Estimates on the number of Fq–rational solutions of variants of diagonal equations over finite fields

In this paper we study the set of

-rational solutions of equations defined by polynomials evaluated in power-sum polynomials with coefficients in

. This is done by means of applying a methodology which relies on the study of the geometry of the set of common zeros of symmetric polynomials over the algebraic closure of

. We provide improved estimates and existence results of

-rational solutions to the following equations: deformed diagonal equations, generalized Markoff-Hurwitz-type equations and Carlitz's equations. We extend these techniques to more general variants of diagonal equations over finite fields.     

On a class of diagonal equations over finite fields

Using properties of Gauss and Jacobi sums, we derive explicit formulas for the number of solutions to a diagonal equation of the form $x_1^{2^m}+\cdots +x_n^{2^m}=0$ over a finite field of characteristic $p\equiv \pm 3(\mathrm{mod}8)$. All of the evaluations are effected in terms of parameters occurring in quadratic partitions of some powers of p.     

On the number of solutions of certain diagonal equations over finite fields

We use character sums over finite fields to give formulas for the number of solutions of certain diagonal equations of the form$a_1{x}_1^{m_1}+a_2{x}_2^{m_2}+\cdots +a_n{x}_n^{m_n}=c.$ We also show that if the value distribution of character sums ${\sum }_{x\in {\mathbb{F}}_q}\chi (a{x}^m+bx)$, $a,b\in {\mathbb{F}}_q$, is known, then one can obtain the number of solutions of the system of equations$\{\begin{array}{cc}\hfill x_1+x_2+\cdots +x_n\hfill & \hfill =\alpha \hfill \\ \hfill x_1^m+x_2^m+\cdots +x_n^m\hfill & \hfill =\beta ,\hfill \end{array}$ for some particular m. We finally apply our results to induce some facts about Waring's problems and the covering radius of certain cyclic codes.     

Factorization formulae on counting zeros of diagonal equations over finite fields

Let be the number of solutions of the equation over the finite field , and let be the number of solutions of the equation . If , let be the least integer represented by . and play important roles in estimating . Based on a partition of , we obtain the factorizations of and , respectively. All these factorizations can simplify the corresponding calculations in most cases or give the explicit formulae for in some special cases.

     

Paley-like graphs over finite fields from vector spaces

Motivated by the well-known Paley graphs over finite fields and their generalizations, in this paper we explore a natural multiplicative-additive analogue of such graphs arising from vector spaces over finite fields. Namely, if $n\ge 2$ and $U⊊{\mathbb{F}}_{q^n}$ is an ${\mathbb{F}}_q$-vector space, $G_U$ is the (undirected) graph with vertex set $V(G_U)={\mathbb{F}}_{q^n}$ and edge set $E(G_U)=\{(a,b)\in {\mathbb{F}}_{q^n}^2|a\ne b,ab\in U\}$. We describe the structure of an arbitrary maximal clique in $G_U$ and provide bounds on the clique number $\omega (G_U)$ of $G_U$. In particular, we compute the largest possible value of $\omega (G_U)$ for arbitrary q and n. Moreover, we obtain the exact value of $\omega (G_U)$ when $U⊊{\mathbb{F}}_{q^n}$ is any ${\mathbb{F}}_q$-vector space of dimension $d_U\in \{1,2,n-1\}$.     

Solving sparse linear systems of equations over finite fields using bit-flipping algorithm

Let _Fq${\mathbb{F}}_q$ be the finite field with q   elements. We give an algorithm for solving sparse linear systems of equations over _Fq${\mathbb{F}}_q$ when the coefficient matrix of the system has a specific structure, here called relatively connected. This algorithm is based on a well-known decoding algorithm for low-density parity-check codes called bit-flipping algorithm. We modify and extend this hard decision decoding algorithm. The complexity of this algorithm is linear in terms of the number of columns n and the number of nonzero coefficients ω of the matrix per iteration. The maximum number of iterations is bounded above by m, the number of equations.     

Universal octonary diagonal forms over some real quadratic fields

In this paper, we will prove there are infinitely many integers n such that n ²— 1 is square-free and admits universal octonary diagonal quadratic forms. Received: November 2, 1998.     

Simultaneous solutions to diagonal equations over finite fields

An elementary proof of the Weil conjectures is given for the special case of a non-singular pair of diagonal equations over a finite field. The number of simultaneous solutions to an arbitrary number of diagonal equations over GF(q) is also estimated by the same classical methods.