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Journals and Conferences
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)
We construct classes of permutation polynomials over FQ2 by exhibiting classes of low-degree rational functions over FQ2 which induce bijections on the set of (Q + 1)-th roots of unity. As a consequence, we prove two conjectures about permutation trinomials from a recent paper by Tu, Zeng, Hu and Li.
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 SA, SB have size at most 3. We prove that the following are equivalent: (1) there… (More)
In a recent paper, Akbary and Wang gave a sufficient condition for x + x to permute Fq, in terms of the period of a certain sequence involving sums of cosines. As an application they gave necessary and sufficient conditions in case u, r, q satisfy certain special properties. We show that the Akbary-Wang sufficient condition follows from a more general… (More)
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. Dedicated to Mel Nathanson on the occasion of his sixtieth birthday
We give necessary and sufficient conditions for a polynomial of the form x(1 + x + x + · · · + x) 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.
In the 1920’s, Ritt studied the operation of functional composition g ◦ h(x) = g(h(x)) on complex rational functions. In the case of polynomials, he described all the ways in which a polynomial can have multiple ‘prime factorizations’ with respect to this operation. Despite significant effort by Ritt and others, little progress has been made towards solving… (More)
We prove that if xm + axn permutes the prime field Fp, where m > n > 0 and a ∈ Fp, 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 xm + axn if and only if gcd(m,n, q − 1) = 1.
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… (More)
We present a general technique for obtaining permutation polynomials over a finite field from permutations of a subfield. 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)