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- DOUG WIEDEMANN, MICHAEL E. ZIEVE
- 2007

We consider n-by-n circulant matrices having entries 0 and 1. Such matrices can be identified with sets of residues mod n, corresponding to the columns in which the top row contains an entry 1. Let A and B be two such matrices, and suppose that the corresponding residue sets S A , S B have size at most 3. We prove that the following are equivalent: (1)… (More)

- MICHAEL E. ZIEVE
- 2008

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 José Marcos.

- MICHAEL E. ZIEVE
- 2008

We give necessary and sufficient conditions for a polynomial of the form x r (1 + x v + x 2v + · · · + x kv) t to permute the elements of the finite field F q. Our results yield especially simple criteria in case (q − 1)/ gcd(q − 1, v) is a small prime.

Ritt studied the functional decomposition of a univariate complex polynomial f into prime (indecomposable) polynomials, f = u1 • u2 • · · · • ur. His main achievement was a procedure for obtaining any decomposition of f from any other by repeatedly applying certain transformations. However, Ritt's results provide no control on the number of times one must… (More)

- ANDREW KRESCH, MICHAEL E. ZIEVE, Chris Skinner, William Stein, Jeff Vanderkam
- 1999

Let N q (g) denote the maximal number of F q-rational points on any curve of genus g over F q. Ihara (for square q) and Serre (for general q) proved that lim sup g→∞ N q (g)/g > 0 for any fixed q. In their proofs they constructed curves with many points in infinitely many genera; however, their sequences of genera are somewhat sparse. In this paper, we… (More)

- H. W. Lenstra, M. Zieve
- 1996

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 PSL 2 (3 k), where k≥3 is odd. This realizes one of the few possibilities for such a family… (More)

- XANDER FABER, MICHAEL E. ZIEVE
- 2008

For any elements a, c of a number field K, let Γ(a, c) denote the backwards orbit of a under the map fc : C → C given by fc(x) = x 2 + c. We prove an upper bound on the number of elements of Γ(a, c) whose degree over K is at most some constant B. This bound depends only on a, [K : Q], and B, and is valid for all a outside an explicit finite set. We also… (More)

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 dynamical analogue of the Mordell-Lang conjecture.

We prove that if x m + ax n permutes the prime field Fp, 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 Fq of the form x m + ax n if and only if gcd(m, n, q − 1) = 1.

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)