Related papers: Permutation Rational Functions over Quadratic Exte…
A sequence of functions {f_n(q)}_{n=1}^{\infty} satisfies the functional equation for multiplication of quantum integers if f_{mn}(q) = f_m(q)f_n(q^m) for all positive integers m and n. This paper describes the structure of all sequences of…
In this paper we study the following type of functions $f: \mathcal{Q}_{\mathbb{R}_{3}} \to \mathbb{R}_{3}$, where $ \mathcal{Q}_{\mathbb{R}_3}$ is the quadratic cone of the algebra $\mathbb{R}_{3}$. From the fact that it is possible to…
Permutation polynomials over finite fields have wide applications in many areas of science and engineering. In this paper, we present six new classes of permutation trinomials over $\mathbb{F}_{2^n}$ which have explicit forms by determining…
For the quantum integer [n]_q = 1+q+q^2+... + q^{n-1} there is a natural polynomial multiplication such that [mn]_q = [m]_q \otimes_q [n]_q. This multiplication is given by the functional equation f_{mn}(q) = f_m(q) f_n(q^m), defined on a…
Let $\mathbb{F}_{q}$ be the finite field with $q$ elements. In this paper we will describe the dynamics of the map $f(X)=X(X^{q-1}-c)^{q+1},$ with $c\in\mathbb{F}_{q}^{\ast},$ over the finite field $\mathbb{F}_{q^2}$.
After a brief review of existing results on permutation binomials of finite fields, we introduce the notion of equivalence among permutation binomials (PBs) and describe how to bring a PB to its canonical form under equivalence. We then…
The multi-valued quantum $j$-invariant in positive characteristic is studied at quadratic elements. For every quadratic $f$, an explicit expression for each of the values of $j^{\rm qt}(f)$ is given as a limit of rational functions of $f$.…
We study a random polynomial of degree $n$ over the finite field $\mathbb{F}_q$, where the coefficients are independent and identically distributed and uniformly chosen from the squares in $\mathbb{F}_q$. Our main result demonstrates that…
Planar functions over finite fields give rise to finite projective planes and other combinatorial objects. They were originally defined only in odd characteristic, but recently Zhou introduced a definition in even characteristic which…
We use the circle method to count $\mathbb{F}_q(t)$-rational points of bounded naive height on a quadric hypersurface $X\subseteq \mathbb{P}^{n-1}$ defined over $\mathbb{F}_q$, provided that $\mathrm{char}(\mathbb{F}_q)>2$ and $n\ge 3$.…
Let $q=2^e$ with $e$ even, and let $\mathbb{F}_{q^2}$ be the finite field of order $q^2$. Put $d=(q^2+q+1)/3$, and consider the permutation polynomial $$\sigma(X)=X+X^d+X^{dq}\in\mathbb{F}_{q^2}[X].$$ For $\alpha\in\mathbb{F}_q^*$, define…
We describe several families of permutation polynomials obtained using functions with linear translators.
We present an overview of the existing methods for computing functional determinants, and outline a possible way forward for Hamiltonians of higher dimensions without radial symmetry.
Let $q = p^m$, where $p$ is an odd prime number and $m$ is a positive integer. In this paper, we examine the finite field $\mathbb{F}_{q^2}$, which consists of $q^2$ elements. We first present an alternative method to determine the…
For an odd prime power $q$ satisfying $q\equiv 1\pmod 3$ we construct totally $2(q-1) $ permutation polyomials, all giving involutory permutations with exactly $ 1+ \frac{q-1}3$ fixed points. Among them $(q-1)$ polynomials are trinomials,…
In this paper we use algebraic curves and other algebraic number theory methods to show the validity of a permutation polynomial conjecture regarding $f(X)=X^{q(p-1)+1} +\alpha X^{pq}+X^{q+p-1}$, on finite fields $\mathbb{F}_{q^2}, q=p^k$,…
Permutation polynomials with coefficients 1 over finite fields attract researchers' interests due to their simple algebraic form. In this paper, we first construct four classes of fractional permutation polynomials over the cyclic subgroup…
In this paper, we propose linear maps over the space of all polynomials $f(x)$ in $\mathbb{F}_q[x]$ that map $0$ to itself, through their evaluation map. Properties of these linear maps throw up interesting connections with permutation…
In this paper, we prove that a binary definite quadratic form over F_q[t], where q is odd, is completely determined up to equivalence by the polynomials it represents up to degree 3m-2, where m is the degree of its discriminant. We also…
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…