In this paper, following L. Carlitz we consider some special equations of $n$ variables over the finite field of $q$ elements. We obtain explicit formulas for the number of solutions of these equations, under a certain restriction on $n$ and $q$.

Let $S$ be a subset of ${\mathbf{F}}_q$, the field of $q$ elements and $h\in {\mathbf{F}}_q\left[x\right]$ a polynomial of degree $d\>1$ with no roots in $S$. Consider the group generated by the image of $\{x-s\mid s\in S\}$ in the group of units of the ring ${\mathbf{F}}_q\left[x\right]/\left(h\right)$. In this paper we present a number of lower bounds for the size of this group. Our main motivation is an application to the recent polynomial time primality testing algorithm [AKS]. The bounds have also applications to graph theory and to the bounding of the number of rational points on abelian covers of the projective...

We study sums and products in a field. Let $F$ be a field with $\mathrm{ch}\left(F\right)\ne 2$, where $\mathrm{ch}\left(F\right)$ is the characteristic of $F$. For any integer $k\ge 4$, we show that any $x\in F$ can be written as $a_1+\cdots +a_k$ with $a_1,\cdots ,a_k\in F$ and $a_1\cdots a_k=1$, and that for any $\alpha \in F\setminus \left\{0\right\}$ we can write every $x\in F$ as $a_1\cdots a_k$ with $a_1,\cdots ,a_k\in F$ and $a_1+\cdots +a_k=\alpha$. We also prove that for any $x\in F$ and $k\in \{2,3,\cdots \}$ there are $a_1,\cdots ,a_{2k}\in F$ such that $a_1+\cdots +a_{2k}=x=a_1\cdots a_{2k}$.

This work was presented in part at the 8th International Conference on Finite Fields and Applications Fq^8 , Melbourne, Australia, 9-13 July, 2007.In this paper, we study the ratio θ(n) = λ2 (n) / ψ2 (n), where λ2 (n) is the number of primitive polynomials and ψ2 (n) is the number of irreducible polynomials in GF (2)[x] of degree n. Let n = ∏ pi^ri, i=1,..l be the prime factorization of n. We show that, for fixed l and ri , θ(n) is close to 1 and θ(2n) is not less than 2/3 for sufficiently large...

We consider an equation of the type$$a_1^{}{x}_1^2+\cdots +a_n^{}{x}_n^2=b{x}_1\cdots x_n$$over the finite field ${𝔽}_q={𝔽}_{p^s}$. Carlitz obtained formulas for the number of solutions to this equation when $n=3$ and when $n=4$ and $q\equiv 3(mod4)$. In our earlier papers, we found formulas for the number of solutions when $d=gcd(n-2,(q-1)/2)=1$ or $2$ or $4$; and when $d\>1$ and $-1$ is a power of $p$ modulo $2d$. In this paper, we obtain formulas for the number of solutions when $d=2^t$, $t\ge 3$, $p\equiv 3\text{or}5(mod8)$ or $p\equiv 9(mod16)$. For general case, we derive lower bounds for the number of solutions.

In the present paper we investigate distributional properties of sparse sequences modulo almost all prime numbers. We obtain new results for a wide class of sparse sequences which in particular find applications on additive problems and the discrete Littlewood problem related to lower bound estimates of the $L_1$-norm of trigonometric sums.