Galois groups of generalized iterates of generic vectorial polynomials

Abhyankar, Shreeram S. ; Sundaram, Ganapathy S. (2001) Galois groups of generalized iterates of generic vectorial polynomials Finite Fields and Their Applications, 7 (1). pp. 92-109. ISSN 1071-5797

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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 ∈jX jYqm-j where the Xj 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 siTi 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 ≤ n si 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, kq({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).

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Deposited On:16 Sep 2010 06:55
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