Wendt's determinant and finiteness of prime characteristics for permutation binomials

Abstract

For f(X)=X^r(X^s+a)∈𝔽_p^m[X], with m,r,sgeqslant1 and a∈𝔽_p^m^×, put g=gcd(s,p^m−1). We prove that, for each fixed g, only finitely many prime characteristics p admit such permutation binomials, even as m,r,s,a vary. If 6nmid g, every such p divides Wendt's determinant W_g=Res_X(X^g−1,(X+1)^g−1), which is nonzero. When 6| g, this determinant vanishes; inverse power sums and iterated resultants yield an explicit nonzero integer divisible by every such p>2g+1. We determine the exact admissible characteristics for 1leqslant gleqslant10, obtain a further divisibility condition over prime fields, and show that in prime fields g=11 occurs precisely when p=23.

Permutation binomials of the form X^r(X^q−1+a) over finite fields
Junna Ni, Xuan Pang, Jianhua Yu +1
arXiv23 Sep 2026PreprintarXiv licenceFull textAlgebra & number theory Combinatorics & discrete mathematicsarXiv
A complete classification of permutation binomials of the form X^r(X^q−1+a) over finite fields
Xiang Fan
arXiv17 Sep 2026PreprintarXiv licenceFull textAlgebra & number theory Combinatorics & discrete mathematicsarXiv
Finiteness theorems for Cayley permutation classes
Giulio Cerbai, Anders Claesson
arXiv21 Oct 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
Rigidity conditions for binomials
Veronika Kikteva
arXiv28 Sep 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
The Gopakumar--Vafa finiteness conjecture
Aleksander Doan, Eleny-Nicoleta Ionel, Thomas Walpuski
Annals of Mathematics1 Nov 2026Peer-reviewedLicence not statedFull text37 citedGeometry & topologyOpenAlex

Cite

BibTeX RIS

BibTeX

@misc{tang2026wendts,
  title = {{Wendt's determinant and finiteness of prime characteristics for permutation binomials}},
  author = {Zhichao Tang and Xiang Fan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18584},
  eprint = {2610.18584},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18584},
}

RIS

TY  - UNPB
TI  - Wendt's determinant and finiteness of prime characteristics for permutation binomials
AU  - Zhichao Tang
AU  - Xiang Fan
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18584
UR  - https://arxiv.org/abs/2610.18584
AB  - For f(X)=X^r(X^s+a)∈𝔽_p^m[X], with m,r,sgeqslant1 and a∈𝔽_p^m^×, put g=gcd(s,p^m−1). We prove that, for each fixed g, only finitely many prime characteristics p admit such permutation binomials, even as m,r,s,a vary. If 6nmid g, every such p divides Wendt's determinant W_g=Res_X(X^g−1,(X+1)^g−1), which is nonzero. When 6| g, this determinant vanishes; inverse power sums and iterated resultants yield an explicit nonzero integer divisible by every such p>2g+1. We determine the exact admissible characteristics for 1leqslant gleqslant10, obtain a further divisibility condition over prime fields, and show that in prime fields g=11 occurs precisely when p=23.
ER  -

Abstract

Related

Cite