bar z_1,\dots,\bar z_n$, the field $\tilde F_n=\mathbb{C}(\nu_j,\bar\nu_j)_{j\ge 2}$ equals $H(\nu_2)$, where $H=\mathbb{C}(z,\bar z)^{S_n\times S_n}$ is the field of rational functions symmetric in the $z$'s and symmetric in the $\bar z$'s. Thus, once the elementary symmetric functions of the vertex coordinates and of their conjugates are known, the single bilinear form $\nu_2$ determines every higher harmonic and anti-harmonic moment as a rational function. The minimal polynomial of $\nu_2$ over $H$ is $P(t)=\prod_{(\$\sigma$,\tau)\in(S_n\times S_n)/G}(t-\$nu_2^{{(\sigma,\tau)}}$)$, whose degree is $n!(n-1)!$ for odd $n$ and $2((n-1)!)^2$ for even $n$; Lemma 3.4 shows its roots are pairwise distinct for generic vertex data. For the harmonic moments alone, Theorem 1.7 states $F_n=\mathbb{C}(\nu_2,\dots,\nu_{2n-1})$, a purely transcendental extension of $\mathbb{C}$ of transcendence degree $2n-2$, so all higher $\nu_j$ are rational in the first $2n-2$ with denominators that are powers of one determinant $D_n$.

Load-bearing premise

The proof that the harmonic moments generate the field assumes that the Toeplitz matrix $U$ in equation (2.6) is invertible as a matrix of rational functions; the paper never proves that its determinant $D_n$ is not identically zero for $n\ge 4$.

Editorial extensions

If this is right

For a real $n$-gon, the entire logarithmic potential at infinity is determined by the $n$ elementary symmetric functions of the vertices $z_j$, the $n$ elementary symmetric functions of their conjugates $\bar z_j$, and the single number $\nu_2$; the higher moments carry no further information.
The exact integer $d_n$ counts the number of distinct conjugate values of $\nu_2$ under all relabelings of the vertices and anti-vertices, so it quantifies how many different area-type assignments are compatible with one set of symmetric vertex data.
Harmonic-moment inversion for polygons requires fitting only $2n-2$ numbers, namely $\nu_2,\dots,\nu_{2n-1}$; all later harmonic moments are rational functions of these.
All denominators in expressing higher harmonic moments are powers of one fixed determinant $D_n$, and the full ring of harmonic moments is not finitely generated, so the finite description is a field-level phenomenon, not a polynomial-ring phenomenon.
For triangles, the whole system collapses to one explicit resultant equation $L=\mathrm{Res}_S(R,Q)=0$, which gives a complete algebraic test for whether seven moment values come from a triangle.

Reading between the lines

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

A practical reconstruction algorithm could proceed in two steps: recover the separately symmetric data and $\nu_2$ from low-order moments, then evaluate the explicit rational formulas for higher moments; this is an algorithmic route the paper does not develop.
The description of the orbit algebra as a Segre ring suggests that moment measurements of polygons live near a determinantal variety, so rank-constrained fitting could be used to denoise or complete partial moment data.
The genericity assumption behind Lemma 3.4 fails on special loci, for example for highly symmetric or degenerate polygons, where the degree of the extension drops; a stratified classification of these loci would be a natural testable extension.
The unproved nonvanishing of $D_n$ is likely true for all $n$; if it failed for some $n$, the equality $F_n=\mathbb{C}(\nu_2,\dots,\nu_{2n-1})$ would still hold on the open set where $D_n\ne 0$, but the global statement would need a different proof.

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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

This is a good paper that substantially clarifies the algebraic dependencies among harmonic and anti-harmonic moments of polygons. The explicit field-generation theorem for the harmonic moments (2n-2 independent generators), the ring-localization statement, and the one-generator picture over the symmetric-function field are genuinely new, and the degree formulas n!(n-1)! and 2((n-1)!)^2 are clean. The Galois group computation in Section 4 is a real addition. The authors are honest: Remark 2.5 discloses that equation (2.5) coincides with [7], and overlap with the shape-from-moments recurrence is flagged clearly.

The main proofs are standard and mostly complete: Stokes' theorem gives the moment formulas, the Toeplitz recurrence generates the field, and the birationality argument for Theorem 1.9(iii) is sound. The one real soft spot is the unproven generic invertibility of the Toeplitz matrix U in (2.5)-(2.6). To solve for the elementary symmetric polynomials e_j as rational functions of ν_2,...,ν_{2n-1}, the authors write "Assuming that U is invertible" and never show that det U = D_n is not identically zero. This assumption is load-bearing: it underlies Theorem 1.7(ii), the equality F_n = C(ν_2,...,ν_{2n-1}), and the localization Theorem 1.8(ii). The same issue appears in the proof of Theorem 1.9(i)-(ii). For n=3, D_3 = ν_2^3, so it is true; for n≥4 the paper has no argument. The stress-test note says an explicit evaluation makes the determinant nonzero, so I expect the gap is fixable, but it belongs in the proof.

A couple of smaller remarks: the genericity arguments (e.g., that a vector space over C is not a finite union of quadrics) are fine, and the ring non-finite-generation argument is correct. The citation pattern looks fair.

Verdict: this deserves peer review. Send it to a good complex analysis or algebraic geometry journal, and ask the authors to patch the nonvanishing argument for D_n. Once that is supplied, the main results stand.

Referee Report

3 major / 6 minor

Summary. The paper studies the algebraic relations among normalized harmonic and anti-harmonic moments ν_k(z,\bar z) of polygonal measures associated to n-vertex (not necessarily convex) polygons. The main results are: (Theorem 1.7) the field F_n generated by all harmonic moments is C(ν_2,\ldots,\nu_{2n-1}), it contains the field C(z)^{S_n}, and ν_2,\ldots,\nu_{2n-1} are algebraically independent; (Theorem 1.8) the ring R_n generated by harmonic moments is not finitely generated, while its localization at a certain determinant D_n is C[ν_2,\ldots,\nu_{2n-1}][1/D_n]; (Theorem 1.9) the joint field \tilde F_n is generated by the first 4n-5 moments, contains H=C(z,\bar z)^{S_n\times S_n}, and is an algebraic extension of H generated by ν_2 of the stated degree, n!(n-1)! for odd n and 2((n-1)!)^2 for even n. The paper also describes the stabilizer of ν_2 under the natural S_n\times S_n action, computes the Galois group of the Galois closure, and works out the triangle case n=3 explicitly. The proofs use an explicit Stokes-theorem formula for moments, a generating function for the moment sequence, a Toeplitz linear system derived from the denominator of the generating function, and a birationality argument for the geometric approach.

Significance. If the stated results hold, they give a precise and satisfying answer to a natural question: the harmonic moments of an n-gon satisfy exactly 2n-2 algebraically independent relations with all higher moments rational functions of the first 2n-2, and the full harmonic/anti-harmonic system is controlled by the single lowest moment ν_2 up to the symmetric-field extension H. The paper's strengths include explicit polynomial formulas for moments, a clean recurrence argument, a representation-theoretic computation of the stabilizer and Galois group, and a fully worked n=3 example with an explicit resultant relation. The dependence on standard tools (Stanley's recurrence, Wedderburn's theorem, Mumford's birationality criterion) is appropriate. The main load-bearing gap is the unproved generic invertibility of the Toeplitz matrix U in equation (2.5)-(2.6); this is a local, likely fixable issue rather than a fundamental flaw.

major comments (3)

[§2, proof of Theorem 1.7, Eqs. (2.5)–(2.6)] The proof solves the linear system U·E=V after the phrase 'Assuming that U is invertible', but the paper never proves that the determinant D_n = det U is a nonzero polynomial in z and \bar z. This assumption is load-bearing: it is used to conclude that e_j(z) ∈ C(ν_2,\ldots,\nu_{2n-1}), hence C(z)^{S_n} ⊂ F_n, and then F_n = C(ν_2,\ldots,\nu_{2n-1}); Theorem 1.8(ii) additionally localizes at D_n. If D_n were identically zero, the displayed linear system would not imply these conclusions. For n=3 one has D_3 = ν_2^3, so the missing case is n ≥ 4; the authors should supply an explicit nonvanishing argument, for example an evaluation at a suitable choice of (z,\bar z), which a referee check indicates exists for n=4.
[§3, proof of Theorem 1.9(i)–(ii)] The proof begins by saying that assertions (i) and (ii) are proved similarly to the corresponding statements in Theorem 1.7. This inheritance carries the same unproved invertibility of the Toeplitz matrix: the anti-harmonic moments require solving the analogue of (2.5) with \bar z in place of z, and the joint field generation for \tilde F_n depends on that nonvanishing. The proof should explicitly state the anti-harmonic Toeplitz system and its determinant, note that it is D_n(\bar z,z), and reduce its nonvanishing to the same evaluation used for D_n.
[§3, proof of Theorem 1.9(iii), irreducibility of P(t)] The argument that the factor Q(t) has coefficients in H and hence forces the set U to be invariant under S_n\times S_n is correct, but the phrase 'By Proposition 3.2, this implies that U intersects any right coset' is terse. The point is that the left action on right cosets is transitive for any subgroup, so a nonempty invariant subset is the whole set; spelling this out would improve clarity. This is not a technical gap, but it is a place where a reader can easily get stuck.

minor comments (6)

[Abstract] The abstract contains the typo 'bu t' instead of 'but'.
[§2, Eq. (2.7)] In equation (2.7), the summation index is printed as 'm−0' and should be 'm=0'; the product sign is also typeset ambiguously.
[§2, proof of Theorem 1.8(i)] The argument that no ν_j is a polynomial in lower moments is terse; expanding it by observing that a polynomial in ν_2,\ldots,\nu_N has total degree in \bar z equal to the degree of the polynomial and z-degree at most N−1 would remove any ambiguity.
[§3, proof of Lemma 3.4] The statement that a finite-dimensional vector space over C cannot be a finite union of nontrivial quadrics is used without proof; a one-sentence justification, for instance by restriction to a generic line, would be helpful.
[§4, Corollary 4.6] The description of the Galois group as all permutations preserving the relations in Corollary 4.5 is close to the definition of a Galois group; the authors should clarify what additional information this corollary is meant to provide, for example an algorithmic or structural characterization.
[§5, references] Reference [3] is a MathOverflow discussion; since the identity 16M² = −det Ω(S) is verified directly in the proof, the reliance on [3] could be stated as a remark rather than a formal citation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all load-bearing derivations are proved in-paper from Stokes' theorem and standard external results.

full rationale

Walked the derivation chain. The moment formula (1.1) is proved from Stokes' theorem via Theorem 1.1, with the proof given in Section 2. The generating-function identity (2.2), attributed to the authors' own prior work [17], is explicitly re-proved in Proposition 2.1 by substituting (1.1), so that self-citation is not load-bearing. The recurrence (2.4) is derived from the independently proved expansion (2.3), and the linear system (2.5) is acknowledged in Remark 2.5 to coincide with the external paper [7]; the overlap is disclosed rather than concealed. The field-generation claim uses that linear system, and the only identified weakness, namely the unproved nonvanishing of D_n = det U for n >= 4, is a correctness gap in the proof as written, not a circularity: the system is not obtained by assuming the conclusion, and the algebraic-independence argument via the tower (2.12) uses independent transcendence-degree reasoning. Theorem 1.9 defines P(t) from the orbit of nu_2, computes its degree from the group-theoretic stabilizer in Proposition 3.2, and proves irreducibility and generic separability using standard facts; the rational dependence of higher moments on H(nu_2) is established by a birationality argument citing Mumford. No fitted parameter is renamed as a prediction, no ansatz is smuggled in by self-citation, and no known result is merely renamed. The paper is a self-contained proof-based account; score 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on explicit integral formulas (Theorem 1.6) and standard algebraic geometry. No parameters are fitted to data. The main unproved background input is the generic invertibility of the Toeplitz matrix U, which is assumed in the proof of Theorem 1.7. The complexification of vertices is the key domain assumption. No new speculative entities are introduced.

assumptions (5)

standard math Recurrence for coefficients of a rational generating function: if sum f(j) t^j = P(t)/Q(t) with deg P < deg Q, then f(k+d) + alpha_1 f(k+d-1) + ... + alpha_d f(k) = 0.
Stanley, Enumerative Combinatorics, [18] Th. 4.1.1; used in the proof of Theorem 1.7 to derive the linear system (2.5).
standard math The image of an irreducible representation of the group algebra of a finite group is the full matrix algebra of the representation space.
Wedderburn's theorem, [21]; used in Lemma 4.1 to characterize the span M_n of the matrices M(sigma,tau).
standard math A generically one-to-one morphism of algebraic varieties of the same dimension is birational.
Mumford, [14]; used in the proof of Theorem 1.9(iii) to conclude nu_j is a rational function of the symmetric functions and nu_2.
domain assumption The polygonal measure is defined by complexifying the vertices: z_j and zbar_j are independent complex variables, and the measure is pulled back from a fixed triangulated convex n-gon.
Sections 1.2 and 2. The theorems are proven in the complexified setting; the real-vertex case is a specialization. This is a modeling choice, not an empirically verified fact.
ad hoc to paper The Toeplitz moment matrix U in (2.6) is generically invertible: det U = D_n is not the zero polynomial in z and zbar.
Assumed without proof in Section 2 in the sentence 'Assuming that U is invertible'; needed to express the elementary symmetric polynomials as rational functions of the first 2n-2 harmonic moments. For n=3, D_3 = nu_2^3, but for n >= 4 no proof is given.

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Pith. "Pith review of Algebraic relations between moments of plane polygons." pith.science (2026). https://pith.science/paper/Y2XOL2DU

@misc{pith2026190807621,
  author       = {Pith},
  title        = {Pith review of: Algebraic relations between moments of plane polygons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y2XOL2DU}},
  note         = {Machine review of arXiv:1908.07621}
}

read the original abstract

We describe the algebraic relations satisfied by the harmonic and anti-harmonic moments of simply connected, but not necessarily convex planar polygons with a given number of vertices.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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Pith tools

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

[1] [7]

Golib, P

G. Golib, P. Milanfar, and J. Varah, A stable numerical me thod for inverting shape from moments, SIAM J. SCI. Comput. vol. 21(4) (1999), 1222–1243

work page 1999

[2] [1]

Barcanescu, N

S. Barcanescu, N. Manolache, Nombres de Betti d’une sing ularité de Segre–Veronese. (French) C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 4, A237–A239

work page 1979

[3] [2]

Bruns, A

W. Bruns, A. Conca, Gröbner bases and determinantal idea ls. Commutative algebra, singu- larities and computer algebra (Sinaia, 2002), 9–66, NATO Sc i. Ser. II Math. Phys. Chem., 115, Kluwer Acad. Publ., Dordrecht, 2003. ALGEBRAIC RELATIONS BETWEEN MOMENTS OF PLANE POLYGONS 19

work page 2002

[4] [3]

Bryant, https://mathoverﬂow.net/questions/81690 /area-of-triangle-from-coeﬃcients-of- its-cubic

R. Bryant, https://mathoverﬂow.net/questions/81690 /area-of-triangle-from-coeﬃcients-of- its-cubic

work page

[5] [4]

M. A. Brodsky, and V. N. Strakhov, On the uniqueness of the inverse logarithmic potential problem, SIAM Journal on Applied Mathematics, vol. 46(2) (1 986), 324–344

work page

[6] [5]

Conca, S

A. Conca, S. Hoşten, R. Thomas, Nice initial complexes of some classical ideals. Algebraic and geometric combinatorics, 11–42, Contemp. Math., 423, A mer. Math. Soc., Providence, RI, 2006

work page 2006

[7] [6]

Ph. J. Davis. Triangle formulas in the complex plane. Mat h. Comp., vol. 18 (1964), 569–577

work page 1964

[8] [8]

Sh. Goto, K. W atanabe, On graded rings. I. J. Math. Soc. Ja pan 30 (1978), no. 2, 179–213

work page 1978