Related papers: Cubic post-critically finite polynomials defined o…
This paper investigates whether or not polynomials that are irreducible over $\mathbb{Q}$ and $\mathbb{Z}$ can remain irreducible under substitution by all quadratic polynomials. It answers this question in the negative in the degree 2 and…
Let $p \geq 2$ be a prime number and let $\mathbb{C}_p$ be the completion of an algebraic closure of the $p$-adic rational field $\mathbb{Q}_p$. Let $f_c(z)$ be a one-parameter family of rational functions of degree $d\geq 2$, where the…
We present a more general proof that cyclotomic polynomials are irreducible over Q and other number fields that meet certain conditions. The proof provides a new perspective that ties together well-known results, as well as some new…
Up to linear transformations, we give a classification of all permutation polynomials of degree $7$ over $\mathbb{F}_{q}$ for any odd prime power $q$, with the help of the SageMath software.
Some families of linear permutation polynomials of $\mathbb{F}_{q^{ms}}$ with coefficients in $\mathbb{F}_{q^{m}}$ are explicitly described (via conditions on their coefficients) as isomorphic images of classical subgroups of the general…
For each integer $k \in [0,9]$, we count the number of plane cubic curves defined over a finite field $\mathbb{F}_q$ that do not share a common component and intersect in exactly $k\ \mathbb{F}_q$-rational points. We set this up as a…
An endomorphism $f:\mathbb{P}^k\to\mathbb{P}^k$ of degree $d\geq2$ is said to be postcritically finite (or PCF) if its critical set $\mathrm{Crit}(f)$ is preperiodic, i.e. if there are integers $m>n\geq0$ such that…
A hyperoval in the projective plane $\mathbb{P}^2(\mathbb{F}_q)$ is a set of $q+2$ points no three of which are collinear. Hyperovals have been studied extensively since the 1950s with the ultimate goal of establishing a complete…
Permutation polynomials over finite fields have extensive applications in various areas. Particularly, permutation polynomials with simple forms are of great interest. In recent papers, several classes of permutation polynomials of the form…
We propose a polynomial time $f$-algorithm (a deterministic algorithm which uses an oracle for factoring univariate polynomials over $\mathbb{F}_q$) for computing an isomorphism (if there is any) of a finite dimensional…
This is the latest in a series of articles aimed at exploring the relationship between the complexity classes of P and NP. In the previous papers, we have proved that the sat CNF problem is polynomially reduced to the problem of finding a…
In this paper, we further investigate the local criterion and present a class of permutation polynomials and their compositional inverses over $ \mathbb{F}_{q^2}$. Additionally, we demonstrate that linearized polynomial over…
Global polynomial optimization methods typically rely on compactness of the feasible region in order to find solutions. These methods can incur considerable computational expense and most commercially available solvers do not verify the…
In 2019, Xiang Fan \cite{xfan} classified all permutation polynomials of degree $7$ over finite fields of odd characteristics. In this paper, we use this classification to determine the complete list of degree $7$ orthomorphism polynomials…
For a set $S$ of quadratic polynomials over a finite field, let $C$ be the (infinite) set of arbitrary compositions of elements in $S$. In this paper we show that there are examples with arbitrarily large $S$ such that every polynomial in…
We find all unitary perfect polynomials over the prime field $\F_2$ with less than five distinct prime factors.
We present an experimental study of the effects of quantifier alternations on the evaluation of quantified Boolean formula (QBF) solvers. The number of quantifier alternations in a QBF in prenex conjunctive normal form (PCNF) is directly…
Second-order polynomials generalize classical first-order ones in allowing for additional variables that range over functions rather than values. We are motivated by their applications in higher-order computational complexity theory,…
We give necessary and sufficient existence criteria, and methods for finding, continuous solutions of linear equations whose coefficients are polynomials.
Quadratic permutation polynomials (QPPs) have been widely studied and used as interleavers in turbo codes. However, less attention has been given to cubic permutation polynomials (CPPs). This paper proves a theorem which states sufficient…