Classification of the non-coprime-index regime

Determine the permutation behavior of the binomials X^r(X^{d(q-1)}+a) over finite fields when η=gcd(d,ℓ_e(q))>1, and classify how the resulting η-to-one quotient geometry interacts with the exponent r and coefficient a.

Background

The paper completely classifies the family Xr(X{d(q-1)}+a) in the coprime-index case gcd(d,ℓ_e(q))=1 by reducing it to the main classification of Xr(X{q-1}+a). When η=gcd(d,ℓ_e(q))>1, the map y↦yd on the group of ℓ_e(q)-th roots of unity is no longer bijective: it has kernel of order η. Consequently, the reduction used in the coprime-index case fails, and the interaction between this η-to-one quotient structure, r, and a remains unresolved.

The authors explicitly identify η>1 as the first genuinely new and open regime and indicate that a general classification should be organized according to η.

References

Thus the first genuinely new regime begins when \eta>1: the map y\mapsto yd then has kernel of order \eta on \boldsymbol{\mu}_{\ell_e(q)}, so the preceding reduction to Theorem~\ref{thm:main} is no longer available. A natural next step is to organize the general classification by \eta=\gcd(d,\ell_e(q)) and analyze how this \eta-to-one quotient geometry interacts with r and a.

— A complete classification of permutation binomials of the form $X^r(X^{q-1}+a)$ over finite fields  (2609.20354 - Fan, 17 Sep 2026) in Section 5, Section 5 opening and concluding paragraph