1,415 research outputs found
Reversed Dickson polynomials
We investigate fixed points and cycle types of permutation polynomials and
complete permutation polynomials arising from reversed Dickson polynomials of
the first kind and second kind over . We also study the
permutation behaviour of reversed Dickson polynomials of the first kind and
second kind over . Moreover, we prove two special cases of a
conjecture on the permutation behaviour of reversed Dickson polynomials over
.Comment: 40 page
Fixed Points of the Dickson Polynomials of the Second Kind
The permutation behavior of Dickson polynomials of the first kind has been extensively studied, while such behavior for Dickson polynomials of the second kind is less known. Necessary and sufficient conditions for a polynomial of the second kind to be a permutation over some finite fields have been established by Cohen, Matthew, and Henderson. We introduce a new way to define these polynomials and determine the number of their fixed points
Value sets of bivariate Chebyshev maps over finite fields
We determine the cardinality of the value sets of bivariate Chebyshev maps
over finite fields. We achieve this using the dynamical properties of these
maps and the algebraic expressions of their fixed points in terms of roots of
unity.Comment: 11 pages, 2 figure
Convolution equations on lattices: periodic solutions with values in a prime characteristic field
These notes are inspired by the theory of cellular automata. A linear
cellular automaton on a lattice of finite rank or on a toric grid is a discrete
dinamical system generated by a convolution operator with kernel concentrated
in the nearest neighborhood of the origin. In the present paper we deal with
general convolution operators. We propose an approach via harmonic analysis
which works over a field of positive characteristic. It occurs that a standard
spectral problem for a convolution operator is equivalent to counting points on
an associate algebraic hypersurface in a torus according to the torsion orders
of their coordinates.Comment: 30 pages, a new editio
Chebyshev Action on Finite Fields
Given a polynomial f and a finite field F one can construct a directed graph
where the vertices are the values in the finite field, and emanating from each
vertex is an edge joining the vertex to its image under f. When f is a
Chebyshev polynomial of prime degree, the graphs display an unusual degree of
symmetry. In this paper we provide a complete description of these graphs, and
also provide some examples of how these graphs can be used to determine the
decomposition of primes in certain field extensions
Elliptic curves and explicit enumeration of irreducible polynomials with two coefficients prescribed
Let be a finite field of characteristic . We give the number of
irreducible polynomials x^m+a_{m-1}x^{m-1}+...+a_0\in\F_q[x] with
and prescribed for any given if , and with and
prescribed for if .Comment: 17 pages, Part of the results was presented at the Polynomials over
Finite Fields and Applications workshop at Banff International Research
Station, Canad
The Graph Structure of Chebyshev Polynomials over Finite Fields and Applications
We completely describe the functional graph associated to iterations of
Chebyshev polynomials over finite fields. Then, we use our structural results
to obtain estimates for the average rho length, average number of connected
components and the expected value for the period and preperiod of iterating
Chebyshev polynomials
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