On the number of solutions of some Kummer equations over finite fields

Let l be a prime number and let $k={\mathbb{F}}_q$ be a finite field of characteristic $p\ne l$ with $q=p^f$ elements. Let $n\ge 0$. We determine the number N of solutions $(x,y)$ in k of the Kummer equation$y^l=x(x^{l^n}-1),$ in terms of the trace of a certain Jacobi sum.     

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.     

有限域上一类方程解数的注记

设Fq是含有q个元素的有限域,其中q=pr,r≥1,p是奇素数.研究了有限域Fq上Markoff-Hurwitz类型方程,并给出了当其增广次数矩阵在剩余类环Z/(q-1)Z中可逆时其解数公式的组合证明方法.进一步研究了其推广形式,并得到了此推广形式的方程在特殊条件下的解数公式.     

A lower regulator bound for number fields

Lower bounds for regulators of algebraic number fields are very important for a variety of applications. Good estimates depend at least on the degree and the discriminant of the considered field. In this paper we present an improved bound which is obtained from more specific field data, e.g. the size of small T₂-values of the integers of the field. This is of considerable interest for computations in practice, for example, of fundamental units.     

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.     

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.     

S-unit equations over number fields

Oblatum 1-XI-1989 &; 24-I-1990     

Graph colourings and solutions of systems of equations over finite fields

A congruence (p is a prime) is said to be strong homogeneous if it has the form

     

A removal lemma for systems of linear equations over finite fields

We prove a removal lemma for systems of linear equations over finite fields: let X ₁, …, X _m be subsets of the finite field F _q and let A be a (k × m) matrix with coefficients in F _q ; if the linear system Ax = b has o(q ^m−k ) solutions with x _i ∈ X _i , then we can eliminate all these solutions by deleting o(q) elements from each X _i . This extends a result of Green [Geometric and Functional Analysis 15 (2) (2005), 340–376] for a single linear equation in abelian groups to systems of linear equations. In particular, we also obtain an analogous result for systems of equations over integers, a result conjectured by Green. Our proof uses the colored version of the hypergraph Removal Lemma.     

Solving resultant form equations over number fields

We give an efficient algorithm for solving resultant form equations over number fields. This is the first time that such equations are completely solved by reducing them to unit equations in two variables.

     

Linearly recurring solution sequences for equations over finite fields

Methods based on character sums are used to study conditions under which the sequence {N_t} of numbers of solutions to sequences of equations, parametrized by their number t of variables, over a finite field satisfies a linear recurrence.     

Bounds for the number of points on curves over finite fields

Let U be a bounded open subset of ?^d, d ≥ 2 and f ∈ C(?U). The Dirichlet solution f^CU of the Dirichlet problem associated with the Laplace equation with a boundary condition f is not continuous on the closure ū of U in general if U is not regular but it is always Baire-one.Let H(U) be the space of all functions continuous on the closure ū and harmonic on U and F(H(U)) be the space of uniformly bounded absolutely convergent series of functions in H(U). We prove that f^CU can be obtained as a uniform limit of a sequence of functions in F(H(U)). Thus f^CU belongs to the subclass B_1/2 of Baire-one functions studied for example in [8]. This is not only an improvement of the result obtained in [10] but it also shows that the Dirichlet solution on the closure ū can share better properties than to be only a Baire-one function. Moreover, our proof is more elementary than that in [10].A generalization to the abstract context of simplicial function space on a metrizable compact space is provided.We conclude the paper with a brief discussion on the solvability of the abstract Dirichlet problem with a boundary condition belonging to the space of differences of bounded semicontinuous functions complementing the results obtained in [17].