Galois groups of generalized iterates of generic vectorial polynomials

Abstract

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 J⁺ be the set of all m− v where v is either 0 or a divisor of m different from m. Let s(T)=∑ 0 ≤ i ≤ n s_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 ≤ n s_i E^*[i]_m,q (Y), where E^*[i] _m,q (Y), is the ordinary ith iterate. We prove that if J⁺ ⊂,j^*, m is square-free, and GCD(m,n)=1=GCD(mnu,2p), then Gal(E^[s]_m,q, k_q({X _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).