Related papers: Permutation Polynomials of $\mathbb{F}_{q^2}$ : A …
In this paper, we present the compositional inverses of several classes permutation polynomials of the form $\sum_{i=1}^kb_i(x^{p^m}+x+\delta)^{s_i}-x$ over $\mathbb{F}_{p^{2m}}$, where for $1\leq i \leq k,$ $s_i, m$ are positive integers,…
We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable vertex-models in lattice statistical mechanics. These involutions correspond respectively to two kinds…
We give necessary and sufficient conditions for a polynomial of the form x^r*(1+x^v+x^(2v)+...+x^(kv))^t to permute the elements of the finite field GF(q). Our results yield especially simple criteria in case (q-1)/gcd(q-1,v) is a small…
We consider chaining multiplicative-inverse operations in finite fields under alternating polynomial bases. When using two distinct polynomial bases to alternate the inverse operation we obtain a partition of $\mathbb F_{p^n}\setminus…
In this paper we introduce and study a class of tableaux which we call permutation tableaux; these tableaux are naturally in bijection with permutations, and they are a distinguished subset of the Le-diagrams of Alex Postnikov. The…
In this paper we extend a previous investigation by us regarding an iterative construction of irreducible polynomials over finite fields of odd characteristic. In particular, we show how it is possible to iteratively construct irreducible…
In this paper we initiate a study on Gauss factorials of polynomials over finite fields, which are the analogues of Gauss factorials of positive integers.
We construct a new family of permutation group polynomials over finite fields of arbitrary characteristic, which are special types of bivariate local permutation polynomials. For this family, we explicitly construct their companion. We also…
Let $G$ be a simply connected and simple algebraic group defined and split over a finite prime field $\mathbb{F}_p$ of $p$ elements. In this paper, using an $\mathbb{F}_p$-linear map splitting Frobenius endomorphism on a hyperalgebra…
We consider the problem of finding a sparse multiple of a polynomial. Given f in F[x] of degree d over a field F, and a desired sparsity t, our goal is to determine if there exists a multiple h in F[x] of f such that h has at most t…
We present the formula for the number of monic irreducible polynomials of degree $n$ over the finite field $\mathbb F_q$ where the coefficients of $x^{n-1}$ and $x$ vanish for $n\ge3$. In particular, we give a relation between rational…
Finding functions, particularly permutations, with good differential properties has received a lot of attention due to their varied applications. For instance, in combinatorial design theory, a correspondence of perfect $c$-nonlinear…
This paper explores a natural action of the group $\mathrm{PGL}_2(\mathbb F_q)$ on the set of monic irreducible polynomials of degree at least two over a finite field $\mathbb F_q$. Our main results deal with the existence and number of…
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…
Motivated by recent results on the constructions of permutation polynomials with few terms over the finite field $\mathbb{F}_{2^n}$, where $n$ is a positive even integer, we focus on the construction of permutation trinomials over…
We give a formula and an estimation for the number of irreducible polynomials in two (or more) variables over a finite field.
We describe an algorithm for splitting permutation representations of finite group over fields of characteristic zero into irreducible components. The algorithm is based on the fact that the components of the invariant inner product in…
Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gau{\ss} count the remaining ones, approximately and exactly. For polynomials in two or more…
We show that up to linear equivalence, there is only finitely many polynomial quadratic mappings $f:\Bbb C^2\to\Bbb C^2$ and $f:\Bbb R^2\to \Bbb R^2.$ We list all possibilities.
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…