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Differential algebra of polytopes and inversion formulas

V. M. Buchstaber, A. P. Veselov

TL;DR

This work develops a differential-algebraic framework on the polytopal Grothendieck ring to derive universal inversion formulas for formal power series from the combinatorics of graph-associahedra. By translating facet-boundary operations into PDEs for generating functions, it recovers Lagrange-type compositional inversions via associahedra and multiplicative inversions via permutohedra, and connects these to Deligne-Mumford moduli spaces, Fa\u00e0 di Bruno theory, and toric geometry. The paper extends the picture to cyclohedra and stellohedra, expressing their motivic interiors through Fa\u00a8a di Bruno-type polynomials and unveiling links to integrable systems and operad formalisms. Overall, it provides a unified combinatorial-geometric mechanism for inversion phenomena across polytope families and their moduli-space interpretations, highlighting deep connections between polytope combinatorics, algebraic geometry, and mathematical physics.

Abstract

We use the differential algebra of polytopes to explain the known remarkable relation of the combinatorics of the associahedra and permutohedra with the universal compositional and multiplicative inversion formulas for the formal power series. This approach allows to single out the associahedra and permutohedra among all graph-associahedra and emphasizes the significance of the differential equations for special sequences of simple polytopes derived earlier by one of the authors. We discuss also the link with the geometry of Deligne-Mumford moduli spaces $\bar M_{0,n}$ and the interpretation of the combinatorics of cyclohedra in relation with the classical Faà di Bruno's formula.

Differential algebra of polytopes and inversion formulas

TL;DR

This work develops a differential-algebraic framework on the polytopal Grothendieck ring to derive universal inversion formulas for formal power series from the combinatorics of graph-associahedra. By translating facet-boundary operations into PDEs for generating functions, it recovers Lagrange-type compositional inversions via associahedra and multiplicative inversions via permutohedra, and connects these to Deligne-Mumford moduli spaces, Fa\u00e0 di Bruno theory, and toric geometry. The paper extends the picture to cyclohedra and stellohedra, expressing their motivic interiors through Fa\u00a8a di Bruno-type polynomials and unveiling links to integrable systems and operad formalisms. Overall, it provides a unified combinatorial-geometric mechanism for inversion phenomena across polytope families and their moduli-space interpretations, highlighting deep connections between polytope combinatorics, algebraic geometry, and mathematical physics.

Abstract

We use the differential algebra of polytopes to explain the known remarkable relation of the combinatorics of the associahedra and permutohedra with the universal compositional and multiplicative inversion formulas for the formal power series. This approach allows to single out the associahedra and permutohedra among all graph-associahedra and emphasizes the significance of the differential equations for special sequences of simple polytopes derived earlier by one of the authors. We discuss also the link with the geometry of Deligne-Mumford moduli spaces and the interpretation of the combinatorics of cyclohedra in relation with the classical Faà di Bruno's formula.
Paper Structure (9 sections, 16 theorems, 84 equations, 4 figures)

This paper contains 9 sections, 16 theorems, 84 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Differential algebra of simple polytopes
  3. PDEs for graph-associahedra and inversion formulas
  4. Inversion formula and the Deligne-Mumford moduli spaces
  5. Inversion formula and permutohedral varieties
  6. Cyclohedra and Faà di Bruno's formula
  7. Equation and formulas for the stellohedra
  8. Concluding remarks
  9. Acknowledgements.

Key Result

Theorem 2.1

The ring $\mathfrak P$ is a polynomial ring generated by indecomposable combinatorial polytopes, so any combinatorial polytope can be uniquely represented as a product of the indecomposable polytopes.

Figures (4)

  • Figure 1: 3D associahedron and its realisation as a truncated cube.
  • Figure 2: 3D permutohedron $\pi_4$.
  • Figure 3: 3D cyclohedron $W_4$.
  • Figure 4: Star-graph and 3D stellohedron $S_4$.

Theorems & Definitions (22)

  • Theorem 2.1: BE
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Lemma 3.1
  • Theorem 3.2: B-2008BP
  • ...and 12 more