Permutation polynomials over finite fields involving x+xq+⋯+xqa−1

Let pp be a prime and q=psq=ps. For integer a≥0a≥0, let Sa=x+xq+⋯+xqa−1∈Fp[x]. We present three constructions of permutation polynomials of FqeFqe involving SaSa which generalize several recent results. When qq 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.