A new mathematical preprint reports arithmetic proofs that two different complex maps have parabolic multiplicity 1 at the fixed point 0 when their multiplier is a primitive n-th root of unity. In this setting, that means the specified complex parameter has n as the first power at which it returns to 1. After the map is applied n times, the first non-identity term appears at the expected order and has a nonzero coefficient.
The result concerns the quadratic polynomial F(z)=ωz(1−z) and the entire map F(z)=ωze−z. The paper also presents a polynomial argument that applies to unicritical polynomials of arbitrary degree, broadening the setting beyond the quadratic example.
This is a preprint rather than a journal publication: the document is listed as arXiv version 1, dated 20 Aug 2026. Its conclusions are drawn from formal maps, their iterates and algebraic or p-adic properties.
Turning a local question into arithmetic
The central calculation concerns the n-th iterate, written in the paper as F^{◦n}(z)=z(1+c z^n+O(z^{n+1})), with c required to be nonzero. That coefficient determines whether the fixed point has multiplicity 1 in the cases under study. The authors recast its nonvanishing as an arithmetic problem involving formal iteration, coefficient polynomials and resultants.
In this setting, a resultant is an integer built from two polynomials; it is used to test whether a coefficient polynomial and a cyclotomic polynomial share a root. If the relevant resultant is nonzero, the coefficient does not vanish at the primitive root-of-unity parameters being examined.
The proof strategy differs between the two main families. In the exponential case, the authors work with arithmetic modulo n−1. In the polynomial case, they use p-adic numbers—an arithmetic system that tracks divisibility by a chosen prime—and select a prime whose multiplicative order of 2 is exactly n. The polynomial proof is described as new in the polynomial setting.
A nonzero integer in the exponential case
For the entire map, the paper defines a resultant called β_n and proves that it is a nonzero integer. The reported base cases are β_1=1 and β_2=1.
The result also gives congruences, which describe how β_n behaves when reduced modulo n−1. If n≥3 is a power of a prime p, then p^{n−1}β_n is congruent to 1 modulo n−1. If n≥5 is not a prime power, β_n is congruent to 1 modulo n−1.
Those arithmetic statements supply the needed nonvanishing for the coefficient at primitive roots of unity. They are propositions about the formal map and its coefficient structure, rather than results from measurements or observations.
The polynomial argument reaches beyond degree two
For unicritical polynomials, the argument is stated for degree d≥2 and iterate index n≥3. Under the stated condition that a prime p≥2 makes d have order n in (Z/pZ)×, the p-adic construction identifies a unique cyclotomic root near d and establishes the bound |c|_p≥|d^n−1|_p≠0. The nonzero bound rules out the coefficient vanishing in this setting.
The general argument does not simply ignore the small or exceptional cases. The paper treats n=1, n=2, and the case d=2 with n=6 by direct computation.
For the d=2, n=6 exception, a computer-assisted calculation reports the resultant Res(Φ6,c6)=10128=24·3·211 and concludes that c6 and Φ6 are coprime. The supplied analysis does not include the computational details or code needed to reproduce that calculation independently.
Divisibility gives the quadratic case extra structure
In the quadratic family, the resultant is called α_n. The paper proves that for every positive integer n and every prime p dividing n, its p-adic valuation satisfies v_p(α_n)≥φ(n)/(p−1). Here φ(n) counts the positive integers up to n that are relatively prime to n. The stated inequality implies that n divides α_n.
The paper also gives a sharper prime-case statement: when n≥2 is prime, α_n is congruent to n modulo n². Together, these results show how divisibility information can replace a direct search for a zero of the coefficient at a primitive root of unity.
Some of the coefficient structure is made explicit through Catalan numbers. The paper identifies a coefficient written as a_n(0)=g_n(0) with the n-th Catalan number, (2n)!/[n!(n+1)!]. It also proves that the 2-adic valuation of a_n equals the 2-adic valuation of its constant coefficient a_n(0).
Several patterns remain open
The authors distinguish proved results from patterns suggested by calculation. They label the observed resultant valuation patterns as conjectures, based on computations through 243, rather than presenting them as general theorems.
The paper also says it cannot prove an assertion used in Yoccoz’s proposed arithmetic proof: a proposition is valid at the root ζ itself, but not for a nearby parameter λ≠ζ. That unresolved point matters when assessing the scope of that prior proof route.
The conclusions are limited to the named complex-dynamical families and the stated integer, root-of-unity, degree and prime conditions. They do not show that every entire map with a root-of-unity multiplier has parabolic multiplicity 1, and they do not establish the conjectural valuation patterns beyond the cases proved in the paper.
The preprint’s main contribution is a change of proof language: a local question about repeated complex maps is attacked with cyclotomic polynomials, resultants, congruences and p-adic estimates. Within its stated scope, that approach establishes multiplicity 1 for the quadratic and exponential maps and extends the polynomial result to unicritical families of arbitrary degree.
Because the work is a formal mathematical study, its uncertainty is about the reach and completeness of the arguments rather than statistical error. The reported n=6 computation lacks reproducibility details in the supplied material, and the valuation conjectures and unresolved assertion remain questions for further proof.
Paper data and sources
Original title: An arithmetic approach to parabolic multiplicity in complex dynamics
Authors: Xavier Buff, Valentin Huguin, Liz Vivas
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text