From r-linearized polynomial equations to rm-linearized polynomial equations

Let r   be a prime power and q=rmq=rm. For 0≤i≤m−10≤i≤m−1, let fi∈Fr[X]fi∈Fr[X] be q  -linearized and ai∈Fqai∈Fq. Assume that z∈F‾r satisfies the equation ∑i=0m−1aifi(z)ri=0, where ∑i=0m−1aifiri∈Fq[X] is an r-linearized polynomial. It is shown that z satisfies a q  -linearized polynomial equation with coefficients in FrFr. This result provides an explanation for numerous permutation polynomials previously obtained through computer search.