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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.
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)
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)
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)
In his Ph.D. Thesis of 1897, Dickson introduced certain permutation polynomials whose Galois groups are essentially the dihedral groups. These are now called Dickson polynomials of the rst kind, to distinguish them from their variations introduced by Schur in 1923 which are now called Dickson polynomials of the second kind. In the last few decades there… (More)
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 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)