Related papers: Complete permutation polynomials induced from comp…

For polynomials of degree two over finite fields, we present an improvement of Fitzgerald's characterization (Finite Fields Appl. 9(1):117-121, 2003). We then use this new characterization to obtain an explicit, complete, and simple…

Let $\mathbb F_q$ denote the finite field with $q$ elements. In this paper we use the relationship between suitable polynomials and number of rational points on algebraic curves to give the exact number of elements $a\in \mathbb F_q$ for…

We consider various counting questions for irreducible binomials over finite fields. We use various results from analytic number theory to investigate these questions.

We explain how to use computer experiments over finite fields to gain heuristic information about the solution set of polynomial equations in characteristic zero. These are notes of a tutorial I gave at the NATO Advanced Study Institute on…

We establish closed-form expansions for the universal edge elimination polynomial of paths and cycles and their generating functions. This includes closed-form expansions for the bivariate matching polynomial, the bivariate chromatic…

In this paper, a method for constructing a near optimal normal basis for algebraic extensions of a finite field is described. In each extension, except for the squares of basis elements, the product of two distinct normal basis elements can…

The first part is expository: it explains how finite fields may be used to prove theorems on infinite fields by a reduction mod p process. The second part gives a variant of P.Smith's fixed point theorem which applies in any characteristic.

We introduce the notion of "quasi-symmetric" polynomials, which is a generalization of the notion of symmetry, and is particularly suited to the setting of polynomial rings over finite fields. The properties of this new class of functions…

In this paper, we present a general framework for the derivation of interesting finite combinatorial sums starting with certain classes of polynomial identities. The sums that can be derived involve products of binomial coefficients and…

We derive inversion formulas involving orthogonal polynomials which can be used to find coefficients of differential equations satisfied by certain generalizations of the classical orthogonal polynomials. As an example we consider special…

We give a formula and an estimation for the number of irreducible polynomials in two (or more) variables over a finite field.

For any finite field $\mathbb{F}$ and any positive integer $n$ we count the number of monic polynomials of degree $n$ over $\mathbb{F}$ with nonzero constant coefficient and a self-reciprocal factor of any specified degree. An application…

Within the framework of finite element systems, we show how spaces of differential forms may be constructed, in such a way that they are equipped with commuting interpolators and contain prescribed functions, and are minimal under these…

We give necessary conditions satisfied by the set of odd prime divisors of binary perfect polynomials. This allows us to get a new characterization of all the known perfect binary polynomials.

Let $f=ax+x^{r(q-1)+1}\in \mathbb{F}_{q^2}^*[x], r\in \{5,7\}.$ We give explicit conditions on the values $(q,a)$ for which $f$ is a permutation polynomials of $\mathbb{F}_{q^2}.$

Let $r$ be a positive integer, $h(X)\in\Bbb F_{q^2}[X]$, and $\mu_{q+1}$ be the subgroup of order $q+1$ of $\Bbb F_{q^2}^*$. It is well known that $X^rh(X^{q-1})$ permutes $\Bbb F_{q^2}$ if and only if $\text{gcd}(r,q-1)=1$ and…

In this paper we look at the automorphisms of the multiplicative group of finite nearfields. We find partial results for the actual automorphism groups. We find counting techniques for the size of all finite nearfields. We then show that…

Let $f=a{\tt x} +b{\tt x}^q+{\tt x}^{2q-1}\in\Bbb F_q[{\tt x}]$. We find explicit conditions on $a$ and $b$ that are necessary and sufficient for $f$ to be a permutation polynomial of $\Bbb F_{q^2}$. This result allows us to solve a related…

We present a general strategy that allows a more flexible method for the construction of fully additive multipartite entanglement monotones than the ones so far reported in the literature of axiomatic entanglement measures. Within this…

We show the existence of and explicitly construct generic polynomials for various groups, over fields of positive characteristic. The methods we develop apply to a broad class of connected linear algebraic groups defined over finite fields…