Michael E. Zieve

Intersections of polynomial orbits, and a dynamical Mordell–Lang conjecture

We prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection, then the polynomials have a common iterate. We also prove a special case of a conjectured dynamical analogue of the Mordell–Lang conjecture.

On some permutation polynomials over Fq of the form x^r h(x^{(q-1)/d})

Several recent papers have given criteria for certain polynomials to permute F q , in terms of the periods of certain generalized Lucas sequences. We show that these results follow from a more general criterion which does not involve such sequences.

Permutation binomials over finite fields

We prove that if x m + ax n permutes the prime field F p , where m > n > 0 and a ∈ F * p , then gcd(m - n,p ― 1) > √/p ― 1. Conversely, we prove that if q > 4 and m > n > 0 are fixed and satisfy gcd(m - n, q ― 1) > 2q(log log q)/ log q, then there exist permutation binomials over F q of the form x m + ax n if and only if gcd(m, n, q - 1) = 1.

Linear relations between polynomial orbits

We study the orbits of a polynomial f 2CŒX , namely, the sets 1 ;f . /; f .f . //; : : :o with 2 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 with a d -tuple of orbits of nonlinear polynomials, and we formulate a question which generalizes both this result and the Mordell–Lang conjecture.

SOME FAMILIES OF PERMUTATION POLYNOMIALS OVER FINITE FIELDS

We give necessary and sufficient conditions for a polynomial of the form xr(1 + xv + x2v + ⋯ + xkv)t to permute the elements of the finite field 𝔽q. Our results yield especially simple criteria in case (q - 1)/gcd(q - 1, v) is a small prime.

Curves of every genus with many points, II: Asymptotically good families

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 non-negative integer g, there is a genus-g curve over F_q with at least c_q * g rational points over F_q. Moreover, we show that there exists a positive constant d such that for every q we can choose c_q = d * (log q). We show also that there is a constant c > 0 such that for every q and every n > 0, and for every sufficiently large g, there is a genus-g curve over F_q that has at least c*g/n rational points and whose Jacobian contains a subgroup of rational points isomorphic to (Z/nZ)^r for some r > c*g/n.

Classes of Permutation Polynomials Based on Cyclotomy and an Additive Analogue

I present a construction of permutation polynomials based on cyclotomy, an additive analogue of this construction, and a generalization of this additive analogue which appears to have no multiplicative analogue. These constructions generalize recent results of Jose Marcos.

A family of exceptional polynomials in characteristic three

We present a family of indecomposable polynomials of non prime-power degree over the finite field of three elements which are permutation polynomials over infinitely many finite extensions of the field. The associated geometric monodromy groups are the simple groups PSL2(3 ), where k≥3 is odd. This realizes one of the few possibilities for such a family which remain following the deep work of Fried, Guralnick and Saxl. Acknowledgements. The first author was supported by NSF grant 9224205 and by a bezoekersbeurs of the Nederlandse organisatie voor wetenschappelijk onderzoek (NWO).

Exceptional Covers and Bijections on Rational Points

We show that if f : X −→ Y is a finite, separable morphism of smooth curves defined over a finite field Fq, where q is larger than an explicit constant depending only on the degree of f and the genus of X, then f maps X(Fq) surjectively onto Y (Fq) if and only if f maps X(Fq) injectively into Y (Fq). Surprisingly, the bounds on q for these two implications have different orders of magnitude. The main tools used in our proof are the Chebotarev density theorem for covers of curves over finite fields, the Castelnuovo genus inequality, and ideas from Galois theory.

Planar functions and perfect nonlinear monomials over finite fields

The study of finite projective planes involves planar functions, namely, functions