Related papers: Periodicity and Dynamical Systems of Dickson Polyn…
Let $D_n(x;a)$ and $E_n(x;a)\in\mathbb F_q[x]$ be Dickson polynomials of first and second kind respectively, where $\mathbb F_q$ is a finite field with $q$ elements. In this article we show explicitly the irreducible factors these…
In this paper we study the monomial dynamical systems of dimension one over finite fields from the viewpoints of arithmetic and graph theory. We give formulas for the number of periodic points with period r and cycles with length r. Then we…
We study the algebraic dynamical systems generated by triangular systems of rational functions and estimate the height growth of iterations generated by such systems. Further, using a result on the reduction modulo primes of systems of…
In this paper, we study the periodicity structure of finite field linear recurring sequences whose period is not necessarily maximal and determine necessary and sufficient conditions for the characteristic polynomial~\(f\) to have exactly…
In this paper, we use the method developed previously by Hong, Qin and Zhao to obtain several results on the permutational behavior of the reversed Dickson polynomial $D_{n,k}(1,x)$ of the $(k+1)$-th kind over the finite field ${\mathbb…
We discuss the properties and the permutation behaviour of the reversed Dickson polynomials of the $(k+1)$-th kind $D_{n,k}(1,x)$ over finite fields. The results in this paper unify and generalize several recently discovered results on…
Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…
The analysis of observable phenomena (for instance, in biology or physics) allows the detection of dynamical behaviors and, conversely, starting from a desired behavior allows the design of objects exhibiting that behavior in engineering.…
In this paper, we obtain several results on the permutational behavior of the reversed Dickson polynomial $D_{n,3}(1,x)$ of the fourth kind over the finite field ${\mathbb F}_{q}$. Particularly, we present the explicit evaluation of the…
A famous theorem of Nakaoka asserts that the cohomology of the symmetric group stabilizes. The first author generalized this theorem to non-trivial coefficient systems, in the form of $\mathrm{FI}$-modules over a field, though one now…
We give new characterizations of the algebra $\mathscr{L}_n(\mathbb{F}_{q^n})$ formed by all linearized polynomials over the finite field $\mathbb{F}_{q^n}$ after briefly surveying some known ones. One isomorphism we construct is between…
We prove recurrence relations and modulo periodic properties of multiple derivatives of Fibonacci polynomials. We apply the obtained results to present the dynamic structures of Fibonacci polynomials over the ring of 2-adic integers by…
The $k$th Dickson polynomial of the first kind, $D_k(x) \in {\mathbb Z}[x]$, is determined by the formula: $D_k(u+1/u) = u^k + 1/u^k$, where $k \ge 0$ and $u$ is an indeterminate. These polynomials are closely related to Chebyshev…
Let $\mathbb{F}_q$ be a finite field with $q$ elements and let $n$ be a positive integer. In this paper, we study the digraph associated to the map $x\mapsto x^n h(x^{\frac{q-1}{m}})$, where $h(x)\in\mathbb{F}_q[x].$ We completely determine…
This paper is devoted to several new results concerning (standard) octonion polynomials. The first is the determination of the roots of all right scalar multiples of octonion polynomials. The roots of left multiples are also discussed,…
Motivated by a question of van der Poorten about the existence of infinite chain of prime numbers (with respect to some base), in this paper we advance the study of sequences of consecutive polynomials whose coefficients are chosen…
Explicit formulas are obtained for the number of periodic points and maximum tail length of split polynomial maps over finite fields for affine and projective space. This work includes a detailed analysis of the structure of the directed…
The periods of polynomials can be used to characterize discrete structures such as algebraic error control codes and feedback shift registers. We study trinomial $x^h+x+1$ over GF(2), which has the maximum number of consecutive zero…
We study degree preserving maps over the set of irreducible polynomials over a finite field. In particular, we show that every permutation of the set of irreducible polynomials of degree $k$ over $\mathbb{F}_q$ is induced by an action from…
We study the computational complexity of decomposing finite discrete dynamical systems (FDDSs) in terms of the semiring operations of alternative and synchronous execution, which is useful for the analysis of discrete phenomena in science…