有限域上四次对角方程ax^4+by^4=c解的存在性

设Fq是特征为p的有限域,d为正整数.对任意的a,b∈F*q,c∈Fq方程.axd+byd=c在Fq上是否恒有解这一问题长期吸引着大量研究者的关注.当d=2时,Cauchy给出了肯定结论.当d=3时,Skolem证明,对任意的素数p≠7,方程.ax3+by3=c在Fq上恒有解;Singh证明,对任意的素数方幂q≠4,方程.ax3+by3=c在Fq上恒有解.本文研究d=4的情形,给出了该方程解的存在性,即当q≠5,9,13,17,25,29时,对任意的a,b∈F*q,c∈Fq,方程.ax4+by4=c在Fq上恒有解.

On the existence of solutions of diagonal quartic equation ax^4+by^4=c over finite field

Let ${\mathbb F}_q$ be the finite field with\ $q$ elements, where\ $q$ is a power of a prime $p$, and $d$ be a positive number. Given any~$a,b \in \mathbb{F}_q^{\ast}$~and $c\in\mathbb{F}_q$,~the problem whether there is always any solution over $\mathbb{F}_q$ for the diagonal equation~$ax^d+by^d=c$~has been attracting significant attentions by scholars. When~$d=2$,~Cauchy gave an affirmative answer. When~$d=3$~, Skolem showed that given any~$p\ne 7$~, there are always solutions over $\mathbb{F}_p$ for~$ax^3+by^3=c$~with~$a,b \in \mathbb{F}_p^{\ast}$~and $c\in\mathbb{F}_p$; Singh extended the result to the general finite fields $\mathbb{F}_q$ for $q \ne 4$. In this paper, we mainly study the case for $d=4$.~We prove that when\ $q\not=5,9,13,17,25,29$, for every~$a,b \in \mathbb F_q^{\ast}$,~$c \in {\mathbb F}_q$, there always exist solutions for the diagonal quartic equation~$ax^4+by^4=c$ over $\mathbb{F}_q$.