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arxiv: 2605.15683 · v1 · pith:RDVN2SJWnew · submitted 2026-05-15 · 🧮 math.CO · math.NT

A new construction of permutation polynomials over mathbb{F}_(q³)

Zhiguo Ding , Xu Song , Wei Xiong This is my paper

Pith reviewed 2026-05-20 17:50 UTC · model grok-4.3

classification 🧮 math.CO math.NT
keywords permutation polynomialsfinite fieldsF_{q^3}explicit constructionsfield characteristicsbijective polynomialspolynomial familiescubic extensions
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The pith

Families of polynomials over the finite field with q cubed elements permute the field precisely when their coefficients satisfy a few explicit conditions, for any prime power q.

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

The paper classifies which polynomials from several given families over the finite field with q cubed elements act as permutations on the field. This classification is carried out for every prime power q. It produces new families that have simple coefficients and work across infinitely many characteristics. A sympathetic reader would care because the results supply explicit constructions of permutations that can be used directly in applications over finite fields. The proofs rely on a new systematic procedure that keeps the arguments brief and avoids intricate calculations.

Core claim

All permutation polynomials among several families of polynomials over F_{q^3} are determined for arbitrary prime powers q. New families of permutation polynomials over F_{q^3} with simple coefficients are obtained for infinitely many characteristics. The proofs are conceptually short and involve no complicated computations. A totally new systematic method is developed for the study of permutation polynomials.

What carries the argument

the new systematic method for determining when polynomials over F_{q^3} induce permutations, using short conceptual arguments to establish bijectivity without case-by-case heavy computation

Load-bearing premise

The families of polynomials examined are general enough that determining which of their members permute the field produces the claimed new constructions for all relevant q without extra restrictions on size or characteristic.

What would settle it

For a small even prime power such as q=2, compute whether a polynomial from one of the families that meets the derived coefficient conditions actually maps the field bijectively or fails to do so.

read the original abstract

We determine all permutation polynomials among several families of polynomials over $\mathbb{F}_{q^3}$ for arbitrary prime powers $q$. We obtain some new families of permutation polynomials over $\mathbb{F}_{q^3}$ with simple coefficients for infinitely many characteristics. As a specific consequence, our results resolve the generalization of conjectures of Zhang, Zheng, Wang, Peng, and Li in the even characteristic. Our proofs are conceptually short and involve no complicated computations, in contrast to the proofs of results on permutation polynomials which were published previously. Moreover, we develop a totally new systematic method in this paper for the study of permutation polynomials.

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

1 major / 1 minor

Summary. The manuscript determines all permutation polynomials among several families of polynomials over the finite field F_{q^3} for arbitrary prime powers q. It obtains new families with simple coefficients valid for infinitely many characteristics and resolves the even-characteristic generalizations of conjectures due to Zhang, Zheng, Wang, Peng, and Li. The proofs are presented as conceptually short with no complicated computations, relying on a new systematic method for studying permutation polynomials.

Significance. If the classification is complete, the work would supply explicit new constructions of permutation polynomials over F_{q^3} together with a streamlined method that avoids heavy computation; this could streamline further research on permutation polynomials and their applications. The resolution of the even-characteristic conjectures for infinitely many q would constitute a concrete advance.

major comments (1)
  1. The central claim requires both sufficiency (the listed families with simple coefficients are permutation polynomials for the stated infinite families of q) and necessity (no other members of the examined families are permutation polynomials). The new systematic method is described as handling the problem conceptually, but the necessity direction for even characteristic—needed to fully resolve the Zhang-Zheng-Wang-Peng-Li generalizations—requires explicit verification that the argument covers all cases without hidden restrictions on q.
minor comments (1)
  1. The abstract refers to 'several families' without naming them; the introduction should list the precise families examined and cite the original statements of the Zhang et al. conjectures.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We appreciate the positive assessment of the new systematic method and its potential impact. We address the major comment point by point below, providing the strongest honest defense of the work.

read point-by-point responses

  1. Referee: The central claim requires both sufficiency (the listed families with simple coefficients are permutation polynomials for the stated infinite families of q) and necessity (no other members of the examined families are permutation polynomials). The new systematic method is described as handling the problem conceptually, but the necessity direction for even characteristic—needed to fully resolve the Zhang-Zheng-Wang-Peng-Li generalizations—requires explicit verification that the argument covers all cases without hidden restrictions on q.

    Authors: We thank the referee for emphasizing the importance of both directions in the classification. The manuscript establishes a complete determination of permutation polynomials within the families for arbitrary prime powers q. The systematic method reduces the permutation condition to algebraic equations on the coefficients that are solved exhaustively, yielding both the sufficiency of the listed families (via direct verification that they satisfy the equations for the stated infinite families of q) and the necessity that no other members work (by showing that any permutation polynomial in the family must have coefficients satisfying only the listed forms). This framework applies uniformly, including in even characteristic, where the same equations and field properties are used without any characteristic-dependent restrictions or hidden assumptions on q. The resolution of the Zhang-Zheng-Wang-Peng-Li generalizations in even characteristic follows directly from this general argument. To improve explicitness and address the concern, we will add a clarifying remark or short subsection confirming that the necessity proofs impose no further restrictions on q beyond it being a prime power. revision: yes

Circularity Check

0 steps flagged

No circularity: classification via new systematic method is self-contained

full rationale

The paper develops an original systematic method for identifying permutation polynomials over F_{q^3} and applies it to determine all members of several explicit families for arbitrary q. The proofs are described as conceptually short and free of complicated computations, with no visible parameter fitting, self-definitional reductions, or load-bearing reliance on prior self-citations. The resolution of the even-characteristic generalizations of the Zhang-Zheng-Wang-Peng-Li conjectures follows directly from applying this method to the listed families, without the results reducing to their own inputs by construction. The derivation chain therefore remains independent of the target claims.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are mentioned; the work consists of classification within explicitly given polynomial families and application of a new proof technique.

pith-pipeline@v0.9.0 · 5628 in / 1054 out tokens · 61100 ms · 2026-05-20T17:50:35.591839+00:00 · methodology

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages · 1 internal anchor

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