Related papers: Dynamic Structures of 2-adic Fibonacci Polynomials

When the Poincar\'{e} map associated with a periodic orbit of a hybrid dynamical system has constant-rank iterates, we demonstrate the existence of a constant-dimensional invariant subsystem near the orbit which attracts all nearby…

Chebyshev polynomials and their modifications are attributes of various fields of mathematics. In particular, they are generating functions of the rows elements of certain Riordan matrices. In paper, we give a selection of some…

Given any two sequences of complex numbers, we establish simple relations between their binomial convolution and the binomial convolution of their individual binomial transforms. We employ these relations to derive new identities involving…

This paper is a short survey of the recent results on examples of periodic two-dimensional continued fractions (in Klein's model). In the last part of this paper we formulate some questions, problems and conjectures on geometrical…

Define the $n$-th fibotomic polynomial to be the product of the monic irredicible factors of the $n$-th Fibonacci polynomial which are not factors of any Fibonacci polynomial of smaller degree. In this paper, we prove a number of properties…

We generalize the polynomial Szemer\'{e}di theorem to intersective polynomials over the ring of integers of an algebraic number field, by which we mean polynomials having a common root modulo every ideal. This leads to the existence of new…

Divisibility sequences are defined by the property that their elements divide each other whenever their indices do. The divisibility sequences that also satisfy a linear recurrence, like the Fibonacci numbers, are generated by polynomials…

We investigate the iterative construction of discrete Laplacians on 2D square lattices, revealing emergent fractal-like patterns shaped by modular arithmetic. While classical 2222-style iterations reproduce known structures such as the…

We prove the dynamical Mordell-Lang conjecture for birational polynomial morphisms on $\mathbb{A}^2$.

We investigate from a statistical perspective the arithmetic properties of the dynamics of polynomials of fixed degree and defined over the field of rational numbers. To start with, ordering their affine conjugacy classes by height, we show…

We obtain results describing the behavior of the action of rotation generators on polynomials over a commutative ring. We also explore harmonic polynomials in a purely algebraic setting.

Using elementary methods, we establish old and new relations between binomial coefficients, Fibonacci numbers, Lucas numbers, and more.

Let $K$ be a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers and $\O$ be its integral ring. The convergent power series with coefficients in $\O$ are studied as dynamical systems on $\O$. A minimal decomposition theorem for…

This survey paper is aimed to describe a relatively new branch of symbolic dynamics which we call Arithmetic Dynamics. It deals with explicit arithmetic expansions of reals and vectors that have a "dynamical" sense. This means precisely…

We consider bivariate polynomials orthogonal on the bicircle with respect to a positive linear functional. The lexicographical and reverse lexicographical orderings are used to order the monomials. Recurrence formulas are derived between…

We survey and prove properties a family of recurrences bears in relation to integer representations, compositions, the Pascal triangle, sums of digits, Nim games and Beatty sequences.

The summatory function of the number of binomial coefficients not divisible by a prime is known to exhibit regular periodic oscillations, yet identifying the less regularly behaved minimum of the underlying periodic functions has been open…

Discrete models have a long tradition in engineering, including finite state machines, Boolean networks, Petri nets, and agent-based models. Of particular importance is the question of how the model structure constrains its dynamics. This…

We study determinants of matrices whose entries are powers of Fibonacci numbers. We then extend the results to include entries that are powers of generalized Fibonacci numbers defined as a second-order linear recurrence relation. These…

Using polynomial evaluation, we give some useful criteria to answer questions about divisibility of polynomials. This allows us to develop interesting results concerning the prime elements in the domain of coefficients. In particular, it is…