Galois Groups of Generalized Iterates of Generic Vectorial Polynomials

Abstract

Let q=pu>1 be a power of a prime p, and let kq 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)=Yqm+∑j∈J*XjYqm−j where the Xj are indeterminates. Let J‡ 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≤nsiTi be an irreducible polynomial of degree n>0 in T with coefficients si 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≤nsiE*[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,kq({Xj: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).