Cyclic polytopes through the lens of iterated integrals

Felix Lotter; Rosa Prei{\ss}

REVIEW 3 major objections 3 minor 1 cited by

Cyclic polytopes through the lens of iterated integrals

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Pith's one-line read Cyclic polytopes carry infinitely many independent volume invariants in every dimension.

desk verdict Genuinely new ring of cyclic-polytope volume invariants, but the printed antipode sign is wrong in a load-bearing way and the even-d case is conditional on a companion paper. read the letter →

arxiv 2412.11283 v2 pith:KVPVEJ6J submitted 2024-12-15 math.RA math.CO

Felix Lotter , Rosa Prei{ss} This is my paper · ORCID

classification math.RAmath.CO MSC 60L1013A5052B05

keywords iteratedintegralspiecewiselinearpathsshufflealgebrapermutationgroupactionsignedvolumeinvariantspositivematricescyclicpolytopes

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 shows that cyclic polytopes have not just a volume but a whole infinite family of independent geometric attributes, all computed from iterated integrals along paths that run through the polytope's vertices. The volume of a cyclic $d$-polytope equals the signed iterated integral of a piecewise linear path through its vertices, and different paths are related by a subgroup of combinatorial automorphisms. The paper characterizes the linear combinations of iterated integrals invariant under this subgroup and proves that in every dimension there are infinitely many algebraically independent invariants with respect to the shuffle product. Consequently the ring of volume invariants is infinitely generated, so cyclic polytopes carry a large supply of canonical features rather than a single numerical invariant.

What carries the argument

The machinery is the signature of a piecewise linear path, a linear form $S(X)$ on words in $d$ letters whose value on a word $i_1\cdots i_k$ is the iterated integral of the corresponding coordinate differentials over the simplex $0\le t_1\le\cdots\le t_k\le 1$. The concatenation identity for signatures makes the signature multiplicative under path concatenation, and the shuffle identity makes the map $H^d_n$ from the shuffle algebra to polynomials an algebra homomorphism. The signed volume is the value on the word $vol_d=\sum_{\sigma\in S_d} sgn(\sigma)\sigma(1)\cdots\sigma(d)$. The invariance subgroup $C^d_n$ is the stabilizer of positive matrices, hence the group of combinatorial automorphisms of the cyclic polytope that preserve the sign of all maximal minors. The proof describes these groups with the evenness criterion for cyclic polytopes, identifies the $n\ge d+3$ invariants as time-reversal and loop-closure invariants via the antipode and cyclic rotations, and uses the weak Chen–Chow theorem to convert equalities of signature polynomials into equalities of words.

What would settle it

For a fixed $d$, exhibit a finite generating set for $Inv^d_{\ge d+3}\cap I(PL^d_{d+2})$ as a shuffle algebra; Theorem 4.9 asserts that no such finite set exists, so a finite generation proof or a finite degree bound on its algebraically independent elements would refute the main claim. Alternatively, for even $d$, find a counterexample to either of the two imported statements from the companion paper, since the even case depends entirely on them.

Watch

Extended reading notes

Core claim

The central claim is Theorem 4.9: for every dimension $d$, the intersection $Inv^d_{\ge d+3} \cap I(PL^d_{d+2})$, and in particular the full ring of volume invariants $Inv^d$, contains infinitely many algebraically independent elements with respect to the shuffle product. Concretely, no finite list of iterated-integral expressions generates all invariant polynomial functions on cyclic $d$-polytopes. For odd $d$ the paper gives the large-$n$ invariant ring explicitly: it is either the whole word algebra when $(d+1)/2$ is odd or the subalgebra of time-reversal invariants when $(d+1)/2$ is even. For even $d$ it identifies the invariant ring with loop-closure invariants, possibly intersecting time-reversal invariants, and imports from a companion paper the fact that this loop-closure ring contains infinitely many algebraically independent elements. Because the independent family lies inside the vanishing ideal of paths with $d+2$ control points, it belongs to $Inv^d$ itself while being invisible on the minimal nontrivial number of vertices.

Load-bearing premise

For even dimensions, the proof imports two facts from the companion paper: that the large-$n$ invariant ring consists exactly of signature quantities unchanged by closing a path into a loop (intersected with time-reversal invariants in one parity case), and that this loop-closure ring contains infinitely many algebraically independent elements. If either imported fact is false, the even-dimensional half of the main theorem is not derived in this paper.

Editorial extensions

If this is right

In every dimension $d$, the shuffle subalgebra $Inv^d$ is not finitely generated, so any finite collection of iterated-integral signatures is insufficient to capture all invariant features of cyclic polytopes.
For odd $d$ with $(d+1)/2$ even, the large-$n$ invariant ring is exactly the ring of time-reversal invariants; for odd $d$ with $(d+1)/2$ odd, every signature word is invariant for paths with at least $d+3$ control points.
For even $d$, invariance under all cyclic rotations of control points for all $n$ is equivalent to loop-closure invariance, which ties cyclic-polytope invariants to signatures of loops and to the equivalence relations studied in the companion paper.
Since the infinite independent family lies in $I(PL^d_{d+2})$, all these new invariants vanish on paths with $d+2$ control points and first become visible for polytopes with at least $d+3$ vertices.

Reading between the lines

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

If the even-dimensional dependence on the companion paper could be removed by a direct construction of infinitely many algebraically independent loop-closure invariants, Theorem 4.9 would become self-contained for all $d$ rather than conditional on an external result.
Restricting the volume invariants to $SL_d$ orbits yields functions on the positive Grassmannian; Theorem 4.9 then suggests the positive Grassmannian carries infinitely many algebraically independent functions of this signature type, beyond the volume itself.
The induced equivalence relation on cyclic polytopes is probably much finer than the three explicit moves (permutations, translations, deleting collinear vertices); if Conjecture 4.1 holds, all $d$-polytopes with $d+2$ vertices of equal volume would be equivalent under the full invariant ring, a rigidity statement one could test numerically for small $d$.

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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

Here's the short version: this paper puts a new object on the table—the ring Inv^d of volume invariants of cyclic polytopes, built from iterated-integral signatures—and shows it is infinitely generated in every dimension. That is a real contribution, and the paper is mostly careful. But as printed, Definition 4.3 contains a sign error that makes the paper internally inconsistent, and it needs to be fixed before I can trust the main theorem.

The good parts first. The stabilizer computation in Section 3 is solid and well-organized: Propositions 3.8 and 3.10 give a full description of C^d_n, and Proposition 3.12 cleanly characterizes when the signed volume is invariant under a permutation. The structural identifications in Section 4—Inv^d_{≥d+3} = TimeRevInv^d or LoopClosureInv^d ∩ TimeRevInv^d depending on parity—are sensible and well-motivated. The overall strategy for Theorem 4.9, using infinite algebraic independence in the shuffle algebra and then pushing it into the quotient by I(PL^d_{d+2}), is sound in spirit.

The soft spots are real, though. The antipode definition in Definition 4.3 sets A(w) = (-1)^{d+1} w^rev. That sign depends on the ambient dimension, not on the length of w. For d=3, A(vol3) = +vol3^rev = -vol3, so vol3 is not in TimeRevInv^3. But Proposition 3.12 puts vol3 in Inv^3, hence in Inv^3_{≥6}, and Proposition 4.5 identifies Inv^3_{≥6} with TimeRevInv^3. So the written definitions contradict each other exactly at the point where the proof of Theorem 4.9 relies on TimeRevInv. This is a load-bearing defect, though almost certainly a typo: replacing (-1)^{d+1} with (-1)^{|w|}, the standard antipode sign, repairs it.

Second, the even-dimensional half of Theorem 4.9 leans on two external results from [19], a companion preprint by one of the present authors, and from [11] with overlapping authors. I could not verify those from the text here. The transcendence-degree argument in the last paragraph of the proof is also compressed: the injection R[s_{N+1},...] → R[x_1,...,x_{d+2}] is asserted, not proved, but it is plausible. Third, Example 4.2 reports Macaulay2 output without the code, which is a reproducibility gap, though minor.

Bottom line: this is a paper a serious referee should engage with, not desk-reject. The construction is new, the computations are mostly careful, and the defects are repairable. I'd send it back for revision, asking for the sign fix and for a fuller statement of exactly which parts of Theorem 4.9 rest on [19]. I'd cite the corrected version.

Referee Report

3 major / 3 minor

Summary. The paper studies linear combinations of iterated-integral signatures of piecewise linear paths through the vertices of a cyclic polytope, focusing on those combinations invariant under the stabilizer C^d_n of positive matrices. The authors describe C^d_n explicitly (Propositions 3.8 and 3.10), prove that the signed volume is the basic invariant (Proposition 3.12), and then characterize the intersection Inv^d_{\ge d+3} in terms of time-reversal invariants for odd d (Proposition 4.5) and loop-closure invariants for even d (Proposition 4.8). The main result, Theorem 4.9, claims that Inv^d_{\ge d+3} \cap I(PL^d_{d+2}), and in particular the ring Inv^d of volume invariants, contains infinitely many algebraically independent elements with respect to the shuffle product, hence is infinitely generated.

Significance. If the main theorem is correct, it establishes that the ring of volume invariants of cyclic polytopes is infinite-dimensional in every dimension, a genuinely new structural result that connects rough-path theory, convex geometry, and invariant theory. The paper carefully computes the stabilizer groups C^d_n, gives a clean formulation of the volume as an iterated integral, and provides explicit invariant polynomials in low-degree cases; these computations are valuable and no numerical fitting is involved. The heavy reliance on the companion paper [19] and a compressed transcendence-degree argument, however, mean that the main theorem is not fully self-contained as written.

major comments (3)

[Definition 4.3] Definition 4.3 defines the antipode A by A(w)=(-1)^{d+1} w^rev, with the sign depending on the ambient dimension d rather than on the length |w| of the word. The standard antipode of the concatenation Hopf algebra is A(w)=(-1)^{|w|} w^rev, and it is this length-dependent map that is adjoint to time reversal of paths. With the printed sign, for d=3 one has A(vol3)=-vol3, so vol3 is not in TimeRevInv^3. But Proposition 3.12 implies vol3 lies in Inv^3_n for all n\ge 4, hence in Inv^3_{\ge 6}, while Proposition 4.5 asserts Inv^3_{\ge 6}=TimeRevInv^3. The written definitions are therefore internally inconsistent at exactly the point used in the proof of Theorem 4.9, both in the odd case and in the even-dimensional intersection. This is repairable by replacing (-1)^{d+1} with (-1)^{|w|} throughout Definition 4.3 and its subsequent uses, but as it stands it is a load-bearing defect.
[Theorem 4.9 and Proposition 4.8] The even-dimensional part of the main theorem depends on three results imported from the companion paper [19]: the identification of rotation-invariant signature elements with LoopClosureInv^d (cited as [19, Prop. 4.3]), the adjoint property of left and right loop closures (cited as [19, Lemma 4.7]), and the statement that LoopClosureInv^d contains an infinite algebraically independent subset. None of these is proved in the present manuscript. Since these statements carry half of the proof of Theorem 4.9, the main claim is conditional on [19]. The authors should either prove these results in an appendix or quote them in full with explicit statements and hypotheses so that the even-dimensional case can be verified independently.
[Proof of Theorem 4.9, final paragraph] The transcendence-degree argument in the last paragraph of the proof is too compressed to be checked. In particular, the claim that, if the kernel of the composition R[s_1,s_2,\ldots]\to R\langle 1,\ldots,d\rangle/I(PL^d_{d+2}) is algebraic over R[s_1,\ldots,s_N], then one obtains an injection R[s_{N+1},s_{N+2},\ldots]\to R[x_1,\ldots,x_{d+2}] does not follow: the composition may have a nonzero kernel on the tail variables, and the image need not contain a polynomial subring on infinitely many variables. A rigorous argument using the Krull dimension or transcendence degree of the image is needed to conclude that the kernel contains infinitely many algebraically independent elements. This step is load-bearing for the theorem's main claim, so it must be repaired or expanded.

minor comments (3)

[Example 4.2] The example states that a Macaulay2 computation shows the invariants of degree at most 6 in Inv^3_4 are spanned by 18 elements, but no code or reproducible script is provided. Since the example is used to motivate Conjecture 4.1, please supply the computation or a precise description of the algorithm used.
[Section 4, before Theorem 4.9] The displayed direct-sum decomposition of Inv^d is garbled as typeset: both summands appear to lie inside I(PL^d_{d+2}), so the displayed equality cannot hold. Please replace it with a correct statement, for instance by explicitly separating the part inside the kernel I(PL^d_{d+2}) from a complementary summand.
[Outlook, last paragraph of the even-d subsection] The sentence 'Inv4_{\ge d+3} \cap I(PL^4_6)' uses a parameter d that is already fixed as 4; it should read 'Inv^4_{\ge 7} \cap I(PL^4_6)'.

Circularity Check

1 steps flagged · score 6.0 of 10

Even-dimensional half of the main theorem is imported from the same authors' companion paper [19] rather than derived here.

self citation load bearing [Theorem 4.9 proof, final paragraph (Section 4); also Proposition 4.8]
"In [19] it is shown that LoopClosureInvd contains an infinite algebraically independent subset and by a similar argument as above it follows that the intersection LoopClosureInvd ∩ TimeRevInvd also does, using that w − Aw is an element of LoopClosureInvd for every w ∈ LoopClosureInvd since Aw is a loop closure invariant if w is (which follows from the definition)."

The even-d case of the main theorem is not proved in this paper. Once Inv^d_{\ge d+3} is identified with LoopClosureInvd (or its intersection with TimeRevInvd) via Proposition 4.8, the required infinite algebraically independent family is taken verbatim from [19], a companion preprint whose authors include R. Preiß, an author of the present paper. Proposition 4.8 itself depends on [19, Proposition 4.3 and Lemma 4.7]. Thus the central existence claim for even d reduces, in this text, to the authors' own prior work. Unless [19] is independently verified, this is a load-bearing self-citation rather than a self-contained derivation.

full rationale

The odd-d case of Theorem 4.9 is essentially self-contained: it uses the characterization Inv^d_{\ge d+3} = TimeRevInvd (or the full algebra) and derives the infinite algebraically independent family from Lyndon words and the antipode. The weak Chen-Chow theorem has independent support through the cited proof in [12, Lemma 8]. The one genuinely load-bearing dependence on same-author prior work is the even-d case, where the existence of infinitely many algebraically independent loop-closure invariants is imported from [19], and the identification of Inv^d_{\ge d+3} itself relies on [19, Proposition 4.3 and Lemma 4.7]. This is a citation of a companion result, not a definitional equivalence, so it is not circular in the strongest 'by construction' sense; however, it does make the even-dimensional half of the central claim contingent on the same group's prior work without independent verification in the present text. A separate correctness concern about the dimension-dependent antipode sign in Definition 4.3 is not a circularity and is not counted in the score.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper contains no fitted constants or hand-chosen numerical parameters. It relies on standard signature theory, on cited automorphism-group classification, and, for the even-dimensional main case, on a companion preprint by overlapping authors. The ad hoc assumption is that companion result.

assumptions (7)

domain assumption Automorphism groups of cyclic polytopes are as in Theorem 3.7 (Kaibel-Wassmer).
Used to compute C^d_n in Propositions 3.8 and 3.10; the theorem is cited, not proved.
domain assumption Weak Chen-Chow: pathwise equality of all signature evaluations implies equality of words (Theorem 4.4).
Cited to [11, Lemma 5.2]; used in Propositions 4.5 and 4.8 to pass from S(X)-equalities to word identities.
ad hoc to paper LoopClosureInv results in [19]: invariants under rotations equal loop-closure invariants, and LoopClosureInv contains an infinite algebraically independent subset.
Backbone of Proposition 4.8 and the even-d case of Theorem 4.9; [19] is a companion preprint by Preiß and coauthors, not proved here.
standard math The shuffle algebra on d letters is freely generated by Lyndon words.
Cited to [21, Theorem 6.1]; used in Theorem 4.9 proof.
standard math Signature determines a path up to translation, reparametrization and tree-like equivalence, and is invariant under reparametrization.
Standard signature theory [4,6]; used to justify well-definedness of H^d_n and control-point reductions.
standard math Positive matrices with positive maximal minors form a Zariski-dense subset of R^{d×n}.
Used in Proposition 3.12 to extend polynomial identities from positive matrices to all matrices.
standard math Chen's identity and Ree's shuffle identity for iterated integrals.
Used to define H^d_n recursively and to make H^d_n a shuffle-algebra homomorphism.

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Cite this review

Pith. "Pith review of Cyclic polytopes through the lens of iterated integrals." pith.science (2026). https://pith.science/paper/KVPVEJ6J

@misc{pith2026241211283,
  author       = {Pith},
  title        = {Pith review of: Cyclic polytopes through the lens of iterated integrals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KVPVEJ6J}},
  note         = {Machine review of arXiv:2412.11283}
}

read the original abstract

The volume of a cyclic polytope can be obtained by forming an iterated integral along a suitable piecewise linear path running through its edges. Different choices of such a path are related by the action of a subgroup of the combinatorial automorphisms of the polytope. Motivated by this observation, we look for other linear combinations of iterated integrals that are invariant under the subgroup action. This yields interesting polynomial attributes of the cyclic polytope. We prove that there are infinitely many of these invariants which are algebraically independent in the shuffle algebra.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Conjugation, loop and closure invariants of the iterated-integrals signature
math.RA 2024-12 conditional novelty 7.0 of 10

Conjugation invariants of the path signature are exactly the cyclic-rotation sums of words; loop and closure invariants are characterized through a Lie-algebraic condition.

Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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work page Pith review arXiv 2002

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work page 2019

[4] [2]

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Carlos Am ´endola, Darrick Lee, and Chiara Meroni. Convex hulls of curves: V ol- umes and signatures. arXiv:2301.09405, 2023

work page Pith review arXiv 2023

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Nima Arkani-Hamed and Jaroslav Trnka. The amplituhedron. Journal of High Energy Physics, 2014(10):1–33, 2014

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Horatio Boedihardjo, Xi Geng, Terry Lyons, and Danyu Yang. The signature of a rough path: Uniqueness. Advances in Mathematics, 293:720–737, 2016

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Kuo-Tsai Chen. Iterated Integrals and Exponential Homomorphisms. Proceedings of the London Mathematical Society, s3-4(1):502–512, 1954

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Integration of Paths – A Faithful Representation of Paths by Non- commutative Formal Power Series

Kuo-Tsai Chen. Integration of Paths – A Faithful Representation of Paths by Non- commutative Formal Power Series. Transactions of the American Mathematical Society, 89(2):395–407, 1958

work page 1958