Permutation binomials of the form X^r(X^q−1+a) over finite fields

@misc{ni2026permutation,
  title = {{Permutation binomials of the form $X^r(X^q-1+a)$ over finite fields}},
  author = {Junna Ni and Xuan Pang and Jianhua Yu and Pingzhi Yuan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28595},
  eprint = {2609.28595},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28595},
}

TY  - UNPB
TI  - Permutation binomials of the form X^r(X^q−1+a) over finite fields
AU  - Junna Ni
AU  - Xuan Pang
AU  - Jianhua Yu
AU  - Pingzhi Yuan
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.28595
UR  - https://arxiv.org/abs/2609.28595
AB  - This paper is devoted to studying permutation binomial F_r,a(X)=X^r(X^q−1+a)∈𝔽_q^e[X] with a∈𝔽_q^e^*. We present a complete characterization for F_r,a to be a permutation of 𝔽_q^e. This yields a complete proof of the conjecture proposed by Masuda--Rubio--Santiago \cite{masuda2022permutation}. More precisely, we show that F_r,a permutes 𝔽_q^e if and only if (−a)^(q^e−1)/(q−1)≠1, gcd(r,q−1)=1, and there exists 1≤ h<e with gcd(h,e)=1 such that r(q^h−1)≡ q−1pmodq^e−1.
ER  -