Michael E. Zieve

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We study the orbits of a polynomial f ∈ C[X], namely the sets {α, f (α), f (f (α)),. .. } with α ∈ C. We prove that if two nonlinear complex polynomials f, g have orbits with infinite intersection, then f and g have a common iterate. More generally, we describe the intersection of any line in C d with a d-tuple of orbits of nonlinear polynomials, and we(More)
We produce a new family of polynomials f (X) over fields k of characteristic 2 which are exceptional, in the sense that f (X) − f (Y) has no absolutely irreducible factors in k[X, Y ] except for scalar multiples of X −Y ; when k is finite, this condition is equivalent to saying that the map α → f (α) induces a bijection on an infinite algebraic extension of(More)
Let k be a field of characteristic p > 0, and let q be a power of p. We determine all polynomials f ∈ k[t] \ k[t p ] of degree q(q − 1)/2 such that Gal(f (t) − u, k(u)) has a transitive normal subgroup isomor-phic to PSL2(q), subject to a certain ramification hypothesis. As a consequence, we describe all polynomials f ∈ k[t] such that deg(f) is not a power(More)
We present a general technique for obtaining permutation polynomials over a finite field from permutations of a sub-field. By applying this technique to the simplest classes of permutation polynomials on the subfield, we obtain several new families of permutation polynomials. Some of these have the additional property that both f (x) and f (x) + x induce(More)
We resolve a 1983 question of Serre by constructing curves with many points of every genus over every finite field. More precisely, we show that for every prime power q there is a positive constant c q with the following property: for every integer g ≥ 0, there is a genus-g curve over F q with at least c q g rational points over F q. Moreover, we show that(More)
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