Related papers: Wilson's Theorem for Finite Fields

In this paper, we strengthen a result by Green about an analogue of Sarkozy's theorem in the setting of polynomial rings $\mathbb{F}_q[x]$. In the integer setting, for a given polynomial $F \in \mathbb{Z}[x]$ with constant term zero, (a…

Let $F$ be a field of $q$ elements, where $q$ is a power of an odd prime. Fix $n = (q+1)/2$. For each $s \in F$, we describe all the irreducible factors over $F$ of the polynomial $g_s(y): = y^n + (1-y)^n -s$, and we give a necessary and…

In this paper we prove that the nonzero elements of a finite field with odd characteristic can be partitioned into pairs with prescribed difference (maybe, with some alternatives) in each pair. The algebraic and topological approaches to…

We prove, using combinatorics and Kloosterman sum technology that if $A \subset {\Bbb F}_q$, a finite field with $q$ elements, and $q^{{1/2}} \lesssim |A| \lesssim q^{{7/10}}$, then $\max \{|A+A|, |A \cdot A|\} \gtrsim…

Let $P_1,\dots,P_m\in\mathbb{Z}[y]$ be any linearly independent polynomials with zero constant term. We show that there exists a $\gamma>0$ such that any subset of $\mathbb{F}_q$ of size at least $q^{1-\gamma}$ contains a nontrivial…

Let $\F$ be the finite field of odd prime power order $q$, We find explicit expressions for the number of triples $\{\al-1,\al,\al+1 \}$ of consecutive non-zero squares in $\F$ and similarly for the number of triples of consecutive…

Let $q>2$, and let $a$ and $b$ be two elements of the finite field $\mathbb{F}_q$ with $a\ne 0$. Carlitz represented the transposition $(0a)$ by a polynomial of degree $(q-2)^3$. In this note, we represent the transposition $(ab)$ by a…

By recent work of the author, Wilson's theorem as well as the Wilson quotient can be described by supercongruences of power sums of Fermat quotients modulo every higher prime power. We translate these congruences into congruences of power…

Freiman's $3k-4$ Theorem states that if a subset $A$ of $k$ integers has a Minkowski sum $A+A$ of size at most $3k-4$, then it must be contained in a short arithmetic progression. We prove a function field analogue that is also a…

Assume $n\geq 2$. Consider the elementary symmetric polynomials $e_k(y_1,y_2,\ldots, y_n)$ and denote by $E_0,E_1,\ldots,E_{n-1}$ the elementary symmetric polynomials in reverse order \begin{align*}…

We discuss the problem of constructing a small subset of a finite field containing primitive elements of the field. Given a finite field, $\mathbb{F}_{q^n}$, small $q$ and large $n$, we show that the set of all low degree polynomials…

In this paper we find exact formulas for the numbers of partitions and compositions of an element into $m$ parts over a finite field, i.e. we find the number of nonzero solutions of the equation $x_1+x_2+...+x_m=z$ over a finite field when…

We construct explicitly in any finite field of the form Fq[x]/(x^m-a) elements with multiplicative order at least 2^{(2m)^(1/2)}

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 estimate mixed character sums of polynomial values over elements of a finite field $\mathbb F_{q^r}$ with sparse representations in a fixed ordered basis over the subfield $\mathbb F_q$. First we use a combination of the…

Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements, and let $m_1$ and $m_2$ be positive integers. Given polynomials $f_1(x), f_2(x) \in \mathbb{F}_q[x]$ with $\textrm{deg}(f_i(x)) \leq m_i$, for $i = 1, 2$, and such that the…

We provide a proof of Wilson's Theorem and Wolstenholme's Theorem based on a direct approach by Lagrange requiring only basic properties of the primes and the Binomial theorem. The goal is to show how similar the two theorems are by…

Let $p$ be a prime, let $s \geq 3$ be a natural number and let $A \subseteq \mathbb{F}_p$ be a non-empty set satisfying $|A| \ll p^{1/2}$. Denoting $J_s(A)$ to be the number of solutions to the system of equations \[ \sum_{i=1}^{s} (x_i -…

We prove finiteness results on integral points on complements of large divisors in projective varieties over finitely generated fields of characteristic zero. To do so, we prove a function field analogue of arithmetic finiteness results of…

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…