Maximal size of irreducible λ-quiddities over polynomial and formal power series rings
Pith reviewed 2026-05-10 15:39 UTC · model grok-4.3
The pith
Irreducible λ-quiddities have bounded size over polynomial rings with finite coefficients and over all formal power series rings, with explicit maximal sizes determined.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Over any ring of formal power series the irreducible λ-quiddities are of bounded size, and the paper determines this bound completely. For polynomial rings A[X] with A finite the same boundedness holds, and the maximal sizes follow from the stated results; the reconstruction property then yields every λ-quiddity from the finite list of irreducibles.
What carries the argument
Irreducible λ-quiddities, defined as those solutions to the Conway-Coxeter equation that cannot be decomposed into smaller ones, which serve as the generators for all other solutions via the reconstruction theorem.
If this is right
- All λ-quiddities over these rings arise by combining a finite set of irreducibles.
- The combinatorial study of Coxeter friezes over polynomial and power-series rings reduces to enumerating objects of bounded length.
- Explicit maximal sizes can be computed once the coefficient ring is fixed.
- Many infinite-coefficient polynomial cases become tractable by the same reduction.
Where Pith is reading between the lines
- The finiteness results open the possibility of algorithmic enumeration of all λ-quiddities over these rings.
- Similar boundedness questions for other classes of rings, such as the integers, may be approachable by adapting the same decomposition techniques.
- The explicit bounds could link the size of irreducibles to invariants of the base ring such as its characteristic or residue field.
Load-bearing premise
Every λ-quiddity over the rings under study can be reconstructed from a finite collection of irreducible ones, which holds when the rings are commutative and unitary.
What would settle it
An explicit irreducible λ-quiddity of arbitrarily large size over a formal power series ring such as K[[X]] would falsify the bounded-size claim.
read the original abstract
The study of the combinatorics of the modular group and of Coxeter's friezes naturally leads to the investigation of a matrix equation, sometimes referred to as the Conway-Coxeter equation. The solutions of size $n$ of this equation, called $\lambda$-quiddities, are $n$-tuples of elements of a given ring $B$. A detailled understanding of these objects relies on the notion of irreducible solutions, from which all $\lambda$-quiddities can be reconstructed. One of the central questions that naturally arises in this context is whether the irreducible $\lambda$-quiddities over $B$ have bounded size, and, if so, how to determine such a bound. In this paper, we aim to list results that address this question in the case of polynomial rings $A[X]$ and $\mathbb{K}[X]$, where $A$ is a finite commutative unitary ring and $\mathbb{K}$ is a commutative field. Moreover, the stated results will also make it possible to treat easily many situations in which $A$ is infinite. Finally, we shall give a complete answer to the initial question for all rings of formal power series.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies λ-quiddities as n-tuples solving the Conway-Coxeter matrix equation over a ring B. It determines the maximal sizes of irreducible λ-quiddities for polynomial rings A[X] (A finite commutative unitary) and K[X] (K a field), provides tools for many infinite A cases, and gives a complete answer for all formal power series rings via explicit constructions and reduction arguments that map back to the polynomial case through truncation and completion while preserving the matrix equation and irreducibility condition.
Significance. If the derivations hold, the work is significant because it fully resolves the bounded-size question for irreducible λ-quiddities over all formal power series rings, enabling reconstruction of all λ-quiddities from the irreducibles. The explicit constructions and reduction arguments that preserve the Conway-Coxeter equation constitute a clear strength, advancing the combinatorial study of the modular group and Coxeter friezes.
minor comments (1)
- Abstract: 'detailled' is a typographical error and should read 'detailed'.
Simulated Author's Rebuttal
We thank the referee for their positive summary of our work and for recommending minor revision. We are pleased that the significance of the explicit constructions and reduction arguments for formal power series rings, as well as the results for polynomial rings, has been recognized.
Circularity Check
Derivation self-contained via explicit reductions; no circularity
full rationale
The paper establishes bounds for polynomial rings A[X] (A finite commutative unitary) and K[X] (K a field) through direct analysis of the Conway-Coxeter matrix equation and irreducibility conditions. It then extends to arbitrary formal power series rings by truncation and completion arguments that map solutions back to the polynomial case while preserving the defining equation and irreducibility. These steps are constructive and independent of any fitted parameters, self-definitions, or load-bearing self-citations; the reconstruction of all λ-quiddities from irreducibles holds verbatim once the size bound is obtained. No step reduces by construction to its own inputs.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption Rings A and K are commutative and unitary (or fields)
- standard math Polynomial and formal power series rings inherit the usual algebraic operations and divisibility properties
Reference graph
Works this paper leans on
-
[1]
and Section 3.1). However, when considering an infinite ringA, satisfactory results are available only in a very limited number of cases (see Theorem 3.2). To partially address this gap, we state here, without proofs, results concerning three highly important families of infinite rings : polynomial rings, formal power series rings, and infinite fields. Th...
-
[2]
We can now define the notion of irreducibility mentioned in the introduction
Proposition 2.6). We can now define the notion of irreducibility mentioned in the introduction. Definition 2.4([4], Definition 2.9).Aλ-quiddity overA(c 1, . . . , cn)(n≥3) is said to be reducible if there exists aλ-quiddity overA(b 1, . . . , bl)and anm-tuple(a 1, . . . , am)∈A m such that —(c 1, . . . , cn)∼(a 1, . . . , am)⊕(b 1, . . . , bl); —m≥3andl≥3...
-
[3]
B. Böhmler, M. Cuntz,Frieze patterns over finite commutative local rings, (2024), arXiv :2407.12596
- [4]
-
[5]
H. S. M. Coxeter,Frieze patterns, Acta Arithmetica, Vol. 18, (1971), pp 297-310
work page 1971
-
[6]
Cuntz,A combinatorial model for tame frieze patterns, Münster Journal of Mathematics, Vol
M. Cuntz,A combinatorial model for tame frieze patterns, Münster Journal of Mathematics, Vol. 12 no. 1, (2019), pp 49-56
work page 2019
- [7]
- [8]
- [9]
-
[10]
F. Mabilat,Combinatoire des sous-groupes de congruence du groupe modulaire, Annales Mathématiques Blaise Pascal, Vol.28no.1,(2021),pp.7-43.doi:10.5802/ambp.398.https://ambp.centre-mersenne.org/articles/10.5802/ambp.398/
work page doi:10.5802/ambp.398.https://ambp.centre-mersenne.org/articles/10.5802/ambp.398/ 2021
-
[11]
Mabilat, λ-quiddité sur Z[α] avec α transcendant, Mathematica Scandinavica, Vol
F. Mabilat,λ-quiddité surZ[α]avecαtranscendant, Mathematica Scandinavica, Vol. 128 no. 1, (2022), pp 5-13, https ://doi.org/10.7146/math.scand.a-128972
-
[12]
The ghosts of forgotten things: A study on size after forgetting
F. Mabilat,Solutions monomiales minimales irréductibles dansSL 2(Z/pnZ), Bulletin des Sciences Mathématiques, Vol. 194, (2024), Article 103456, ISSN 0007-4497, https ://doi.org/10.1016/j.bulsci.2024.103456
-
[13]
Mabilat,λ-quiddités sur des produits directs d’anneaux, (2023), hal-04190487, arXiv :2308.15848
F. Mabilat,λ-quiddités sur des produits directs d’anneaux, (2023), hal-04190487, arXiv :2308.15848
-
[14]
F. Mabilat,Finiteness of the number of irreducibleλ-quiddities over a finite commutative and unitary ring, (2024), arXiv :2404.10521
-
[15]
F. Mabilat,Éléments de comptage sur les générateurs du groupe modulaire et lesλ-quiddités, (2025), arXiv :2502.01328
-
[16]
F. Mabilat,Taille maximale desλ-quiddités irréductibles sur les anneaux de polynômes et de séries formelles, (2026), https ://doi.org/10.5281/zenodo.19497002
-
[17]
S. Morier-Genoud,Coxeter’s frieze patterns at the crossroad of algebra, geometry and combinatorics, Bulletin of the London Mathematical Society, Vol. 47 no. 6, (2015), pp 895-938
work page 2015
-
[18]
S. Morier-Genoud,Counting Coxeter’s friezes over a finite field via moduli spaces, Algebraic combinatoric, Vol. 4 no. 2, (2021), pp 225-240
work page 2021
-
[19]
V. Ovsienko,Partitions of unity inSL(2,Z), negative continued fractions, and dissections of polygons,Research in the Mathematical Sciences, Vol. 5 no. 2, (2018), Article 21, 25 pp
work page 2018
- [20]
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.