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arxiv: 2601.06814 · v2 · submitted 2026-01-11 · 🧮 math.AG · math.AT· math.CO

Algebraic topology of the Lagrange inversion

Victor M. Buchstaber , Alexander P. Veselov This is my paper

Pith reviewed 2026-05-16 15:47 UTC · model grok-4.3

classification 🧮 math.AG math.ATmath.CO
keywords Lagrange inversionChern numberscomplex projective spacecomplex cobordismChern-Dold charactertheta divisorsCatalan numbersHirzebruch genus
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The pith

Lagrange inversion formula follows from Chern numbers of complex projective space.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the classical Lagrange inversion formula for power series has a topological interpretation using the Chern numbers of complex projective spaces. This comes from applying the Chern-Dold character in complex cobordism theory, which also gives a new derivation of the formula itself. A parallel result interprets the multiplicative inversion formula through Chern numbers of smooth theta divisors. The authors introduce a formal group law based on the Catalan numbers and give it a topological meaning via the associated Hirzebruch genus. They further examine when the Chern numbers of an algebraic variety are all divisible by its Euler characteristic.

Core claim

The Lagrange inversion formula admits a natural topological interpretation in terms of the Chern numbers of the complex projective space. The proof relies on the Chern-Dold character in complex cobordism theory and provides a new derivation of the formula. Similarly, the multiplicative inversion formula receives an interpretation via Chern numbers of smooth theta divisors, accompanied by a new formal group defined by the Catalan numbers whose Hirzebruch genus has a topological meaning.

What carries the argument

The Chern-Dold character in complex cobordism theory, which connects cobordism classes to Chern numbers and allows the inversion formulas to be read off from the geometry of projective spaces and theta divisors.

If this is right

  • The Lagrange inversion formula can be derived purely from topological data of complex projective spaces.
  • A new formal group law arises from the Catalan numbers with a corresponding Hirzebruch genus.
  • The multiplicative inversion formula corresponds to Chern numbers on smooth theta divisors.
  • Conditions exist under which all Chern numbers of an algebraic variety are divisible by the Euler characteristic.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar topological interpretations might apply to other classical inversion formulas in combinatorics.
  • The divisibility condition could be tested on other classes of varieties beyond those considered.
  • Connections between formal group laws and cobordism might extend to other genera in algebraic topology.

Load-bearing premise

The earlier established properties of the Chern-Dold character in complex cobordism must transfer directly to produce the inversion coefficients from the Chern numbers.

What would settle it

Compute the relevant Chern numbers for low-dimensional complex projective spaces and check if they reproduce the known coefficients of the Lagrange inversion series up to that order; mismatch would falsify the interpretation.

Figures

Figures reproduced from arXiv: 2601.06814 by Alexander P. Veselov, Victor M. Buchstaber.

Figure 1.5
Figure 1.5. Figure 1.5: 3-dimensional associahedron and the corresponding graph. to a single application of the associative law. This explains the name associahedron for the polytope PΓ of this example; we shall denote it Asn. Example 1.5.26. Proposition 1.5.11 describes the associahedron as the result of consecutive hyperplane cuts of a simplex. In the case n =3, consider the graph shown in [PITH_FULL_IMAGE:figures/full_fig_p… view at source ↗
Figure 1
Figure 1. Figure 1: Left: The permutahedra Figure 2. 3D permuto [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
read the original abstract

The Lagrange inversion formula for power series is one of the classical formulas from analysis and combinatorics. A nice geometric interpretation of this formula in terms of the Stasheff polytopes was discovered by Loday. We show that it also admits a natural topological interpretation in terms of the Chern numbers of the complex projective space. The proof is based on our earlier work on the Chern-Dold character in complex cobordism theory and leads to a new derivation of the Lagrange inversion formula. We provide a similar interpretation of the multiplicative inversion formula in terms of Chern numbers of the smooth theta divisors. In this relation we introduce a new formal group defined by the Catalan numbers and explain the topological meaning of the corresponding Hirzebruch genus. Finally, we discuss a related general problem of when all Chern numbers of an algebraic variety are divisible by its Euler characteristic.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to provide a natural topological interpretation of the Lagrange inversion formula in terms of the Chern numbers of complex projective space CP^n, using the Chern-Dold character from complex cobordism theory. This leads to a new derivation of the formula. It extends this to the multiplicative inversion formula via Chern numbers of smooth theta divisors, introducing a new formal group law defined by the Catalan numbers and discussing the topological meaning of the corresponding Hirzebruch genus. Additionally, it addresses when all Chern numbers of an algebraic variety are divisible by its Euler characteristic.

Significance. If the central claims hold, the paper offers a significant bridge between combinatorial analysis and algebraic topology, providing geometric insights into classical formulas through cobordism and formal group laws. The new Catalan formal group law could have broader applications in topology. The discussion on Chern number divisibility adds to the understanding of algebraic varieties. However, the strength is tempered by the dependence on prior results, which if confirmed, would enhance the paper's contribution to the field.

major comments (2)
  1. [§3] §3 (main derivation): The mapping from the cobordism class of CP^n via the Chern-Dold character to the Lagrange inversion coefficients invokes properties from the authors' earlier work without restating the precise theorems applied or verifying their direct applicability here; this dependence is load-bearing for the claimed new derivation.
  2. [§5] §5 (multiplicative case): The new formal group law defined by Catalan numbers and its identification with theta-divisor Chern numbers requires an explicit check that the law satisfies the formal group axioms and that the Hirzebruch genus extraction reproduces the inversion series without additional assumptions.
minor comments (2)
  1. [Abstract] Abstract: The phrase 'our earlier work' should include a specific citation to the Chern-Dold character paper for clarity.
  2. [Introduction] Notation throughout: Define the new Catalan formal group law explicitly in an early section to distinguish it from standard examples like the multiplicative group.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and will revise the manuscript to strengthen the clarity of the derivations.

read point-by-point responses

  1. Referee: [§3] §3 (main derivation): The mapping from the cobordism class of CP^n via the Chern-Dold character to the Lagrange inversion coefficients invokes properties from the authors' earlier work without restating the precise theorems applied or verifying their direct applicability here; this dependence is load-bearing for the claimed new derivation.

    Authors: We agree that the derivation in §3 relies on results from our earlier work on the Chern-Dold character. In the revised manuscript we will restate the relevant theorems explicitly and verify their applicability to the cobordism class of CP^n, making the new derivation self-contained. revision: yes

  2. Referee: [§5] §5 (multiplicative case): The new formal group law defined by Catalan numbers and its identification with theta-divisor Chern numbers requires an explicit check that the law satisfies the formal group axioms and that the Hirzebruch genus extraction reproduces the inversion series without additional assumptions.

    Authors: We acknowledge the need for explicit verification. In the revision we will add a direct check confirming that the Catalan formal group law satisfies the formal group axioms and that the associated Hirzebruch genus reproduces the multiplicative inversion series without additional assumptions. revision: yes

Circularity Check

1 steps flagged

Self-citation on Chern-Dold character is load-bearing for the derivation

specific steps
  1. self citation load bearing [Abstract]
    "The proof is based on our earlier work on the Chern-Dold character in complex cobordism theory and leads to a new derivation of the Lagrange inversion formula."

    The topological interpretation and new derivation of the Lagrange inversion are justified by applying properties established only in the authors' previous papers on the Chern-Dold character; the present work does not supply independent verification of those properties.

full rationale

The manuscript's central claim is a topological derivation of the Lagrange inversion formula via Chern numbers of CP^n using the Chern-Dold character. This rests explicitly on the authors' prior results in complex cobordism theory, which are invoked without re-derivation or external benchmarks in the present text. The application to the inversion coefficients and the extension to the multiplicative case via Catalan formal group appear to add a direct mapping step, so the central claim retains independent content beyond pure self-reference. No self-definitional reduction or fitted-input prediction is exhibited in the abstract or described chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Claims rest on domain assumptions from complex cobordism theory and introduce one new entity without external falsifiable evidence in the abstract.

axioms (1)
  • domain assumption Chern-Dold character maps complex cobordism to cohomology with the required ring homomorphism properties
    Invoked as the basis for the new derivation of Lagrange inversion.
invented entities (1)
  • New formal group law defined by Catalan numbers no independent evidence
    purpose: To give topological meaning to the multiplicative inversion formula and associated Hirzebruch genus
    Introduced in the paper; no independent evidence or falsifiable prediction supplied in the abstract.

pith-pipeline@v0.9.0 · 5439 in / 1264 out tokens · 45425 ms · 2026-05-16T15:47:56.349763+00:00 · methodology

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Reference graph

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