Galois groups of generalized iterates of generic vectorial polynomials

Let q = p " > 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 + Sigma X-j epsilonJ*(j) Y-qm=j where the X-j are indetermin ates. Let J(double dagger) be the set of all m - nu where nu is either 0 or a divisor of m different from m. Let s(T) = Sigma (0 less than or equal to i less than or equal to n) s(i) T-i be an irreducible polynomial of degree n > O in T with coefficients si in GF(q). Let E-m,E-q*([s])(Y) be the gene ralized sth iterate of E-m,E-q*(Y); i.e., E-m,E-q*([s])(Y) = Sigma (0 less than or equal to i less than or equal to n)s(i)E(m,q)*([i])(Y), where E-m,E -q*([i])(Y), is the ordinary ith iterate. We prove that if J(double dagger) subset of J*, m is square-free, and GCD(m, n) = 1 = GCD(mnu, 2p), then Gal (E-m,E-q*([s]), k(q) {X-j: j epsilon J*}) = GL(m, q(n)). The proof is based on CT ( = the Classification Theorem of Finite Simple Groups) in its incar nation as CPT (= the Classification of Projectively Transitive Permutation Groups, i.e., subgroups of GL acting transitively on nonzero vectors). (C) 2000 Academic Press.