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Some classes of permutation pentanomials

T0 review · 1 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read For every prime p other than 3, this paper gives two four-parameter families of five-term polynomials that permute the field F_{q^2} exactly when two simple gcd conditions hold, unifying 76 prior results and resolving an open problem for…

desk verdict Genuinely new permutation pentanomial families in all odd characteristics, with one fixable proof typo; the paper should be refereed and accepted after minor revision. read the letter →

arxiv 2501.04115 v1 pith:JQ5PJFN2 submitted 2025-01-07 math.NT

Zhiguo Ding , Michael E. Zieve This is my paper · ORCID
classification math.NT MSC 11T06
keywords permutationpolynomialspentanomialsfinitefieldsrootsofunityMöbiustransformationsgcdcriteriacharacteristictwoopenproblem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that two large families of sparse polynomials over the field $F_{q^2}$ are permutation polynomials (bijections of the field to itself) under a simple arithmetical condition. For every prime $p$ other than 3 and every power $q=p^k$, the authors give five-term polynomials $X^r B(X^{q-1})$ with $B$ having coefficients in $\{1,-1\}$, and show they permute $F_{q^2}$ if and only if two gcd conditions hold. This unifies 76 earlier results and conjectures, all of which lived in characteristic 2, and for the first time produces such polynomials in odd characteristic, resolving an open problem. The proof is short and computation-free, replacing earlier case-by-case computer verification with a single conceptual reduction to fractional linear maps on roots of unity.

What carries the argument

The central object is the pair of Möbius transformations $\rho(X)=(X-\omega)/(-\omega X+1)$ and $\eta(X)=(X+\omega)/(\omega X+1)$, where $\omega$ is an element of order 3 in $F_{q^2}^*$. Proposition 2.5 shows these maps either permute the $(q+1)$-th roots of unity $\mu_{q+1}$ (when $q\equiv 1 \pmod 3$) or interchange that set with the projective line $P^1(F_q)$ (when $q\equiv 2 \pmod 3$). The proof composes these maps with the monomial $X^n$ to build the rational function $g=\rho\circ X^n\circ\eta$, and shows the ratio $B_{3-z}(X)/B_z(X)$ on $\mu_{q+1}$ equals $g$ or $1/g$. By Lemma 2.2, permutation of $F_{q^2}$ is equivalent to permutation of $\mu_{q+1}$ by a derived function, so the whole problem reduces to asking when a monomial permutes $\mu_{q+1}$ or $P^1(F_q)$, which is exactly the stated gcd condition.

What would settle it

Exhaustively evaluate $f(X)=X\cdot B_1(X^4)$ over $F_{25}$ for the data $p=5$, $q=5$, $Q=5$, $R=1$, $S=25$, $r=1$, where $B_1$ comes from Theorem 1.1 and Table 1; the theorem predicts a permutation, so finding any repeated value would refute it.

Watch

Extended reading notes

Core claim

The central discovery is a necessary-and-sufficient criterion for permutation pentanomials of the form $X^r B_z(X^{q-1})$ over $F_{q^2}$. Theorem 1.1 states that when $Q,R,S$ are powers of $p$ and $B_1,B_2$ are the five-term polynomials from Table 1, the polynomial permutes $F_{q^2}$ iff $\gcd(r,q-1)=1$ and $\gcd(Q+R+S, q+e)=1$, where $e=1$ if $q\equiv 1 \pmod 3$ and $e=-1$ if $q\equiv 2 \pmod 3$. Theorem 1.3 gives a second family that permutes iff $q\equiv 1 \pmod 3$ and $\gcd(Q-R+S, q+1)=1$. Because the arguments go through for every $p\neq 3$, the special role of characteristic 2 in all 76 earlier results is shown to be an artifact of the literature, not of the mathematics.

Load-bearing premise

Everything rests on the lemma that a polynomial $X^r B(X^{q-1})$ permutes $F_{q^2}$ exactly when $\gcd(r,q-1)=1$ and the associated function permutes the $(q+1)$-th roots of unity; if that reduction fails in any case, the 'if and only if' conclusions of both theorems fail with it.

Editorial extensions

If this is right

  • Only the gcd conditions need checking: for any allowed $q$, $Q$, $R$, $S$ and any $r$ congruent to $Q+R+S$ mod $q+1$, the same two-line test decides whether the corresponding pentanomial permutes $F_{q^2}$.
  • All 76 prior permutation trinomial and pentanomial results and conjectures in characteristic 2 listed in Tables 3–6 are special cases of the two theorems, and three of them receive implicit corrections to their stated conditions.
  • Odd-characteristic permutation pentanomials of this form now exist for every prime other than 3, resolving the open problem noted in the literature.
  • When $r=Q+R+S$, the permutation is $F_q$-linearly equivalent to a simple monomial map on $F_{q^2}$ or on $F_q\times F_q$, giving a one-page alternative proof for that case and subsuming the two longest prior papers on the topic.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same fractional-linear reduction may apply to other sparse shapes: the proof only needs the ratio of the two 'companion' polynomials to be a conjugate of a monomial, so trinomials or heptanomials built from the same $\omega$ may satisfy analogous gcd criteria.
  • Characteristic 3 is excluded because no order-3 element $\omega$ with the required properties exists there; a modified construction using a different root-of-unity subgroup might fill that gap, though the present methods do not.
  • A companion result promised by the authors would likely complete a systematic classification of low-weight permutation polynomials of the form $X^r B(X^{q-1})$, with the two papers together covering the full landscape of such constructions.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper constructs two large families of permutation pentanomials over F_{q^2}, of the form X^r B(X^{q-1}) where B(X) has at most five terms and (under a pairwise-distinctness hypothesis) coefficients in {1,-1}, for every prime p different from 3 and every q=p^k. Theorem 1.1 gives an if-and-only-if criterion in terms of gcd(r,q-1)=1 and gcd(Q+R+S,q+e)=1, where e is determined by q mod 3; Theorem 1.3 gives a second family requiring q≡1 mod 3 and gcd(Q-R+S,q+1)=1. The proofs use Zieve's reduction (Lemma 2.2) and a proposition about the Möbius maps ρ and η (Proposition 2.5). Theorem 1.7 provides F_q-linear equivalence reformulations when r=Q+R+S. The paper also includes tables identifying 76 prior results and conjectures from the characteristic-2 literature as instances of the new families, and notes implicit corrections to three of them.

Significance. If the results are correct, this is a substantial contribution: it unifies and vastly generalizes 76 existing characteristic-2 results and conjectures, and it provides the first permutation pentanomials of this shape in odd characteristic, resolving an open problem (Open Problem 2 of [5]). The proofs are refreshingly short and conceptual, avoiding the heavy case computations of the earlier literature, and the if-and-only-if form of the criteria is clean and falsifiable. The explicit subsumption tables and the stated corrections to prior results are valuable scholarly service. The only substantive flaw I found is a typo in the proof of Theorem 1.1 that is local and easily fixed; the theorem statements themselves have the correct gcd conditions.

major comments (1)
  1. [Section 3, proof of Theorem 1.1, q≡1 mod 3 branch] The line reading "X^{Q+R+S} permutes μ_{q+1}, or equivalently gcd(Q−R+S,q+1)=1" is incorrect: the correct condition is gcd(Q+R+S,q+1)=1. As printed, the proof does not establish the criterion for the q≡1 branch; for example, with q=4, Q=1, R=4, S=8, one has gcd(Q−R+S,5)=5 whereas gcd(Q+R+S,5)=1. The theorem statement (Theorem 1.1) has the correct condition, and the immediately preceding equivalence with X^{Q+R+S} permuting μ_{q+1} supports the corrected expression, so this is almost certainly a typographical error. Nevertheless it is load-bearing in the proof-as-written and must be corrected.
minor comments (4)
  1. [Abstract and Section 1] The abstract and introduction state that B(X) has coefficients in {1,-1} without mentioning the hypothesis that Q, R, S are pairwise distinct. The theorem statements include this hypothesis, so the abstract should be qualified to avoid overstating the coefficient condition.
  2. [Section 5] The proof of Theorem 1.7 omits the details of several "routine verifications" and asks the reader to check them. Since this theorem is not needed for the main permutation results, this is acceptable, but a short derivation or an appendix would improve reproducibility.
  3. [References] Reference [16] (Wu, Yuan, Ding, Ma, arXiv:2209.04762) does not appear to be cited in the body of the paper. Please either cite it where relevant or remove it from the bibliography.
  4. [Section 3] The phrase "One sees by inspection that U(X)=C2(X) and V(X)=C1(X)" is terse; a brief indication of how the identity follows from the definitions of C1 and C2 would help the reader verify this key identification.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: main derivation is self-contained modulo independent prior lemmas; an internal gcd typo in the q≡1 branch is a proof error, not circularity.

full rationale

The paper's derivation chain is not circular. The main theorems reduce (via Lemma 2.2, cited from Zieve 2008) the permutation property of X^r B_z(X^{q-1}) over F_{q^2} to the permutation property of a rational function g(X)=rho(X) ∘ X^{Q±R+S} ∘ eta(X) on the (q+1)-th roots of unity. That rational function is then analyzed using Proposition 2.5, which is proved in the paper from Lemmas 2.3 and 2.4 (Zieve 2013). None of these cited lemmas states or assumes the pentanomial permutation conclusion; they are general facts about the reduction X^r B(X^{q-1}) on F_{q^2} and about Möbius transformations on μ_{q+1} and P^1(F_q). The polynomials B_z(X) are explicit constructions, not fitted parameters, and no subset of data is used to predict a closely related quantity. The gcd conditions are derived, not assumed, from the standard criterion that X^n permutes a cyclic group of order m iff gcd(n,m)=1. I also checked the self-citations: Lemma 2.2 and Lemmas 2.3–2.4 are independent published results with stated hypotheses that do not include the target theorem, so under the review rules they are real evidence and do not raise the circularity score. Two issues in the manuscript should be noted separately as non-circularity concerns. First, in the proof of Theorem 1.1, Section 3, the q≡1 branch states that X^{Q+R+S} permutes μ_{q+1} 'or equivalently gcd(Q−R+S,q+1)=1', but the equivalence requires gcd(Q+R+S,q+1)=1, as in the theorem statement. This is an internal typo or misprint that would make that branch of the proof as printed fail, but it is a correctness error, not circularity: it does not make the conclusion an input of the derivation. Second, the abstract's unqualified claim that coefficients lie in {1,−1} omits the pairwise-distinct hypothesis present in Theorems 1.1 and 1.3; this is an overstatement, not circularity. Finally, Section 5 explicitly omits routine verifications, saying 'we decided not to write out the details of some routine verifications'; this concerns the auxiliary structural result Theorem 1.7, not the main permutation criteria, and omitting details is not circularity. Overall, the central claims are genuinely derived from lemmas that do not contain the conclusions, so the paper is largely self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The construction has no fitted parameters; Q, R, S, and r are universal inputs in the theorem statements. The proof imports standard published lemmas from the second author's earlier work and the standard monomial-permutation criterion. No new entities are introduced. The only caveat is reliance on self-cited lemmas, but these are independent of the target result.

assumptions (4)
  • standard math Existence of an element omega of order 3 in F_{q^2} whenever p is not 3
    Used to define rho, eta, and the polynomials C_z in Theorems 1.1 and 1.3; follows from 3 dividing q^2-1 for p not 3.
  • standard math Lemma 2.2: X^r B(X^{q-1}) permutes F_{q^2} iff gcd(r,q-1)=1 and X^r B(X)^{q-1} permutes mu_{q+1}
    Central reduction used in Sections 3 and 4; quoted from Zieve 2008 and not re-proved in this paper.
  • standard math Lemmas 2.3 and 2.4 on Mobius maps permuting mu_{q+1} or mapping it bijectively to P_1(F_q)
    Used to prove Proposition 2.5, which controls the behavior of rho and eta; cited from Zieve 2013.
  • standard math A monomial X^n permutes F_{q^i} iff gcd(n, q^i-1)=1
    Used to translate monomial permutation into gcd conditions in Theorem 1.7 and in the main proofs.

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Cite this review

Pith. "Pith review of Some classes of permutation pentanomials." pith.science (2026). https://pith.science/paper/JQ5PJFN2

@misc{pith2026250104115,
  author       = {Pith},
  title        = {Pith review of: Some classes of permutation pentanomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JQ5PJFN2}},
  note         = {Machine review of arXiv:2501.04115}
}
read the original abstract

For each prime p other than 3, and each power q=p^k, we present two large classes of permutation polynomials over F_{q^2} of the form X^r B(X^{q-1}) which have at most five terms, where B(X) is a polynomial with coefficients in {1,-1}. The special case p=2 of our results comprises a vast generalization of 76 recent results and conjectures in the literature. In case p>2, no instances of our permutation polynomials have appeared in the literature, and the construction of such polynomials had been posed as an open problem. Our proofs are short and involve no computations, in contrast to the proofs of many of the special cases of our results which were published previously.

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Reference graph

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