论文标题
在有限场上的对角线立方形式的零数
On the number of zeros of diagonal cubic forms over finite fields
论文作者
论文摘要
令$ {\ mathbb f} _q $为有限字段，$ q = p^k $元素，$ p $为prime，$ k $是一个正整数。 For any $y, z\in\mathbb{F}_q$, let $N_s(z)$ and $T_s(y)$ denote the numbers of zeros of $x_1^{3}+\cdots+x_s^3=z$ and $x_1^3+\cdots+x_{s-1}^3+yx_s^3=0$, respectively.高斯证明，如果$ q = p，p \ equiv1 \ pmod3 $和$ y $是非立方体，则$ t_3（y）= p^2+\ frac {1} {2} {2} {2}（p-1）（ - c+9d）$，$ c $和$ c $ whene $ c $ and $ c $ 4p = c^c^c^2+27D+27d+27d+27d+27d^2+27d^2， $ D $。 1978年，Chowla，Cowles和Cowles确定了$ 2 $的$ D $的标志，为$ {\ Mathbb f} _p $的非潜水元素。但是，对于$ {\ mathbb f} _p $的剩余案例，$ 2 $的剩余案例均保持了符号问题。在本文中，我们通过确定$ d $的标志在$ {\ mathbb f} _p $时解决此标志问题。此外，我们表明生成函数$ \ sum_ {s = 1}^{\ infty} n_ {s}（z）x^{s} $和$ \ sum_ {s = 1}^{\ infty} f_q^*：= \ Mathbb f_q \ setMinus \ {0 \} $，$ y $在$ {\ mathbb f} _q $上是非立方体，也给出他们的显式表达式。这扩展了Myerson的定理以及Chowla，Cowles和Cowles的定理。
Let ${\mathbb F}_q$ be the finite field with $q=p^k$ elements with $p$ being a prime and $k$ be a positive integer. For any $y, z\in\mathbb{F}_q$, let $N_s(z)$ and $T_s(y)$ denote the numbers of zeros of $x_1^{3}+\cdots+x_s^3=z$ and $x_1^3+\cdots+x_{s-1}^3+yx_s^3=0$, respectively. Gauss proved that if $q=p, p\equiv1\pmod3$ and $y$ is non-cubic, then $T_3(y)=p^2+\frac{1}{2}(p-1)(-c+9d)$, where $c$ and $d$ are uniquely determined by $4p=c^2+27d^2,~c\equiv 1 \pmod 3$ except for the sign of $d$. In 1978, Chowla, Cowles and Cowles determined the sign of $d$ for the case of $2$ being a non-cubic element of ${\mathbb F}_p$. But the sign problem is kept open for the remaining case of $2$ being cubic in ${\mathbb F}_p$. In this paper, we solve this sign problem by determining the sign of $d$ when $2$ is cubic in ${\mathbb F}_p$. Furthermore, we show that the generating functions $\sum_{s=1}^{\infty} N_{s}(z) x^{s}$ and $\sum_{s=1}^{\infty} T_{s}(y)x^{s}$ are rational functions for any $z, y\in\mathbb F_q^*:=\mathbb F_q\setminus \{0\}$ with $y$ being non-cubic over ${\mathbb F}_q$ and also give their explicit expressions. This extends the theorem of Myerson and that of Chowla, Cowles and Cowles.
