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arXiv 2108.00396math.NT
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关于有限域上对角四次型的零点个数

On the number of zeros of diagonal quartic forms over finite fields

Junyong Zhao, Yulu Feng, Shaofang Hong, Chaoxi Zhu

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22 pages
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摘要：

设$\mathbb{F}_q$是由$q=p^m\equiv 1\pmod 4$元素构成的有限域，$p$是奇素数，$m$是正整数.对于具有$y\in\mathbb{F}_q^*$非四次曲线的$c, y \in\mathbb{F}_q$，令$N_n(c)$和$M_n(y)$分别为$x_1^4+...+x_n^4=c$和$x_1^4+...+x_{n-1}^4+yx_n^4=0$的零点个数。1979年，Myerson利用Gauss和和指数和证明了$x$中的母函数$\sum_{n=1}^{\infty}N_n(0)x^n$是有理函数，并给出了它的显式表达式。本文利用分圆理论和指数和证明了生成函数$\sum_{n=1}^{\infty}N_n(c)x^n$和$\sum_{n=1}^{\infty}M_{n+1}(y)x^n$是$x$中的有理函数。我们还得到了这些生成函数的显式表达式。我们的结果推广了1979年的Myerson定理。

英文摘要：

Let $\mathbb{F}_q$ be the finite field of $q=p^m\equiv 1\pmod 4$ elements with $p$ being an odd prime and $m$ being a positive integer. For $c, y \in\mathbb{F}_q$ with $y\in\mathbb{F}_q^*$ non-quartic, let $N_n(c)$ and $M_n(y)$ be the numbers of zeros of $x_1^4+...+x_n^4=c$ and $x_1^4+...+x_{n-1}^4+yx_n^4=0$, respectively. In 1979, Myerson used Gauss sum and exponential sum to show that the generating function $\sum_{n=1}^{\infty}N_n(0)x^n$ is a rational function in $x$ and presented its explicit expression. In this paper, we make use of the cyclotomic theory and exponential sums to show that the generating functions $\sum_{n=1}^{\infty}N_n(c)x^n$ and $\sum_{n=1}^{\infty}M_{n+1}(y)x^n$ are rational functions in $x$. We also obtain the explicit expressions of these generating functions. Our result extends Myerson's theorem gotten in 1979.

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