Permutation polynomials over finite fields involving

Let p be a prime and q = p s . For integer a ≥ 0 , let S a = x + x q + ⋯ + x q a − 1 ∈ F p [ x ] . We present three constructions of permutation polynomials of F q e involving S a which generalize several recent results. When q is even, the third construction produces a large class of Dembowski–Ostrom permutation polynomials. We also discuss an interesting linear algebraic problem arising from the third construction.