A complete classification of permutation binomials of the form X^r(X^q−1+a) over finite fields

Abstract

We classify, for every prime power q and every egeqslant2, the permutation binomials X^r(X^q−1+a) over 𝔽_q^e. Writing ℓ_j(q)=(q^j−1)/(q−1), such a binomial is a permutation if and only if gcd(r,q−1)=1, (−a)^ℓ_e(q)≠1, and rℓ_h(q)≡1pmodℓ_e(q) for some 1leqslant h<e coprime to e. This proves a conjecture of Masuda, Rubio, and Santiago: every permutation binomial of this form arises from (X^{q^h}+aX)∘ X^r for a suitable h. We also determine the exact number of distinct permutation functions represented by this family. As a further consequence, we completely classify the broader family X^r(X^d(q−1)+a) in the coprime-index case gcd(d,ℓ_e(q))=1. The new ingredient in the main classification is the necessity argument: selected Hermite power sums are organized so that Lucas' theorem turns their coefficients into digit conditions; a Farey-guided local argument then forces successive base-q digits, and cyclic rotations yield the inverse congruence. In characteristic $2$, a mod-$4$ lift to an auxiliary ring retains endpoint information lost modulo $2$.

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BibTeX RIS

BibTeX

@misc{fan2026complete,
  title = {{A complete classification of permutation binomials of the form $X^r(X^q-1+a)$ over finite fields}},
  author = {Xiang Fan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20354},
  eprint = {2609.20354},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20354},
}

RIS

TY  - UNPB
TI  - A complete classification of permutation binomials of the form X^r(X^q−1+a) over finite fields
AU  - Xiang Fan
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.20354
UR  - https://arxiv.org/abs/2609.20354
AB  - We classify, for every prime power q and every egeqslant2, the permutation binomials X^r(X^q−1+a) over 𝔽_q^e. Writing ℓ_j(q)=(q^j−1)/(q−1), such a binomial is a permutation if and only if gcd(r,q−1)=1, (−a)^ℓ_e(q)≠1, and rℓ_h(q)≡1pmodℓ_e(q) for some 1leqslant h<e coprime to e. This proves a conjecture of Masuda, Rubio, and Santiago: every permutation binomial of this form arises from (X^q^h+aX)∘ X^r for a suitable h. We also determine the exact number of distinct permutation functions represented by this family. As a further consequence, we completely classify the broader family X^r(X^d(q−1)+a) in the coprime-index case gcd(d,ℓ_e(q))=1. The new ingredient in the main classification is the necessity argument: selected Hermite power sums are organized so that Lucas' theorem turns their coefficients into digit conditions; a Farey-guided local argument then forces successive base-q digits, and cyclic rotations yield the inverse congruence. In characteristic $2$, a mod-$4$ lift to an auxiliary ring retains endpoint information lost modulo $2$.
ER  -

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