Galois Groups of Generalized Iterates of Generic Vectorial Polynomials

Let q=p^u>1 be a power of a prime p, and let k_q be an overfield of GF(q). Let m>0 be an integer, let J^* be a subset of {1,…,m}, and let E^*_{m, q}(Y)=Y^qm+∑_j∈J^*X_jY^qm−j where the X_j are indeterminates. Let J³ be the set of all m−ν where ν is either 0 or a divisor of m different from m. Let s(T)=∑_0≤i≤ns_iTⁱ be an irreducible polynomial of degree n>0 in T with coefficients s_i in GF(q). Let E^*s]_{m, q}(Y) be the generalized sth iterate of E^*_{m, q}(Y); i.e., E^*s]_{m, q}(Y)=∑_0≤i≤ns_iE^*i]_{m, q}(Y), where E^*i]_{m, q}(Y), is the ordinary ith iterate. We prove that if J³⊂J^*, m is square-free, and GCD(m,n)=1=GCD(mnu,2p), then Gal(E^*s]_{m, q},k_q({X_j:j∈j^*})=GL(m, qⁿ). The proof is based on CT (=the Classification Theorem of Finite Simple Groups) in its incarnation as CPT (=the Classification of Projectively Transitive Permutation Groups, i.e., subgroups of GL acting transitively on nonzero vectors).